Integer n, Integer m, Integer ncnln, Integer nonln, Integer njnln, Integer iobj, Integer nnz, double a[], const Integer ha[], const Integer ka[], double bl[], double bu[], double xs[], Integer *ninf, double *sinf, double *objf, Nag_Comm *comm, Nag_E04_Opt *options, NagError *fail)

The function may be called by the names: e04ugc, nag_opt_nlp1_sparse_solve or nag_opt_nlp_sparse.

3 Description

e04ugc is designed to solve a class of nonlinear programming problems that are assumed to be stated in the following general form:

\underset{x \in R^{n}}{minimize} ​ ​ f (x) subject to l \leq {\begin{matrix} x \\ F (x) \\ G x \end{matrix}} \leq u,

(1)

where

x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}

is a set of variables,

f (x)

is a smooth scalar objective function,

l

and

u

are constant lower and upper bounds,

F (x)

is a vector of smooth nonlinear constraint functions

{F_{i} (x)}

and

G

is a sparse matrix.

The constraints involving

F

and

G x

are called the general constraints. Note that upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the

j

th constraint can be defined as an equality by setting

l_{j} = u_{j}

. If certain bounds are not present, the associated elements of

l

u

can be set to special values that will be treated as

- \infty

+ \infty

. (See the description of the optional parameter

options . inf_bound

in Section 12.2).

e04ugc converts the upper and lower bounds on the

m

elements of

F

and

G x

to equalities by introducing a set of slack variables

s

, where

s = {(s_{1}, s_{2}, \dots, s_{m})}^{T}

. For example, the linear constraint

5 \leq 2 x_{1} + 3 x_{2} \leq + \infty

is replaced by

2 x_{1} + 3 x_{2} - s_{1} = 0

, together with the bounded slack

5 \leq s_{1} \leq + \infty

. The problem defined by (1) can, therefore, be re-written in the following equivalent form:

\underset{x \in R^{n}, s \in R^{m}}{minimize} f (x) subject to {\begin{matrix} F (x) \\ G x \end{matrix}} - s = 0, l \leq {\begin{matrix} x \\ s \end{matrix}} \leq u .

(2)

Since the slack variables

s

are subject to the same upper and lower bounds as the elements of

F

and

G x

, the bounds on

F

and

G x

can simply be thought of as bounds on the combined vector

(x, s)

. The elements of

x

and

s

are partitioned into basic, nonbasic and superbasic variables defined as follows (see Section 11 for more details):

A basic variable is a variable associated with a column of a square nonsingular basis matrix $B$ .
A nonbasic variable is a variable that is temporarily fixed at its current value (usually its upper or lower bound).
A superbasic variable is a non basic variable which is not at one of its bounds and which is free to move in any desired direction (namely one that will improve the value of the objective function or reduce the sum of infeasibilities). At each step, basic variables are adjusted depending on the values of superbasic variables.

For example, in the simplex method (see Gill et al. (1981)) the elements of

x

can be partitioned at each vertex into a set of

m

basic variables (all non-negative) and a set of

(n - m)

nonbasic variables (all zero). This is equivalent to partitioning the columns of the constraint matrix as

(B ∣ N)

, where

B

contains the

m

columns that correspond to the basic variables and

N

contains the

(n - m)

columns that correspond to the nonbasic variables.

The optional parameter

options . direction

(default value

options . direction = Nag_Minimize

) may be used to specify an alternative problem in which

f (x)

is maximized (setting

options . direction = Nag_Maximize

), or to only find a feasible point (setting

options . direction = Nag_FeasiblePoint

). If the objective function is nonlinear and all the constraints are linear,

F

is absent and the problem is said to be linearly constrained. In general, the objective and constraint functions are structured in the sense that they are formed from sums of linear and nonlinear functions. This structure can be exploited by the function during the solution process as follows.

Consider the following nonlinear optimization problem with four variables

(u, v, z, w)

\underset{u, v, z, w}{minimize} {(u + v + z)}^{2} + 3 z + 5 w

subject to the constraints

\begin{matrix} u^{2} + v^{2} + z = 2 \\ u^{4} + v^{4} + w = 4 \\ 2 u + 4 v \geq 0 \end{matrix}

and to the bounds

\begin{matrix} z \geq 0 \\ w \geq 0 \end{matrix} .

This problem has several characteristics that can be exploited by the function:

the objective function is nonlinear. It is the sum of a nonlinear function of the variables $(u, v, z)$ and a linear function of the variables $(z, w)$ ;
the first two constraints are nonlinear. The third is linear;
each nonlinear constraint function is the sum of a nonlinear function of the variables $(u, v)$ and a linear function of the variables $(z, w)$ .

The nonlinear terms are defined by the user-supplied subroutines objfun and confun (see Section 5), which involve only the appropriate subset of variables.

For the objective, we define the function

f (u, v, z) = {(u + v + z)}^{2}

to include only the nonlinear part of the objective. The three variables

(u, v, z)

associated with this function are known as the nonlinear objective variables. The number of them is given by nonln (see Section 5), and they are the only variables needed in objfun. The linear part

3 z + 5 w

of the objective is stored in row iobj (see Section 5) of the (constraint) Jacobian matrix

A

(see below).

Thus, if

x^{'}

and

y^{'}

denote the nonlinear and linear objective variables, respectively, the objective may be re-written in the form

f (x^{'}) + c^{T} x^{'} + d^{T} y^{'},

where

f (x^{'})

is the nonlinear part of the objective and

c

and

d

are constant vectors that form a row of

A

. In this example,

x^{'} = (u, v, z)

and

y^{'} = w

Similarly for the constraints, we define a vector function

F (u, v)

to include just the nonlinear terms. In this example,

F_{1} (u, v) = u^{2} + v^{2}

and

F_{2} (u, v) = u^{4} + v^{4}

, where the two variables

(u, v)

are known as the nonlinear Jacobian variables. The number of them is given by njnln (see Section 5), and they are the only variables needed in confun. Thus, if

x^{''}

and

y^{''}

denote the nonlinear and linear Jacobian variables, respectively, the constraint functions and the linear part of the objective have the form

(\begin{matrix} F (x^{''}) & A_{2} y^{''} \\ A_{3} x^{''} & A_{4} y^{''} \end{matrix}),

(3)

where

x^{''} = (u, v)

and

y^{''} = (z, w)

in this example. This ensures that the Jacobian is of the form

A = (\begin{matrix} J (x^{''}) & A_{2} \\ A_{3} & A_{4} \end{matrix})

where

J (x^{''}) = \frac{\partial F (x^{''})}{\partial x}

. Note that

J (x^{''})

always appears in the top left-hand corner of

A

The inequalities

l_{1} \leq F (x^{''}) + A_{2} y^{''} \leq u_{1}

and

l_{2} \leq A_{3} x^{''} + A_{4} y^{''} \leq u_{2}

implied by the constraint functions in (3) are known as the nonlinear and linear constraints, respectively. The nonlinear constraint vector

F (x^{''})

in (3) and (optionally) its partial derivative matrix

J (x^{''})

are set in confun. The matrices

A_{2}

A_{3}

and

A_{4}

contain any (constant) linear terms. Along with the sparsity pattern of

J (x^{''})

they are stored in the arrays a, ha and ka (see Section 5).

In general, the vectors

x^{'}

and

x^{''}

have different dimensions, but they must always overlap, in the sense that the shorter vector should always be the beginning of the other. In the above example, the nonlinear Jacobian variables

(u, v)

are an ordered subset of the nonlinear objective variables

(u, v, z)

. In other cases it could be the other way round. Note that in some cases it might be necessary to add variables to

x^{'}

x^{''}

(whichever is the most convenient), but the first way keeps

J (x^{''})

as small as possible. Thus, the nonlinear objective function

f (x^{'})

may involve either a subset or superset of the variables appearing in the nonlinear constraint functions

F (x^{''})

, and

nonln \leq njnln

(or vice-versa). Sometimes the objective and constraints may really involve disjoint sets of nonlinear variables. In such cases the variables should be ordered so that

nonln > njnln

and

x^{'} = (x^{''}, x^{'''})

, where the objective is nonlinear in just the last vector

x^{'''}

. The first njnln elements of the gradient array objgrad (corresponding to

x^{''}

) should then be set to zero in objfun. This is illustrated in Section 10.

If there are no nonlinear constraints in (1) and

f (x)

is linear or quadratic, then it may be simpler and/or more efficient to use e04nkc to solve the resulting linear or quadratic programming problem, or one of e04mfc, e04ncc or e04nfc if

G

is a dense matrix. If the problem is dense and does have nonlinear constraints, then e04ucc should be used instead.

You must supply an initial estimate of the solution to (1), together with versions of objfun and confun that define

f (x^{'})

and

F (x^{''})

, respectively, and as many first partial derivatives as possible. Note that if there are any nonlinear constraints, then the first call to confun will precede the first call to objfun.

e04ugc is based on the SNOPT package described in Gill et al. (1997), which in turn utilizes routines from the MINOS package (see Murtagh and Saunders (1995)). It incorporates a Sequential Quadratic Programming (SQP) method that obtains search directions from a sequence of Quadratic Programming (QP) subproblems. Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to a linearization of the constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. Further details can be found in Section 11.

Throughout this document the symbol

ε

is used to represent the machine precision (see X02AJC).

4 References

Conn A R (1973) Constrained optimization using a nondifferentiable penalty function SIAM J. Numer. Anal. 10 760–779

Eldersveld S K (1991) Large-scale sequential quadratic programming algorithms PhD Thesis Department of Operations Research, Stanford University, Stanford

Fletcher R (1984) An

l_{1}

penalty method for nonlinear constraints Numerical Optimization 1984 (eds P T Boggs, R H Byrd and R B Schnabel) 26–40 SIAM Philadelphia

Fourer R (1982) Solving staircase linear programs by the simplex method Math. Programming 23 274–313

Gill P E, Murray W and Saunders M A (1997) SNOPT: an SQP algorithm for large-scale constrained optimization Numerical Analysis Report 97–2 Department of Mathematics, University of California, San Diego

Gill P E, Murray W and Saunders M A (2002) SNOPT: An SQP Algorithm for Large-scale Constrained Optimization 12 979–1006 SIAM J. Optim.

Gill P E, Murray W, Saunders M A and Wright M H (1986) Users' guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming Report SOL 86-2 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization Math. Programming 45 437–474

Gill P E, Murray W, Saunders M A and Wright M H (1992) Some theoretical properties of an augmented Lagrangian merit function Advances in Optimization and Parallel Computing (ed P M Pardalos) 101–128 North Holland

Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press

Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems 187 Springer–Verlag

Murtagh B A and Saunders M A (1995) MINOS 5.4 users' guide Report SOL 83-20R Department of Operations Research, Stanford University

Ortega J M and Rheinboldt W C (1970) Iterative Solution of Nonlinear Equations in Several Variables Academic Press

Powell M J D (1974) Introduction to constrained optimization Numerical Methods for Constrained Optimization (eds P E Gill and W Murray) 1–28 Academic Press

5 Arguments

1: $confun$ – function, supplied by the user External Function

confun must calculate the vector

F (x)

of nonlinear constraint functions and (optionally) its Jacobian (

= \frac{\partial F}{\partial x}

) for a specified njnln

(\leq n)

element vector

x

. If there are no nonlinear constraints (i.e.,

ncnln = 0

), confun will never be called by e04ugc and the NAG defined null void function pointer, NULLFN, can be supplied in the call to e04ugc. If there are nonlinear constraints, the first call to confun will occur before the first call to objfun.

The specification of confun is:

void	confun (Integer ncnln, Integer njnln, Integer nnzjac, const double x[], double conf[], double conjac[], Nag_Comm *comm)

1: $ncnln$ – Integer Input

On entry: the number of nonlinear constraints. These must be the first ncnln constraints in the problem.

2: $njnln$ – Integer Input

On entry: the number of nonlinear variables. These must be the first njnln variables in the problem.

3: $nnzjac$ – Integer Input

On entry: the number of nonzero elements in the constraint Jacobian. Note that nnzjac will always be less than, or equal to,

ncnln \times njnln

4: $x [njnln]$ – const double Input

On entry:

x

, the vector of nonlinear Jacobian variables at which the nonlinear constraint functions and/or all available elements of the constraint Jacobian are to be evaluated.

5: $conf [ncnln]$ – double Output

On exit: if

comm \to flag = 0

2

conf [i - 1]

must contain the value of

F_{i} (x)

, the

i

th nonlinear constraint at

x

6: $conjac [\dim]$ – double Output

On exit: if

comm \to flag = 1

2

, conjac must return the available elements of

J (x)

, the constraint Jacobian evaluated at

x

. These elements must be stored in conjac in exactly the same positions as implied by the definitions of the arrays a, ha and ka described below, remembering that

J (x)

always appears in the top left-hand corner of

A

. Note that the function does not perform any internal checks for consistency (except indirectly via the optional parameter

options . verify_grad

), so great care is essential.

If all elements of the constraint Jacobian are known, i.e., the optional parameter

options . con_deriv = Nag_TRUE

(the default), any constant elements of the Jacobian may be assigned to a at the start of the optimization if desired. If an element of conjac is not assigned in confun, the corresponding value from a is used. See also the description for a.

options . con_deriv = Nag_FALSE

, then any available partial derivatives of

c_{i} (x)

must be assigned to the elements of conjac; the remaining elements must remain unchanged. It must be emphasized that, in that case, unassigned elements of conjac are not treated as constant; they are estimated by finite differences, at non-trivial expense.

7: $comm$ – Nag_Comm *

Pointer to a structure of type Nag_Comm; the following members are relevant to confun.

flag – IntegerInput/Output: On entry: confun is called with $comm \to flag$ set to $0$ , 1 or 2.
If $comm \to flag = 0$ , only conf has to be referenced.

If $comm \to flag = 1$ , only conjac has to be referenced.

If $comm \to flag = 2$ , both conf and conjac are referenced.

On exit: if confun resets $comm \to flag$ to $−1$ , e04ugc will terminate with the error indicator NE_CANNOT_CALCULATE, unless this occurs during the linesearch; in this case, the linesearch will shorten the step and try again. If confun resets $comm \to flag$ to a value smaller or equal to $−2$ ,, e04ugc will terminate immediately with the error indicator NE_USER_STOP. In both cases, if fail is supplied to e04ugc, $fail . errnum$ will be set to your setting of $comm \to flag$ .

first – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the first call to confun and Nag_FALSE for all subsequent calls. This argument setting allows you to save computation time if certain data must be read or calculated only once.

last – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the last call to confun and Nag_FALSE for all other calls. This argument setting allows you to perform some additional computation on the final solution.

user – double *
iuser – Integer *
p – Pointer: The type Pointer is void *.

Before calling e04ugc these pointers may be allocated memory and initialized with various quantities for use by confun when called from e04ugc.

Note: confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04ugc. If your code inadvertently does return any NaNs or infinities, e04ugc is likely to produce unexpected results.

Note: confun should be tested separately before being used in conjunction with e04ugc. The optional parameters

options . verify_grad

and

options . major_iter_lim

can be used to assist this process (see Section 12.2). The array x must not be changed by confun.

If confun does not calculate all of the Jacobian constraint elements then the optional parameter

options . con_deriv

should be set to Nag_FALSE.

2: $objfun$ – function, supplied by the user External Function

objfun must calculate the nonlinear part of the objective

f (x)

and (optionally) its gradient (

= \frac{\partial f}{\partial x}

) for a specified nonln

(\leq n)

element vector

x

. If there are no nonlinear objective variables (i.e.,

nonln = 0

), objfun will never be called by e04ugc and the NAG defined null void function pointer, NULLFN, can be supplied in the call to e04ugc.

The specification of objfun is:

void	objfun (Integer nonln, const double x[], double objf, double objgrad[], Nag_Comm comm)

1: $nonln$ – Integer Input

On entry: the number of nonlinear objective variables. These must be the first nonln variables in the problem.

2: $x [nonln]$ – const double Input

On entry: the vector

x

of nonlinear variables at which the nonlinear part of the objective function and/or all available elements of its gradient are to be evaluated.

3: $objf$ – double * Output

On exit: if

comm \to flag = 0

2

, objfun must set objf to the value of the nonlinear part of the objective function at

x

. If it is not possible to evaluate the objective function at

x

, then objfun should assign

−1

comm \to flag

; e04ugc will then terminate, unless this occurs during the linesearch; in this case, the linesearch will shorten the step and try again.

4: $objgrad [nonln]$ – double Output

On exit: if

comm \to flag = 1

2

, objgrad must return the available elements of the gradient

\frac{\partial f}{\partial x}

evaluated at the current point

x

If the optional parameter

options . obj_deriv = Nag_TRUE

(the default), all elements of objgrad must be set; if

options . obj_deriv = Nag_FALSE

, any available elements of the Jacobian matrix must be assigned to the elements of objgrad; the remaining elements must remain unchanged.

5: $comm$ – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to objfun.

flag – IntegerInput/Output: On entry: objfun is called with $comm \to flag$ set to $0$ , 1 or 2.
If $comm \to flag = 0$ then only objf has to be referenced.

If $comm \to flag = 1$ then only objgrad has to be referenced.

If $comm \to flag = 2$ then both objf and objgrad are referenced.

On exit: if objfun resets $comm \to flag$ to $−1$ , then e04ugc will terminate with the error indicator NE_CANNOT_CALCULATE, unless this occurs during the linesearch; in this case, the linesearch will shorten the step and try again. If objfun resets $comm \to flag$ to a value smaller or equal to $−2$ , then e04ugc will terminate immediately with the error indicator NE_USER_STOP. In both cases, if fail is supplied to e04ugc $fail . errnum$ will then be set to your setting of $comm \to flag$ .

first – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the first call to objfun and Nag_FALSE for all subsequent calls. This argument setting allows you to save computation time if certain data must be read or calculated only once.

last – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the last call to objfun and Nag_FALSE for all other calls. This argument setting allows you to perform some additional computation on the final solution.

nf – IntegerInput: On entry: the number of evaluations of the objective function; this value will be equal to the number of calls made to objfun including the current one.

user – double *
iuser – Integer *
p – Pointer: The type Pointer is void *.

Before calling e04ugc these pointers may be allocated memory and initialized with various quantities for use by objfun when called from e04ugc.

Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04ugc. If your code inadvertently does return any NaNs or infinities, e04ugc is likely to produce unexpected results.

Note: objfun should be tested separately before being used in conjunction with e04ugc. The optional parameters

options . verify_grad

and

options . major_iter_lim

can be used to assist this process (see Section 12.2). The array x must not be changed by objfun.

If the function objfun does not calculate all of the Jacobian elements then the optional parameter

options . obj_deriv

should be set to Nag_FALSE.

3: $n$ – Integer Input

On entry:

n

, the number of variables (excluding slacks). This is the number of columns in the full Jacobian matrix

A

Constraint:

n \geq 1

4: $m$ – Integer Input

On entry:

m

, the number of general constraints (or slacks). This is the number of rows in

A

, including the free row (if any; see iobj). Note that

A

must contain at least one row. If your problem has no constraints, or only upper and lower bounds on the variables, then you must include a dummy ‘free’ row consisting of a single (zero) element subject to ‘infinite’ upper and lower bounds. Further details can be found under the descriptions for iobj, nnz, a, ha, ka, bl and bu.

Constraint:

m \geq 1

5: $ncnln$ – Integer Input

On entry: the number of nonlinear constraints. These correspond to the leading ncnln rows of

A

Constraint:

0 \leq ncnln \leq m

6: $nonln$ – Integer Input

On entry: the number of nonlinear objective variables. If the objective function is nonlinear, the leading nonln columns of

A

belong to the nonlinear objective variables. (See also the description for njnln.)

Constraint:

0 \leq nonln \leq n

7: $njnln$ – Integer Input

On entry: the number of nonlinear Jacobian variables. If there are any nonlinear constraints, the leading njnln columns of

A

belong to the nonlinear Jacobian variables. If

nonln > 0

and

njnln > 0

, the nonlinear objective and Jacobian variables overlap. The total number of nonlinear variables is given by

\bar{n} = \max (nonln, njnln)

Constraints:

if $ncnln = 0$ , $njnln = 0$ ;
if $ncnln > 0$ , $1 \leq njnln \leq n$ .

8: $iobj$ – Integer Input

On entry: if

iobj > ncnln

, row iobj of

A

is a free row containing the nonzero elements of the linear part of the objective function.

$iobj = 0$: There is no free row.
$iobj = −1$: There is a dummy ‘free’ row.

Constraints:

if $iobj > 0$ , $ncnln < iobj \leq m$ ;
otherwise $iobj \geq −1$ .

9: $nnz$ – Integer Input

On entry: the number of nonzero elements in

A

(including the Jacobian for any nonlinear constraints,

J

). If

iobj = −1

, set

nnz = 1

Constraint:

1 \leq nnz \leq n \times m

10: $a [nnz]$ – double Input/Output

On entry: the nonzero elements of the Jacobian matrix

A

, ordered by increasing column index. Note that elements with the same row and column index are not allowed. Since the constraint Jacobian matrix

J (x^{''})

must always appear in the top left-hand corner of

A

, those elements in a column associated with any nonlinear constraints must come before any elements belonging to the linear constraint matrix

G

and the free row (if any; see iobj).

In general,

A

is partitioned into a nonlinear part and a linear part corresponding to the nonlinear variables and linear variables in the problem. Elements in the nonlinear part may be set to any value (e.g., zero) because they are initialized at the first point that satisfies the linear constraints and the upper and lower bounds. If the optional parameter

options . con_deriv = Nag_TRUE

(the default), the nonlinear part may also be used to store any constant Jacobian elements. Note that if confun does not define the constant Jacobian element

conjac [i]

, the missing value will be obtained directly from the corresponding element of a. The linear part must contain the nonzero elements of

G

and the free row (if any). If

iobj = −1

, set

a [0] = β

, say, where

| β | < bigbnd

and

bigbnd

is the value of the optional parameter

options . inf_bound

(default value

= 10^{20}

). Elements with the same row and column indices are not allowed. (See also the descriptions for ha and ka.)

On exit: elements in the nonlinear part corresponding to nonlinear Jacobian variables are overwritten.

11: $ha [nnz]$ – const Integer Input

On entry:

ha [i - 1]

must contain the row index of the nonzero element stored in

a [i - 1]

, for

i = 1, 2, \dots, nnz

. The row indices for a column may be supplied in any order subject to the condition that those elements in a column associated with any nonlinear constraints must appear before those elements associated with any linear constraints (including the free row, if any). Note that confun must define the Jacobian elements in the same order. If

iobj = −1

, set

ha [0] = 1

Constraint:

1 \leq ha [i - 1] \leq m

, for

i = 1, 2, \dots, nnz

12: $ka [n + 1]$ – const Integer Input

On entry:

ka [j - 1]

must contain the index in a of the start of the

j

th column, for

j = 1, 2, \dots, n

. To specify the

j

th column as empty, set

ka [j] = ka [j - 1]

. Note that the first and last elements of ka must be such that

ka [0] = 0

and

ka [n] = nnz

. If

iobj = −1

, set

ka [j] = 1

, for

j = 1, 2, \dots, n

Constraints:

$ka [0] = 0$ ;
$ka [j - 1] \geq 0$ , for $j = 2, 3, \dots, n$ ;
$ka [n] = nnz$ ;
$0 \leq ka [j] - ka [j - 1] \leq m$ , for $j = 1, 2, \dots, n$ .

13: $bl [n + m]$ – double Input/Output

14: $bu [n + m]$ – double Input/Output

On entry: bl must contain the lower bounds

l

and bu the upper bounds

u

, for all the variables and general constraints, in the following order. The first n elements of bl must contain the bounds on the variables x, the next ncnln elements the bounds for the nonlinear constraints

F (x)

(if any) and the next

(m - ncnln)

elements the bounds for the linear constraints

G x

and the free row (if any). To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j - 1] \leq - options . inf_bound

, and to specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

bu [j - 1] \geq options . inf_bound

, where

options . inf_bound

is one of the optional parameters (default value

10^{20}

, see Section 12.2). To specify the

j

th constraint as an equality, set

bl [j - 1] = bu [j - 1] = β

, say, where

| β | < options . inf_bound

. Note that the lower bound corresponding to

iobj \neq 0

must be set to

- \infty

and stored in

bl [n + | iobj | - 1]

; similarly, the upper bound must be set to

+ \infty

and stored in

bu [n + | iobj | - 1]

On exit: the elements of bl and bu may have been modified internally, but they are restored on exit.

Constraints:

$bl [j] \leq bu [j]$ , for $j = 0, 1, \dots, n + m - 1$ ;
if $bl [j] = bu [j] = β$ , $| β | < options . inf_bound$ ;
if $ncnln < iobj \leq m$ or $iobj = −1$ , $bl [n + | iobj | - 1] \leq - options . inf_bound$ and $bu [n + | iobj | - 1] \geq options . inf_bound$ .

15: $xs [n + m]$ – double Input/Output

On entry:

xs [j - 1]

, for

j = 1, 2, \dots, n

, must contain the initial values of the variables,

x

. In addition, if a ‘warm start’ is specified by means of the optional parameter

options . start

(see Section 12.2) the elements

xs [n + i - 1]

, for

i = 1, 2, \dots, m

, must contain the initial values of the slack variables,

s

On exit: the final values of the variables and slacks

(x, s)

16: $ninf$ – Integer * Output

On exit: the number of constraints that lie outside their bounds by more than the value of the optional parameter

options . minor_feas_tol

(default value

= \sqrt{ε}

If the linear constraints are infeasible, the sum of the infeasibilities of the linear constraints is minimized subject to the upper and lower bounds being satisfied. In this case, ninf contains the number of elements of

G x

that lie outside their upper or lower bounds. Note that the nonlinear constraints are not evaluated.

Otherwise, the sum of the infeasibilities of the nonlinear constraints is minimized subject to the linear constraints and the upper and lower bounds being satisfied. In this case, ninf contains the number of elements of

F (x)

that lie outside their upper or lower bounds.

17: $sinf$ – double * Output

On exit: the sum of the infeasibilities of constraints that lie outside their bounds by more than the value of the optional parameter

options . minor_feas_tol

(default value

= \sqrt{ε}

If the linear constraints are infeasible, sinf contains the sum of the infeasibilities of the linear constraints. Otherwise, sinf contains the sum of the infeasibilities of the nonlinear constraints.

18: $objf$ – double * Output

On exit: the value of the objective function at the final iterate.

19: $comm$ – Nag_Comm * Input/Output

Note: comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

On entry/exit: structure containing pointers for communication to the user-supplied functions objfun and confun; see the description of objfun and confun for details. If you do not need to make use of this communication feature the null pointer NAGCOMM_NULL may be used in the call to e04ugc; comm will then be declared internally for use in calls to user-supplied functions.

20: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04ugc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given below in Section 12. Some of the results returned in options can be used by e04ugc to perform a ‘warm start’ (see the optional parameter

options . start

in Section 12.2).

If any of these optional parameters are required then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to e04ugc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

21: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_CONS: On entry, $njnln = ⟨ value ⟩$ while $ncnln = ⟨ value ⟩$ . These arguments must satisfy $njnln = 0$ when $ncnln = 0$ .
NE_2_INT_OPT_ARG_CONS: On entry, $options . con_check_start = ⟨ value ⟩$ while $options . con_check_stop = ⟨ value ⟩$ . These arguments must satisfy $options . con_check_start \leq options . con_check_stop$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckCon$ or $Nag_CheckObjCon$ .)

On entry, $options . obj_check_start = ⟨ value ⟩$ while $options . obj_check_stop = ⟨ value ⟩$ . These arguments must satisfy $options . obj_check_start \leq options . obj_check_stop$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckObj$ or $Nag_CheckObjCon$ .)
NE_3_INT_ARG_CONS: On entry, $ncnln = ⟨ value ⟩$ , $iobj = ⟨ value ⟩$ and $m = ⟨ value ⟩$ . These arguments must satisfy $ncnln < iobj \leq m$ when $iobj > 0$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_CONS: The contents of array ka are not valid.
Constraint: $0 \leq ka [i + 1] - ka [i] \leq m$ , for $0 \leq i < n$ .

The contents of array ka are not valid.
Constraint: $ka [0] = 0$ .

The contents of array ka are not valid.
Constraint: $ka [n] = nnz$ .
NE_BAD_PARAM: On entry, argument $options . crash$ had an illegal value.

On entry, argument $options . direction$ had an illegal value.

On entry, argument $options . hess_storage$ had an illegal value.

On entry, argument $options . minor_print_level$ had an illegal value.

On entry, argument $options . print_deriv$ had an illegal value.

On entry, argument $options . print_level$ had an illegal value.

On entry, argument $options . start$ had an illegal value.

On entry, argument $options . verify_grad$ had an illegal value.
NE_BASIS_SINGULAR: The basis is singular after 15 attempts to factorize it (and adding slacks when necessary). Either the problem is badly scaled or the value of the optional parameter $options . lu_factor_tol$ (default value $= 5.0$ or 100.0) is too large.
NE_BOUND: The lower bound for variable $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ ) is greater than the upper bound.
NE_BOUND_EQ: The lower bound and upper bound for variable $⟨ value ⟩$ (array elements $bl [⟨ value ⟩]$ and $bu [⟨ value ⟩]$ ) are equal but they are greater than or equal to $options . inf_bound$ .
NE_BOUND_EQ_LCON: The lower bound and upper bound for linear constraint $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ and $bu [⟨ value ⟩]$ ) are equal but they are greater than or equal to $options . inf_bound$ .
NE_BOUND_EQ_NLCON: The lower bound and upper bound for nonlinear constraint $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ and $bu [⟨ value ⟩]$ ) are equal but they are greater than or equal to $options . inf_bound$ .
NE_BOUND_LCON: The lower bound for linear constraint $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ ) is greater than the upper bound.
NE_BOUND_NLCON: The lower bound for nonlinear constraint $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ ) is greater than the upper bound.
NE_CANNOT_CALCULATE: The objective and/or constraint functions could not be calculated.
NE_CON_DERIV_ERRORS: Subroutine confun appears to be giving incorrect gradients.
The user-provided derivatives of the nonlinear constraint functions computed by confun appear to be incorrect. Check that confun has been coded correctly and that all relevant elements of the nonlinear constraint Jacobian have been assigned their correct values.
NE_DUPLICATE_ELEMENT: Duplicate sparse matrix element found in row $⟨ value ⟩$ , column $⟨ value ⟩$ .
NE_INT_ARG_LT: On entry, $iobj = ⟨ value ⟩$ .
Constraint: $iobj \geq −1$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m \geq 1$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .
NE_INT_ARRAY_1: Value $⟨ value ⟩$ given to $ka [⟨ value ⟩]$ not valid. Correct range for elements of ka is $\geq 0$ .
NE_INT_ARRAY_2: Value $⟨ value ⟩$ given to $ha [⟨ value ⟩]$ is not valid. Correct range for elements of ha is 1 to m.
NE_INT_OPT_ARG_GT: On entry, $options . con_check_start = ⟨ value ⟩$ .
Constraint: $options . con_check_start \leq nonln$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckCon$ or $Nag_CheckObjCon$ .)

On entry, $options . con_check_stop = ⟨ value ⟩$ .
Constraint: $options . con_check_stop \leq nonln$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckCon$ or $Nag_CheckObjCon$ .)

On entry, $options . obj_check_start = ⟨ value ⟩$ .
Constraint: $options . obj_check_start \leq nonln$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckObj$ or $Nag_CheckObjCon$ .)

On entry, $options . obj_check_stop = ⟨ value ⟩$ .
Constraint: $options . obj_check_stop \leq nonln$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckObj$ or $Nag_CheckObjCon$ .)
NE_INT_OPT_ARG_LT: On entry, $options . con_check_start = ⟨ value ⟩$ .
Constraint: $options . con_check_start \geq 1$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckCon$ or $Nag_CheckObjCon$ .)

On entry, $options . con_check_stop = ⟨ value ⟩$ .
Constraint: $options . con_check_stop \geq 1$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckCon$ or $Nag_CheckObjCon$ .)

On entry, $options . expand_freq = ⟨ value ⟩$ .
Constraint: $options . expand_freq \geq 0$ .

On entry, $options . factor_freq = ⟨ value ⟩$ .
Constraint: $options . factor_freq \geq 0$ .

On entry, $options . fcheck = ⟨ value ⟩$ .
Constraint: $options . fcheck \geq 0$ .

On entry, $options . obj_check_start = ⟨ value ⟩$ .
Constraint: $options . obj_check_start \geq 1$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckObj$ or $Nag_CheckObjCon$ .)

On entry, $options . obj_check_stop = ⟨ value ⟩$ .
Constraint: $options . obj_check_stop \geq 1$ .
(Note that this error may only occur when $options . verify_grad = Nag_CheckObj$ or $Nag_CheckObjCon$ .)
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_INT_RANGE_1: Value $⟨ value ⟩$ given to ncnln is not valid. Correct range is 0 to m.

Value $⟨ value ⟩$ given to njnln is not valid. Correct range is (when $ncnln > 0$ ) 1 to n.

Value $⟨ value ⟩$ given to nnz is not valid. Correct range is 1 to $n \times m$ .

Value $⟨ value ⟩$ given to nonln is not valid. Correct range is 0 to n.

Value $⟨ value ⟩$ given to $options . hess_freq$ is not valid. Correct range is $options . hess_freq > 0$ .

Value $⟨ value ⟩$ given to $options . hess_update$ is not valid. Correct range is $options . hess_update \geq 0$ .

Value $⟨ value ⟩$ given to $options . iter_lim$ is not valid. Correct range is $options . iter_lim > 0$ .

Value $⟨ value ⟩$ given to $options . major_iter_lim$ is not valid. Correct range is $options . major_iter_lim \geq 0$ .

Value $⟨ value ⟩$ given to $options . max_sb$ is not valid. Correct range is $options . max_sb > 0$ .

Value $⟨ value ⟩$ given to $options . minor_iter_lim$ is not valid. Correct range is $options . minor_iter_lim \geq 0$ .

Value $⟨ value ⟩$ given to $options . nsb$ is not valid. Correct range is: if $options . start = Nag_Warm$ , then $options . nsb \geq 0$ .

Value $⟨ value ⟩$ given to $options . part_price$ is not valid. Correct range is $options . part_price > 0$ .
NE_INVALID_INT_RANGE_2: Value $⟨ value ⟩$ given to $options . scale_opt$ is not valid. Correct range is $0 \leq options . scale_opt \leq 2$ .
NE_INVALID_REAL_RANGE_E: Value $⟨ value ⟩$ given to $options . elastic_wt$ is not valid. Correct range is $options . elastic_wt > 0.0$ .

Value $⟨ value ⟩$ given to $options . inf_bound$ is not valid. Correct range is $options . inf_bound > 0.0$ .

Value $⟨ value ⟩$ given to $options . inf_step$ is not valid. Correct range is $options . inf_step > 0.0$ .

Value $⟨ value ⟩$ given to $options . lu_den_tol$ is not valid. Correct range is $options . lu_den_tol \geq 0.0$ .

Value $⟨ value ⟩$ given to $options . lu_factor_tol$ is not valid. Correct range is $options . lu_factor_tol \geq 1.0$ .

Value $⟨ value ⟩$ given to $options . lu_sing_tol$ is not valid. Correct range is $options . lu_sing_tol > 0.0$ .

Value $⟨ value ⟩$ given to $options . lu_update_tol$ is not valid. Correct range is $options . lu_update_tol \geq 1.0$ .

Value $⟨ value ⟩$ given to $options . major_feas_tol$ is not valid. Correct range is $options . major_feas_tol > ε$ .

Value $⟨ value ⟩$ given to $options . major_opt_tol$ is not valid. Correct range is $options . major_opt_tol > 0.0$ .

Value $⟨ value ⟩$ given to $options . major_step_lim$ is not valid. Correct range is $options . major_step_lim > 0.0$ .

Value $⟨ value ⟩$ given to $options . minor_feas_tol$ is not valid. Correct range is $options . minor_feas_tol > ε$ .

Value $⟨ value ⟩$ given to $options . minor_opt_tol$ is not valid. Correct range is $options . minor_opt_tol > 0.0$ .

Value $⟨ value ⟩$ given to $options . nz_coef$ is not valid. Correct range is $options . nz_coef \geq 1.0$ .

Value $⟨ value ⟩$ given to $options . pivot_tol$ is not valid. Correct range is $options . pivot_tol > 0.0$ .

Value $⟨ value ⟩$ given to $options . unbounded_obj$ is not valid. Correct range is $options . unbounded_obj > 0.0$ .

Value $⟨ value ⟩$ given to $options . violation_limit$ is not valid. Correct range is $options . violation_limit > 0.0$ .
NE_INVALID_REAL_RANGE_EE: Value $⟨ value ⟩$ given to $options . c_diff_int$ is not valid. Correct range is $ε \leq options . c_diff_int < 1.0$ .

Value $⟨ value ⟩$ given to $options . crash_tol$ is not valid. Correct range is $0.0 \leq options . crash_tol < 1.0$ .

Value $⟨ value ⟩$ given to $options . f_diff_int$ is not valid. Correct range is $ε \leq options . f_diff_int < 1.0$ .

Value $⟨ value ⟩$ given to $options . f_prec$ is not valid. Correct range is $ε \leq options . f_prec < 1.0$ .

Value $⟨ value ⟩$ given to $options . linesearch_tol$ is not valid. Correct range is $0.0 \leq options . linesearch_tol < 1.0$ .

Value $⟨ value ⟩$ given to $options . scale_tol$ is not valid. Correct range is $0.0 < options . scale_tol < 1.0$ .
NE_LIN_NOT_FEASIBLE: No feasible point was found for the linear constraints. Sum of infeasibilities: $⟨ value ⟩$ .
The problem is infeasible. The linear constraints cannot all be satisfied to within the values of the optional parameter $options . minor_feas_tol$ (default value $= \sqrt{ε}$ ).
NE_MAYBE_UNBOUNDED: Violation limit exceeded. The problem may be unbounded.
Check the values of the optional parameters $options . unbounded_obj$ (default value $= 10^{15}$ ) and $options . inf_step$ (default value $= \max (bigbnd, 10^{20})$ ) are not too small. This exit also implies that the objective function is not bounded below (or above in the case of maximization) in the feasible region defined by expanding the bounds by the value of the optional parameter $options . violation_limit$ (default value $= 10.0$ ).
NE_NAME_TOO_LONG: The string pointed to by $options . crnames [⟨ value ⟩]$ is too long. It should be no longer than 8 characters.
NE_NO_IMPROVE: The current point cannot be improved on.
Check that objfun and confun have been coded correctly and that they are consistent with the values of the optional parameters $options . obj_deriv$ and $options . con_deriv$ (default value $= Nag_TRUE$ ).
NE_NONLIN_NOT_FEASIBLE: No feasible point was found for the nonlinear constraints. Sum of infeasibilities: $⟨ value ⟩$ .
The problem is infeasible. The nonlinear constraints cannot all be satisfied to within the values of the optional parameter $options . major_feas_tol$ (default value $= \sqrt{ε}$ ).
NE_NOT_APPEND_FILE: Cannot open file $⟨ string ⟩$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $⟨ string ⟩$ .
NE_NOT_REQUIRED_ACC: Feasible solution, but required accuracy could not be achieved.
Check that the value of the optional parameter $options . major_opt_tol$ (default value $= \sqrt{ε}$ ) is not too small.
NE_OBJ_BOUND: Invalid lower bound for objective row. Bound should be $\leq - options . inf_bound$ .

Invalid upper bound for objective row. Bound should be $\geq options . inf_bound$ .
NE_OBJ_DERIV_ERRORS: Subroutine objfun appears to be giving incorrect gradients.
The user-provided derivatives of the objective function computed by objfun appear to be incorrect. Check that objfun has been coded correctly and that all relevant elements of the objective gradient have been assigned their correct values.
NE_OPT_NOT_INIT: Options structure not initialized.
NE_OUT_OF_WORKSPACE: There is insufficient workspace for the basis factors, and the maximum allowed number of reallocation attempts, as specified by max_restart, has been reached.
NE_STATE_VAL: $options . state [⟨ value ⟩]$ is out of range. $options . state [⟨ value ⟩] = ⟨ value ⟩$ .
NE_SUPERBASICS_LIMIT: Too many superbasic variables ( $options . max_sb = ⟨ value ⟩$ ).
The value of the optional parameter $options . max_sb$ (default value $= \min$ (500, $\bar{n} + 1$ ,n)) is too small and should be increased.
NE_TOO_MANY_ITER: Iteration limit ( $options . iter_lim = ⟨ value ⟩$ ) exceeded.
NE_TOO_MANY_MAJOR_ITER: Major iteration limit ( $options . major_iter_lim = ⟨ value ⟩$ ) exceeded.
NE_TOO_MANY_MINOR_ITER: Minor iteration limit ( $options . minor_iter_lim = ⟨ value ⟩$ ) exceeded.
NE_UNBOUNDED: Solution appears to be unbounded.
The problem is unbounded (or badly scaled). The objective function is not bounded below (or above in the case of maximization) in the feasible region because a nonbasic variable can apparently be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Add an upper or lower bound to the variable (whose index is printed by default) and rerun e04ugc.
NE_USER_STOP: User requested termination, user flag value $= ⟨ value ⟩$ .
This exit occurs if you set $comm \to flag$ to a negative value in objfun or confun. If fail is supplied the value of $fail . errnum$ will be the same as your setting of $comm \to flag$ .

7 Accuracy

options . major_feas_tol

is set to

10^{- d}

(default value

= \sqrt{ε}

) and

fail . code = NE_NOERROR

on exit, then the final value of

f (x)

should have approximatively

d

correct digits.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04ugc is not threaded in any implementation.

9 Further Comments

9.1 Termination Criteria

If e04ugc returns with

fail . code = NE_NOERROR

, the iterates have converged to a point

x

that satisfies the first-order Kuhn–Karesh–Tucker conditions (see Section 11.1) to the accuracy requested by the optional parameters

options . major_feas_tol

(default value

= \sqrt{ε}

) and

options . major_opt_tol

(default value

= \sqrt{ε}

10 Example

This example is a reformulation of Problem 74 from Hock and Schittkowski (1981) and involves minimization of the nonlinear function

f (x) = 10^{−6} x_{3}^{3} + \frac{2}{3} \times 10^{−6} x_{4}^{3} + 3 x_{3} + 2 x_{4}

subject to the bounds

\begin{matrix} - 0.55 \leq x_{1} \leq 0.55 \\ - 0.55 \leq x_{2} \leq 0.55 \\ 5 \leq x_{3} \leq 1200 \\ 5 \leq x_{4} \leq 1200 \end{matrix}

to the nonlinear constraints

\begin{matrix} 1000 \sin (- x_{1} - 0.25) + 1000 \sin (- x_{2} - 0.25) - x_{3} & = - 894.8 \\ 1000 \sin (x_{1} - 0.25) + 1000 \sin (x_{1} - x_{2} - 0.25) - x_{4} & = - 894.8 \\ 1000 \sin (x_{2} - 0.25) + 1000 \sin (x_{2} - x_{1} - 0.25) & = - 1294.8 \end{matrix}

and to the linear constraints

\begin{matrix} - x_{1} + x_{2} \geq - 0.55 \\ x_{1} - x_{2} \geq - 0.55 \end{matrix}

The initial point, which is infeasible, is

x_{0} = {(0, 0, 0, 0)}^{T},

and

f (x_{0}) = 0

The optimal solution (to five figures) is

x^{*} = {(0.11887, - 0.39623, 679.94, 1026.0)}^{T},

and

f (x^{*}) = 5126.4

. All the nonlinear constraints are active at the solution.

The use of the interface to e04ugc for this particular example is briefly illustrated below. First, note that because of the constraints on the definitions of nonlinear Jacobian variables and nonlinear objective variables in the interface to e04ugc, the first objective variables

x_{1}

and

x_{2}

are considered as nonlinear objective variables. Thus,

nonln = 4

, and there are

njnln = 2

nonlinear Jacobian variables (

x_{1}

and

x_{2}

). (The alternative would have consisted in reordering the problem to have

nonln = 2

nonlinear objective variables and

njnln = 4

nonlinear constraint variables, but, as mentioned earlier, it is preferable to keep the size of the nonlinear Jacobian

(J)

small, having

nonln > njnln

The Jacobian matrix

A

is the

m = 6 \times n = 4

matrix below

A = (\begin{matrix} conjac [0] & conjac [3] & −1 & 0 \\ conjac [1] & conjac [4] & 0 & −1 \\ conjac [2] & conjac [5] & 0 & 0 \\ −1 & 1 & 0 & 0 \\ 1 & −1 & 0 & 0 \\ 0 & 0 & 3 & 2 \end{matrix}),

where zeros are not stored, each column represents a variable, each row a constraint (except the free row), and the

conjac [i]

entries reflect the structure of the Jacobian

J

corresponding to the nonlinear constraints. The first 3 rows correspond to the

ncnln = 3

nonlinear constraints, rows 4 and 5 define the 2 linear constraints and there is finally an

iobj = 6

th free row defining the linear part of the objective function,

3 x_{3} + 2 x_{4}

A

contains

nnz = 14

nonzero elements of which six entries define the structure of

J

. In this case all entries in

J

are defined in the supplied function confun and there is no constant value that we want to pass only once via

A

, so all entries in the corresponding array a corresponding to

J

can just be initialized to dummy values (here

1.0e+25

). Effective Jacobian values will be provided in the argument

conjac [i - 1]

, for

i = 1, 2, \dots, nnzjac

nnzjac = 6

, in the function confun. Note also that in this simple example,

J

is indeed full; otherwise, the structure of

A

should reflect the sparsity of

J

This example includes source code to store the matrix

A

in the arrays a, ha, ka, based on the simple format from the data file.

Finally, the lower and upper bounds are defined by

bl = {(- 0.55, - 0.55, 0.0, 0.0, - 894.6, - 894.6, - 1294.8, - 0.55, - 0.55, - 1.0e+25)}^{T}

, and

bu = {(0.55, 0.55, 1200.0, 1200.0, - 894.6, - 894.6, - 1294.8, 1.0e+25, 1.0e+25, 1.0e+25)}^{T} .

The first

n = 4

elements of bl and bu are simple bounds on the variables; the next 3 elements are bounds on the nonlinear constraints; the next 2 elements are bounds on the linear constraints; and finally, the last (unbounded) element corresponds to the free row.

The options structure is declared and initialized by e04xxc. The

options . crnames

member is assigned to the array of character strings into which the column and row names were read and the

options . major_iter

member is assigned a value of

100

. Two options are read from the data file by use of e04xyc. Note that, unlike for some other optimization functions, optional parameters to e04ugc are not checked inside e04xyc; they are checked inside the main call to e04ugc.

On return from e04ugc, the solution is perturbed slightly and some further options set, selecting a warm start and a reduced level of printout. e04ugc is then called for a second time. Finally, the memory freeing function e04xzc is used to free the memory assigned by e04ugc to the pointers in the options structure. You must not use the standard C function free() for this purpose.

11 Further Description

e04ugc implements a Sequential Quadratic Programming (SQP) method that obtains search directions from a sequence of Quadratic Programming (QP) subproblems. This section gives a detailed description of the algorithms used by e04ugc. This, and possibly the next section, Section 12, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

11.1 Overview

Here we briefly summarise the main features of the method and introduce some terminology. Where possible, explicit reference is made to the names of variables that are arguments of the function or appear in the printed output. Further details can be found in Gill et al. (2002).

At a solution of (1), some of the constraints will be active, i.e., satisfied exactly. Let

r (x) = (\begin{matrix} x \\ F (x) \\ G x \end{matrix}),

and

G

denote the set of indices of

r (x)

corresponding to active constraints at an arbitrary point

x

. Let

r_{j}^{'} (x)

denote the usual derivative of

r_{j} (x)

, which is the row vector of first partial derivatives of

r_{j} (x)

(see Ortega and Rheinboldt (1970)). The vector

r_{j}^{'} (x)

comprises the

j

th row of

r^{'} (x)

so that

r^{'} (x) = (\begin{matrix} I \\ J (x) \\ G \end{matrix}),

where

J (x)

is the Jacobian of

F (x)

A point

x

is a first-order Kuhn–Karesh–Tucker (KKT) point for (1) (see, e.g., Powell (1974)) if the following conditions hold:

(a) $x$ is feasible;
(b)there exists a vector $λ$ (the Lagrange multiplier vector for the bound and general constraints) such that

$g (x) = r^{'} {(x)}^{T} λ = (\begin{matrix} I & J {(x)}^{T} & G^{T} \end{matrix}) λ,$ (4)

where $g$ is the gradient of $f$ evaluated at $x$ ;
(c)the Lagrange multiplier $λ_{j}$ associated with the $j$ th constraint satisfies $λ_{j} = 0$ if $l_{j} < r_{j} (x) < u_{j}$ ; $λ_{j} \geq 0$ if $l_{j} = r_{j} (x)$ ; $λ_{j} \leq 0$ if $r_{j} (x) = u_{j}$ ; and $λ_{j}$ can have any value if $l_{j} = u_{j}$ .

An equivalent statement of the condition (4) is

Z^{T} g (x) = 0,

where

Z

is a matrix defined as follows. Consider the set

N

of vectors orthogonal to the gradients of the active constraints, i.e.,

N = {z ∣ r_{j}^{'} (x) z = 0 for all j \in G} .

The columns of

Z

may then be taken as any basis for the vector space

N

. The vector

Z^{T} g

is termed the reduced gradient of

f

x

. Certain additional conditions must be satisfied in order for a first-order KKT point to be a solution of (1) (see, e.g., Powell (1974)).

The basic structure of e04ugc involves major and minor iterations. The major iterations generate a sequence of iterates

{x_{k}}

that satisfy the linear constraints and converge to a point

x^{*}

that satisfies the first-order KKT optimality conditions. At each iterate a QP subproblem is used to generate a search direction towards the next iterate

(x_{k + 1})

. The constraints of the subproblem are formed from the linear constraints

G x - s_{L} = 0

and the nonlinear constraint linearization

F (x_{k}) + F^{'} (x_{k}) (x - x_{k}) - s_{N} = 0,

where

F^{'} (x_{k})

denotes the Jacobian matrix, whose rows are the first partial derivatives of

F (x)

evaluated at the point

x_{k}

. The QP constraints, therefore, comprise the

m

linear constraints

\begin{matrix} F^{'} (x_{k}) x & - s_{N} & = - F (x_{k}) + F^{'} (x_{k}) x_{k}, \\ G x & - s_{L} & = 0, \end{matrix}

where

x

and

s = {(s_{N}, s_{L})}^{T}

are bounded above and below by

u

and

l

as before. If the

m \times n

matrix

A

and

m

element vector

b

are defined as

A = (\begin{matrix} F^{'} (x_{k}) \\ G \end{matrix}) and b = (\begin{matrix} - F (x_{k}) + F^{'} (x_{k}) x_{k} \\ 0 \end{matrix}),

then the QP subproblem can be written as

\underset{x, s}{minimize} q (x) subject to A x - s = b, l \leq {\begin{matrix} x \\ s \end{matrix}} \leq u,

(5)

where

q (x)

is a quadratic approximation to a modified Lagrangian function (see Gill et al. (2002)).

The linear constraint matrix

A

is stored in the arrays a, ha and ka (see Section 5). This allows you to specify the sparsity pattern of nonzero elements in

F^{'} (x)

and

G

, and identify any nonzero elements that remain constant throughout the minimization.

Solving the QP subproblem is itself an iterative procedure, with the minor iterations of an SQP method being the iterations of the QP method. At each minor iteration, the constraints

A x - s = b

are (conceptually) partitioned into the form

{B x}_{B} + {S x}_{S} + {N x}_{N} = b,

where the basis matrix

B

is square and nonsingular. The elements of

x_{B}

x_{S}

and

x_{N}

are called the basic, superbasic and nonbasic variables respectively; they are a permutation of the elements of

x

and

s

. At a QP solution, the basic and superbasic variables will lie somewhere between their bounds, while the nonbasic variables will be equal to one of their upper or lower bounds. At each minor iteration,

x_{S}

is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the QP objective function

q (x)

or sum of infeasibilities (as appropriate). The basic variables are then adjusted in order to ensure that

(x, s)

continues to satisfy

A x - s = b

. The number of superbasic variables (

n_{S}

say), therefore, indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms,

n_{S}

is a measure of how nonlinear the problem is. In particular,

n_{S}

will always be zero if there are no nonlinear constraints in (1) and

f (x)

is linear.

If it appears that no improvement can be made with the current definition of

B

S

and

N

, a nonbasic variable is selected to be added to

S

, and the process is repeated with the value of

n_{S}

increased by one. At all stages, if a basic or superbasic variable encounters one of its bounds, the variable is made nonbasic and the value of

n_{S}

decreased by one.

Associated with each of the

m

equality constraints

A x - s = b

is a dual variable

π_{i}

. Similarly, each variable in

(x, s)

has an associated reduced gradient

d_{j}

(also known as a reduced cost). The reduced gradients for the variables

x

are the quantities

g - A^{T} π

, where

g

is the gradient of the QP objective function

q (x)

; and the reduced gradients for the slack variables

s

are the dual variables

π

. The QP subproblem (5) is optimal if

d_{j} \geq 0

for all nonbasic variables at their lower bounds,

d_{j} \leq 0

for all nonbasic variables at their upper bounds and

d_{j} = 0

for other variables (including superbasics). In practice, an approximate QP solution is found by slightly relaxing these conditions on

d_{j}

(see the description of the optional parameter

options . minor_opt_tol

After a QP subproblem has been solved, new estimates of the solution to (1) are computed using a linesearch on the augmented Lagrangian merit function

M (x, s, π) = f (x) - π^{T} (F (x) - s_{N}) + \frac{1}{2} {(F (x) - s_{N})}^{T} D (F (x) - s_{N}),

(6)

where

D

is a diagonal matrix of penalty parameters. If

(x_{k}, s_{k}, π_{k})

denotes the current estimate of the solution and

(\hat{x}, \hat{s}, \hat{π})

denotes the optimal QP solution, the linesearch determines a step

α_{k}

(where

0 < α_{k} \leq 1

) such that the new point

(\begin{matrix} x_{k + 1} \\ s_{k + 1} \\ π_{k + 1} \end{matrix}) = (\begin{matrix} x_{k} \\ s_{k} \\ π_{k} \end{matrix}) + α_{k} (\begin{matrix} {\hat{x}}_{k} - x_{k} \\ {\hat{s}}_{k} - s_{k} \\ {\hat{π}}_{k} - π_{k} \end{matrix})

produces a sufficient decrease in the merit function (6). When necessary, the penalties in

D

are increased by the minimum-norm perturbation that ensures descent for

M

(see Gill et al. (1992)). As in e04ucc,

s_{N}

is adjusted to minimize the merit function as a function of

s

prior to the solution of the QP subproblem. Further details can be found in Eldersveld (1991) and Gill et al. (1986).

11.2 Treatment of Constraint Infeasibilities

e04ugc makes explicit allowance for infeasible constraints. Infeasible linear constraints are detected first by solving a problem of the form

\underset{x, v, w}{minimize} e^{T} (v + w) subject to l \leq {\begin{matrix} x \\ G x - u + w \end{matrix}} \leq u, v \geq 0, w \geq 0,

(7)

where

e = {(1, 1, \dots, 1)}^{T}

. This is equivalent to minimizing the sum of the general linear constraint violations subject to the simple bounds. (In the linear programming literature, the approach is often called elastic programming.)

If the linear constraints are infeasible (i.e.,

v \neq 0

w \neq 0

), the function terminates without computing the nonlinear functions.

If the linear constraints are feasible, all subsequent iterates will satisfy the linear constraints. (Such a strategy allows linear constraints to be used to define a region in which

f (x)

and

F (x)

can be safely evaluated.) The function proceeds to solve (1) as given, using search directions obtained from a sequence of QP subproblems (5). Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to linearized constraints. An augmented Lagrangian merit function (6) is reduced along each search direction to ensure convergence from any starting point.

The function enters ‘elastic’ mode if the QP subproblem proves to be infeasible or unbounded (or if the dual variables

π

for the nonlinear constraints become ‘large’) by solving a problem of the form

\underset{x, v, w}{minimize} \bar{f} (x, v, w) subject to l \leq {\begin{matrix} x \\ F (x) - v + w \\ G x \end{matrix}} \leq u, v \geq 0, w \geq 0,

(8)

where

\bar{f} (x, v, w) = f (x) + γ e^{T} (v + w)

(9)

is called a composite objective and

γ

is a non-negative argument (the elastic weight). If

γ

is sufficiently large, this is equivalent to minimizing the sum of the nonlinear constraint violations subject to the linear constraints and bounds. A similar

l_{1}

formulation of (1) is fundamental to the S

l_{1}

QP algorithm of Fletcher (1984). See also Conn (1973).

12 Optional Parameters

A number of optional input and output arguments to e04ugc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04ugc; the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.

After return from e04ugc, the options structure may only be re-used for future calls of e04ugc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.

Option settings may also be read from a text file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of memory to pointers in the options structure is required, then this must be done directly in the calling program; they cannot be assigned using e04xyc.

12.1 Optional Parameter Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for e04ugc together with their default values where relevant. The number

ε

is a generic notation for machine precision (see X02AJC).

Nag_Start start	$Nag_Cold$
Boolean list	Nag_TRUE
Boolean print_80ch	Nag_TRUE
Nag_PrintType print_level	$Nag_Soln_Iter$
Nag_PrintType minor_print_level	$Nag_NoPrint$
Nag_DPrintType print_deriv	$Nag_D_Print$
char outfile[512]	stdout
char **crnames	NULL
Boolean obj_deriv	Nag_TRUE
Boolean con_deriv	Nag_TRUE
Nag_GradChk verify_grad	$Nag_SimpleCheck$
Integer obj_check_start	1
Integer obj_check_stop	nonln
Integer con_check_start	1
Integer con_check_stop	njnln
double f_diff_int	$\sqrt{options . f_prec}$
double c_diff_int	${options . f_prec}^{0.33}$
Nag_CrashType crash	$Nag_NoCrash$ or $Nag_CrashThreeTimes$
Integer expand_freq	10000
Integer factor_freq	50 or 100
Integer fcheck	60
Integer hess_freq	99999999
Integer hess_update	20
Integer iter_lim	10000
Integer major_iter_lim	1000
Integer minor_iter_lim	500
Integer part_price	1 or 10
Integer scale_opt	1 or 2
Integer max_sb	$\min (500, n, \bar{n} + 1)$
double crash_tol	0.1
double elastic_wt	$1.0$ or 100.0
double f_prec	$ε^{0.8}$
double inf_bound	$10^{20}$
double linesearch_tol	0.9
double lu_den_tol	0.6
double lu_sing_tol	$ε^{0.67}$
double lu_factor_tol	$5.0$ or 100.0
double lu_update_tol	$5.0$ or 10.0
double major_feas_tol	$\sqrt{ε}$
double major_opt_tol	$\sqrt{ε}$
double major_step_lim	2.0
double minor_feas_tol	$\sqrt{ε}$
double minor_opt_tol	$\sqrt{ε}$
double nz_coef	5.0
double pivot_tol	$ε^{0.67}$
double scale_tol	0.9
double unbounded_obj	$10^{15}$
double inf_step	$\max (options . inf_bound, 10^{20})$
double violation_limit	10.0
Boolean deriv_linesearch	Nag_TRUE
Boolean feas_exit	Nag_FALSE
Nag_HessianType hess_storage	$Nag_HessianFull$ or $Nag_HessianLimited$
Nag_DirectionType direction	$Nag_Minimize$
Integer *state	size $n + m$
double *lambda	size $n + m$
Integer iter
Integer major_iter
Integer nsb
Integer nf

12.2 Description of the Optional Parameters

start – Nag_Start

Default

= Nag_Cold

On entry: indicates how a starting basis is to be obtained.

options . start = Nag_Cold

, then an initial Crash procedure will be used to choose an initial basis.

options . start = Nag_Warm

, then you must provide a valid definition of the optional parameters

options . state

and

options . nsb

. (These may be the output of a previous call.)

A warm start will be advantageous if a good estimate of the initial working set is available – for example, when e04ugc is called repeatedly to solve related problems.

Constraint:

options . start = Nag_Cold

Nag_Warm

list – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . list = Nag_TRUE

the argument settings in the call to e04ugc will be printed. The actual options printed can vary depending on the problem type being solved.

print_80ch – Nag_Boolean

Default

= Nag_TRUE

On entry: controls the maximum length of each line of output produced by major and minor iterations and by the printing of the solution.

options . print_80ch = Nag_TRUE

(the default), then a maximum of 80 characters per line is printed.

options . print_80ch = Nag_FALSE

, then a maximum of 120 characters per line is printed.

(See also

options . print_level

and

options . minor_print_level

below.)

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04ugc at each major iteration, as indicated below. A detailed description of the printed output is given in Section 12.3. (See also

options . minor_print_level

, below.)

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution only.
$Nag_Iter$	One line of output for each major iteration (no printout of the final solution).
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Full$	The final solution, one line of output for each major iteration, matrix statistics (initial status of rows and columns, number of elements, density, biggest and smallest elements, etc.), details of the scale factors resulting from the scaling procedure (if $options . scale_opt = 1$ or 2; see below), basis factorization statistics and details of the initial basis resulting from the Crash procedure (if $options . start = Nag_Cold$ and $options . crash \neq Nag_NoCrash$ ).

Note that the output for each line of major iteration and for the solution printout contains a maximum of 80 characters if

options . print_80ch = Nag_TRUE

, and a maximum of 120 characters otherwise. However, if

options . print_level = Nag_Soln_Iter_Full

, some printout may exceed 80 characters even when

options . print_80ch = Nag_TRUE

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

Nag_Soln_Iter_Full

minor_print_level – Nag_PrintType

Default

= Nag_NoPrint

On entry: controls the amount of printout produced by the minor iterations of e04ugc (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in Section 9.1 (default output at each iteration) and in Section 12.3. (See also

options . print_level

options . minor_print_level = Nag_NoPrint

, no output is produced.

options . minor_print_level = Nag_Iter

, the following output is produced for each minor iteration:

if $options . print_80ch = Nag_TRUE$ , one line of summary output ( $\leq 80$ characters);
if $options . print_80ch = Nag_FALSE$ , one long line of output ( $\leq 120$ characters).

Constraint:

options . minor_print_level = Nag_NoPrint

Nag_Iter

print_deriv – Nag_DPrintType

Default

= Nag_D_Print

On entry: controls whether the results of any derivative checking are printed out (see also the optional parameter

options . verify_grad

If a component derivative check has been carried out, then full details will be printed if

options . print_deriv = Nag_D_Print

. If only a simple derivative check is requested,

Nag_D_Print

will produce a statement indicating failure or success. To prevent any printout from a derivative check, set

options . print_deriv = Nag_D_NoPrint

Constraint:

options . print_deriv = Nag_D_NoPrint

Nag_D_Print

outfile – const char[512]

Default

= stdout

On entry: the name of the file to which results should be printed. If

options . outfile [0] =' \0'

then the stdout stream is used.

crnames – char **

Default

=

NULL

On entry: if

options . crnames

is not NULL then it must point to an array of

n + m

character strings with maximum string length

8

, containing the names of the columns and rows (i.e., variables and constraints) of the problem. Thus,

options . crnames [j - 1]

contains the name of the

j

th column (variable), for

j = 1, 2, \dots, n

, and

options . crnames [n + i - 1]

contains the names of the

i

th row (constraint), for

i = 1, 2, \dots, m

. If supplied, the names are used in the solution output (see Section 9.1 and Section 12.3).

Constraint:

options . crnames = NULL ​ or ​ strlen (options . crnames [i - 1]) \leq 8

, for

i = 1, 2, \dots, n + m

obj_deriv – Nag_Boolean

Default

= Nag_TRUE

On entry: this argument indicates whether all elements of the objective gradient are provided in function objfun. If none or only some of the elements are being supplied by objfun then

options . obj_deriv

should be set to Nag_FALSE.

Whenever possible all elements should be supplied, since e04ugc is more reliable and will usually be more efficient when all derivatives are exact.

options . obj_deriv = Nag_FALSE

, e04ugc will approximate unspecified elements of the objective gradient using finite differences. The computation of finite difference approximations usually increases the total run-time, since a call to objfun is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of Gill et al. (2002), for a discussion of limiting accuracy).

At times, central differences are used rather than forward differences, in which case twice as many calls to objfun are needed. (The switch to central differences is not under your control.)

con_deriv – Nag_Boolean

Default

= Nag_TRUE

On entry: this argument indicates whether all elements of the constraint Jacobian are provided in function confun (or possibly directly in a for constant elements). If none or only some of the derivatives are being supplied then

options . con_deriv

should be set to Nag_FALSE.

Whenever possible all elements should be supplied, since e04ugc is more reliable and will usually be more efficient when all derivatives are exact.

options . con_deriv = Nag_FALSE

, e04ugc will approximate unspecified elements of the constraint Jacobian. One call to confun will be needed for each variable for which partial derivatives are not available. For example, if the sparsity of the constraint Jacobian has the form

(\begin{matrix} * & * & * \\ ? & ? \\ * & ? \\ * & * \end{matrix})

where ‘

*

’ indicates a provided element and ‘? ’ indicates an unspecified element, e04ugc will call confun twice: once to estimate the missing element in column

2

, and again once to estimate the two missing elements in column

3

. (Since columns 1 and 4 are known, they require no calls to confun.)

At times, central differences are used rather than forward differences, in which case twice as many calls to confun are needed. (The switch to central differences is not under your control.)

verify_grad – Nag_GradChk

Default

= Nag_SimpleCheck

On entry: specifies the level of derivative checking to be performed by e04ugc on the gradient elements computed by the user-supplied functions objfun and confun. Gradients are verified at the first point that satisfies the linear constraints and the upper and lower bounds. Unspecified gradient elements are not checked, and hence they result in no overhead.

The following values are available:

$Nag_NoCheck$	No derivative checking is performed.
$Nag_SimpleCheck$	Only a cheap test is performed, requiring three calls to objfun and two calls to confun. Note that no checks are carried out if every column of the constraint gradients (Jacobian) contains a missing element.
$Nag_CheckObj$	Individual objective gradient elements will be checked using a reliable (but more expensive) test. If $options . print_deriv = Nag_D_Print$ , a key of the form OK or BAD? indicates whether or not each element appears to be correct.
$Nag_CheckCon$	Individual columns or the constraint gradient (Jacobian) will be checked using a reliable (but more expensive) test. If $options . print_deriv = Nag_D_Print$ , a key of the form OK or BAD? indicates whether or not each element appears to be correct.
$Nag_CheckObjCon$	Check both constraint and objective gradients (in that order) as described for $Nag_CheckCon$ and $Nag_CheckObj$ (respectively).

This component check will be made in the range specified by the optional parameters

options . obj_check_start

and

options . obj_check_stop

for the objective gradient, with default values

1

and nonln, respectively. For the constraint gradient the range is specified by

options . con_check_start

and

options . con_check_stop

, with default values

1

and njnln.

Constraint:

options . verify_grad = Nag_NoCheck

Nag_SimpleCheck

Nag_CheckObj

Nag_CheckCon

Nag_CheckObjCon

obj_check_start – Integer

Default

= 1

obj_check_stop – Integer

Default

= nonln

These options take effect only when

options . verify_grad = Nag_CheckObj

Nag_CheckObjCon

On entry: these arguments may be used to control the verification of gradient elements computed by the function objfun. For example, if the first 30 elements of the objective gradient appear to be correct in an earlier run, so that only element 31 remains questionable, it is reasonable to specify

options . obj_check_start = 31

Constraint:

1 \leq options . obj_check_start \leq options . obj_check_stop \leq nonln

con_check_start – Integer

Default

= 1

con_check_stop – Integer

Default

= njnln

These options take effect only when

options . verify_grad = Nag_CheckCon

Nag_CheckObjCon

On entry: these arguments may be used to control the verification of the Jacobian elements computed by the function confun. For example, if the first 30 columns of the constraint Jacobian appeared to be correct in an earlier run, so that only column 31 remains questionable, it is reasonable to specify

options . con_check_start = 31

Constraint:

1 \leq options . con_check_start \leq options . con_check_stop \leq njnln

f_diff_int – double

Default

= \sqrt{options . f_prec}

This option does not apply when both

options . obj_deriv

and

options . con_deriv

are true.

On entry:

options . f_diff_int

defines an interval used to estimate derivatives by finite differences in the following circumstances:

(a)For verifying the objective and/or constraint gradients (see the description of the optional parameter $options . verify_grad$ ).
(b)For estimating unspecified elements of the objective and/or the constraint Jacobian matrix.

Using the notation

r = options . f_diff_int

, a derivative with respect to

x_{j}

is estimated by perturbing that element of

x

to the value

x_{j} + r (1 + | x_{j} |)

, and then evaluating

f (x)

and/or

F (x)

(as appropriate) at the perturbed point. If the functions are well scaled, the resulting derivative approximation should be accurate to O

(r)

. Judicious alteration of

options . f_diff_int

may sometimes lead to greater accuracy. See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.

Constraint:

ε \leq options . f_diff_int < 1.0

c_diff_int – double

Default

= {options . f_prec}^{0.33}

This option does not apply when both

options . obj_deriv

and

options . con_deriv

are true.

On entry:

options . c_diff_int

is used near an optimal solution in order to obtain more accurate (but more expensive) estimates of gradients. This requires twice as many function evaluations as compared to using forward difference (see the optional parameter

options . f_diff_int

). Using the notation

r = options . c_diff_int

, the interval used for the

j

th variable is

h_{j} = r (1 + | x_{j} |)

. If the functions are well scaled, the resultant gradient estimates should be accurate to O

(r^{2})

. The switch to central differences (not under user-control) is indicated by C at the end of each line of intermediate printout produced by the major iterations (see Section 12.3). The use of finite differences is discussed under the option

options . f_diff_int

Constraint:

ε \leq options . c_diff_int < 1.0

crash – Nag_CrashType

Default

= Nag_NoCrash

Nag_CrashThreeTimes

This option does not apply when

options . start = Nag_Warm

On entry: the default value of

options . crash = Nag_NoCrash

if there are any nonlinear constraints, and

options . crash = Nag_CrashThreeTimes

otherwise.

options . start = Nag_Cold

, an internal Crash procedure is used to select an initial basis from the various rows and columns of the constraint matrix

(\begin{matrix} A & - I \end{matrix})

. The value of

options . crash

determines which rows and columns of

A

are initially eligible for the basis, and how many times the Crash procedure is called. Columns of

- I

are used to pad the basis where necessary. The possible choices for

options . crash

are as follows:

$Nag_NoCrash$	The initial basis contains only slack variables: $B = I$ .
$Nag_CrashOnce$	The Crash procedure is called once (looking for a triangular basis in all rows and columns of $A$ ).
$Nag_CrashTwice$	The Crash procedure is called twice (if there are any nonlinear constraints). The first call looks for a triangular basis in linear rows, and the iteration proceeds with simplex iterations until the linear constraints are satisfied. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (whilst retaining the current basis for linear rows).
$Nag_CrashThreeTimes$	The Crash procedure is called up to three times (if there are any nonlinear constraints). The first two calls treat linear equality constraints and linear inequality constraints separately. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (whilst retaining the current basis for linear rows).

options . crash \neq Nag_NoCrash

, certain slacks on inequality rows are selected for the basis first. (If

options . crash = Nag_CrashTwice

Nag_CrashThreeTimes

, numerical values are used to exclude slacks that are close to a bound.) The Crash procedure then makes several passes through the columns of

A

, searching for a basis matrix that is essentially triangular. A column is assigned to ‘pivot’ on a particular row if the column contains a suitably large element in a row that has not yet been assigned. (The pivot elements ultimately form the diagonals of the triangular basis.) For remaining unassigned rows, slack variables are inserted to complete the basis.

Constraint:

options . crash = Nag_NoCrash

Nag_CrashOnce

Nag_CrashTwice

Nag_CrashThreeTimes

expand_freq – Integer

Default

= 10000

On entry: this option is part of the EXPAND anti-cycling procedure due to Gill et al. (1989), which is designed to make progress even on highly degenerate problems.

For linear models, the strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of

options . minor_feas_tol

(see below) is

δ

. Over a period of

options . expand_freq

iterations, the feasibility tolerance actually used by e04ugc (i.e., the working feasibility tolerance) increases from

0.5

δ

(in steps of

0.5 δ / options . expand_freq

For nonlinear models, the same procedure is used for iterations in which there is only one superbasic variable. (Cycling can only occur when the current solution is at a vertex of the feasible region.) Thus, zero steps are allowed if there is more than one superbasic variable, but otherwise positive steps are enforced.

Increasing the value of

options . expand_freq

helps reduce the number of slightly infeasible nonbasic basic variables (most of which are eliminated during the resetting procedure). However, it also diminishes the freedom to choose a large pivot element (see

options . pivot_tol

below).

options . expand_freq = 0

, the value

99999999

is used and effectively no anti-cycling procedure is invoked.

Constraint:

options . expand_freq \geq 0

factor_freq – Integer

Default

= 50

or 100

On entry:

options . factor_freq

specifies the maximum number of basis changes that will occur between factorizations of the basis matrix. The default value of

options . factor_freq

is 50 if there are any nonlinear constraints, and 100 otherwise.

For linear problems, the basis factors are usually updated at every iteration. The default value

options . factor_freq = 100

is reasonable for typical problems, particularly those that are extremely sparse and well-scaled.

When the objective function is nonlinear, fewer basis updates will occur as the solution is approached. The number of iterations between basis factorizations will, therefore, increase. During these iterations a test is made regularly according to the value of the optional parameter

options . fcheck

(see below) to ensure that the general constraints are satisfied. If necessary, the basis will be refactorized before the limit of

options . factor_freq

updates is reached.

Constraint:

options . factor_freq \geq 0

fcheck – Integer

Default

= 60

On entry: every

options . fcheck

th iteration after the most recent basis iteration, a numerical test is made to see if the current solution

(x, s)

satisfies the general linear constraints (including any linearized nonlinear constraints). The constraints are of the form

A x - s = b

, where

s

is the set of slack variables. If the largest element of the residual vector

r = b - A x + s

is judged to be too large, the current basis is refactorized and the basic variables recomputed to satisfy the general constraints more accurately. If

options . fcheck = 0

, the value

options . fcheck = 99999999

is used and effectively no checks are made.

Constraint:

options . fcheck \geq 0

hess_freq – Integer

Default

= 99999999

This option only takes effect when

options . hess_storage = Nag_HessianFull

On entry: this option forces the approximate Hessian formed from

options . hess_freq

BFGS updates to be reset to the identity matrix upon completion of a major iteration.

Constraint:

options . hess_freq > 0

hess_update – Integer

Default

= 20

This option only takes effect when

options . hess_storage = Nag_HessianLimited

On entry: if

options . hess_storage = Nag_HessianLimited

(see below), this option defines the maximum number of pairs of Hessian update vectors that are to be used to define the quasi-Newton approximate Hessian. Once the limit of

options . hess_update

updates is reached, all but the diagonal elements of the accumulated updates are discarded and the process starts again. Broadly speaking, the more updates that are stored, the better the quality of the approximate Hessian. On the other hand, the more vectors that are stored, the greater the cost of each QP iteration.

The default value of

options . hess_update

is likely to give a robust algorithm without significant expense, but faster convergence may often be obtained with far fewer updates (e.g.,

options . hess_update = 5

Constraint:

options . hess_update \geq 0

iter_lim – Integer

Default

= 10000

On entry: specifies the maximum number of minor iterations allowed (i.e., iterations of the simplex method or the QP algorithm), summed over all major iterations. (See also

options . major_iter_lim

and

options . minor_iter_lim

below.)

Constraint:

options . iter_lim > 0

major_iter_lim – Integer

Default

= 1000

On entry: specifies the maximum number of major iterations allowed before termination. It is intended to guard against an excessive number of linearizations of the nonlinear constraints. Setting

options . major_iter_lim = 0

and

options . print_deriv = Nag_D_Print

means that the objective and constraint gradients will be checked if

options . verify_grad \neq Nag_NoCheck

, but no iterations will be performed.

Constraint:

options . major_iter_lim \geq 0

minor_iter_lim – Integer

Default

= 500

On entry: specifies the maximum number of iterations allowed between successive linearizations of the nonlinear constraints. A value in the range

10 \leq i \leq 50

prevents excessive effort being expended on early major iterations, but allows later QP subproblems to be solved to completion. Note that an extra

m

minor iterations are allowed if the first QP subproblem to be solved starts with the all-slack basis

B = I

. (See

options . crash

In general, it is unsafe to specify values as small as 1 or 2 (because even when an optimal solution has been reached, a few minor iterations may be needed for the corresponding QP subproblem to be recognised as optimal).

Constraint:

options . minor_iter_lim \geq 0

part_price – Integer

Default

= 1

or 10

On entry: this option is recommended for large problems that have significantly more variables than constraints (i.e.,

n ≫ m

). The default value of

options . part_price

is 1 if there are any nonlinear constraints, and 10 otherwise. It reduces the work required for each ‘pricing’ operation (i.e., when a nonbasic variable is selected to become superbasic). The possible choices for

options . part_price

are the following.

options . part_price

Meaning

1

All columns of the constraint matrix

(\begin{matrix} A & - I \end{matrix})

are searched.

\geq 2

Both

A

and

I

are partitioned to give

options . part_price

roughly equal segments

A_{j}, I_{j}

, for

j = 1, 2, \dots, p

(modulo

p

). If the previous pricing search was successful on

A_{j}, I_{j}

, the next search begins on the segments

A_{j + 1}, I_{j + 1}

. If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to enter the basis. If nothing is found, the search continues on the next segments

A_{j + 2}, I_{j + 2}

, and so on.

Constraint:

options . part_price > 0

scale_opt – Integer

Default

= 1

or 2

On entry: the default value of

options . scale_opt

is 1 if there are any nonlinear constraints, and 2 otherwise. This option enables you to scale the variables and constraints using an iterative procedure due to Fourer (1982), which attempts to compute row scales

r_{i}

and column scales

c_{j}

such that the scaled matrix coefficients

{\bar{a}}_{i j} = a_{i j} \times (c_{j} / r_{i})

are as close as possible to unity. (The lower and upper bounds on the variables and slacks for the scaled problem are redefined as

{\bar{l}}_{j} = l_{j} / c_{j}

and

{\bar{u}}_{j} = u_{j} / c_{j}

respectively, where

c_{j} \equiv r_{j - n}

j > n

.) The possible choices for

options . scale_opt

are the following.

$options . scale_opt$	Meaning
0	No scaling is performed. This is recommended if it is known that the elements of $x$ and the constraint matrix $A$ (along with its Jacobian) never become large (say, $> 1000$ ).
1	All linear constraints and variables are scaled. This may improve the overall efficiency of the function on some problems.
2	All constraints and variables are scaled. Also, an additional scaling is performed that takes into account columns of $(\begin{matrix} A & - I \end{matrix})$ that are fixed or have positive lower bounds or negative upper bounds. If there are any nonlinear constraints present, the scale factors depend on the Jacobian at the first point that satisfies the linear constraints and the upper and lower bounds. The setting $options . scale_opt = 2$ should, therefore, be used only if a ‘good’ starting point is available and the problem is not highly nonlinear.

Constraint:

options . scale_opt = 0

1

2

max_sb – Integer

Default

= \min (500, n, \bar{n} + 1)

This option does not apply to linear problems.

On entry:

options . max_sb

places a limit on the storage allocated for superbasic variables. Ideally, the value of

options . max_sb

should be set slightly larger than the ‘number of degrees of freedom’ expected at the solution.

For nonlinear problems, the number of degrees of freedom is often called the ‘number of independent variables’. Normally, the value of

options . max_sb

need not be greater than

\bar{n} + 1

(where

\bar{n} = \max (nonln, njnln)

), but for many problems it may be considerably smaller. (This will save storage if

\bar{n}

is very large.)

Constraint:

options . max_sb > 0

crash_tol – double

Default

= 0.1

On entry: this option allows the Crash procedure to ignore certain ‘small’ nonzero elements in the columns of

A

while searching for a triangular basis. If

a_{\max}

is the largest element in the

j

th column, other nonzeros

a_{i j}

in the column are ignored if

| a_{i j} | \leq a_{\max} \times options . crash_tol

When

options . crash_tol > 0

, the basis obtained by the Crash procedure may not be strictly triangular, but it is likely to be nonsingular and almost triangular. The intention is to obtain a starting basis containing more columns of

A

and fewer (arbitrary) slacks. A feasible solution may be reached earlier on some problems.

Constraint:

0.0 \leq options . crash_tol < 1.0

elastic_wt – double

Default

= 1.0

or 100.0

On entry: this option defines the initial weight

γ

associated with problem (8). The default value of

options . elastic_wt

100.0

if there are any nonlinear constraints, and

1.0

otherwise.

At any given major iteration

k

, elastic mode is entered if the QP subproblem is infeasible or the QP dual variables (Lagrange multipliers) are larger in magnitude than

options . elastic_wt \times (1 + {‖ g (x_{k}) ‖}_{2})

, where

g

is the objective gradient. In either case, the QP subproblem is re-solved in elastic mode with

γ = options . elastic_wt \times (1 + {‖ g (x_{k}) ‖}_{2})

Constraint:

options . elastic_wt \geq 0.0

f_prec – double

Default

= ε^{0.8}

On entry: this option defines the relative function precision

ε_{R}

, which is intended to be a measure of the relative accuracy with which the nonlinear functions can be computed. For example, if

f (x)

(or

F_{i} (x)

) is computed as

1000.56789

for some relevant

x

and the first 6 significant digits are known to be correct, the appropriate value for

ε_{R}

would be

10^{−6}

Ideally the functions

f (x)

F_{i} (x)

should have magnitude of order

1

. If all functions are substantially less than 1 in magnitude,

ε_{R}

should be the absolute precision. For example, if

f (x)

(or

F_{i} (x)

) is computed as

1.23456789 \times 10^{−4}

for some relevant

x

and the first 6 significant digits are known to be correct, the appropriate value for

ε_{R}

would be

10^{−10}

The choice of

ε_{R}

can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy.

In some cases the function values will be the result of extensive computation, possibly involving an iterative procedure that can provide few digits of precision at reasonable cost. Specifying an appropriate value of

options . f_prec

may, therefore, lead to savings, by allowing the linesearch procedure to terminate when the difference between function values along the search direction becomes as small as the absolute error in the values.

Constraint:

ε \leq options . f_prec < 1.0

inf_bound – double

Default

= 10^{20}

On entry:

options . inf_bound

defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to

options . inf_bound

will be regarded as

+ \infty

(and similarly any lower bound less than or equal to

- options . inf_bound

will be regarded as

- \infty

Constraint:

options . inf_bound > 0.0

linesearch_tol – double

Default

= 0.9

On entry: this option controls the accuracy with which a steplength will be located along the direction of search at each iteration. At the start of each linesearch a target directional derivative for the Lagrangian merit function is identified. The value of

options . linesearch_tol

, therefore, determines the accuracy to which this target value is approximated.

The default value

options . linesearch_tol = 0.9

requests an inaccurate search, and is appropriate for most problems, particularly those with any nonlinear constraints.

If the nonlinear functions are cheap to evaluate, a more accurate search may be appropriate; try

options . linesearch_tol = 0.1

0.01

0.001

. The number of major iterations required to solve the problem might decrease.

If the nonlinear functions are expensive to evaluate, a less accurate search may be appropriate. If all derivatives are available (

options . con_deriv

and

options . obj_deriv

are both true), try

options . linesearch_tol = 0.99

. (The number of major iterations required to solve the problem might increase, but the total number of function evaluations may decrease enough to compensate.)

If some derivatives are not available (at least one of

options . con_deriv

options . obj_deriv

is false), a moderately accurate search may be appropriate; try

options . linesearch_tol = 0.5

. Each search will (typically) require only

1 - 5

function values, but many function calls will then be needed to estimate the missing gradients for the next iteration.

Constraint:

0.0 \leq options . linesearch_tol < 1.0

lu_den_tol – double

Default

= 0.6

On entry: this option defines the density tolerance used during the

L U

factorization of the basis matrix. Columns of

L

and rows of

U

are formed one at a time, and the remaining rows and columns of the basis are altered appropriately. At any stage, if the density of the remaining matrix exceeds

options . lu_den_tol

, the Markowitz strategy for choosing pivots is terminated. The remaining matrix is then factorized using a dense

L U

procedure. Increasing the value of

options . lu_den_tol

towards unity may give slightly sparser

L U

factors, with a slight increase in factorization time.

Constraint:

options . lu_den_tol \geq 0.0

lu_sing_tol – double

Default

= ε^{0.67}

On entry: this option defines the singularity tolerance used to guard against ill-conditioned basis matrices. Whenever the basis is refactorized, the diagonal elements of

U

are tested as follows. If

| u_{j j} | \leq options . lu_sing_tol

| u_{j j} | < options . lu_sing_tol \times \max_{i} | u_{i j} |

, the

j

th column of the basis is replaced by the corresponding slack variable. This is most likely to occur when

options . start = Nag_Warm

, or at the start of a major iteration.

In some cases, the Jacobian matrix may converge to values that make the basis exactly singular (e.g., a whole row of the Jacobian matrix could be zero at an optimal solution). Before exact singularity occurs, the basis could become very ill-conditioned and the optimization could progress very slowly (if at all). Setting

options . lu_sing_tol = 0.00001

(say) may, therefore, help cause a judicious change of basis in such situations.

Constraint:

options . lu_sing_tol > 0.0

lu_factor_tol – double

Default

= 5.0

or 100.0

lu_update_tol – double

Default

= 5.0

or 10.0

On entry:

options . lu_factor_tol

and

options . lu_update_tol

affect the stability and sparsity of the basis factorization

B = L U

, during refactorization and updates respectively. The default values are

options . lu_factor_tol = options . lu_update_tol = 5.0

if there are any nonlinear constraints, and

options . lu_factor_tol = 100.0

and

options . lu_update_tol = 10.0

otherwise.

The lower triangular matrix

L

can be seen as a product of matrices of the form

(\begin{matrix} 1 \\ μ & 1 \end{matrix})

where the multipliers

μ

satisfy

| μ | < options . lu_factor_tol

during refactorization or

| μ | < options . lu_update_tol

during update. The default values of

options . lu_factor_tol

and

options . lu_update_tol

usually strike a good compromise between stability and sparsity. Smaller values of

options . lu_factor_tol

and

options . lu_update_tol

favour stability, while larger values favour sparsity. For large and relatively dense problems, setting

options . lu_factor_tol

10.0

5.0

(say) may give a marked improvement in sparsity without impairing stability to a serious degree. Note that for problems involving band matrices it may be necessary to reduce

options . lu_factor_tol

and/or

options . lu_update_tol

in order to achieve stability.

Constraints:

$options . lu_factor_tol \geq 1.0$ ;
$options . lu_update_tol \geq 1.0$ .

major_feas_tol – double

Default

= \sqrt{ε}

On entry: this option specifies how accurately the nonlinear constraints should be satisfied. The default value is appropriate when the linear and nonlinear constraints contain data to approximately that accuracy. A larger value may be appropriate if some of the problem functions are known to be of low accuracy.

Let rowerr be defined as the maximum nonlinear constraint violation normalized by the size of the solution. It is required to satisfy

rowerr = \max_{i} \frac{{viol}_{i}}{‖ (x, s) ‖} \leq options . major_feas_tol,

where

{viol}_{i}

is the violation of the

i

th nonlinear constraint.

Constraint:

options . major_feas_tol > ε

major_opt_tol – double

Default

= \sqrt{ε}

On entry: this option specifies the final accuracy of the dual variables. If e04ugc terminates with

fail . code = NE_NOERROR

, a primal and dual solution

(x, s, π)

will have been computed such that

maxgap = \max_{j} \frac{{g a p}_{j}}{‖ π ‖} \leq options . major_opt_tol,

where

{g a p}_{j}

is an estimate of the complementarity gap for the

j

th variable and

‖ π ‖

is a measure of the size of the QP dual variables (or Lagrange multipliers) given by

‖ π ‖ = \max (\frac{σ}{\sqrt{m}}, 1),   where σ = \sum_{i = 1}^{m} | π_{i} | .

It is included to make the tests independent of a scale factor on the objective function. Specifically,

{g a p}_{j}

is computed from the final QP solution using the reduced gradients

d_{j} = g_{j} - π^{T} a_{j}

, where

g_{j}

is the

j

th element of the objective gradient and

a_{j}

is the associated column of the constraint matrix

(\begin{matrix} A & - I \end{matrix})

{g a p}_{j} = {\begin{matrix} d_{j} \min (x_{j} - l_{j}, 1) & if d_{j} \geq 0; \\ - d_{j} \min (u_{j} - x_{j}, 1) & if d_{j} < 0 . \end{matrix}

Constraint:

options . major_opt_tol > 0.0

major_step_lim – double

Default

= 2.0

On entry: this option limits the change in

x

during a linesearch. It applies to all nonlinear problems once a ‘feasible solution’ or ‘feasible subproblem’ has been found.

A linesearch determines a step

α

in the interval

0 < α \leq β

, where

β = 1

if there are any nonlinear constraints, or the step to the nearest upper or lower bound on

x

if all the constraints are linear. Normally, the first step attempted is

α_{1} = \min (1, β)

In some cases, such as

f (x) = {a e}^{b x}

f (x) = {a x}^{b}

, even a moderate change in the elements of

x

can lead to floating-point overflow. The optional parameter

options . major_step_lim

is, therefore, used to define a step limit

\bar{β}

given by

\bar{β} = \frac{options . major_step_lim (1 + {‖ x ‖}_{2})}{{‖ p ‖}_{2}},

where

p

is the search direction and the first evaluation of

f (x)

is made at the (potentially) smaller step length

α_{1} = \min (1, \bar{β}, β)

Wherever possible, upper and lower bounds on

x

should be used to prevent evaluation of nonlinear functions at meaningless points.

options . major_step_lim

provides an additional safeguard. The default value

options . major_step_lim = 2.0

should not affect progress on well-behaved functions, but values such as

options . major_step_lim = 0.1

0.01

may be helpful when rapidly varying functions are present. If a small value of

options . major_step_lim

is selected, a ‘good’ starting point may be required. An important application is to the class of nonlinear least squares problems.

Constraint:

options . major_step_lim > 0.0

minor_feas_tol – double

Default

= \sqrt{ε}

On entry: this option attempts to ensure that all variables eventually satisfy their upper and lower bounds to within the tolerance

options . minor_feas_tol

. Since this includes slack variables, general linear constraints should also be satisfied to within

options . minor_feas_tol

. Note that feasibility with respect to nonlinear constraints is judged by the value of

options . major_feas_tol

and not by

options . minor_feas_tol

If the bounds and linear constraints cannot be satisfied to within

options . minor_feas_tol

, the problem is declared infeasible. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise

options . minor_feas_tol

by a factor of 10 or

100

. Otherwise, some error in the data should be suspected.

options . scale_opt \geq 1

, feasibility is defined in terms of the scaled problem (since it is more likely to be meaningful).

Nonlinear functions will only be evaluated at points that satisfy the bounds and linear constraints. If there are regions where a function is undefined, every effort should be made to eliminate these regions from the problem. For example, if

f (x_{1}, x_{2}) = \sqrt{x_{1}} + \log (x_{2})

, it is essential to place lower bounds on both

x_{1}

and

x_{2}

. If the value

options . minor_feas_tol = 10^{−6}

is used, the bounds

x_{1} \geq 10^{−5}

and

x_{2} \geq 10^{−4}

might be appropriate. (The log singularity is more serious; in general, you should attempt to keep

x

as far away from singularities as possible.)

In reality,

options . minor_feas_tol

is used as a feasibility tolerance for satisfying the bounds on

x

and

s

in each QP subproblem. If the sum of infeasibilities cannot be reduced to zero, the QP subproblem is declared infeasible and the function is then in elastic mode thereafter (with only the linearized nonlinear constraints defined to be elastic). (See also

options . elastic_wt

Constraint:

options . minor_feas_tol > ε

minor_opt_tol – double

Default

= \sqrt{ε}

On entry: this option is used to judge optimality for each QP subproblem. Let the QP reduced gradients be

d_{j} = g_{j} - π^{T} a_{j}

, where

g_{j}

is the

j

th element of the QP gradient,

a_{j}

is the associated column of the QP constraint matrix and

π

is the set of QP dual variables.

By construction, the reduced gradients for basic variables are always zero. The QP subproblem will be declared optimal if the reduced gradients for nonbasic variables at their upper or lower bounds satisfy

\frac{d_{j}}{‖ π ‖} \geq - options . minor_opt_tol or \frac{d_{j}}{‖ π ‖} \leq options . minor_opt_tol

respectively, and if

\frac{| d_{j} |}{‖ π ‖} \leq options . minor_opt_tol

for superbasic variables.

Note that

‖ π ‖

is a measure of the size of the dual variables. It is included to make the tests independent of a scale factor on the objective function. (The value of

‖ π ‖

actually used is defined in the description of the optional parameter

options . major_opt_tol

If the objective is scaled down to be very small, the optimality test reduces to comparing

d_{j}

against

options . minor_opt_tol

Constraint:

options . minor_opt_tol > 0.0

nz_coef – double

Default

= 5.0

This option is ignored if

options . hess_storage = Nag_HessianFull

On entry:

options . nz_coef

defines how much memory is initially allocated for the basis factors: by default, e04ugc allocates approximatively

nnz \times options . nz_coef

reals and

2 \times nnz \times options . nz_coef

integers in order to compute and store the basis factors. If at some point this appears not to be enough, an internal warm restart with more memory is automatically attempted, so that e04ugc should complete anyway. Thus this option generally does not need to be modified.

However, if a lot of memory is available, it is possible to increase the value of

options . nz_coef

such as to limit the number of compressions of the work space and possibly avoid internal restarts. On the other hand, for large problems where memory might be critical, decreasing the value of

options . nz_coef

can sometimes save some memory.

Constraint:

options . nz_coef \geq 1.0

pivot_tol – double

Default

= ε^{0.67}

On entry: this option is used during the solution of QP subproblems to prevent columns entering the basis if they would cause the basis to become almost singular.

When

x

changes to

x + α p

for some specified search direction

p

, a ‘ratio test’ is used to determine which element of

x

reaches an upper or lower bound first. The corresponding element of

p

is called the pivot element. Elements of

p

are ignored (and, therefore, cannot be pivot elements) if they are smaller than

options . pivot_tol

It is common in practice for two (or more) variables to reach a bound at essentially the same time. In such cases, the optional parameter

options . minor_feas_tol

provides some freedom to maximize the pivot element and thereby improve numerical stability. Excessively small values of

options . minor_feas_tol

should, therefore, not be specified. To a lesser extent, the optional parameter

options . expand_freq

also provides some freedom to maximize the pivot element. Excessively large values of

options . expand_freq

should, therefore, not be specified.

Constraint:

options . pivot_tol > 0.0

scale_tol – double

Default

= 0.9

On entry: this option is used to control the number of scaling passes to be made through the constraint matrix

A

. At least 3 (and at most 10) passes will be made. More precisely, let

a_{p}

denote the largest column ratio (i.e.,

\frac{' biggest' ​ element}{' smallest' ​ element}

in some sense) after the

p

th scaling pass through

A

. The scaling procedure is terminated if

a_{p} \geq a_{p - 1} \times options . scale_tol

for some

p \geq 3

. Thus, increasing the value of

options . scale_tol

from

0.9

0.99

(say) will probably increase the number of passes through

A

Constraint:

0.0 < options . scale_tol < 1.0

unbounded_obj – double

Default

= 10^{15}

inf_step – double

Default

= \max (options . inf_bound, 10^{20})

On entry: these options are intended to detect unboundedness in nonlinear problems. During the linesearch, the objective function

f

is evaluated at points of the form

x + α p

, where

x

and

p

are fixed and

α

varies. If

| f |

exceeds

options . unbounded_obj

α

exceeds

options . inf_step

, the iterations are terminated and the function returns with

fail . code = NE_MAYBE_UNBOUNDED

If singularities are present, unboundedness in

f (x)

may manifest itself by a floating-point overflow during the evaluation of

f (x + α p)

, before the test against

options . unbounded_obj

can be made.

Unboundedness in

x

is best avoided by placing finite upper and lower bounds on the variables.

Constraints:

$options . unbounded_obj > 0.0$ ;
$options . inf_step > 0.0$ .

violation_limit – double

Default

= 10.0

On entry: this option defines an absolute limit on the magnitude of the maximum constraint violation after the linesearch. Upon completion of the linesearch, the new iterate

x_{k + 1}

satisfies the condition

v_{i} (x_{k + 1}) \leq options . violation_limit \times \max (1, v_{i} (x_{0})),

where

x_{0}

is the point at which the nonlinear constraints are first evaluated and

v_{i} (x)

is the

i

th nonlinear constraint violation

v_{i} (x) = \max (0, l_{i - F} {i - F}_{i} (x), F_{i} (x) - u_{i})

The effect of the violation limit is to restrict the iterates to lie in an expanded feasible region whose size depends on the magnitude of

options . violation_limit

. This makes it possible to keep the iterates within a region where the objective function is expected to be well-defined and bounded below (or above in the case of maximization). If the objective function is bounded below (or above in the case of maximization) for all values of the variables, then

options . violation_limit

may be any large positive value.

Constraint:

options . violation_limit > 0.0

deriv_linesearch – Nag_Boolean

Default

= Nag_TRUE

On entry: at each major iteration, a linesearch is used to improve the value of the Lagrangian merit function (6). The default linesearch uses safeguarded cubic interpolation and requires both function and gradient values in order to compute estimates of the step

α_{k}

. If some analytic derivatives are not provided or

options . deriv_linesearch = Nag_FALSE

is specified, a linesearch based upon safeguarded quadratic interpolation (which does not require the evaluation or approximation of any gradients) is used instead.

A nonderivative linesearch can be slightly less robust on difficult problems, and it is recommended that the default be used if the functions and their derivatives can be computed at approximately the same cost. If the gradients are very expensive to compute relative to the functions however, a nonderivative linesearch may result in a significant decrease in the total run-time.

options . deriv_linesearch = Nag_FALSE

is selected, e04ugc signals the evaluation of the linesearch by calling objfun and confun with

comm \to flag = 0

. Once the linesearch is complete, the nonlinear functions are re-evaluated with

comm \to flag = 2

. If the potential savings offered by a nonderivative linesearch are to be fully realised, it is essential that objfun and confun be coded so that no derivatives are computed when

comm \to flag = 0

Constraint:

options . deriv_linesearch = Nag_TRUE ​ or ​ Nag_FALSE

feas_exit – Nag_Boolean

Default

= Nag_FALSE

This option is ignored if the value of

options . major_iter_lim

is exceeded, or the linear constraints are infeasible.

On entry: if termination is about to occur at a point that does not satisfy the nonlinear constraints and

options . feas_exit = Nag_TRUE

is selected, this option requests that additional iterations be performed in order to find a feasible point (if any) for the nonlinear constraints. This involves solving a feasible point problem in which the objective function is omitted.

Otherwise, this option requests no additional iterations be performed.

Constraint:

options . feas_exit = Nag_TRUE ​ or ​ Nag_FALSE

hess_storage – Nag_HessianType

Default

= Nag_HessianFull

Nag_HessianLimited

On entry: this option specifies the method for storing and updating the quasi-Newton approximation to the Hessian of the Lagrangian function. The default is

Nag_HessianFull

if the number of nonlinear variables

\bar{n}

(

= \max (nonln, njnln)

)

< 75

, and

Nag_HessianLimited

otherwise.

options . hess_storage = Nag_HessianFull

, the approximate Hessian is treated as a dense matrix, and BFGS quasi-Newton updates are applied explicitly. This is most efficient when the total number of nonlinear variables is not too large (say,

\bar{n} < 75

). In this case, the storage requirement is fixed and you can expect one-step Q-superlinear convergence to the solution.

options . hess_storage = Nag_HessianLimited

should only be specified when

\bar{n}

is very large. In this case a limited memory procedure is used to update a diagonal Hessian approximation

H_{r}

a limited number of times. (Updates are accumulated as a list of vector pairs. They are discarded at regular intervals after

H_{r}

has been reset to their diagonal.)

Note that if

options . hess_freq = 20

is used in conjunction with

options . hess_storage = Nag_HessianFull

, the effect will be similar to using

options . hess_storage = Nag_HessianLimited

in conjunction with

options . hess_update = 20

, except that the latter will retain the current diagonal during resets.

Constraint:

options . hess_storage = Nag_HessianLimited

Nag_HessianFull

direction – Nag_DirectionType

Default

= Nag_Minimize

On entry: if

options . direction = Nag_FeasiblePoint

, e04ugc attempts to find a feasible point (if any) for the nonlinear constraints by omitting the objective function. It can also be used to check whether the nonlinear constraints are feasible.

Otherwise,

options . direction

specifies the direction of optimization. It applies to both linear and nonlinear terms (if any) in the objective function. Note that if two problems are the same except that one minimizes

f (x)

and the other maximizes

- f (x)

, their solutions will be the same but the signs of the dual variables

π_{i}

and the reduced gradients

d_{j}

will be reversed.

Constraint:

options . direction = Nag_FeasiblePoint

Nag_Minimize

Nag_Maximize

state – Integer *

Default memory

= n + m

On entry:

options . state

need not be set if the default option of

options . start = Nag_Cold

is used as

n + m

values of memory will be automatically allocated by e04ugc.

If the optional parameter

options . start = Nag_Warm

has been chosen,

options . state

must point to a minimum of

n + m

elements of memory. This memory will already be available if the options structure has been used in a previous call to e04ugc from the calling program, with

options . start = Nag_Cold

and the same values of n and m. If a previous call has not been made you must allocate sufficient memory.

If you supply a

options . state

vector and

options . start = Nag_Cold

, then the first n elements of

options . state

must specify the initial states of the problem variables. (The slacks

s

need not be initialized.) An internal Crash procedure is then used to select an initial basis matrix

B

. The initial basis matrix will be triangular (neglecting certain small elements in each column). It is chosen from various rows and columns of

(\begin{matrix} A & - I \end{matrix})

. Possible values for

options . state [j - 1]

, for

j = 1, 2, \dots, n

, are:

$options . state [j - 1]$	State of $xs [j - 1]$ during Crash procedure
0 or 1	Eligible for the basis
2	Ignored
3	Eligible for the basis (given preference over 0 or 1)
4 or 5	Ignored

If nothing special is known about the problem, or there is no wish to provide special information, you may set

options . state [j - 1] = 0

(and

xs [j - 1] = 0.0

), for

j = 1, 2, \dots, n

. All variables will then be eligible for the initial basis. Less trivially, to say that the

j

th variable will probably be equal to one of its bounds, you should set

options . state [j - 1] = 4

and

xs [j - 1] = bl [j - 1]

options . state [j - 1] = 5

and

xs [j - 1] = bu [j - 1]

as appropriate.

Following the Crash procedure, variables for which

options . state [j - 1] = 2

are made superbasic. Other variables not selected for the basis are then made nonbasic at the value

xs [j - 1]

bl [j - 1] \leq xs [j - 1] \leq bu [j - 1]

, or at the value

bl [j - 1]

bu [j - 1]

closest to

xs [j - 1]

When

options . start = Nag_Warm

options . state

and xs must specify the initial states and values, respectively, of the variables and slacks

(x, s)

. If e04ugc has been called previously with the same values of n and m,

options . state

already contains satisfactory information.

Constraints:

if $options . start = Nag_Cold$ , $0 \leq options . state [j - 1] \leq 5$ , for $j = 1, 2, \dots, n$ ;
if $options . start = Nag_Warm$ , $0 \leq options . state [j - 1] \leq 3$ , for $j = 1, 2, \dots, n + m$ .

On exit: the final states of the variables and slacks

(x, s)

. The significance of each possible value of

options . state

is as follows:

$options . state [j - 1]$	State of variable $j$	Normal value of $xs [j - 1]$
0	Nonbasic	$bl [j - 1]$
1	Nonbasic	$bu [j - 1]$
2	Superbasic	Between $bl [j - 1]$ and $bu [j - 1]$
3	Basic	Between $bl [j - 1]$ and $bu [j - 1]$

If the problem is feasible (i.e.,

ninf = 0

), basic and superbasic variables may be outside their bounds by as much as the optional parameter

options . minor_feas_tol

. Note that unless the optional parameter

options . scale_opt = 0

options . minor_feas_tol

applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as

0.1

outside their bounds, but this is unlikely unless the problem is very badly scaled.

Very occasionally some nonbasic variables may be outside their bounds by as much as

options . minor_feas_tol

, and there may be some nonbasic variables for which

xs [j - 1]

lies strictly between its bounds.

If the problem is infeasible (i.e.,

ninf > 0

), some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used

(options . scale_opt = 0)

lambda – double *

Default memory

= n + m

On entry: if

options . start = Nag_Cold

, you do not need to provide memory for

options . lambda

, as

n + m

values of memory will be automatically allocated by e04ugc. This is the recommended method of use of

options . lambda

. However you may supply memory from the calling program.

If the option

options . start = Nag_Warm

has been chosen,

options . lambda

must point to a minimum of

n + m

elements of memory. This memory will already be available if the options structure has been used in a previous call to e04ugc from the calling program, with

options . start = Nag_Cold

and the same values of n and m. If a previous call has not been made, you must allocate sufficient memory.

When a ‘warm start’ is chosen

options . lambda [j - 1]

must contain a multiplier estimate for each nonlinear constraint for

j = n + 1, n + 2, \dots, n + ncnln

. The remaining elements need not be set. If nothing is known about the problem, or there is no wish to provide special information, you may set

options . lambda [j - 1] = 0

for

j = n + 1, n + 2, \dots, n + ncnln

On exit: a set of Lagrange multipliers for the bound constraints on the variables (reduced costs) and the general constraints (shadow costs). More precisely, the first n elements contain the multipliers for the bound constraints on the variables, the next ncnln elements contain the multipliers for the nonlinear constraints

F (x)

(if any) and the next

m - ncnln

elements contain the multipliers for the linear constraints

G x

and the free row (if any).

iter – Integer

On exit: the total number of minor iterations (summed over all major iterations).

major_iter – Integer

On exit: the number of major iterations that have been performed in e04ugc.

nsb – Integer

On entry: the number of superbasic variables. It need not be specified if

options . start = Nag_Cold

but must retain its value from a previous call when

options . start = Nag_Warm

Constraint: if

options . start = Nag_Warm

options . nsb \geq 0

On exit: the final number of superbasic variables.

nf – Integer

On exit: the number of calls to objfun.

ncon – Integer

On exit: the number of calls to confun.

12.3 Description of Printed Output

This section describes the intermediate printout and final printout produced by e04ugc. The level of printed output can be controlled with the structure members

options . list

options . print_deriv

options . print_level

options . minor_print_level

options . print_80ch

, and

options . outfile

(see Section 12.2). If

options . list = Nag_TRUE

then the argument values to e04ugc are listed, followed by the result of any derivative check when

options . print_deriv = Nag_D_Print

. The printout of results is then governed by the values of

options . print_80ch

options . print_level

and

options . minor_print_level

. The default of

options . print_level = Nag_Soln_Iter

options . minor_print_level = Nag_NoPrint

, and

options . print_80ch = Nag_TRUE

produces a single line of output at each major iteration and the final result.

The following line of summary output (

\leq 80

characters) is produced at every major iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Maj	is the major iteration count.
Mnr	is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 1 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 11).
Step	is the step taken along the computed search direction. On reasonably well-behaved problems, the unit step will be taken as the solution is approached.
Merit function	is the value of the augmented Lagrangian merit function (6) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 11.2). As the solution is approached, Merit function will converge to the value of the objective function at the solution. In elastic mode (see Section 11.2), the merit function is a composite function involving the constraint violations weighted by the value of the optional parameter $options . elastic_wt$ (default value $= 1.0$ or 100.0). If there are no nonlinear constraints present, this entry contains Objective, the value of the objective function $f (x)$ . In this case, $f (x)$ will decrease monotonically to its optimal value.
Feasibl	is the value of rowerr, the largest element of the scaled nonlinear constraint residual vector defined in the description of the optional parameter $options . major_feas_tol$ . The solution is regarded as ‘feasible’ if Feasibl is less than (or equal to) the $options . major_feas_tol$ (default value $= \sqrt{ε}$ ). Feasibl will be approximately zero in the neighbourhood of a solution. If there are no nonlinear constraints present, all iterates are feasible and this entry is not printed.
Optimal	is the value of maxgap, the largest element of the maximum complementarity gap vector defined in the description of the optional parameter $options . major_opt_tol$ . The Lagrange multipliers are regarded as ‘optimal’ if Optimal is less than (or equal to) the optional parameter $options . major_opt_tol$ (default value $= \sqrt{ε}$ ). Optimal will be approximately zero in the neighbourhood of a solution.
Cond Hz	is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of an upper triangular matrix $R$ . This constitutes a lower bound on the condition number of the matrix $R^{T} R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.
PD	is a two-letter indication of the status of the convergence tests involving the feasibility and optimality of the iterates defined in the descriptions of the optional parameters $options . major_feas_tol$ and $options . major_opt_tol$ . Each letter is T if the test is satisfied, and F otherwise. The tests indicate whether the values of Feasibl and Optimal are sufficiently small. For example, TF or TT is printed if there are no nonlinear constraints present (since all iterates are feasible).
M	is printed if an extra evaluation of objfun and confun was needed in order to define an acceptable positive definite quasi-Newton update to the Hessian of the Lagrangian. This modification is only performed when there are nonlinear constraints present.
m	is printed if, in addition, it was also necessary to modify the update to include an augmented Lagrangian term.
s	is printed if a self-scaled BFGS (Broyden–Fletcher–Goldfarb–Shanno) update was performed. This update is always used when the Hessian approximation is diagonal, and hence always follows a Hessian reset.
S	is printed if, in addition, it was also necessary to modify the self-scaled update in order to maintain positive-definiteness.
n	is printed if no positive definite BFGS update could be found, in which case the approximate Hessian is unchanged from the previous iteration.
r	is printed if the approximate Hessian was reset after 10 consecutive major iterations in which no BFGS update could be made. The diagonal elements of the approximate Hessian are retained if at least one update has been performed since the last reset. Otherwise, the approximate Hessian is reset to the identity matrix.
R	is printed if the approximate Hessian has been reset by discarding all but its diagonal elements. This reset will be forced periodically by the values of the optional parameters $options . hess_freq$ (default value $= 99999999$ ) and $options . hess_update$ (default value $= 20$ ). However, it may also be necessary to reset an ill-conditioned Hessian from time to time.
l	is printed if the change in the variables was limited by the value of the optional parameter $options . major_step_lim$ (default value $= 2.0$ ). If this output occurs frequently during later iterations, it may be worthwhile increasing the value of $options . major_step_lim$ .
c	is printed if central differences have been used to compute the unknown elements of the objective and constraint gradients. A switch to central differences is made if either the linesearch gives a small step, or $x$ is close to being optimal. In some cases, it may be necessary to re-solve the QP subproblem with the central difference gradient and Jacobian.
u	is printed if the QP subproblem was unbounded.
t	is printed if the minor iterations were terminated because the number of iterations specified by the value of the optional parameter $options . minor_iter_lim$ (default value $= 500$ ) was reached.
i	is printed if the QP subproblem was infeasible when the function was not in elastic mode. This event triggers the start of nonlinear elastic mode, which remains in effect for all subsequent iterations. Once in elastic mode, the QP subproblems are associated with the elastic problem (8) (see Section 11.2). It is also printed if the minimizer of the elastic subproblem does not satisfy the linearized constraints when the function is already in elastic mode. (In this case, a feasible point for the usual QP subproblem may or may not exist.)
w	is printed if a weak solution of the QP subproblem was found.

The final printout includes a listing of the status of every variable and constraint.

The following describes the printout for each variable.

Variable

gives the name of the variable. If the optional parameter

options . crnames =

NULL, a default name is assigned to the

j

th variable, for

j = 1, 2, \dots, n

. Otherwise, the name supplied in

options . crnames [j - 1]

is assigned to the

j

th variable.

State

gives the state of the variable (LL if nonbasic on its lower bound, UL if nonbasic on its upper bound, EQ if nonbasic and fixed, FR if nonbasic and strictly between its bounds, BS if basic and SBS if superbasic).

A key is sometimes printed before State to give some additional information about the state of a variable. Note that unless the optional parameter

options . scale_opt = 0

(default value

= 1

or 2) is specified, the tests for assigning a key are applied to the variables of the scaled problem.

A	Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the Lagrange multipliers might also change.
D	Degenerate. The variable is basic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. The variable is basic and is currently violating one of its bounds by more than the value of the optional parameter $options . minor_feas_tol$ (default value $= \sqrt{ε}$ ).
N	Not precisely optimal. The variable is nonbasic. Its reduced gradient is larger than the value of the optional parameter $options . major_feas_tol$ (default value $= \sqrt{ε}$ ).

Value

is the value of the variable at the final iterate.

Lower Bound

is the lower bound specified for the variable. None indicates that

bl [j - 1] \leq - options . inf_bound

Upper Bound

is the upper bound specified for the variable. None indicates that

bu [j - 1] \geq options . inf_bound

Lagr Mult

is the Lagrange multiplier for the associated bound. This will be zero if State is FR. If

x

is optimal, the multiplier should be non-negative if State is LL, non-positive if State is UL, and zero if State is BS or SBS.

Residual

is the difference between the variable Value and the nearer of its (finite) bounds

bl [j - 1]

and

bu [j - 1]

. A blank entry indicates that the associated variable is not bounded (i.e.,

bl [j - 1] \leq - options . inf_bound

and

bu [j - 1] \geq options . inf_bound

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,

n

replaced by

m

options . crnames [j - 1]

) replaced by

options . crnames [n + j - 1]

bl [j - 1]

and

bu [j - 1]

replaced by

bl [n + j - 1]

and

bu [n + j - 1]

respectively, and with the following change in the heading:

Constrnt

gives the name of the general constraint.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

The following describes all of the possible other levels of results printout available from e04ugc.

If a simple derivative check,

options . verify_grad = Nag_SimpleCheck

, is requested then a statement indicating success or failure is given. The largest error found in the objective and the constraint Jacobian are also output.

When a component derivative check (see the optional parameter

options . verify_grad

in Section 12.2) is selected, the element with the largest relative error is identified for the objective and the constraint Jacobian.

options . print_deriv = Nag_D_Print

then the following results are printed for each component:

x[j-1]	the element of $x$ .
dx[j-1]	the finite difference interval.
Jacobian value	the nonlinear Jacobian element.
g[j-1]	the objective gradient element.
Difference approx.	the finite difference approximation.

The indicator, OK or BAD?, states whether the derivative provided and the finite difference approximation are in agreement. If the derivatives are believed to be in error e04ugc will exit with fail set to either NE_CON_DERIV_ERRORS or NE_OBJ_DERIV_ERRORS, depending on whether the error was detected in the constraint Jacobian or in the objective gradient.

When

options . print_level = Nag_Iter

Nag_Soln_Iter

Nag_Soln_Iter_Full

, and

options . print_80ch = Nag_FALSE

, the following line of intermediate printout (

\leq 120

characters) is sent at every major iteration to

options . outfile

. Unless stated otherwise, the values of the quantities printed are those in effect on completion of the given iteration.

Major	is the major iteration count.
Minor	is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Minor will be 1 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 11).
Step	is the step taken along the computed search direction. On reasonably well-behaved problems, the unit step will be taken as the solution is approached.
nObj	is the number of times objfun has been called to evaluate the nonlinear part of the objective function. Evaluations needed for the estimation of the gradients by finite differences are not included. nObj is printed as a guide to the amount of work required for the linesearch.
nCon	is the number of times confun has been called to evaluate the nonlinear constraint functions (not printed if ncnln is zero).
Merit	is the value of the augmented Lagrangian merit function (6) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 11.1). As the solution is approached, Merit will converge to the value of the objective function at the solution. In elastic mode (see Section 11.2), the merit function is a composite function involving the constraint violations weighted by the value of the optional parameter $options . elastic_wt$ (default value $= 1.0$ or 100.0). If there are no nonlinear constraints present, this entry contains Objective, the value of the objective function $f (x)$ . In this case, $f (x)$ will decrease monotonically to its optimal value.
Feasibl	is the value of rowerr, the largest element of the scaled nonlinear constraint residual vector defined in the description of the optional parameter $options . major_feas_tol$ . The solution is regarded as ‘feasible’ if Feasibl is less than (or equal to) $options . major_feas_tol$ (default value $= \sqrt{ε}$ ). Feasibl will be approximately zero in the neighbourhood of a solution. If there are no nonlinear constraints present, all iterates are feasible and this entry is not printed.
Optimal	is the value of maxgap, the largest element of the maximum complementarity gap vector defined in the description of the optional parameter $options . major_opt_tol$ . The Lagrange multipliers are regarded as ‘optimal’ if Optimal is less than (or equal to) $options . major_opt_tol$ (default value $= \sqrt{ε}$ ). Optimal will be approximately zero in the neighbourhood of a solution.
nS	is the current number of superbasic variables.
Penalty	is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian function (not printed if ncnln is zero).
LU	is the number of nonzeros representing the basis factors $L$ and $U$ on completion of the QP subproblem. If there are nonlinear constraints present, the basis factorization $B = L U$ is computed at the start of the first minor iteration. At this stage, $LU = lenL + lenU$ , where lenL is the number of subdiagonal elements in the columns of a lower triangular matrix and lenU is the number of diagonal and superdiagonal elements in the rows of an upper triangular matrix. As columns of $B$ are replaced during the minor iterations, the value of LU may fluctuate up or down (but in general will tend to increase). As the solution is approached and the number of minor iterations required to solve each QP subproblem decreases towards zero, LU will reflect the number of nonzeros in the $L U$ factors at the start of each QP subproblem. If there are no nonlinear constraints present, refactorization is subject only to the value of the optional parameter $options . factor_freq$ (default value $= 50$ or 100) and hence LU will tend to increase between factorizations.
Swp	is the number of columns of the basis matrix $B$ that were swapped with columns of $S$ in order to improve the condition number of $B$ (not printed if ncnln is zero). The swaps are determined by an $L U$ factorization of the rectangular matrix $B_{S} = {(B S)}^{T}$ , with stability being favoured more than sparsity.
Cond Hz	is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of the upper triangular matrix $R$ . This constitutes a lower bound on the condition number of the matrix $R^{T} R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.
PD	is a two-letter indication of the status of the convergence tests involving the feasibility and optimality of the iterates defined in the descriptions of the optional parameters $options . major_feas_tol$ and $options . major_opt_tol$ . Each letter is T if the test is satisfied, and F otherwise. The tests indicate whether the values of Feasibl and Optimal are sufficiently small. For example, TF or TT is printed if there are no nonlinear constraints present (since all iterates are feasible).
M	is printed if an extra evaluation of objfun and confun was needed in order to define an acceptable positive definite quasi-Newton update to the Hessian of the Lagrangian. This modification is only performed when there are nonlinear constraints present.
m	is printed if, in addition, it was also necessary to modify the update to include an augmented Lagrangian term.
s	is printed if a self-scaled BFGS (Broyden–Fletcher–Goldfarb–Shanno) update was performed. This update is always used when the Hessian approximation is diagonal, and hence always follows a Hessian reset.
S	is printed if, in addition, it was also necessary to modify the self-scaled update in order to maintain positive-definiteness.
n	is printed if no positive definite BFGS update could be found, in which case the approximate Hessian is unchanged from the previous iteration.
r	is printed if the approximate Hessian was reset after 10 consecutive major iterations in which no BFGS update could be made. The diagonal elements of the approximate Hessian are retained if at least one update has been performed since the last reset. Otherwise, the approximate Hessian is reset to the identity matrix.
R	is printed if the approximate Hessian has been reset by discarding all but its diagonal elements. This reset will be forced periodically by the values of the optional parameters $options . hess_freq$ (default value $= 99999999$ ) and $options . hess_update$ (default value $= 20$ ). However, it may also be necessary to reset an ill-conditioned Hessian from time to time.
l	is printed if the change in the variables was limited by the value of the optional parameter $options . major_step_lim$ (default value $= 2.0$ ). If this output occurs frequently during later iterations, it may be worthwhile increasing the value of $options . major_step_lim$ .
c	is printed if central differences have been used to compute the unknown elements of the objective and constraint gradients. A switch to central differences is made if either the linesearch gives a small step, or $x$ is close to being optimal. In some cases, it may be necessary to re-solve the QP subproblem with the central difference gradient and Jacobian.
u	is printed if the QP subproblem was unbounded.
t	is printed if the minor iterations were terminated because the number of iterations specified by the value of the optional parameter $options . minor_iter_lim$ (default value $= 500$ ) was reached.
i	is printed if the QP subproblem was infeasible when the function was not in elastic mode. This event triggers the start of nonlinear elastic mode, which remains in effect for all subsequent iterations. Once in elastic mode, the QP subproblems are associated with the elastic problem (8) (see Section 11.2). It is also printed if the minimizer of the elastic subproblem does not satisfy the linearized constraints when the function is already in elastic mode. (In this case, a feasible point for the usual QP subproblem may or may not exist.)
w	is printed if a weak solution of the QP subproblem was found.

When

options . minor_print_level = Nag_Iter

and

options . print_80ch = Nag_TRUE

, the following line of intermediate printout (

\leq 80

characters) is sent at every minor iteration to

options . outfile

. Unless stated otherwise, the values of the quantities printed are those in effect on completion of the given iteration.

Itn	is the iteration count.
Step	is the step taken along the computed search direction.
Ninf	is the number of infeasibilities. This will not increase unless the iterations are in elastic mode. Ninf will be zero during the optimality phase.
Sinf	is the value of the sum of infeasibilities if Ninf is nonzero. This will be zero during the optimality phase.
Objective	is the value of the current QP objective function when Ninf is zero and the iterations are not in elastic mode. The switch to elastic mode is indicated by a change in the heading to Composite Obj (see below).
Composite Obj	is the value of the composite objective function (9) when the iterations are in elastic mode. This function will decrease monotonically at each iteration.
Norm rg	is the Euclidean norm of the reduced gradient of the QP objective function. During the optimality phase, this norm will be approximately zero after a unit step.

When

options . minor_print_level = Nag_Iter

and

options . print_80ch = Nag_FALSE

, the following line of intermediate printout (

\leq 120

characters) is sent at every minor iteration to

options . outfile

. Unless stated otherwise, the values of the quantities printed are those in effect on completion of the given iteration.

In the description below, a ‘pricing’ operation is defined to be the process by which a nonbasic variable is selected to become superbasic (in addition to those already in the superbasic set). If the problem is purely linear, the variable selected will usually become basic immediately (unless it happens to reach its opposite bound and return to the nonbasic set).

Itn	is the iteration count.
pp	is the partial price indicator. The variable selected by the last pricing operation came from the ppth partition of $A$ and $- I$ . Note that pp is reset to zero whenever the basis is refactorized.
dj	is the value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration.
+SBS	is the variable selected by the pricing operation to be added to the superbasic set.
-SBS	is the variable chosen to leave the superbasic set. It has become basic if the entry under -B is nonzero; otherwise it has become nonbasic.
-BS	is the variable removed from the basis (if any) to become nonbasic.
-B	is the variable removed from the basis (if any) to swap with a slack variable made superbasic by the latest pricing operation. The swap is done to ensure that there are no superbasic slacks.
Step	is the value of the step length $α$ taken along the current search direction $p$ . The variables $x$ have just been changed to $x + α p$ . If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During the optimality phase, the step can be greater than unity only if the reduced Hessian is not positive definite.
Pivot	is the $r$ th element of a vector $y$ satisfying $B y = a_{q}$ whenever $a_{q}$ (the $q$ th column of the constraint matrix $(\begin{matrix} A & - I \end{matrix})$ ) replaces the $r$ th column of the basis matrix $B$ . Wherever possible, Step is chosen so as to avoid extremely small values of Pivot (since they may cause the basis to be nearly singular). In extreme cases, it may be necessary to increase the value of the optional parameter $options . pivot_tol$ (default value $= ε^{0.67}$ ) to exclude very small elements of $y$ from consideration during the computation of Step.
Ninf	is the number of infeasibilities. This will not increase unless the iterations are in elastic mode. Ninf will be zero during the optimality phase.
Sinf/Objective	is the value of the current objective function. If $x$ is infeasible, Sinf gives the value of the sum of infeasibilities at the start of the current iteration. It will usually decrease at each nonzero value of Step, but may occasionally increase if the value of Ninf decreases by a factor of 2 or more. However, in elastic mode this entry gives the value of the composite objective function (9), which will decrease monotonically at each iteration. If $x$ is feasible, Objective is the value of the current QP objective function.
L	is the number of nonzeros in the basis factor $L$ . Immediately after a basis factorization $B = L U$ , this entry contains lenL. Further nonzeros are added to L when various columns of $B$ are later replaced. (Thus, L increases monotonically.)
U	is the number of nonzeros in the basis factor $U$ . Immediately after a basis factorization $B = L U$ , this entry contains lenU. As columns of $B$ are replaced, the matrix $U$ is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general, it will tend to increase.
Ncp	is the number of compressions required to recover workspace in the data structure for $U$ . This includes the number of compressions needed during the previous basis factorization. Normally, Ncp should increase very slowly. If it does not, e04ugc will attempt to expand the internal workspace allocated for the basis factors.

The following items are printed only if the problem is nonlinear or the superbasic set is non-empty (i.e., if the current solution is nonbasic).

Norm rg	is the Euclidean norm of the reduced gradient at the start of the current iteration. During the optimality phase, this norm will be approximately zero after a unit step.
nS	is the current number of superbasic variables.
Cond Hz	is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of an upper triangular matrix $R$ . This constitutes a lower bound on the condition number of the matrix $R^{T} R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.

When

options . print_level = Nag_Soln_Iter_Full

, the following lines of intermediate printout (

\leq 120

characters) are sent to

options . outfile

whenever the matrix

B

B_{S} = {(B S)}^{T}

is factorized. Gaussian elimination is used to compute a sparse

L U

factorization of

B

B_{S}

, where

{P L P}^{T}

is a lower triangular matrix and

P U Q

is an upper triangular matrix for some permutation matrices

P

and

Q

. The factorization is stabilized in the manner described under the optional parameter

options . lu_factor_tol

(default value

= 5.0

or 100.0).

Note that

B_{S}

may be factorized at the beginning of just some of the major iterations. It is immediately followed by a factorization of

B

itself. Note also that factorizations can occur during the solution of a QP problem.

Factorize

is the factorization count.

Demand

is a code giving the reason for the present factorization as follows:

Code	Meaning
$0$	First $L U$ factorization.
$1$	The number of updates reached the value of the optional parameter $options . factor_freq$ (default value $= 50$ or 100).
$2$	The number of nonzeros in the updated factors has increased significantly.
$7$	Not enough storage to update factors.
$10$	Row residuals too large (see the description for the optional parameter $options . fcheck$ ).
$11$	Ill-conditioning has caused inconsistent results.

Iteration

is the iteration count.

Nonlinear

is the number of nonlinear variables in the current basis

B

(not printed if

B_{S}

is factorized).

Linear

is the number of linear variables in

B

(not printed if

B_{S}

is factorized).

Slacks

is the number of slack variables in

B

(not printed if

B_{S}

is factorized).

Elems

is the number of nonzeros in

B

(not printed if

B_{S}

is factorized).

Density

is the percentage nonzero density of

B

(not printed if

B_{S}

is factorized). More precisely,

Density = 100 \times {Elems / (Nonlinear + Linear + Slacks)}^{2}

Compressns

is the number of times the data structure holding the partially factorized matrix needed to be compressed, in order to recover unused workspace. Ideally, it should be zero.

Merit

is the average Markowitz merit count for the elements chosen to be the diagonals of

P U Q

. Each merit count is defined to be

(c - 1) (r - 1)

, where

c

and

r

are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of m such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization.

lenL

is the number of nonzeros in

L

lenU

is the number of nonzeros in

U

Increase

is the percentage increase in the number of nonzeros in

L

and

U

relative to the number of nonzeros in

B

. More precisely,

Increase = 100 \times (lenL + lenU - Elems) / Elems

is the number of rows in the problem. Note that

m = Ut + Lt + bp

is the number of triangular rows of

B

at the top of

U

is the number of columns remaining when the density of the basis matrix being factorized reached 0.3.

Lmax

is the maximum subdiagonal element in the columns of

L

. This will not exceed the value of the optional parameter

options . lu_factor_tol

(default value

= 5.0

or 100.0).

Bmax

is the maximum nonzero element in

B

(not printed if

B_{S}

is factorized).

BSmax

is the maximum nonzero element in

B_{S}

(not printed if

B

is factorized).

Umax

is the maximum nonzero element in

U

, excluding elements of

B

that remain in

U

unchanged. (For example, if a slack variable is in the basis, the corresponding row of

B

will become a row of

U

without modification. Elements in such rows will not contribute to Umax. If the basis is strictly triangular, none of the elements of

B

will contribute, and Umax will be zero.)

Ideally, Umax should not be significantly larger than Bmax. If it is several orders of magnitude larger, it may be advisable to reset the

options . lu_factor_tol

to some value nearer unity.

Umax is not printed if

B_{S}

is factorized.

Umin

is the magnitude of the smallest diagonal element of

P U Q

Growth

is the value of the ratio

Umax / Bmax

, which should not be too large.

Providing Lmax is not large (say

< 10.0

), the ratio

\max (Bmax, Umax) / Umin

is an estimate of the condition number of

B

. If this number is extremely large, the basis is nearly singular and some numerical difficulties might occur. (However, an effort is made to avoid near-singularity by using slacks to replace columns of

B

that would have made Umin extremely small, and the modified basis is refactorized.)

is the number of triangular columns of

B

at the left of

L

is the size of the ‘bump’ or block to be factorized nontrivially after the triangular rows and columns of

B

have been removed.

is the number of columns remaining when the density of the basis matrix being factorized has reached 0.6.

When

options . print_level = Nag_Soln_Iter_Full

, and

options . crash \neq Nag_NoCrash

(default value

options . crash = Nag_NoCrash

Nag_CrashThreeTimes

), the following lines of intermediate printout are sent to

options . outfile

whenever

options . start = Nag_Cold

. They refer to the number of columns selected by the Crash procedure during each of several passes through

A

while searching for a triangular basis matrix.

Slacks	is the number of slacks selected initially.
Free cols	is the number of free columns in the basis, including those whose bounds are rather far apart.
Preferred	is the number of ‘preferred’ columns in the basis (i.e., $options . state [j - 1] = 3$ for some $j \leq n$ ). It will be a subset of the columns for which $options . state [j - 1] = 3$ was specified.
Unit	is the number of unit columns in the basis.
Double	is the number of columns in the basis containing two nonzeros.
Triangle	is the number of triangular columns in the basis with three (or more) nonzeros.
Pad	is the number of slacks used to pad the basis (to make it a nonsingular triangle).

When

options . print_level = Nag_Soln

Nag_Soln_Iter

, and

options . print_80ch = Nag_FALSE

, the following lines of final printout (

\leq 120

characters) are sent to

options . outfile

Let

x_{j}

denote the

j

th ‘column variable’, for

j = 1, 2, \dots, n

. We assume that a typical variable

x_{j}

has bounds

α \leq x_{j} \leq β

The following describes the printout for each column (or variable).

Number

is the column number

j

. (This is used internally to refer to

x_{j}

in the intermediate output.)

Column

gives the name of

x_{j}

State

gives the state of

x_{j}

relative to the bounds

α

and

β

. The various possible states are as follows:

LL	$x_{j}$ is nonbasic at its lower limit, $α$ .
UL	$x_{j}$ is nonbasic at its upper limit, $β$ .
EQ	$x_{j}$ is nonbasic and fixed at the value $α = β$ .
FR	$x_{j}$ is nonbasic at some value strictly between its bounds: $α < x_{j} < β$ .
BS	$x_{j}$ is basic. Usually $α < x_{j} < β$ .
SBS	$x_{j}$ is superbasic. Usually $α < x_{j} < β$ .

A key is sometimes printed before State to give some additional information about the state of

x_{j}

. Note that unless the optional parameter

options . scale_opt = 0

(default value

= 1

or 2) is specified, the tests for assigning a key are applied to the variables of the scaled problem.

A	Alternative optimum possible. $x_{j}$ is nonbasic, but its reduced gradient is essentially zero. This means that if $x_{j}$ were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the Lagrange multipliers might also change.
D	Degenerate. $x_{j}$ is basic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. $x_{j}$ is basic and is currently violating one of its bounds by more than the value of the optional parameter $options . minor_feas_tol$ (default value $= \sqrt{ε}$ ).
N	Not precisely optimal. $x_{j}$ is nonbasic. Its reduced gradient is larger than the value of the optional parameter $options . major_feas_tol$ (default value $= \sqrt{ε}$ ).

Activity

is the value of

x_{j}

at the final iterate.

Obj Gradient

is the value of

g_{j}

at the final iterate. (If any

x_{j}

is infeasible,

g_{j}

is the gradient of the sum of infeasibilities.)

Lower Limit

α

, the lower bound specified for

x_{j}

. None indicates that

bl [j - 1] \leq - options . inf_bound

Upper Limit

β

, the upper bound specified for

x_{j}

. None indicates that

bu [j - 1] \geq options . inf_bound

Reduced Gradnt

is the value of

d_{j}

at the final iterate.

m + j

is the value of

m + j

General linear constraints take the form

l \leq A x \leq u

. Let

a_{i}^{T}

denote the

i

th row of

A

, for

i = 1, 2, \dots, n

. The

i

th constraint is, therefore, of the form

α \leq a_{i}^{T} x \leq β

, and the value of

a_{i}^{T} x

is called the row activity. Internally, the linear constraints take the form

A x - s = 0

, where the slack variables

s

should satisfy the bounds

l \leq s \leq u

. For the

i

th ‘row’, it is the slack variable

s_{i}

that is directly available, and it is sometimes convenient to refer to its state. Slacks may be basic or nonbasic (but not superbasic).

Nonlinear constraints

α \leq F_{i} (x) + a_{i}^{T} x \leq β

are treated similarly, except that the row activity and degree of infeasibility are computed directly from

F_{i} (x) + a_{i}^{T} x

rather than from

s_{i}

The following describes the printout for each row (or constraint).

Number

is the value of

n + i

. (This is used internally to refer to

s_{i}

in the intermediate output.)

Row

gives the name of the

i

th row.

State

gives the state of the

i

th row relative to the bounds

α

and

β

. The various possible states are as follows:

LL	The row is at its lower limit, $α$ .
UL	The row is at its upper limit, $β$ .
EQ	The limits are the same $(α = β)$ .
BS	The constraint is not binding. $s_{i}$ is basic.

A key is sometimes printed before State to give some additional information about the state of

s_{i}

. Note that unless the optional parameter

options . scale_opt = 0

(default value

= 1

or 2) is specified, the tests for assigning a key are applied to the variables of the scaled problem.

A	Alternative optimum possible. $s_{i}$ is nonbasic, but its reduced gradient is essentially zero. This means that if $s_{i}$ were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the dual variables (or Lagrange multipliers) might also change.
D	Degenerate. $s_{i}$ is basic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. $s_{i}$ is basic and is currently violating one of its bounds by more than the value of the optional parameter $options . minor_feas_tol$ (default value $= \sqrt{ε}$ ).
N	Not precisely optimal. $s_{i}$ is nonbasic. Its reduced gradient is larger than the value of the optional parameter $options . major_feas_tol$ (default value $= \sqrt{ε}$ ).

Activity

is the value of

a_{i}^{T x}

(or

F_{i} (x) + a_{i}^{T} x

for nonlinear rows) at the final iterate.

Slack Activity

is the value by which the row differs from its nearest bound. (For the free row (if any), it is set to Activity.)

Lower Limit

α

, the lower bound specified for the

i

th row. None indicates that

bl [n + i - 1] \leq - options . inf_bound

Upper Limit

β

, the upper bound specified for the

i

th row. None indicates that

bu [n + i - 1] \geq options . inf_bound

Dual Activity

is the value of the dual variable

π_{i}

gives the index

i

of the

i

th row.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

NAG Library Manual, Mark 28.6

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04ug: FL CL CPP AD PY MB

NAG CL Interfacee04ugc (nlp1_​sparse_​solve)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Termination Criteria

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Further Description

11.1 Overview

11.2 Treatment of Constraint Infeasibilities

12 Optional Parameters

12.1 Optional Parameter Checklist and Default Values

12.2 Description of the Optional Parameters

12.3 Description of Printed Output

NAG CL Interface
e04ugc (nlp1_sparse_solve)