const double a[], const Integer ha[], const Integer ka[], const double bl[], const double bu[], double xs[], Integer *ninf, double *sinf, double *obj, Nag_E04_Opt *options, Nag_Comm *comm, NagError *fail)

The function may be called by the names: e04nkc, nag_opt_qpconvex1_sparse_solve or nag_opt_sparse_convex_qp.

3 Description

e04nkc is designed to solve a class of quadratic 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 \\ A x \end{matrix}\} \leq u,

(1)

where

x

is a set of variables,

A

is an

m

n

matrix and the objective function

f (x)

may be specified in a variety of ways depending upon the particular problem to be solved. The optional parameter

options . minimize

(see Section 12.2) may be used to specify an alternative problem in which

f (x)

is maximized. The possible forms for

f (x)

are listed in Table 1 below, in which the prefixes FP, LP and QP stand for ‘feasible point’, ‘linear programming’ and ‘quadratic programming’ respectively,

c

is an

n

element vector and

H

is the

n

n

second-derivative matrix

\nabla^{2} f (x)

(the Hessian matrix).

Problem Type	Objective Function $f (x)$	Hessian Matrix $H$
FP	Not applicable	Not applicable
LP	$c^{T} x$	Not applicable
QP	$c^{T} x + \frac{1}{2} x^{T} H x$	Symmetric positive semidefinite

Table 1

For LP and QP problems, the unique global minimum value of

f (x)

is found. For FP problems,

f (x)

is omitted and the function attempts to find a feasible point for the set of constraints. For QP problems, a function must also be provided to compute

H x

for any given vector

x

. (

H

need not be stored explicitly.)

e04nkc is intended to solve large-scale linear and quadratic programming problems in which the constraint matrix

A

is sparse (i.e., when the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations and storage involving zero elements). e04nkc also takes advantage of sparsity in

c

. (Sparsity in

H

can be exploited in the function that computes

H x

.) For problems in which

A

can be treated as a dense matrix, it is usually more efficient to use e04mfc, e04ncc or e04nfc.

H

is positive definite, then the final

x

will be unique. If e04nkc detects that

H

is indefinite, it terminates immediately with an error condition (see Section 6). In that case, it may be more appropriate to call e04ugc instead. If

H

is the zero matrix, the function will still solve the resulting LP problem; however, this can be accomplished more efficiently by setting the argument

ncolh = 0

(see Section 5).

The upper and lower bounds on the

m

elements of

A x

are said to define the general constraints of the problem. Internally, e04nkc converts the general constraints 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 A x - s = 0, l \leq \{\begin{matrix} x \\ s \end{matrix}\} \leq u .

Since the slack variables

s

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

A x

, the bounds on

A x

and

x

can simply be thought of as bounds on the combined vector

(x, s)

. (In order to indicate their special role in QP problems, the original variables

x

are sometimes known as ‘column variables’, and the slack variables

s

are known as ‘row variables’.)

Each LP or QP problem is solved using an active-set method. This is an iterative procedure with two phases: a feasibility phase, in which the sum of infeasibilities is minimized to find a feasible point; and an optimality phase, in which

f (x)

is minimized by constructing a sequence of iterations that lies within the feasible region.

A constraint is said to be active or binding at

x

if the associated element of either

x

A x

is equal to one of its upper or lower bounds. Since an active constraint in

A x

has its associated slack variable at a bound, the status of both simple and general upper and lower bounds can be conveniently described in terms of the status of the variables

(x, s)

. A variable is said to be nonbasic if it is temporarily fixed at its upper or lower bound. It follows that regarding a general constraint as being active is equivalent to thinking of its associated slack as being nonbasic.

At each iteration of an active-set method, the constraints

A x - s = 0

are (conceptually) partitioned into the form

{B x}_{B} + {S x}_{S} + {N x}_{N} = 0,

where

x_{N}

consists of the nonbasic elements of

(x, s)

and the basis matrix

B

is square and nonsingular. The elements of

x_{B}

and

x_{S}

are called the basic and superbasic variables respectively; with

x_{N}

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 upper or lower bounds, while the nonbasic variables will be equal to one of their bounds. At each 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 objective function (or sum of infeasibilities). The basic variables are then adjusted in order to ensure that

(x, s)

continues to satisfy

A x - s = 0

. 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 for FP and LP problems.

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}

is decreased by one.

Associated with each of the

m

equality constraints

A x - s = 0

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; and the reduced gradients for the slack variables

s

are the dual variables

π

. The QP subproblem 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 all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on

d_{j}

(see the description of the optional parameter

options . optim_tol

in Section 12.2).

The process of computing and comparing reduced gradients is known as pricing (a term first introduced in the context of the simplex method for linear programming). To ‘price’ a nonbasic variable

x_{j}

means that the reduced gradient

d_{j}

associated with the relevant active upper or lower bound on

x_{j}

is computed via the formula

d_{j} = g_{j} - a^{T} π

, where

a_{j}

is the

j

th column of

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

. (The variable selected by such a process and the corresponding value of

d_{j}

(i.e., its reduced gradient) are the quantities +S and dj in the detailed printed output from e04nkc; see Section 12.3.) If

A

has significantly more columns than rows (i.e.,

n ≫ m

), pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute and compare only a subset of the

d_{j}

's.

e04nkc is based on SQOPT, which is part of the SNOPT package described in Gill et al. (2002), which in turn utilizes routines from the MINOS package (see Murtagh and Saunders (1995)). It uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse

L U

factors of the basis matrix

B

), a practical anti-degeneracy procedure, efficient handling of linear constraints and bounds on the variables (by an active-set strategy), as well as automatic scaling of the constraints. Further details can be found in Section 9.

4 References

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

Gill P E and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming 14 349–372

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 (1987) Maintaining LU factors of a general sparse matrix Linear Algebra and its Applics. 88/89 239–270

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 (1991) Inertia-controlling methods for general quadratic programming SIAM Rev. 33 1–36

Hall J A J and McKinnon K I M (1996) The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling Report MS 96–100 Department of Mathematics and Statistics, University of Edinburgh

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

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of variables (excluding slacks). This is the number of columns in the linear constraint matrix

A

Constraint:

n \geq 1

2: $m$ – Integer Input

On entry:

m

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

A

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

Constraint:

m \geq 1

3: $nnz$ – Integer Input

On entry: the number of nonzero elements in

A

Constraint:

1 \leq nnz \leq n \times m

4: $iobj$ – Integer Input

On entry: if

iobj > 0

, row iobj of

A

is a free row containing the nonzero elements of the vector

c

appearing in the linear objective term

c^{T} x

iobj = 0

, there is no free row – i.e., the problem is either an FP problem (in which case iobj must be set to zero), or a QP problem with

c = 0

Constraint:

0 \leq iobj \leq m

5: $ncolh$ – Integer Input

On entry:

n_{H}

, the number of leading nonzero columns of the Hessian matrix

H

. For FP and LP problems, ncolh must be set to zero.

Constraint:

0 \leq ncolh \leq n

6: $qphx$ – function, supplied by the user External Function

qphx must be supplied for QP problems to compute the matrix product

H x

. If

H

has zero rows and columns, it is most efficient to order the variables

x = {(y z)}^{T}

so that

H x = (\begin{matrix} H_{1} & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix} y \\ z \end{matrix}) = (\begin{matrix} H_{1} y \\ 0 \end{matrix}),

where the nonlinear variables

y

appear first as shown. For FP and LP problems, qphx will never be called and the NAG defined null function pointer, NULLFN, can be supplied in the call to e04nkc.

The specification of qphx is:

void	qphx (Integer ncolh, const double x[], double hx[], Nag_Comm *comm)

1: $ncolh$ – Integer Input

On entry: the number of leading nonzero columns of the Hessian matrix

H

, as supplied to e04nkc.

2: $x [ncolh]$ – const double Input

On entry: the first ncolh elements of x.

3: $hx [ncolh]$ – double Output

On exit: the product

H x

4: $comm$ – Nag_Comm *

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

first – Nag_BooleanInput: On entry: will be set to Nag_TRUE on the first call to qphx and Nag_FALSE for all subsequent calls.

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void * with a C compiler that defines void * or char *.

Before calling e04nkc these pointers may be allocated memory and initialized with various quantities for use by qphx when called from e04nkc.

Note: qphx should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04nkc. If your code inadvertently does return any NaNs or infinities, e04nkc is likely to produce unexpected results.

Note: qphx should be tested separately before being used in conjunction with e04nkc. The array x must not be changed by qphx.

7: $a [nnz]$ – const double Input

On entry: the nonzero elements of

A

, ordered by increasing column index. Note that elements with the same row and column indices are not allowed. The row and column indices are specified by arguments ha and ka (see below).

8: $ha [nnz]$ – const Integer Input

On entry:

ha [i]

must contain the row index of the nonzero element stored in

a [i]

, for

i = 0, 1, \dots, nnz - 1

. Note that the row indices for a column may be supplied in any order.

Constraint:

1 \leq ha [i] \leq m

, for

i = 0, 1, \dots, nnz - 1

9: $ka [n + 1]$ – 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 - 1] = ka [j]

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

ka [0] = 0

and

ka [n] = nnz

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 $j = 1, 2, \dots, n$ .

10: $bl [n + m]$ – const double Input

11: $bu [n + m]$ – const double Input

On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first

n

elements of each array must contain the bounds on the variables, and the next

m

elements the bounds for the general linear constraints

A 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, for LP and QP problems, the lower bound corresponding to the free row 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]

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 $iobj > 0$ , $bl [n + iobj - 1] \leq - options . inf_bound$ and
$bu [n + iobj - 1] \geq options . inf_bound$ .

12: $xs [n + m]$ – 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)

13: $ninf$ – Integer * Output

On exit: the number of infeasibilities. This will be zero if an optimal solution is found, i.e., if e04nkc exits with

fail . code = NE_NOERROR

or NW_SOLN_NOT_UNIQUE.

14: $sinf$ – double * Output

On exit: the sum of infeasibilities. This will be zero if

ninf = 0

. (Note that e04nkc does attempt to compute the minimum value of sinf in the event that the problem is determined to be infeasible, i.e., when e04nkc exits with

fail . code = NW_NOT_FEASIBLE

15: $obj$ – double * Output

On exit: the value of the objective function.

ninf = 0

, obj includes the quadratic objective term

\frac{1}{2} x^{T} H x

(if any).

ninf > 0

, obj is just the linear objective term

c^{T} x

(if any).

For FP problems, obj is set to zero.

16: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04nkc. 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 e04nkc to perform a ‘warm start’ (see the member

options . start

in Section 12.2).

The options structure also allows names to be assigned to the columns and rows (i.e., the variables and constraints) of the problem, which are then used in solution output.

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 e04nkc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

17: $comm$ – 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 function, qphx, and the optional user-defined printing function; see the description of qphx and Section 12.3.1 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 e04nkc; comm will then be declared internally for use in calls to user-supplied functions.

18: $fail$ – 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_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 $options . print_level$ had an illegal value.

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

On entry, argument $options . start$ had an illegal value.
NE_BASIS_ILL_COND: Numerical error in trying to satisfy the general constraints. The basis is very ill conditioned.
NE_BASIS_SINGULAR: The basis is singular after 15 attempts to factorize it.

The basis is singular after 15 attempts to factorize it (adding slacks where necessary). Either the problem is badly scaled or the value of the optional parameter $options . lu_factor_tol$ is too large; see Section 12.2.
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 elements $bl [〈value〉]$ 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_DUPLICATE_ELEMENT: Duplicate sparse matrix element found in row $〈value〉$ , column $〈value〉$ .
NE_HESS_INDEF: The Hessian matrix $H$ appears to be indefinite.

The Hessian matrix $Z^{T} H Z$ (see Section 11.2) appears to be indefinite – normally because $H$ is indefinite. Check that function qphx has been coded correctly. If qphx is coded correctly with $H$ symmetric positive (semi-)definite, then the problem may be due to a loss of accuracy in the internal computation of the reduced Hessian. Try to reduce the values of the optional parameters $options . lu_factor_tol$ and $options . lu_update_tol$ (see Section 12.2).
NE_HESS_TOO_BIG: Reduced Hessian exceeds assigned dimension. $options . max_sb = 〈value〉$ .

The reduced Hessian matrix $Z^{T} H Z$ (see Section 11.2) exceeds its assigned dimension. The value of the optional parameter $options . max_sb$ is too small; see Section 12.2.
NE_INT_ARG_LT: 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 $ha [〈value〉]$ not valid. Correct range for elements of ha is 1 to m.
NE_INT_OPT_ARG_LT: On entry, $options . factor_freq = 〈value〉$ .
Constraint: $options . factor_freq \geq 1$ .

On entry, $options . fcheck = 〈value〉$ .
Constraint: $options . fcheck \geq 1$ .

On entry, $options . max_iter = 〈value〉$ .
Constraint: $options . max_iter \geq 0$ .

On entry, $options . max_sb = 〈value〉$ .
Constraint: $options . max_sb \geq 1$ .

On entry, $options . nsb = 〈value〉$ .
Constraint: $options . nsb \geq 0$ .

On entry, $options . partial_price = 〈value〉$ .
Constraint: $options . partial_price \geq 1$ .
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 iobj is not valid. Correct range is $0 \leq iobj \leq m$ .

Value $〈value〉$ given to ncolh is not valid. Correct range is $0 \leq ncolh \leq n$ .

Value $〈value〉$ given to nnz is not valid. Correct range is $1 \leq nnz \leq n \times m$ .
NE_INVALID_INT_RANGE_2: Value $〈value〉$ given to $options . reset_ftol$ is not valid. Correct range is $0 < options . reset_ftol < 10000000$ .
NE_INVALID_REAL_RANGE_F: Value $〈value〉$ given to $options . ftol$ is not valid. Correct range is $options . ftol \geq ε$ .

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 $options . lu_factor_tol$ 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 $options . optim_tol$ is not valid. Correct range is $options . optim_tol \geq ε$ .

Value $〈value〉$ given to $options . pivot_tol$ is not valid. Correct range is $options . pivot_tol > 0.0$ .
NE_INVALID_REAL_RANGE_FF: 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 $options . scale_tol$ is not valid. Correct range is $0.0 < options . scale_tol < 1.0$ .
NE_NAME_TOO_LONG: The string pointed to by $options . crnames [〈value〉]$ is too long. It should be no longer than 8 characters.
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_NULL_QPHX: Since argument ncolh is nonzero, the problem is assumed to be of type QP. However, the argument qphx is a null function. qphx must be non-null for QP problems.
NE_OBJ_BOUND: Invalid lower bound for objective row. Bound should be $\leq 〈value〉$ .

Invalid upper bound for objective row. Bound should be $\geq 〈value〉$ .
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 options.max_restart, has been reached.
NE_STATE_VAL: $options . state [〈value〉]$ is out of range. $options . state [〈value〉] = 〈value〉$ .
NE_UNBOUNDED: Solution appears to be unbounded.

The problem is unbounded (or badly scaled). The objective function is not bounded below in the feasible region.
NE_WRITE_ERROR: Error occurred when writing to file $〈string〉$ .
NW_NOT_FEASIBLE: No feasible point was found for the linear constraints.

The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the value of the optional parameter $options . ftol$ ; see Section 12.2.
NW_SOLN_NOT_UNIQUE: Optimal solution is not unique.

Weak solution found. The final $x$ is not unique, although $x$ gives the global minimum value of the objective function.
NW_TOO_MANY_ITER: The maximum number of iterations, $〈value〉$ , have been performed.

Too many iterations. The value of the optional parameter $options . max_iter$ is too small; see Section 12.2.

7 Accuracy

e04nkc implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

8 Parallelism and Performance

e04nkc is not threaded in any implementation.

9 Further Comments

None.

10 Example

To minimize the quadratic function

f (x) = c^{T} x + \frac{1}{2} x^{T} H x

, where

c = {(- 200, - 2000, - 2000, - 2000, - 2000, 400, 400)}^{T}

H = (\begin{matrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 \end{matrix})

subject to the bounds

\begin{matrix} 0 \leq x_{1} \leq 200 \\ 0 \leq x_{2} \leq 2500 \\ 400 \leq x_{3} \leq 800 \\ 100 \leq x_{4} \leq 700 \\ 0 \leq x_{5} \leq 1500 \\ 0 \leq x_{6} \\ 0 \leq x_{7} \end{matrix}

and the general constraints

\begin{matrix} x_{1} + & x_{2} + & x_{3} + & x_{4} + & x_{5} + & x_{6} + & x_{7} = & 2000 \\ 0.15 x_{1} + & 0.04 x_{2} + & 0.02 x_{3} + & 0.04 x_{4} + & 0.02 x_{5} + & 0.01 x_{6} + & 0.03 x_{7} \leq & 60 \\ 0.03 x_{1} + & 0.05 x_{2} + & 0.08 x_{3} + & 0.02 x_{4} + & 0.06 x_{5} + & 0.01 x_{6} & \leq & 100 \\ 0.02 x_{1} + & 0.04 x_{2} + & 0.01 x_{3} + & 0.02 x_{4} + & 0.02 x_{5} & \leq & 40 \\ 0.02 x_{1} + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & 30 \\ 1500 \leq & 0.70 x_{1} + & 0.75 x_{2} + & 0.80 x_{3} + & 0.75 x_{4} + & 0.80 x_{5} + & 0.97 x_{6} \\ 250 \leq & 0.02 x_{1} + & 0.06 x_{2} + & 0.08 x_{3} + & 0.12 x_{4} + & 0.02 x_{5} + & 0.01 x_{6} + & 0.97 x_{7} \leq & 300 \end{matrix}

The initial point, which is infeasible, is

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

The optimal solution (to five figures) is

x^{*} = {(0.0, 349.40, 648.85, 172.85, 407.52, 271.36, 150.02)}^{T} .

One bound constraint and four linear constraints are active at the solution. Note that the Hessian matrix

H

is positive semidefinite.

The function to calculate

H x

(qphx in the argument list; see Section 5) is qphess.

The example program shows the use of the options and comm structures. The data for the example include a set of user-defined column and row names, and data for the Hessian in a sparse storage format (see Section 10.2 for further details).

The options structure is initialized by e04xxc and the

options . crnames

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

comm \to p

member of comm is used to pass the Hessian into e04nkc for use by the function qphess.

On return from e04nkc, the Hessian data is perturbed slightly and two further options set, selecting a warm start and a reduced level of printout. e04nkc is then called for a second time. Finally, the memory freeing function e04xzc is used to free the memory assigned by e04nkc to the pointers in the options structure. You must not use the standard C function free() for this purpose.

The sparse storage scheme used for the Hessian in this example is similar to that which e04nkc uses for the constraint matrix a, but since the Hessian is symmetric we need only store the lower triangle (including the diagonal) of the matrix. Thus, an array hess contains the nonzero elements of the lower triangle arranged in order of increasing column index. The array khess contains the indices in hess of the first element in each column, and the array hhess contains the row index associated with each element in hess. To allow the data to be passed via the

comm \to p

member of comm, a struct HessianData is declared, containing pointer members which are assigned to the three arrays defining the Hessian. Alternative approaches would have been to use the

comm \to user

and

comm \to iuser

members of comm to pass suitably partitioned arrays to qphess, or to avoid the use of comm altogether and declare the Hessian data as global. The storage scheme suggested here is for illustrative purposes only.

11 Further Description

This section gives a detailed description of the algorithm used in e04nkc. 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

e04nkc is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is similar to that of Gill and Murray (1978), and is described in detail by Gill et al. (1991). Here we briefly summarise the main features of the method. Where possible, explicit reference is made to the names of variables that are arguments of the function or appear in the printed output.

The method used has two distinct phases: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and minimizing the quadratic objective function within the feasible region (the optimality phase). The computations in both phases are performed by the same code. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities (the quantity Sinf described in Section 12.3) to the quadratic objective function (the quantity Objective, see Section 12.3).

In general, an iterative process is required to solve a quadratic program. Given an iterate

(x, s)

in both the original variables

x

and the slack variables

s

, a new iterate

(\bar{x}, \bar{s})

is defined by

(\begin{matrix} \bar{x} \\ \bar{s} \end{matrix}) = (\begin{matrix} x \\ s \end{matrix}) + α p,

(2)

where the step length

α

is a non-negative scalar (the printed quantity Step, see Section 12.3), and

p

is called the search direction. (For simplicity, we shall consider a typical iteration and avoid reference to the index of the iteration.) Once an iterate is feasible (i.e., satisfies the constraints), all subsequent iterates remain feasible.

11.2 Definition of the Working Set and Search Direction

At each iterate

(x, s)

, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the value of the optional parameter

options . ftol

; see Section 12.2). The working set is the current prediction of the constraints that hold with equality at a solution of the LP or QP problem. Let

m_{W}

denote the number of constraints in the working set (including bounds), and let

W

denote the associated

m_{W}

(n + m)

working set matrix consisting of the

m_{W}

gradients of the working set constraints.

The search direction is defined so that constraints in the working set remain unaltered for any value of the step length. It follows that

p

must satisfy the identity

W p = 0 .

(3)

This characterisation allows

p

to be computed using any

n

n_{Z}

full-rank matrix

Z

that spans the null space of

W

. (Thus,

n_{Z} = n - m_{W}

and

W Z = 0

.) The null space matrix

Z

is defined from a sparse

L U

factorization of part of

W

(see (6) and (7) below). The direction

p

will satisfy (3) if

p = {Z p}_{Z},

(4)

where

p_{Z}

is any

n_{Z}

-vector.

The working set contains the constraints

A x - s = 0

and a subset of the upper and lower bounds on the variables

(x, s)

. Since the gradient of a bound constraint

x_{j} \geq l_{j}

x_{j} \leq u_{j}

is a vector of all zeros except for

\pm 1

in position

j

, it follows that the working set matrix contains the rows of

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

and the unit rows associated with the upper and lower bounds in the working set.

The working set matrix

W

can be represented in terms of a certain column partition of the matrix

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

. As in Section 3 we partition the constraints

A x - s = 0

so that

{B x}_{B} + {S x}_{S} + {N x}_{N} = 0,

(5)

where

B

is a square nonsingular basis and

x_{B}

x_{S}

and

x_{N}

are the basic, superbasic and nonbasic variables respectively. The nonbasic variables are equal to their upper or lower bounds at

(x, s)

, and the superbasic variables are independent variables that are chosen to improve the value of the current objective function. The number of superbasic variables is

n_{S}

(the quantity Ns in the detailed printed output; see Section 12.3). Given values of

x_{N}

and

x_{S}

, the basic variables

x_{B}

are adjusted so that

(x, s)

satisfies (5).

P

is a permutation matrix such that

(\begin{matrix} A & - I \end{matrix}) P = (\begin{matrix} B & S & N \end{matrix})

, then the working set matrix

W

satisfies

W P = (\begin{matrix} B & S & N \\ 0 & 0 & I_{N} \end{matrix}),

(6)

where

I_{N}

is the identity matrix with the same number of columns as

N

The null space matrix

Z

is defined from a sparse

L U

factorization of part of

W

. In particular,

Z

is maintained in ‘reduced gradient’ form, using the LUSOL package (see Gill et al. (1987)) to maintain sparse

L U

factors of the basis matrix

B

that alters as the working set

W

changes. Given the permutation

P

, the null space basis is given by

Z = P (\begin{matrix} {- B}^{- 1} S \\ I \\ 0 \end{matrix}) .

(7)

This matrix is used only as an operator, i.e., it is never computed explicitly. Products of the form

Z v

and

Z^{T} g

are obtained by solving with

B

B^{T}

. This choice of

Z

implies that

n_{Z}

, the number of ‘degrees of freedom’ at

(x, s)

, is the same as

n_{S}

, the number of superbasic variables.

Let

g_{Z}

and

H_{Z}

denote the reduced gradient and reduced Hessian of the objective function:

g_{Z} = Z^{T} g and H_{Z} = Z^{T} H Z,

(8)

where

g

is the objective gradient at

(x, s)

. Roughly speaking,

g_{Z}

and

H_{Z}

describe the first and second derivatives of an

n_{S}

-dimensional unconstrained problem for the calculation of

p_{Z}

. (The condition estimator of

H_{Z}

is the quantity Cond Hz in the detailed printed output; see Section 12.3.)

At each iteration, an upper triangular factor

R

is available such that

H_{Z} = R^{T} R

. Normally,

R

is computed from

R^{T} R = Z^{T} H Z

at the start of the optimality phase and then updated as the QP working set changes. For efficiency, the dimension of

R

should not be excessive (say,

n_{S} \leq 1000

). This is guaranteed if the number of nonlinear variables is ‘moderate’.

If the QP problem contains linear variables,

H

is positive semidefinite and

R

may be singular with at least one zero diagonal element. However, an inertia-controlling strategy is used to ensure that only the last diagonal element of

R

can be zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.)

If the initial

R

is singular, enough variables are fixed at their current value to give a nonsingular

R

. This is equivalent to including temporary bound constraints in the working set. Thereafter,

R

can become singular only when a constraint is deleted from the working set (in which case no further constraints are deleted until

R

becomes nonsingular).

11.3 The Main Iteration

If the reduced gradient is zero,

(x, s)

is a constrained stationary point on the working set. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that

x

minimizes the quadratic objective function when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers

λ

are defined from the equations

W^{T} λ = g (x) .

(9)

A Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set is said to be optimal if

λ_{j} \leq σ

when the associated constraint is at its upper bound, or if

λ_{j} \geq - σ

when the associated constraint is at its lower bound, where

σ

depends on the value of the optional parameter

options . optim_tol

(see Section 12.2). If a multiplier is non-optimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by continuing the minimization with the corresponding constraint excluded from the working set. (This step is sometimes referred to as ‘deleting’ a constraint from the working set.) If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is nonzero, there is no feasible point and the function terminates immediately with

fail . code = NW_NOT_FEASIBLE

(see Section 6).

The special form (6) of the working set allows the multiplier vector

λ

, the solution of (9), to be written in terms of the vector

d = (\begin{matrix} g \\ 0 \end{matrix}) - {(\begin{matrix} A & - I \end{matrix})}^{T} π = (\begin{matrix} g - A^{T} π \\ π \end{matrix}),

(10)

where

π

satisfies the equations

B^{T} π = g_{B}

, and

g_{B}

denotes the basic elements of

g

. The elements of

π

are the Lagrange multipliers

λ_{j}

associated with the equality constraints

A x - s = 0

. The vector

d_{N}

of nonbasic elements of

d

consists of the Lagrange multipliers

λ_{j}

associated with the upper and lower bound constraints in the working set. The vector

d_{S}

of superbasic elements of

d

is the reduced gradient

g_{Z}

in (8). The vector

d_{B}

of basic elements of

d

is zero, by construction. (The Euclidean norm of

d_{S}

and the final values of

d_{S}

g

and

π

are the quantities Norm rg, Reduced Gradnt, Obj Gradient and Dual Activity in the detailed printed output; see Section 12.3.)

If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction is given by

p = {Z p}_{Z}

(see (7) and (11)). The step length is chosen to maintain feasibility with respect to the satisfied constraints.

There are two possible choices for

p_{Z}

, depending on whether or not

H_{Z}

is singular. If

H_{Z}

is nonsingular,

R

is nonsingular and

p_{Z}

in (4) is computed from the equations

R^{T} R p_{Z} = - g_{Z},

(11)

where

g_{Z}

is the reduced gradient at

x

. In this case,

(x, s) + p

is the minimizer of the objective function subject to the working set constraints being treated as equalities. If

(x, s) + p

is feasible,

α

is defined to be unity. In this case, the reduced gradient at

(\bar{x}, \bar{s})

will be zero, and Lagrange multipliers are computed at the next iteration. Otherwise,

α

is set to

α_{M}

, the step to the ‘nearest’ constraint along

p

. This constraint is added to the working set at the next iteration.

H_{Z}

is singular, then

R

must also be singular, and an inertia-controlling strategy is used to ensure that only the last diagonal element of

R

is zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.) In this case,

p_{Z}

satisfies

p_{Z}^{T} H_{Z} p_{Z} = 0 and g_{Z}^{T} p_{Z} \leq 0,

(12)

which allows the objective function to be reduced by any step of the form

(x, s) + α p

, where

α > 0

. The vector

p = {Z p}_{Z}

is a direction of unbounded descent for the QP problem in the sense that the QP objective is linear and decreases without bound along

p

. If no finite step of the form

(x, s) + α p

(where

α > 0

) reaches a constraint not in the working set, the QP problem is unbounded and the function terminates immediately with

fail . code = NE_UNBOUNDED

(see Section 6). Otherwise,

α

is defined as the maximum feasible step along

p

and a constraint active at

(x, s) + α p

is added to the working set for the next iteration.

11.4 Miscellaneous

If the basis matrix is not chosen carefully, the condition of the null space matrix

Z

in (7) could be arbitrarily high. To guard against this, the function implements a ‘basis repair’ feature in which the LUSOL package (see Gill et al. (1987)) is used to compute the rectangular factorization

{(\begin{matrix} B & S \end{matrix})}^{T} = L U,

(13)

returning just the permutation

P

that makes

P L P^{T}

unit lower triangular. The pivot tolerance is set to require

{|P L P^{T}|}_{i j} \leq 2

, and the permutation is used to define

P

in (6). It can be shown that

‖Z‖

is likely to be little more than unity. Hence,

Z

should be well conditioned regardless of the condition of

W

. This feature is applied at the beginning of the optimality phase if a potential

B - S

ordering is known.

The EXPAND procedure (see Gill et al. (1989)) is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. Although there is no absolute guarantee that cycling will not occur, the probability of cycling is extremely small (see Hall and McKinnon (1996)). The main feature of EXPAND is that the feasibility tolerance is increased at the start of every iteration. This allows a positive step to be taken at every iteration, perhaps at the expense of violating the bounds on

(x, s)

by a small amount.

Suppose that the value of the optional parameter

options . ftol

(see Section 12.2) is

δ

. Over a period of

K

iterations (where

K

is the value of the optional parameter

options . reset_ftol

; see Section 12.2), the feasibility tolerance actually used by e04nkc (i.e., the working feasibility tolerance) increases from

0.5 δ

δ

(in steps of

0.5 δ / K

At certain stages the following ‘resetting procedure’ is used to remove small constraint infeasibilities. First, all nonbasic variables are moved exactly onto their bounds. A count is kept of the number of non-trivial adjustments made. If the count is nonzero, the basic variables are recomputed. Finally, the working feasibility tolerance is reinitialized to

0.5 δ

If a problem requires more than

K

iterations, the resetting procedure is invoked and a new cycle of iterations is started. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with

δ

The resetting procedure is also invoked when e04nkc reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any non-trivial adjustments are made, iterations are continued.

The EXPAND procedure not only allows a positive step to be taken at every iteration, but also provides a potential choice of constraints to be added to the working set. All constraints at a distance

α

(where

α \leq α_{M}

) along

p

from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. This strategy helps keep the basis matrix

B

well conditioned.

12 Optional Parameters

A number of optional input and output arguments to e04nkc 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 e04nkc; 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 e04nkc, the options structure may only be re-used for future calls of e04nkc 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 functions and 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 e04nkc 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
Nag_PrintType print_level	Nag_Soln_Iter
char outfile[512]	stdout
void (*print_fun)()	NULL
char prob_name[9]	' $\ 0$ '
char obj_name[9]	' $\ 0$ '
char rhs_name[9]	' $\ 0$ '
char range_name[9]	' $\ 0$ '
char bnd_name[9]	' $\ 0$ '
char **crnames	NULL
Boolean minimize	Nag_TRUE
Integer max_iter	$\max (50, 5 (n + m))$
Nag_CrashType crash	$Nag_CrashTwice$
double crash_tol	0.1
Nag_ScaleType scale	$Nag_ExtraScale$
double scale_tol	0.9
double optim_tol	$\max (10^{- 6}, \sqrt{ε})$
double ftol	$\max (10^{- 6}, \sqrt{ε})$
Integer reset_ftol	10000
Integer fcheck	60
Integer factor_freq	100
Integer partial_price	10
double pivot_tol	$ε^{0.67}$
double lu_factor_tol	100.0
double lu_update_tol	10.0
double lu_sing_tol	$ε^{0.67}$
Integer max_sb	$\min (ncolh + 1 n)$
double inf_bound	$10^{20}$
double inf_step	$\max (options . inf_bound, 10^{20})$
Integer *state	size $n + m$
double *lambda	size $n + m$
Integer nsb
Integer iter
Integer nf

12.2 Description of the Optional Parameters

start – Nag_Start

Default

= Nag_Cold

On entry: specifies how the initial working set is to be chosen.

$options . start = Nag_Cold$: An internal Crash procedure will be used to choose an initial basis matrix, $B$ .
$options . start = Nag_Warm$: You must provide a valid definition of every array element of the optional parameter $options . state$ (see below), probably obtained from a previous call of e04nkc, while, for QP problems, the optional parameter $options . nsb$ (see below) must retain its value from a previous call.

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 e04nkc will be printed.

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04nkc. The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Iter_Long$	A longer line of output for each iteration with more information (line exceeds 80 characters).
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Long$	The final solution and one long line of output for each iteration (line exceeds 80 characters).
$Nag_Soln_Iter_Full$	As $Nag_Soln_Iter_Long$ with the matrix statistics (initial status of rows and columns, number of elements, density, biggest and smallest elements, etc.), factors resulting from the scaling procedure (if $options . scale = Nag_RowColScale$ or $Nag_ExtraScale$ ; see below), basis factorization statistics and details of the initial basis resulting from the Crash procedure (if $options . start = Nag_Cold$ ).

Details of each level of results printout are described in Section 12.3.

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Full

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.

print_fun – pointer to function

Default

=

NULL

On entry: printing function defined by you; the prototype of

options . print_fun

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 12.3.1 below for further details.

prob_name – const char

Default:

options . prob_name [0] =' \ 0'

obj_name – const char

Default:

options . obj_name [0] =' \ 0'

rhs_name – const char

Default:

options . rhs_name [0] =' \ 0'

range_name – const char

Default:

options . range_name [0] =' \ 0'

bnd_name – const char

Default:

options . bnd_name [0] =' \ 0'

On entry: these options contain the names associated with the so-called MPSX form of the problem. The arguments contain, respectively, the names of: the problem; the objective (or free) row; the constraint right-hand side; the ranges, and the bounds. They are used in the detailed output when optional parameter

options . print_level = Nag_Soln_Iter_Full

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 12.3).

minimize – Nag_Boolean

Default

= Nag_TRUE

On entry:

options . minimize

specifies the required 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}

(see Section 11.3) will be reversed.

max_iter – Integer

Default

= \max (50, 5 (n + m))

On entry:

options . max_iter

specifies the maximum number of iterations allowed before termination.

If you wish to check that a call to e04nkc is correct before attempting to solve the problem in full then

options . max_iter

may be set to

0

. No iterations will then be performed but all initialization prior to the first iteration will be done and a listing of argument settings will be output, if optional parameter

options . list = Nag_TRUE

(the default setting).

Constraint:

options . max_iter \geq 0

crash – Nag_CrashType

Default

= Nag_CrashTwice

This option does not apply when optional parameter

options . start = Nag_Warm

On entry: if

options . start = Nag_Cold

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

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

. The value of

options . crash

determines which rows and columns are initially eligible for the basis, and how many times the Crash procedure is called.

options . crash = Nag_NoCrash

, the all-slack basis

B = - I

is chosen.

$options . crash = Nag_CrashOnce$: The Crash procedure is called once (looking for a triangular basis in all rows and columns of the linear constraint matrix $A$ ).
$options . crash = Nag_CrashTwice$: The Crash procedure is called twice (looking at any equality constraints first followed by any inequality constraints).

options . crash = Nag_CrashOnce

Nag_CrashTwice

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

options . crash = Nag_CrashTwice

, 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

crash_tol – double

Default

= 0.1

On entry:

options . crash_tol

allows the Crash procedure to ignore certain ‘small’ nonzero elements in the constraint matrix

A

while searching for a triangular basis. For each column of

A

, if

a_{\max}

is the largest element in the column, other nonzeros in that column are ignored if they are less than (or equal to)

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 with more column variables and fewer (arbitrary) slacks. A feasible solution may be reached earlier for some problems.

Constraint:

0.0 \leq options . crash_tol < 1.0

scale – Nag_ScaleType

Default

= Nag_ExtraScale

On entry: this option enables the scaling of 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. This may improve the overall efficiency of the function on some problems. (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

$options . scale = Nag_NoScale$: No scaling is performed.
$options . scale = Nag_RowColScale$: All rows and columns of the constraint matrix $A$ are scaled.
$options . scale = Nag_ExtraScale$: An additional scaling is performed that may be helpful when the solution $x$ is large; it takes into account columns of $(\begin{matrix} A & - I \end{matrix})$ that are fixed or have positive lower bounds or negative upper bounds.

Constraint:

options . scale = Nag_NoScale

Nag_RowColScale

Nag_ExtraScale

scale_tol – double

Default

= 0.9

This option does not apply when optional parameter

options . scale = Nag_NoScale

On entry:

options . scale_tol

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., ('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

optim_tol – double

Default

= \max (10^{- 6}, \sqrt{ε})

On entry:

options . optim_tol

is used to judge the size of the reduced gradients

d_{j} = g_{j} - π^{T} a_{j}

. By definition, the reduced gradients for basic variables are always zero. Optimality is declared if the reduced gradients for any nonbasic variables at their lower or upper bounds satisfy

- options . optim_tol \times \max (1, |π|) \leq d_{j} \leq options . optim_tol \times \max (1, |π|)

, and if

|d_{j}| \leq options . optim_tol \times \max (1, |π|)

for any superbasic variables.

Constraint:

options . optim_tol \geq ε

ftol – double

Default

= \max (10^{- 6}, \sqrt{ε})

On entry:

options . ftol

defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point (including slack variables). For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify

options . ftol

10^{- 6}

e04nkc attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the problem is assumed to be infeasible. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise

options . ftol

by a factor of 10 or

100

. Otherwise, some error in the data should be suspected. Note that e04nkc does not attempt to find the minimum value of Sinf.

If the constraints and variables have been scaled (see optional parameter

options . scale

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

Constraint:

options . ftol \geq ε

reset_ftol – Integer

Default

= 5

On entry: this option is part of an anti-cycling procedure designed to guarantee progress even on highly degenerate problems (see Section 11.4).

For LP problems, 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 the optional parameter

options . ftol

δ

. Over a period of

options . reset_ftol

iterations, the feasibility tolerance actually used by e04nkc (i.e., the working feasibility tolerance) increases from

0.5 δ

δ

(in steps of

0.5 δ / options . reset_ftol

For QP problems, 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 . reset_ftol

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).

Constraint:

0 < options . reset_ftol < 10000000

fcheck – Integer

Default

= 60

On entry: every

options . fcheck

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

(x, s)

satisfies the linear constraints

A x - s = 0

. If the largest element of the residual vector

r = A x - s

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

Constraint:

options . fcheck \geq 1

factor_freq – Integer

Default

= 100

On entry: at most

options . factor_freq

basis changes will occur between factorizations of the basis matrix. For LP problems, the basis factors are usually updated at every iteration. For QP problems, 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 optional parameter

options . fcheck

to ensure that the linear constraints

A x - s = 0

are satisfied. If necessary, the basis will be refactorized before the limit of

options . factor_freq

updates is reached.

Constraint:

options . factor_freq \geq 1

partial_price – Integer

Default

= 10

This option does not apply to QP problems.

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

n ≫ m

). It reduces the work required for each pricing operation (i.e., when a nonbasic variable is selected to enter the basis). If

options . partial_price = 1

, all columns of the constraint matrix

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

are searched. If

options . partial_price > 1

A

and

- I

are partitioned to give

options . partial_price

roughly equal segments

A_{j}, K_{j}

, for

j = 1, 2, \dots, p

(modulo

p

). If the previous pricing search was successful on

A_{j - 1}, K_{j - 1}

, the next search begins on the segments

A_{j}, K_{j}

. 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 + 1}, K_{j + 1}

, and so on.

Constraint:

options . partial_price \geq 1

pivot_tol – double

Default

= ε^{0.67}

On entry:

options . pivot_tol

is used to prevent columns entering the basis if they would cause the basis to become almost singular.

Constraint:

options . pivot_tol > 0.0

lu_factor_tol – double

Default

= 100.0

lu_update_tol – double

Default

= 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 lower triangular matrix

L

is a product of matrices of the form

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

where the multipliers

μ

will 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. For large and relatively dense problems, setting

options . lu_factor_tol

and

options . lu_update_tol

to 25 (say) may give a marked improvement in sparsity without impairing stability to a serious degree. Note that for band matrices it may be necessary to set

options . lu_factor_tol

in the range

1 \leq options . lu_factor_tol < 2

in order to achieve stability.

Constraints:

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

lu_sing_tol – double

Default

= ε^{0.67}

On entry:

options . lu_sing_tol

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.

Constraint:

options . lu_sing_tol > 0.0

max_sb – Integer

Default

= \min (ncolh + 1, n)

This option does not apply to FP or LP problems.

On entry:

options . max_sb

places an upper bound on the number of variables which may enter the set of superbasic variables (see Section 11.2). If the number of superbasics exceeds this bound then e04nkc will terminate with

fail . code = NE_HESS_TOO_BIG

. In effect,

options . max_sb

specifies ‘how nonlinear’ the QP problem is expected to be.

Constraint:

options . max_sb > 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

inf_step – double

Default

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

On entry:

options . inf_step

specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is not positive definite.) If the change in

x

during an iteration would exceed the value of

options . inf_step

, the objective function is considered to be unbounded below in the feasible region.

Constraint:

options . inf_step > 0.0

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 e04nkc.

If the option

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 e04nkc 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:

$options . state [j]$	State of $xs [j]$ 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] = 0

(and

xs [j] = 0.0

), for

j = 0, 1, \dots, n - 1

. 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] = 4

and

xs [j] = bl [j]

options . state [j] = 5

and

xs [j] = bu [j]

as appropriate.

Following the Crash procedure, variables for which

options . state [j] = 2

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

xs [j]

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

, or at the value

bl [j]

bu [j]

closest to

xs [j]

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 e04nkc has been called previously with the same values of n and m,

options . state

already contains satisfactory information.

Constraints:

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

On exit: the final states of the variables and slacks

(x, s)

. The significance of each possible value of

options . state

is as follows:

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

If the problem is feasible (i.e.,

ninf = 0

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

options . ftol

. Note that unless the optional parameter

options . scale = Nag_NoScale

options . ftol

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 . ftol

, and there may be some nonbasic variables for which

xs [j]

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

options . scale = Nag_NoScale

lambda – double *

Default memory

= n + m

On entry:

n + m

values of memory will be automatically allocated by e04nkc and this is the recommended method of use of

options . lambda

. However you may supply memory from the calling program.

On exit: the values of the multipliers for each constraint with respect to the current working set. The first n elements contain the multipliers (reduced costs) for the bound constraints on the variables, and the next m elements contain the Lagrange multipliers (shadow prices) for the general linear constraints.

nsb – Integer

On entry:

n_{S}

, the number of superbasics. For QP problems,

options . nsb

need not be specified if optional parameter

options . start = Nag_Cold

, but must retain its value from a previous call when

options . start = Nag_Warm

. For FP and LP problems,

options . nsb

is not referenced.

Constraint:

options . nsb \geq 0

On exit: the final number of superbasics. This will be zero for FP and LP problems.

iter – Integer

On exit: the total number of iterations performed.

nf – Integer

On exit: the number of times the product

H x

has been calculated (i.e., number of calls of qphx).

12.3 Description of Printed Output

The level of printed output can be controlled with the structure members

options . list

and

options . print_level

(see Section 12.2). If

options . list = Nag_TRUE

then the argument values to e04nkc are listed, whereas the printout of results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single short line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04nkc.

When

options . print_level = Nag_Iter

Nag_Soln_Iter

the output produced at each iteration is as follows:

Itn	is the iteration count.
Step	is the step taken along the computed search direction.
Ninf	is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective	is the current value of the objective function. If $x$ is not feasible, Sinf gives the sum of magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
	During the optimality phase, the value of the objective function will be non-increasing. During the feasibility phase, the number of constraint infeasibilities will not increase until either a feasible point is found, or the optimality of the multipliers implies that no feasible point exists.
Norm rg	is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see Section 11.3). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.

In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

options . print_level = Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Full

the following, more detailed, line of output is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

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.
+S	is the variable selected by the pricing operation to be added to the superbasic set.
-S	is the variable chosen to leave the superbasic set.
-B	is the variable removed from the basis (if any) to become nonbasic.
-B	is the variable chosen to leave the set of basics (if any) in a special basic $\leftrightarrow$ superbasic swap. The entry under -S has become basic if this entry is nonzero, and nonbasic otherwise. The swap is done to ensure that there are no superbasic slacks.
Step	is the value of the steplength $α$ taken along the computed search direction $p$ . The variables $x$ have been changed to $x + α p$ . If a variable is made superbasic during the current iteration (i.e., +S 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 $= ε^{0.67}$ , where $ε$ is the machine precision; see Section 12.2) to exclude very small elements of $y$ from consideration during the computation of Step.
Ninf	is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective	is the current value of the objective function. If $x$ is not feasible, Sinf gives the sum of magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
	During the optimality phase, the value of the objective function will be non-increasing. During the feasibility phase, the number of constraint infeasibilities will not increase until either a feasible point is found, or the optimality of the multipliers implies that no feasible point exists.
L	is the number of nonzeros in the basis factor $L$ . Immediately after a basis factorization $B = L U$ , this is lenL, the number of subdiagonal elements in the columns of a lower triangular matrix. 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, this is lenU, the number of diagonal and superdiagonal elements in the rows of an upper triangular matrix. 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, e04nkc will attempt to expand the internal workspace allocated for the basis factors.
Norm rg	is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see Section 11.3). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.
Ns	is the current number of superbasic variables. For FP and LP problems, Ns is not printed.
Cond Hz	is a lower bound on the condition number of the reduced Hessian (see Section 11.2). The larger this number, the more difficult the problem. For FP and LP problems, Cond Hz is not printed.

When

options . print_level = Nag_Soln_Iter_Full

the following intermediate printout (

< 120

characters) is produced whenever the matrix

B

B_{S} = {(\begin{matrix} B & S \end{matrix})}^{T}

is factorized. Gaussian elimination is used to compute an

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

(see Section 12.2).

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.
$1$	Number of updates reached the value of the optional parameter $options . factor_freq$ (see Section 12.2).
$2$	Excessive nonzeros in updated factors.
$7$	Not enough storage to update factors.
$10$	Row residuals too large (see the description for the optional parameter $options . fcheck$ in Section 12.2).
$11$	Ill conditioning has caused inconsistent results.

Iteration

is the iteration count.

Nonlinear

is the number of nonlinear variables in

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.

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

(not printed if

B_{S}

is factorized). This will not exceed the value of the optional parameter

options . lu_factor_tol

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 optional parameter

options . lu_factor_tol

to a value near

1.0

. Umax is not printed if

B_{S}

is factorized.

Umin

is the magnitude of the smallest diagonal element of

P U Q

(not printed if

B_{S}

is factorized).

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 could occur in subsequent computations. (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.)

Growth is not printed if

B_{S}

is factorized.

is the number of triangular columns of

B

at the beginning of

L

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

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

When

options . print_level = Nag_Soln_Iter_Full

the following lines of intermediate printout (

< 80

characters) are produced whenever

options . start = Nag_Cold

(see Section 12.2). They refer to the number of columns selected by the Crash procedure during each of several passes through

A

, whilst 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.
Preferred	is the number of ‘preferred’ columns in the basis (i.e., $options . state [j] = 3$ for some $j < n$ ).
Unit	is the number of unit columns in the basis.
Double	is the number of double columns in the basis.
Triangle	is the number of triangular columns in the basis.
Pad	is the number of slacks used to pad the basis.

When

options . print_level = Nag_Soln_Iter_Full

the following lines of intermediate printout (

< 80

characters) are produced, following the final iteration. They refer to the ‘MPSX names’ stored in the optional parameters

options . prob_name

options . obj_name

options . rhs_name

options . range_name

and

options . bnd_name

(see Section 12.2).

Name	gives the name for the problem (blank if none).
Status	gives the exit status for the problem (i.e., Optimal soln, Weak soln, Unbounded, Infeasible, Excess itns, Error condn or Feasble soln) followed by details of the direction of the optimization (i.e., (Min) or (Max)).
Objective	gives the name of the free row for the problem (blank if none).
RHS	gives the name of the constraint right-hand side for the problem (blank if none).
Ranges	gives the name of the ranges for the problem (blank if none).
Bounds	gives the name of the bounds for the problem (blank if none).

When

options . print_level = Nag_Soln

Nag_Soln_Iter

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 variable

j

, for

j = 1, 2, \dots, n

. If an options structure is supplied to e04nkc, and the

options . crnames

member is assigned to an array of column and row names (see Section 12.2 for details), the name supplied in

options . crnames [j - 1]

is assigned to the

j

th variable. Otherwise, a default name is assigned to the 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 = Nag_NoScale

(default value is

options . scale = Nag_ExtraScale

; see Section 12.2) 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 bound, there would be no change in the value of the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the Lagrange multipliers might also change.
D	Degenerate. The variable is basic or superbasic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. The variable is basic or superbasic and is currently violating one of its bounds by more than the value of the optional parameter $options . ftol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ , where $ε$ is the machine precision; see Section 12.2).
N	Not precisely optimal. The variable is nonbasic or superbasic. If the value of the reduced gradient for the variable exceeds the value of the optional parameter $options . optim_tol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ ; see Section 12.2), the solution would not be declared optimal because the reduced gradient for the variable would not be considered negligible.

Value

is the value of the variable at the final iteration.

Lower Bound

is the lower bound specified for variable

j

. (None indicates that

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

, where

options . inf_bound

is the optional parameter.)

Upper Bound

is the upper bound specified for variable

j

. (None indicates that

bu [j - 1] \geq options . inf_bound

Lagr Mult

is the value of 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 linear constraint.

Note that the movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Residual column to become positive.

When

options . print_level = Nag_Soln_Iter_Long

Nag_Soln_Iter_Full

, the following longer lines of final printout (

< 120

characters) are produced.

Let

a_{j}

denote the

j

th column of

A

, for

j = 1, 2, \dots, n

. 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}

(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

x_{j}

. Note that unless the optional parameter

options . scale = Nag_NoScale

(default value is

options . scale = Nag_ExtraScale

; see Section 12.2) 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 bound, 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. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the Lagrange multipliers might also change.
D	Degenerate. $x_{j}$ is basic or superbasic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. $x_{j}$ is basic or superbasic and is currently violating one of its bounds by more than the value of the optional parameter $options . ftol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ , where $ε$ is the machine precision; see Section 12.2).
N	Not precisely optimal. $x_{j}$ is nonbasic or superbasic. If the value of the reduced gradient for $x_{j}$ exceeds the value of the optional parameter $options . optim_tol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ ; see Section 12.2), the solution would not be declared optimal because the reduced gradient for $x_{j}$ would not be considered negligible.

Activity

is the value of

x_{j}

at the final iterate.

Obj Gradient

is the value of

g_{j}

at the final iterate. For FP problems,

g_{j}

is set to zero.

Lower Bound

is the lower bound specified for

x_{j}

. (None indicates that

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

, where

options . inf_bound

is the optional parameter.)

Upper Bound

is 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 (see Section 11.3). For FP problems,

d_{j}

is set to zero.

m + j

is the value of

m + j

Let

v_{i}

denote the

i

th row of

A

, for

i = 1, 2, \dots, m

. 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

v_{i}

State

gives the state of

v_{i}

(LL if active on its lower bound, UL if active on its upper bound, EQ if active and fixed, BS if inactive when

s_{i}

is basic and SBS if inactive when

s_{i}

is superbasic).

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 = Nag_NoScale

(default value is

options . scale = Nag_ExtraScale

; see Section 12.2) 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 bound, 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. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case, the values of the dual variables (or Lagrange multipliers) might also change.
D	Degenerate. $s_{i}$ is basic or superbasic, but it is equal to (or very close to) one of its bounds.
I	Infeasible. $s_{i}$ is basic or superbasic and is currently violating one of its bounds by more than the value of the optional parameter $options . ftol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ , where $ε$ is the machine precision; see Section 12.2).
N	Not precisely optimal. $s_{i}$ is nonbasic or superbasic. If the value of the reduced gradient for $s_{i}$ exceeds the value of the optional parameter $options . optim_tol$ (default value $= \max (10^{- 6}, \sqrt{ε})$ ; see Section 12.2), the solution would not be declared optimal because the reduced gradient for $s_{i}$ would not be considered negligible.

Activity

is the value of

v_{i}

at the final iterate.

Slack Activity

is the value by which

v_{i}

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

Lower Bound

is the lower bound specified for

v_{j}

. None indicates that

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

, where

options . inf_bound

is the optional parameter.

Upper Bound

is the upper bound specified for

v_{j}

. None indicates that

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

Dual Activity

is the value of the dual variable

π_{i}

(the Lagrange multiplier for

v_{i}

; see Section 11.3). For FP problems,

π_{i}

is set to zero.

gives the index

i

v_{i}

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

options . print_level = Nag_NoPrint

then printout will be suppressed; you can print the final solution when e04nkc returns to the calling program.

12.3.1 Output of results via a user-defined printing function

You may also specify your own print function for output of iteration results and the final solution by use of the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

The rest of this section can be skipped if you wish to use the default printing facilities.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of e04nkc. Calls to the user-defined function are again controlled by means of the

options . print_level

member. Information is provided through st and comm, the two structure arguments to

options . print_fun

comm \to it_prt = Nag_TRUE

then the results from the last iteration of e04nkc are provided through st. Note that

options . print_fun

will be called with

comm \to it_prt = Nag_TRUE

only if

options . print_level = Nag_Iter

Nag_Iter_Long

Nag_Soln_Iter

Nag_Soln_Iter_Long

Nag_Soln_Iter_Full

The following members of st are set:

iter – Integer: The iteration count.

qp – Nag_Boolean: Nag_TRUE if a QP problem is being solved; Nag_FALSE otherwise.

pprice – Integer: The partial price indicator.

rgval – double: The value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration.

sb_add – Integer: The variable selected to enter the superbasic set.

sb_leave – double: The variable chosen to leave the superbasic set.

b_leave – Integer: The variable chosen to leave the basis (if any) to become nonbasic.

bswap_leave – Integer: The variable chosen to leave the basis (if any) in a special basic $\leftrightarrow$ superbasic swap.

step – double: The step length taken along the computed search direction.

pivot – double: 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 $B$ .

ninf – Integer: The number of violated constraints or infeasibilities.

f – double: The current value of the objective function if $st \to ninf$ is zero; otherwise, the sum of the magnitudes of constraint violations.

nnz_l – Integer: The number of nonzeros in the basis factor $L$ .

nnz_u – Integer: The number of nonzeros in the basis factor $U$ .

ncp – Integer: The number of compressions of the basis factorization workspace carried out so far.

norm_rg – double: The Euclidean norm of the reduced gradient at the start of the current iteration. This value is meaningful only if $st \to qp = Nag_TRUE$ .

nsb – Integer: The number of superbasic variables. This value is meaningful only if $st \to qp = Nag_TRUE$ .

cond_hz – double: A lower bound on the condition number of the reduced Hessian. This value is meaningful only if $st \to qp = Nag_TRUE$ .

comm \to sol_prt = Nag_TRUE

then the final results for one row or column are provided through st. Note that

options . print_fun

will be called with

comm \to sol_prt = Nag_TRUE

only if

options . print_level = Nag_Soln

Nag_Soln_Iter

Nag_Soln_Iter_Long

Nag_Soln_Iter_Full

. The following members of st are set (note that

options . print_fun

is called repeatedly, for each row and column):

m – Integer: The number of rows (or general constraints) in the problem.

n – Integer: The number of columns (or variables) in the problem.

col – Nag_Boolean: Nag_TRUE if column information is being provided; Nag_FALSE if row information is being provided.

index – Integer: If $st \to col = Nag_TRUE$ then $st \to index$ is the index $j$ (in the range $1 \leq j \leq n$ ) of the current column (variable) for which the remaining members of st, as described below, are set.

If $st \to col = Nag_FALSE$ then $st \to index$ is the index $i$ (in the range $1 \leq$ i $\leq m$ ) of the current row (constraint) for which the remaining members of st, as described below, are set.

name – char *: The name of row $i$ or column $j$ .

sstate – char *: $st \to sstate$ is a character string describing the state of row $i$ or column $j$ . This may be "LL", "UL", "EQ", "FR", "BS" or "SBS". The meaning of each of these is described in Section 12.3 (State).

key – char *: $st \to key$ is a character string which gives additional information about the current row or column. The possible values of $st \to key$ are: " ", "A", "D", "I" or "N". The meaning of each of these is described in Section 12.3 (State).

val – double: The activity of row $i$ or column $j$ at the final iterate.

blo – double: The lower bound on row $i$ or column $j$ .

bup – double: The upper bound on row $i$ or column $j$ .

lmult – double: The value of the Lagrange multiplier associated with the current row or column (i.e., the dual activity $π_{i}$ for a row, or the reduced gradient $d_{j}$ for a column) at the final iterate.

objg – double: The value of the objective gradient $g_{j}$ at the final iterate. $st \to objg$ is meaningful only when $st \to col = Nag_TRUE$ and should not be accessed otherwise. It is set to zero for FP problems.

The relevant members of the structure comm are:

it_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the final result.

user – double
iuser – Integer
p – Pointer: Pointers for communication of user information. If used they must be allocated memory either before entry to e04nkc or during a call to qphess or $options . print_fun$ . The type Pointer will be void * with a C compiler that defines void * and char * otherwise.