e05ucc is designed to find the global minimum of an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) by generating a number of different starting points and performing a local search from each using sequential quadratic programming.

2 Specification

#include <nag.h>

void

e05ucc (Integer n, Integer nclin, Integer ncnln, const double a[], Integer tda, const double bl[], const double bu[],

void	(confun)(Integer mode, Integer ncnln, Integer n, Integer tdcjsl, const Integer needc[], const double x[], double c[], double cjsl[], Integer nstate, Nag_Comm *comm),

void	(objfun)(Integer mode, Integer n, const double x[], double objf, double objgrd[], Integer nstate, Nag_Comm comm),

Integer npts, double x[], Integer ldx,

void	(start)(Integer npts, double quas[], Integer n, Nag_Boolean repeat, const double bl[], const double bu[], Nag_Comm comm, Integer *mode),

Nag_Boolean repeat, Integer nb, double objf[], double objgrd[], Integer ldobjgrd, Integer iter[], double c[], Integer ldc, double cjac[], Integer ldcjac, Integer sdcjac, double r[], Integer ldr, Integer sdr, double clamda[], Integer ldclamda, Integer istate[], Integer ldistate, Integer iopts[], double opts[], Nag_Comm *comm, Integer info[], NagError *fail)

The function may be called by the names: e05ucc or nag_glopt_nlp_multistart_sqp.

Before calling e05ucc, the optional parameter arrays iopts and opts MUST be initialized for use with e05ucc by calling e05zkc with optstr set to ‘Initialize = e05ucc’. Optional parameters may be specified by calling e05zkc before the call to e05ucc.

The declared lenths of iopts and opts must be at least

740

and

485

respectively.

3 Description

The problem is assumed to be stated in the following form:

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

(1)

where

F (x)

(the objective function) is a nonlinear function,

A_{L}

is an

n_{L}

n

linear constraint matrix, and

c (x)

is an

n_{N}

element vector of nonlinear constraint functions. (The matrix

A_{L}

and the vector

c (x)

may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twice-continuously differentiable. (This function will usually solve (1) if there are only isolated discontinuities away from the solution.)

e05ucc solves a user-specified number of local optimization problems with different starting points. You may specify the starting points via the function start. If a random number generator is used to generate the starting points then the argument repeat allows you to specify whether a repeatable set of points are generated or whether different starting points are generated on different calls. The resulting local minima are ordered and the best nb results returned in order of ascending values of the resulting objective function values at the minima. Thus the value returned in position

1

will be the best result obtained. If a sufficient number of different points are chosen then this is likely to be be the global minimum. Please note that the default version of start uses a random number generator to generate the starting points.

4 References

Dennis J E Jr and Moré J J (1977) Quasi-Newton methods, motivation and theory SIAM Rev. 19 46–89

Dennis J E Jr and Schnabel R B (1981) A new derivation of symmetric positive-definite secant updates nonlinear programming (eds O L Mangasarian, R R Meyer and S M Robinson) 4 167–199 Academic Press

Dennis J E Jr and Schnabel R B (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice–Hall

Fletcher R (1987) Practical Methods of Optimization (2nd Edition) Wiley

Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1984) Users' guide for SOL/QPSOL version 3.2 Report SOL 84–5 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1986a) Some theoretical properties of an augmented Lagrangian merit function Report SOL 86–6R Department of Operations Research, Stanford University

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

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

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

Powell M J D (1983) Variable metric methods in constrained optimization Mathematical Programming: the State of the Art (eds A Bachem, M Grötschel and B Korte) 288–311 Springer–Verlag

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of variables.

Constraint:

n > 0

2: $nclin$ – Integer Input

On entry:

n_{L}

, the number of general linear constraints.

Constraint:

nclin \geq 0

3: $ncnln$ – Integer Input

On entry:

n_{N}

, the number of nonlinear constraints.

Constraint:

ncnln \geq 0

4: $a [\dim]$ – const double Input

Note: the dimension, dim, of the array a must be at least

nclin \times tda

On entry: the matrix

A_{L}

of general linear constraints in (1). That is,

a [(i - 1) \times tda + j - 1]

must contain the

j

th coefficient of the

i

th general linear constraint, for

j = 1, 2, \dots, n

and

i = 1, 2, \dots, nclin

. If

nclin = 0

then a may be specified as NULL.

5: $tda$ – Integer Input

On entry: the stride separating matrix column elements in the array a.

Constraints:

if $nclin > 0$ , $tda \geq n$ .

6: $bl [n + nclin + ncnln]$ – const double Input

7: $bu [n + nclin + ncnln]$ – 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, the next

n_{L}

elements the bounds for the general linear constraints (if any) and the next

n_{N}

elements the bounds for the general nonlinear constraints (if any). To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j - 1] \leq - bigbnd

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

u_{j} = + \infty

), set

bu [j - 1] \geq bigbnd

; the default value of

bigbnd

10^{20}

, but this may be changed by the optional parameter

Infinite Bound Size

. To specify the

j

th constraint as an equality, set

bl [j - 1] = bu [j - 1] = β

, say, where

|β| < bigbnd

Constraints:

$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, n + nclin + ncnln$ ;
if $bl [j - 1] = bu [j - 1] = β$ , $|β| < bigbnd$ .

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

confun must calculate the vector

c (x)

of nonlinear constraint functions and (optionally) its Jacobian (

= \frac{\partial c}{\partial x}

) for a specified

n

-element vector

x

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

ncnln = 0

), confun will never be called by e05ucc and If there are nonlinear constraints, the first call to confun will occur before the first call to objfun.

The specification of confun is:

void	confun (Integer mode, Integer ncnln, Integer n, Integer tdcjsl, const Integer needc[], const double x[], double c[], double cjsl[], Integer nstate, Nag_Comm comm)

1: $mode$ – Integer * Input/Output

On entry: indicates which values must be assigned during each call of confun. Only the following values need be assigned, for each value of

i

such that

needc [i - 1] > 0

$mode = 0$: $c [i - 1]$ .
$mode = 1$: All available elements in $CJSL (i, j)$ , for $j = 1, 2, \dots, n$ (see cjsl for the definition of CJSL).
$mode = 2$: $c [i - 1]$ and all available elements in $CJSL (i, j)$ , for $j = 1, 2, \dots, n$ (see cjsl for the definition of CJSL).

On exit: may be set to a negative value if you wish to abandon the solution to the current local minimization problem. In this case e05ucc will move to the next local minimization problem.

2: $ncnln$ – Integer Input

On entry:

n_{N}

, the number of nonlinear constraints.

3: $n$ – Integer Input

On entry:

n

, the number of variables.

4: $tdcjsl$ – Integer Input

On entry: the stride separating matrix column elements in the array cjsl.

5: $needc [ncnln]$ – const Integer Input

On entry: the indices of the elements of c and/or cjsl that must be evaluated by confun. If

needc [i - 1] > 0

c [i - 1]

and/or the available elements of

CJSL (i, j)

, for

j = 1, 2, \dots, n

(see argument mode) must be evaluated at

x

. See cjsl for the definition of CJSL.

6: $x [n]$ – const double Input

On entry:

x

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

7: $c [ncnln]$ – double Output

On exit: if

needc [k - 1] > 0

and

mode = 0

2

c [k - 1]

must contain the value of

c_{k} (x)

. The remaining elements of c, corresponding to the non-positive elements of needc, need not be set.

8: $cjsl [ncnln \times tdcjsl]$ – double Input/Output

Note: where

CJSL (k, j)

appears in this document, it refers to the array element

cjsl [(k - 1) \times n + j - 1]

cjsl may be regarded as a two-dimensional ‘slice’ in row order of the three-dimensional matrix CJAC stored in the array cjac of e05ucc.

On entry: unless

Derivative Level = 2

3

, the elements of cjsl are set to special values which enable e05ucc to detect whether they are changed by confun.

On exit: if

needc [k - 1] > 0

and

mode = 1

2

CJSL (k, j)

, for

j = 1, 2, \dots, n

, must contain the available elements of the vector

\nabla c_{k}

given by

\nabla c_{k} = {(\frac{\partial c_{k}}{\partial x_{1}}, \frac{\partial c_{k}}{\partial x_{2}}, \dots, \frac{\partial c_{k}}{\partial x_{n}})}^{T},

where

\frac{\partial c_{k}}{\partial x_{j}}

is the partial derivative of the

k

th constraint with respect to the

j

th variable, evaluated at the point

x

. See also the argument nstate. The remaining

CJSL (k, j)

, for

j = 1, 2, \dots, n

, corresponding to non-positive elements of needc, need not be set.

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

Derivative Level = 2

3

), any constant elements may be assigned to cjsl one time only at the start of each local optimization. An element of cjsl that is not subsequently assigned in confun will retain its initial value throughout the local optimization. Constant elements may be loaded into cjsl during the first call to confun for the local optimization (signalled by the value

nstate = 1

). The ability to preload constants is useful when many Jacobian elements are identically zero, in which case cjsl may be initialized to zero and nonzero elements may be reset by confun.

Note that constant nonzero elements do affect the values of the constraints. Thus, if

CJSL (k, j)

is set to a constant value, it need not be reset in subsequent calls to confun, but the value

CJSL (k, j) \times x [j - 1]

must nonetheless be added to

c [k - 1]

. For example, if

CJSL (1, 1) = 2

and

CJSL (1, 2) = - 5

then the term

2 \times x [0] - 5 \times x [1]

must be included in the definition of

c [0]

It must be emphasized that, if

Derivative Level = 0

1

, unassigned elements of cjsl are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional parameter

Difference Interval

, an interval for each element of

x

is computed automatically at the start of each local optimization. The automatic procedure can usually identify constant elements of cjsl, which are then computed once only by finite differences.

9: $nstate$ – Integer Input

On entry: if

nstate = 1

then e05ucc is calling confun for the first time on the current local optimization problem. This argument setting allows you to save computation time if certain data must be calculated only once.

10: $comm$ – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling e05ucc you may allocate memory and initialize these pointers with various quantities for use by confun when called from e05ucc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05ucc. If your code inadvertently does return any NaNs or infinities, e05ucc is likely to produce unexpected results.

confun should be tested separately before being used in conjunction with e05ucc. See also the description of the optional parameter

Verify

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

objfun must calculate the objective function

F (x)

and (optionally) its gradient

g (x) = \frac{\partial F}{\partial x}

for a specified

n

-vector

x

The specification of objfun is:

void	objfun (Integer mode, Integer n, const double x[], double objf, double objgrd[], Integer nstate, Nag_Comm *comm)

1: $mode$ – Integer * Input/Output

On entry: indicates which values must be assigned during each call of objfun. Only the following values need be assigned:

$mode = 0$: objf.
$mode = 1$: All available elements of objgrd.
$mode = 2$: objf and all available elements of objgrd.

On exit: may be set to a negative value if you wish to abandon the solution to the current local minimization problem. In this case e05ucc will move to the next local minimization problem.

2: $n$ – Integer Input

On entry:

n

, the number of variables.

3: $x [n]$ – const double Input

On entry:

x

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

4: $objf$ – double * Output

On exit: if

mode = 0

2

, objf must be set to the value of the objective function at

x

5: $objgrd [n]$ – double Input/Output

On entry: the elements of objgrd are set to special values which enable e05ucc to detect whether they are changed by objfun.

On exit: if

mode = 1

2

, objgrd must return the available elements of the gradient evaluated at

x

6: $nstate$ – Integer Input

On entry: if

nstate = 1

then e05ucc is calling objfun for the first time on the current local optimization problem. This argument setting allows you to save computation time if certain data must be calculated only once.

7: $comm$ – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling e05ucc you may allocate memory and initialize these pointers with various quantities for use by objfun when called from e05ucc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05ucc. If your code inadvertently does return any NaNs or infinities, e05ucc is likely to produce unexpected results.

objfun should be tested separately before being used in conjunction with e05ucc. See also the description of the optional parameter

Verify

10: $npts$ – Integer Input

On entry: the number of different starting points to be generated and used. The more points used, the more likely that the best returned solution will be a global minimum.

Constraint:

1 \leq nb \leq npts

11: $x [ldx \times nb]$ – double Output

Note: where

X (j, i)

appears in this document, it refers to the array element

x [(i - 1) \times ldx + j - 1]

On exit:

X (j, i)

contains the final estimate of the

i

th solution, for

j = 1, 2, \dots, n

12: $ldx$ – Integer Input

On entry: the first dimension of X as stored in the array x.

Constraint:

ldx \geq n

13: $start$ – function, supplied by the user External Function

start must calculate the npts starting points to be used by the local optimizer. If you do not wish to write a function specific to your problem then you can specify the NAG defined null void function pointer, NULLFN in the call. In this case, a default function uses the NAG quasi-random number generators to distribute starting points uniformly across the domain. It is affected by the value of repeat.

The specification of start is:

void	start (Integer npts, double quas[], Integer n, Nag_Boolean repeat, const double bl[], const double bu[], Nag_Comm comm, Integer mode)

1: $npts$ – Integer Input

On entry: indicates the number of starting points.

2: $quas [n \times npts]$ – double Input/Output

On entry: all elements of quas will have been set to zero, so only nonzero values need be set subsequently.

On exit: must contain the starting points for the npts local minimizations, i.e.,

quas [(j - 1) \times npts + i - 1]

must contain the

j

th component of the

i

th starting point.

3: $n$ – Integer Input

On entry: the number of variables.

4: $repeat$ – Nag_Boolean Input

On entry: specifies whether a repeatable or non-repeatable sequence of points are to be generated.

5: $bl [n]$ – const double Input

On entry: the lower bounds on the variables. These may be used to ensure that the starting points generated in some sense ‘cover’ the region, but there is no requirement that a starting point be feasible.

6: $bu [n]$ – const double Input

On entry: the upper bounds on the variables. (See bl.)

7: $comm$ – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling e05ucc you may allocate memory and initialize these pointers with various quantities for use by start when called from e05ucc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

8: $mode$ – Integer * Input/Output

On entry: mode will contain

0

On exit: if you set mode to a negative value then e05ucc will terminate immediately with

fail . code =

NE_USER_STOP. Provided fail is not NAGERR_DEFAULT on entry to e05ucc, fail

. errnum

will contain this value of mode.

Note: start should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e05ucc. If your code inadvertently does return any NaNs or infinities, e05ucc is likely to produce unexpected results.

14: $repeat$ – Nag_Boolean Input

On entry: is passed as an argument to start and may be used to initialize a random number generator to a repeatable, or non-repeatable, sequence.

15: $nb$ – Integer Input

On entry: the number of solutions to be returned. The function saves up to nb local minima ordered by increasing value of the final objective function. If the defining criterion for ‘best solution’ is only that the value of the objective function is as small as possible then nb should be set to

1

. However, if you want to look at other solutions that may have desirable properties then setting

nb > 1

will produce nb local minima, ordered by increasing value of their objective functions at the minima.

Constraint:

1 \leq nb \leq npts

16: $objf [nb]$ – double Output

On exit:

objf [i - 1]

contains the value of the objective function at the final iterate for the

i

th solution.

17: $objgrd [ldobjgrd \times nb]$ – double Output

Note: where

OBJGRD (j, i)

appears in this document, it refers to the array element

objgrd [(i - 1) \times ldobjgrd + j - 1]

On exit:

OBJGRD (j, i)

contains the gradient of the objective function for the

i

th solution at the final iterate (or its finite difference approximation), for

j = 1, 2, \dots, n

18: $ldobjgrd$ – Integer Input

On entry: the first dimension of OBJGRD as stored in the array objgrd.

Constraint:

ldobjgrd \geq n

19: $iter [nb]$ – Integer Output

On exit:

iter [i - 1]

contains the number of major iterations performed to obtain the

i

th solution. If less than nb solutions are returned then

iter [nb - 1]

contains the number of starting points that have resulted in a converged solution. If this is close to npts then this might be indicative that fewer than nb local minima exist.

20: $c [ldc \times nb]$ – double Output

Note: where

C (j, i)

appears in this document, it refers to the array element

c [(i - 1) \times ldc + j - 1]

On exit: if

ncnln > 0

C (j, i)

contains the value of the

j

th nonlinear constraint function

c_{j}

at the final iterate, for the

i

th solution, for

j = 1, 2, \dots, ncnln

ncnln = 0

, c is not referenced and may be specified as NULL.

21: $ldc$ – Integer Input

On entry: the first dimension of C as stored in the array c.

Constraint:

ldc \geq ncnln

22: $cjac [\dim]$ – double Output

Note: the dimension, dim, of the array cjac must be at least

ldcjac \times sdcjac \times nb

where

CJAC (k, j, i)

appears in this document, it refers to the array element

cjac [(i - 1) \times ldcjac \times sdcjac + (j - 1) \times ldcjac + k - 1]

On exit: if

ncnln > 0

, cjac contains the Jacobian matrices of the nonlinear constraint functions at the final iterate for each of the returned solutions, i.e.,

CJAC (k, j, i)

contains the partial derivative of the

k

th constraint function with respect to the

j

th variable, for

k = 1, 2, \dots, ncnln

and

j = 1, 2, \dots, n

, for the

i

th solution. (See the discussion of argument cjsl under confun.)

ncnln = 0

, cjac is not referenced and may be specified as NULL.

23: $ldcjac$ – Integer Input

On entry: the first dimension of the matrix CJAC as stored in the array cjac.

Constraint:

ldcjac \geq ncnln

24: $sdcjac$ – Integer Input

On entry: the second dimension of the matrix CJAC as stored in the array cjac.

Constraint: if

ncnln > 0

sdcjac \geq n

25: $r [\dim]$ – double Output

Note: the dimension, dim, of the array r must be at least

ldr \times sdr \times nb

the element

R (i, j, k)

is stored in the array element

r [(k - 1) \times ldr \times sdr + (j - 1) \times ldr + i - 1]

On exit: for each of the nb solutions r will contain a form of the Hessian; for the

i

th returned solution

R (ldr, sdr, i)

contains the Hessian that would be returned from the local minimizer. If

Hessian = NO

, the default, each

R (ldr, sdr, i)

contains the upper triangular Cholesky factor

R

Q^{T} H Q

, an estimate of the transformed and reordered Hessian of the Lagrangian at

x

. If

Hessian = YES

R (ldr, sdr, i)

contains the upper triangular Cholesky factor

R

H

, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.

26: $ldr$ – Integer Input

On entry: the first dimension of the matrix R as stored in the array r.

Constraint:

ldr \geq n

27: $sdr$ – Integer Input

On entry: the second dimension of the matrix R as stored in the array r.

Constraint:

sdr \geq n

28: $clamda [ldclamda \times nb]$ – double Output

Note: where

CLAMDA (j, i)

appears in this document, it refers to the array element

clamda [(i - 1) \times ldclamda + j - 1]

On exit: the values of the QP multipliers from the last QP subproblem solved for the

i

th solution.

CLAMDA (j, i)

should be non-negative if

ISTATE (j, i) = 1

and non-positive if

ISTATE (j, i) = 2

29: $ldclamda$ – Integer Input

On entry: the first dimension of CLAMDA as stored in the array clamda.

Constraint:

ldclamda \geq n + nclin + ncnln

30: $istate [ldistate \times nb]$ – Integer Output

Note: where

ISTATE (j, i)

appears in this document, it refers to the array element

istate [(i - 1) \times ldistate + j - 1]

On exit:

ISTATE (j, i)

contains the status of the constraints in the QP working set for the

i

th solution. The significance of each possible value of

ISTATE (j, i)

is as follows:

$ISTATE (j, i)$	Meaning
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set.
$1$	This inequality constraint is included in the QP working set at its lower bound.
$2$	This inequality constraint is included in the QP working set at its upper bound.
$3$	This constraint is included in the QP working set as an equality. This value of istate can occur only when $bl [j - 1] = bu [j - 1]$ .

31: $ldistate$ – Integer Input

On entry: the first dimension of ISTATE as stored in the array istate.

Constraint:

ldistate \geq n + nclin + ncnln

32: $iopts [740]$ – Integer Communication Array

33: $opts [485]$ – double Communication Array

The arrays iopts and opts MUST NOT be altered between calls to any of the functions e05ucc and e05zkc.

34: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

35: $info [nb]$ – Integer Output

On exit:

info [i - 1]

contains one of

0

1

6

$info [i - 1] = 1$: The final iterate $x$ satisfies the first-order Kuhn–Tucker conditions (see Section 11.1) to the accuracy requested, but the sequence of iterates has not yet converged. The local optimizer was terminated because no further improvement could be made in the merit function (see Section 9.2).
$info [i - 1] = 6$: $x$ does not satisfy the first-order Kuhn–Tucker conditions (see Section 11.1) and no improved point for the merit function (see Section 9.2) could be found during the final linesearch.
This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional parameter $Optimality Tolerance$ ( $default value = ε_{R}^{0.8}$ , where $ε_{R}$ is the value of the optional parameter $Function Precision$ ( $default value = ε^{0.9}$ , where $ε$ is the machine precision)) is too small.

As usual

0

denotes success.

fail . code =

NW_SOME_SOLUTIONS on exit, then not all nb solutions have been found, and

info [nb - 1]

contains the number of solutions actually found.

36: $fail$ – 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.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_BOUND: On entry, $bl [i - 1] > bu [i - 1]$ : $i = 〈value〉$ .
Constraint: $bl [i - 1] \leq bu [i - 1]$ , for all $i$ .
NE_DERIV_ERRORS: User-supplied derivatives probably wrong.
The user-supplied derivatives of the objective function and/or nonlinear constraints appear to be incorrect.
Large errors were found in the derivatives of the objective function and/or nonlinear constraints. This value of fail.code will occur if the verification process indicated that at least one gradient or Jacobian element had no correct figures. You should refer to or enable the printed output to determine which elements are suspected to be in error.
As a first-step, you should check that the code for the objective and constraint values is correct – for example, by computing the function at a point where the correct value is known. However, care should be taken that the chosen point fully tests the evaluation of the function. It is remarkable how often the values $x = 0$ or $x = 1$ are used to test function evaluation procedures, and how often the special properties of these numbers make the test meaningless.
Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed before each function evaluation.
Errors in programming the function may be quite subtle in that the function value is ‘almost’ correct. For example, the function may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the function depends. A common error on machines where numerical calculations are usually performed in double precision is to include even one single precision constant in the calculation of the function; since some compilers do not convert such constants to double precision, half the correct figures may be lost by such a seemingly trivial error.
NE_INITIALIZATION: Failed to initialize optional parameter arrays.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 0$ .

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

On entry, $ncnln = 〈value〉$ .
Constraint: $ncnln \geq 0$ .
NE_INT_2: On entry, $ldc = 〈value〉$ and $ncnln = 〈value〉$ .
Constraint: $ldc \geq ncnln$ .

On entry, $ldcjac = 〈value〉$ and $ncnln = 〈value〉$ .
Constraint: $ldcjac \geq ncnln$ .

On entry, $ldobjgrd = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldobjgrd \geq n$ .

On entry, $ldr = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldr \geq n$ .

On entry, $ldx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldx \geq n$ .

On entry, $nb = 〈value〉$ and $npts = 〈value〉$ .
Constraint: $1 \leq nb \leq npts$ .

On entry, $sdr = 〈value〉$ and $n = 〈value〉$ .
Constraint: $sdr \geq n$ .
NE_INT_3: On entry, $ncnln > 0$ , $sdcjac = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $ncnln > 0$ , $sdcjac \geq n$ .

On entry, $tda = 〈value〉$ , $nclin = 〈value〉$ and $n = 〈value〉$ .
Constraint: $tda \geq n$ .
NE_INT_4: On entry, $ldclamda = 〈value〉$ , $n = 〈value〉$ , $nclin = 〈value〉$ and $ncnln = 〈value〉$ .
Constraint: $ldclamda \geq n + nclin + ncnln$ .

On entry, $ldistate = 〈value〉$ , $n = 〈value〉$ , $nclin = 〈value〉$ and $ncnln = 〈value〉$ .
Constraint: $ldistate \geq n + nclin + ncnln$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_LIN_NOT_FEASIBLE: No solution obtained. Linear constraints may be infeasible.
e05ucc has terminated without finding any solutions. The majority of calls to the local optimizer have failed to find a feasible point for the linear constraints and bounds, which means that either no feasible point exists for the given value of the optional parameter $Linear Feasibility Tolerance$ (default value $\sqrt{ε}$ , where $ε$ is the machine precision), or no feasible point could be found in the number of iterations specified by the optional parameter $Minor Iteration Limit$ . You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision $σ$ , you should ensure that the value of the optional parameter $Linear Feasibility Tolerance$ is greater than $σ$ . For example, if all elements of $A_{L}$ are of order unity and are accurate to only three decimal places, $Linear Feasibility Tolerance$ should be at least $10^{- 3}$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_SOLUTION: No solution obtained. Many potential solutions reach iteration limit.
The $Iteration Limit$ may be changed using e05zkc.
NE_NONLIN_NOT_FEASIBLE: e05ucc has failed to find any solutions. The majority of local optimizations could not find a feasible point for the nonlinear constraints. The problem may have no feasible solution. This behaviour will occur if there is no feasible point for the nonlinear constraints. (However, there is no general test that can determine whether a feasible point exists for a set of nonlinear constraints.)
NE_USER_STOP: User terminated computation from start procedure: $mode = 〈value〉$ .
NW_SOME_SOLUTIONS: Only $〈value〉$ solutions obtained.
Not all nb solutions have been found. $info [nb - 1]$ contains the number actually found.

7 Accuracy

fail . code =

NE_NOERROR on exit and the value of

info [i - 1] = 0

, then the vector returned in the array x for solution

i

is an estimate of the solution to an accuracy of approximately

Optimality Tolerance

8 Parallelism and Performance

e05ucc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library. In these implementations, this function may make calls to the user-supplied functions from within an OpenMP parallel region. Thus OpenMP pragmas within the user functions can only be used if you are compiling the user-supplied function and linking the executable in accordance with the instructions in the Users' Note for your implementation. You must also ensure that you use the NAG communication argument comm in a thread safe manner, which is best achieved by only using it to supply read-only data to the user functions.

e05ucc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

You should be wary of requesting much intermediate output from the local optimizer, since large volumes may be produced if npts is large.

If NULLFN is supplied an actual argument from start then the default NAG function makes use of the NAG quasi-random Sobol generator (g05ylc and g05ymc). If this is used as an argument for start, by specifying NULLFN in the calling sequence (see the description of start) and

repeat = Nag_FALSE

then a randomly chosen value for iskip is used, otherwise iskip is set to

100

. If repeat is set to Nag_FALSE and the program is executed several times, each time producing the same best answer, then there is increased probability that this answer is a global minimum. However, if it is important that identical results be obtained on successive runs, then repeat should be set to Nag_TRUE.

9.1 Description of the Printed Output

This section describes the intermediate printout and final printout produced by e05ucc. The intermediate printout is a subset of the monitoring information produced by the function at every iteration (see Section 13). You can control the level of printed output (see the description of the optional parameters

Major Print Level

and

Minor Print Level

). Note that the intermediate printout and final printout are produced only if

Major Print Level \geq 10

Minor Print Level \geq 10

The following line of summary output (

< 80

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

Maj	is the major iteration count.
Mnr	is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be $1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 11). Note that Mnr may be greater than the optional parameter $Minor Iteration Limit$ if some iterations are required for the feasibility phase.
Step	is the step $α_{k}$ taken along the computed search direction. On reasonably well-behaved local problems, the unit step (i.e., $α_{k} = 1$ ) will be taken as the solution is approached.
Merit Function	is the value of the augmented Lagrangian merit function (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 11.3). As the solution is approached, Merit Function will converge to the value of the objective function at the solution. If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or the local optimizer terminates. Repeated failures will prevent a feasible point being found for the nonlinear constraints. If there are no nonlinear constraints present (i.e., $ncnln = 0$ ) then this entry contains Objective, the value of the objective function $F (x)$ . The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.
Norm Gz	is $‖Z^{T} g_{FR}‖$ , the Euclidean norm of the projected gradient (see Section 11.2). Norm Gz will be approximately zero in the neighbourhood of a solution.
Violtn	is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.
Cond Hz	is a lower bound on the condition number of the projected Hessian approximation $H_{Z}$ ( $H_{Z} = Z^{T} H_{FR} Z = R_{Z}^{T} R_{Z}$ ; see (6)). The larger this number, the more difficult the local problem.
M	is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive definite (see Section 11.4).
I	is printed if the QP subproblem has no feasible point.
C	is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that $x$ is close to a Kuhn–Tucker point (see Section 11.1).
L	is printed if the linesearch has produced a relative change in $x$ greater than the value defined by the optional parameter $Step Limit$ . If this output occurs frequently during later iterations of the run, optional parameter $Step Limit$ should be set to a larger value.
R	is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of $R$ indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, $R$ is modified so that its diagonal condition estimator is bounded.

The final printout includes a listing of the status of every variable and constraint. The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.

Varbl

gives the name (V) and index

j

, for

j = 1, 2, \dots, n

, of the variable.

State

gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the

Feasibility Tolerance

, State will be ++ or -- respectively. (The latter situation can occur only when there is no feasible point for the bounds and linear constraints.)

A key is sometimes printed before State.

A	Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to 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 free, but it is equal to (or very close to) one of its bounds.
I	Infeasible. The variable is currently violating one of its bounds by more than the $Feasibility Tolerance$ .

Value

is the value of the variable at the final iteration.

Lower Bound

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

bl [j - 1] \leq - bigbnd

Upper Bound

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

bu [j - 1] \geq bigbnd

Lagr Mult

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

bl [j - 1] \leq - bigbnd

and

bu [j - 1] \geq bigbnd

, in which case the entry will be blank. If

x

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

Slack

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 - bigbnd

and

bu [j - 1] \geq bigbnd

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

bl [j - 1]

and

bu [j - 1]

are replaced by

bl [n + j - 1]

and

bu [n + j - 1]

respectively, and with the following changes in the heading:

L Con	gives the name (L) and index $j$ , for $j = 1, 2, \dots, n_{L}$ , of the linear constraint.
N Con	gives the name (N) and index ( $j - n_{L}$ ), for $j = n_{L} + 1, \dots, n_{L} + n_{N}$ , of the nonlinear constraint.

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

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

10 Example

This example finds the global minimum of the two-dimensional Schwefel function:

\underset{x \in R^{2}}{minimize} f = \sum_{j = 1}^{2} x_{j} \sin (\sqrt{|x_{j}|})

subject to the constraints:

\begin{matrix} - 10000 < 3.0 x_{1} - 2.0 x_{2} < 10.0, \\ - 1.0 < x_{1}^{2} - x_{2}^{2} + 3.0 x_{1} x_{2} < 500000.0, \\ - 0.9 < \cos ({(x_{1} / 200)}^{2} + (x_{2} / 100)) < 0.9, \\ - 500 \leq x_{1} \leq 500, \\ - 500 \leq x_{2} \leq 500 . \end{matrix}

11 Algorithmic Details

This section contains a detailed description of the method used by e05ucc.

11.1 Overview

e05ucc uses a local optimizer that is essentially identical to the function NPSOL described in Gill et al. (1986b).

For the local optimizer, at a solution of (1), some of the constraints will be active, i.e., satisfied exactly. An active simple bound constraint implies that the corresponding variable is fixed at its bound, and hence the variables are partitioned into fixed and free variables. Let

C

denote the

m

n

matrix of gradients of the active general linear and nonlinear constraints. The number of fixed variables will be denoted by

n_{FX}

, with

n_{FR}

(n_{FR} = n - n_{FX})

the number of free variables. The subscripts ‘FX’ and ‘FR’ on a vector or matrix will denote the vector or matrix composed of the elements corresponding to fixed or free variables.

A point

x

is a first-order Kuhn–Tucker point for (1) (see Powell (1974)) if the following conditions hold:

(i) $x$ is feasible;
(ii)there exist vectors $ξ$ and $λ$ (the Lagrange multiplier vectors for the bound and general constraints) such that
$g = C^{T} λ + ξ$ (2)
where $g$ is the gradient of $F$ evaluated at $x$ , and $ξ_{j} = 0$ if the $j$ th variable is free.
(iii)The Lagrange multiplier corresponding to an inequality constraint active at its lower bound must be non-negative, and non-positive for an inequality constraint active at its upper bound.

Let

Z

denote a matrix whose columns form a basis for the set of vectors orthogonal to the rows of

C_{FR}

; i.e.,

C_{FR} Z = 0

. An equivalent statement of the condition (2) in terms of

Z

Z^{T} g_{FR} = 0 .

The vector

Z^{T} g_{FR}

is termed the projected gradient of

F

x

. Certain additional conditions must be satisfied in order for a first-order Kuhn–Tucker point to be a solution of (1) (see Powell (1974)).

The local optimizer implements a sequential quadratic programming (SQP) method. For an overview of SQP methods, see, for example, Fletcher (1987), Gill et al. (1981) and Powell (1983).

The basic structure of the local optimizer involves major and minor iterations. The major iterations generate a sequence of iterates

\{x_{k}\}

that converge to

x^{*}

, a first-order Kuhn–Tucker point of (1). At a typical major iteration, the new iterate

\bar{x}

is defined by

\bar{x} = x + α p

(3)

where

x

is the current iterate, the non-negative scalar

α

is the step length, and

p

is the search direction. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Also associated with each major iteration are estimates of the Lagrange multipliers and a prediction of the active set.

The search direction

p

in (3) is the solution of a quadratic programming subproblem of the form

\underset{p}{minimize} ​ ​ g^{T} p + \frac{1}{2} p^{T} H p subject to \bar{l} \leq \{\begin{matrix} p \\ A_{L} p \\ A_{N} p \end{matrix}\} \leq \bar{u},

(4)

where

g

is the gradient of

F

x

, the matrix

H

is a positive definite quasi-Newton approximation to the Hessian of the Lagrangian function (see Section 11.4), and

A_{N}

is the Jacobian matrix of

c

evaluated at

x

. (Finite difference estimates may be used for

g

and

A_{N}

; see the optional parameter

Derivative Level

.) Let

l

in (1) be partitioned into three sections:

l_{B}

l_{L}

and

l_{N}

, corresponding to the bound, linear and nonlinear constraints. The vector

\bar{l}

in (4) is similarly partitioned, and is defined as

{\bar{l}}_{B} = l_{B} - x, {\bar{l}}_{L} = l_{L} - A_{L} x,   and {\bar{l}}_{N} = l_{N} - c,

where

c

is the vector of nonlinear constraints evaluated at

x

. The vector

\bar{u}

is defined in an analogous fashion.

The estimated Lagrange multipliers at each major iteration are the Lagrange multipliers from the subproblem (4) (and similarly for the predicted active set). (The numbers of bounds, general linear and nonlinear constraints in the QP active set are the quantities Bnd, Lin and Nln in the monitoring file output of e05ucc; see Section 13.) The local optimizer repeatedly solves as major iterations quadratic programming problems. These are themselves iterative procedures and comprise the minor iterations. (More details about solving the subproblem are given in Section 11.2.)

Certain matrices associated with the QP subproblem are relevant in the major iterations. Let the subscripts ‘FX’ and ‘FR’ refer to the predicted fixed and free variables, and let

C

denote the

m

n

matrix of gradients of the general linear and nonlinear constraints in the predicted active set. First, we have available the

T Q

factorization of

C_{FR}

C_{FR} Q_{FR} = (\begin{matrix} 0 & T \end{matrix}),

(5)

where

T

is a nonsingular

m

m

reverse-triangular matrix (i.e.,

t_{i j} = 0

i + j < m

), and the nonsingular

n_{FR}

n_{FR}

matrix

Q_{FR}

is the product of orthogonal transformations (see Gill et al. (1984)). Second, we have the upper triangular Cholesky factor

R

of the transformed and reordered Hessian matrix

R^{T} R = H_{Q} \equiv Q^{T} \tilde{H} Q,

(6)

where

\tilde{H}

is the Hessian

H

with rows and columns permuted so that the free variables are first, and

Q

is the

n

n

matrix

Q = (\begin{matrix} Q_{FR} \\ I_{FX} \end{matrix})

(7)

with

I_{FX}

the identity matrix of order

n_{FX}

. If the columns of

Q_{FR}

are partitioned so that

Q_{FR} = (\begin{matrix} Z & Y \end{matrix}),

the

n_{Z}

(

n_{Z} \equiv n_{FR} - m

) columns of

Z

form a basis for the null space of

C_{FR}

. The matrix

Z

is used to compute the projected gradient

Z^{T} g_{FR}

at the current iterate. (The values Nz and Norm Gz printed by e05ucc give

n_{Z}

and

‖Z^{T} g_{FR}‖

; see Section 13.)

A theoretical characteristic of SQP methods is that the predicted active set from the QP subproblem (4) is identical to the correct active set in a neighbourhood of

x^{*}

. In the local optimizer underlying e05ucc, this feature is exploited by using the QP active set from the previous iteration as a prediction of the active set for the next QP subproblem, which leads in practice to optimality of the subproblems in only one iteration as the solution is approached. Separate treatment of bound and linear constraints in the local optimizer also saves computation in factorizing

C_{FR}

and

H_{Q}

Once

p

has been computed, the major iteration proceeds by determining a step length

α

that produces a ‘sufficient decrease’ in an augmented Lagrangian merit function (see Section 11.3). Finally, the approximation to the transformed Hessian matrix

H_{Q}

is updated using a modified BFGS quasi-Newton update (see Section 11.4) to incorporate new curvature information obtained in the move from

x

\bar{x}

On entry to the local optimizer, an iterative procedure is executed, starting with the user-supplied initial point, to find a point that is feasible with respect to the bounds and linear constraints (using the tolerance specified by optional parameter

Linear Feasibility Tolerance

). If no feasible point exists for the bound and linear constraints, (1) has no solution and the local optimizer terminates. Otherwise, the problem functions will thereafter be evaluated only at points that are feasible with respect to the bounds and linear constraints. The only exception involves variables whose bounds differ by an amount comparable to the finite difference interval (see the discussion of optional parameter

Difference Interval

). In contrast to the bounds and linear constraints, it must be emphasized that the nonlinear constraints will not generally be satisfied until an optimal point is reached.

Facilities are provided to check whether the user-supplied gradients appear to be correct (see the description of the optional parameter

Verify

). In general, the check is provided at the first point that is feasible with respect to the linear constraints and bounds. However, you may request that the check be performed at the initial point.

In summary, the local method of e05ucc first determines a point that satisfies the bound and linear constraints. Thereafter, each iteration includes:

(a)the solution of a quadratic programming subproblem;
(b)a linesearch with an augmented Lagrangian merit function; and
(c)a quasi-Newton update of the approximate Hessian of the Lagrangian function.

These three procedures are described in more detail in Sections 11.2 to 11.4.

11.2 Solution of the Quadratic Programming Subproblem

The search direction

p

is obtained by solving (4) using a method (see Gill et al. (1986)) that was specifically designed to be used within an SQP algorithm for nonlinear programming. This method is based on a two-phase (primal) quadratic programming method. The two phases of the method are: 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 functions. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function.

In general, a quadratic program must be solved by iteration. Let

p

denote the current estimate of the solution of (4); the new iterate

\bar{p}

is defined by

\bar{p} = p + σ d

(8)

where, as in (3),

σ

is a non-negative step length and

d

is a search direction.

At the beginning of each iteration of the QP method, a working set is defined of constraints (general and bound) that are satisfied exactly. The vector

d

is then constructed so that the values of constraints in the working set remain unaltered for any move along

d

. For a bound constraint in the working set, this property is achieved by setting the corresponding element of

d

to zero, i.e., by fixing the variable at its bound. As before, the subscripts ‘FX’ and ‘FR’ denote selection of the elements associated with the fixed and free variables.

Let

C

denote the sub-matrix of rows of

(\begin{matrix} A_{L} \\ A_{N} \end{matrix})

corresponding to general constraints in the working set. The general constraints in the working set will remain unaltered if

C_{FR} d_{FR} = 0,

(9)

which is equivalent to defining

d_{FR}

d_{FR} = Z d_{Z}

(10)

for some vector

d_{Z}

, where

Z

is the matrix associated with the

T Q

factorization (5) of

C_{FR}

The definition of

d_{Z}

in (10) depends on whether the current

p

is feasible. If not,

d_{Z}

is zero except for an element

γ

in the

j

th position, where

j

and

γ

are chosen so that the sum of infeasibilities is decreasing along

d

. (For further details, see Gill et al. (1986).) In the feasible case,

d_{Z}

satisfies the equations

R_{Z}^{T} R_{Z} d_{Z} = - Z^{T} q_{FR},

(11)

where

R_{Z}

is the Cholesky factor of

Z^{T} H_{FR} Z

and

q

is the gradient of the quadratic objective function

(q = g + H p)

. (The vector

Z^{T} q_{FR}

is the projected gradient of the QP.) With (11),

p + d

is the minimizer of the quadratic objective function subject to treating the constraints in the working set as equalities.

If the QP projected gradient is zero, the current point is a constrained stationary point in the subspace defined by the working set. During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may vanish at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that

p

minimizes the quadratic objective function when the constraints in the working set are treated as equalities. In either case, Lagrange multipliers are computed. Given a positive constant

δ

of the order of the machine precision, the Lagrange multiplier

μ_{j}

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

μ_{j} \leq δ

when the

j

th constraint is at its upper bound, or if

μ_{j} \geq - δ

when the associated constraint is at its lower bound. If any multiplier is nonoptimal, the current objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint from the working set.

If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, no feasible point exists. The QP algorithm will then continue iterating to determine the minimum sum of infeasibilities. At this point, the Lagrange multiplier

μ_{j}

will satisfy

- (1 + δ) \leq μ_{j} \leq δ

for an inequality constraint at its upper bound, and

- δ \leq μ_{j} \leq (1 + δ)

for an inequality at its lower bound. The Lagrange multiplier for an equality constraint will satisfy

|μ_{j}| \leq 1 + δ

The choice of step length

σ

in the QP iteration (8) is based on remaining feasible with respect to the satisfied constraints. During the optimality phase, if

p + d

is feasible,

σ

will be taken as unity. (In this case, the projected gradient at

\bar{p}

will be zero.) Otherwise,

σ

is set to

σ_{M}

, the step to the ‘nearest’ constraint, which is added to the working set at the next iteration.

Each change in the working set leads to a simple change to

C_{FR}

: if the status of a general constraint changes, a row of

C_{FR}

is altered; if a bound constraint enters or leaves the working set, a column of

C_{FR}

changes. Explicit representations are recurred of the matrices

T

Q_{FR}

and

R

, and of the vectors

Q^{T} q

and

Q^{T} g

11.3 The Merit Function

After computing the search direction as described in Section 11.2, each major iteration proceeds by determining a step length

α

in (3) that produces a ‘sufficient decrease’ in the augmented Lagrangian merit function

L (x, λ, s) = F (x) - \sum_{i} λ_{i} (c_{i} (x) - s_{i}) + \frac{1}{2} \sum_{i} ρ_{i} {(c_{i} (x) - s_{i})}^{2},

(12)

where

x

λ

and

s

vary during the linesearch. The summation terms in (12) involve only the nonlinear constraints. The vector

λ

is an estimate of the Lagrange multipliers for the nonlinear constraints of (1). The non-negative slack variables

\{s_{i}\}

allow nonlinear inequality constraints to be treated without introducing discontinuities. The solution of the QP subproblem (4) provides a vector triple that serves as a direction of search for the three sets of variables. The non-negative vector

ρ

of penalty parameters is initialized to zero at the beginning of the first major iteration. Thereafter, selected elements are increased whenever necessary to ensure descent for the merit function. Thus, the sequence of norms of

ρ

(the printed quantity Penalty; see Section 13) is generally nondecreasing, although each

ρ_{i}

may be reduced a limited number of times.

The merit function (12) and its global convergence properties are described in Gill et al. (1986a).

11.4 The Quasi-Newton Update

The matrix

H

in (4) is a positive definite quasi-Newton approximation to the Hessian of the Lagrangian function. (For a review of quasi-Newton methods, see Dennis and Schnabel (1983).) At the end of each major iteration, a new Hessian approximation

\bar{H}

is defined as a rank-two modification of

H

. In the local optimizer used by e05ucc, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton update is used:

\bar{H} = H - \frac{1}{s^{T} H s} H s s^{T} H + \frac{1}{y^{T} s} y y^{T},

(13)

where

s = \bar{x} - x

(the change in

x

In the local optimizer,

H

is required to be positive definite. If

H

is positive definite,

\bar{H}

defined by (13) will be positive definite if and only if

y^{T} s

is positive (see Dennis and Moré (1977)). Ideally,

y

in (13) would be taken as

y_{L}

, the change in gradient of the Lagrangian function

y_{L} = \bar{g} - {\bar{A}}_{N}^{T} μ_{N} - g + A_{N}^{T} μ_{N},

(14)

where

μ_{N}

denotes the QP multipliers associated with the nonlinear constraints of the original problem. If

y_{L}^{T} s

is not sufficiently positive, an attempt is made to perform the update with a vector

y

of the form

y = y_{L} + \sum_{i = 1}^{m_{N}} ω_{i} (a_{i} (\hat{x}) c_{i} (\hat{x}) - a_{i} (x) c_{i} (x)),

where

ω_{i} \geq 0

. If no such vector can be found, the update is performed with a scaled

y_{L}

; in this case, M is printed to indicate that the update was modified.

Rather than modifying

H

itself, the Cholesky factor of the transformed Hessian

H_{Q}

(6) is updated, where

Q

is the matrix from (5) associated with the active set of the QP subproblem. The update (13) is equivalent to the following update to

H_{Q}

{\bar{H}}_{Q} = H_{Q} - \frac{1}{s_{Q}^{T} H_{Q} s_{Q}} H_{Q} s_{Q} s_{Q}^{T} H_{Q} + \frac{1}{y_{Q}^{T} s_{Q}} y_{Q} y_{Q}^{T},

(15)

where

y_{Q} = Q^{T} y

, and

s_{Q} = Q^{T} s

. This update may be expressed as a rank-one update to

R

(see Dennis and Schnabel (1981)).

12 Optional Parameters

Several optional parameters in e05ucc define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e05ucc these optional parameters have associated default values that are appropriate for most problems. Therefore you need only specify those optional parameters whose values are to be different from their default values.

The remainder of this section can be skipped if you wish to use the default values for all optional parameters. The following is a list of the optional parameters available and a full description of each optional parameter is provided in Section 12.2.

Optional parameters may be specified by calling e05zkc before a call to e05ucc. Before calling e05ucc, the optional parameter arrays iopts and opts MUST be initialized for use with e05ucc by calling e05zkc with optstr set to ‘Initialize = e05ucc’.

All optional parameters not specified are set to their default values. Optional parameters specified are unaltered by e05ucc (unless they define invalid values) and so remain in effect for subsequent calls to e05ucc.

12.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted)
a parameter value, where the letters $a$ , $i$ and $r$ denote options that take character, integer and real values respectively
the default value, where the symbol $ε$ is a generic notation for machine precision (see X02AJC), and $ε_{r}$ denotes the relative precision of the objective function $Function Precision$ , and $bigbnd$ signifies the value of $Infinite Bound Size$

Keywords and character values are case insensitive, however they must be separated by at least one whitespace.

Optional parameters used to specify files have type Nag_FileID (see Section 3.1.1 in the Introduction to the NAG Library CL Interface). This ID value must either be set to

0

(the default value) in which case there will be no output, or will be as returned by a call of x04acc.

For e05ucc the maximum length of the argument cvalue used by e05zlc is

11

Central Difference Interval

r

Default values are computed

If the algorithm switches to central differences because the forward-difference approximation is not sufficiently accurate, the value of

r

is used as the difference interval for every element of

x

. The switch to central differences is indicated by C at the end of each line of intermediate printout produced by the major iterations (see Section 9.2). The use of finite differences is discussed further under the optional parameter

Difference Interval

If you supply a value for this optional parameter, a small value between

0.0

and

1.0

is appropriate.

Crash Tolerance

r

Default

= 0.01

This value is used when the local minimizer selects an initial working set. If

0 \leq r \leq 1

, the initial working set will include (if possible) bounds or general inequality constraints that lie within

r

of their bounds. In particular, a constraint of the form

a_{j}^{T} x \geq l

will be included in the initial working set if

|a_{j}^{T} x - l| \leq r (1 + |l|)

. If

r < 0

r > 1

, the default value is used.

Defaults

This special keyword is used to reset all optional parameters to their default values, and any random state stored in state will be destroyed.

Any option value given with this keyword will be ignored. This optional parameter cannot be queried or got.

Derivative Level

i

Default

= 3

This parameter indicates which derivatives are provided in user-supplied functions objfun and confun. The possible choices for

i

are the following.

$i$	Meaning
3	All elements of the objective gradient and the constraint Jacobian are provided.
2	All elements of the constraint Jacobian are provided, but some elements of the objective gradient are not specified.
1	All elements of the objective gradient are provided, but some elements of the constraint Jacobian are not specified.
0	Some elements of both the objective gradient and the constraint Jacobian are not specified.

The value

i = 3

should be used whenever possible, since e05ucc is more reliable (and will usually be more efficient) when all derivatives are exact.

i = 0

2

, e05ucc will estimate the unspecified elements of the objective gradient, using finite differences. The computation of finite difference approximations usually increases the total run-time, since a call to objfun is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of Gill et al. (1981), for a discussion of limiting accuracy).

i = 0

1

, e05ucc will approximate unspecified elements of the constraint Jacobian. One call to confun is needed for each variable for which partial derivatives are not available. For example, if the Jacobian has the form

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

where ‘

*

’ indicates an element provided by you and ‘?’ indicates an unspecified element, the local minimizer will call confun twice: once to estimate the missing element in column 2, and again to estimate the two missing elements in column

3

. (Since columns

1

and

4

are known, they require no calls to confun.)

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

i < 0

i > 3

, the default value is used.

Difference Interval

r

Default values are computed

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

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

In general, a derivative with respect to the

j

th variable is approximated using the interval

δ_{j}

, where

δ_{j} = r (1 + |{\hat{x}}_{j}|)

, with

\hat{x}

the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled, the resulting derivative approximation should be accurate to

O (r)

. See Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.

If a difference interval is not specified, a finite difference interval will be computed automatically for each variable by a procedure that requires up to six calls of confun and objfun for each element. This option is recommended if the function is badly scaled or you wish to have the local minimizer determine constant elements in the objective and constraint gradients (see the descriptions of confun and objfun in Section 5).

If you supply a value for this optional parameter, a small value between

0.0

and

1.0

is appropriate.

Feasibility Tolerance

r

Default

= \sqrt{ε}

The scalar

r

defines the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a constraint is considered satisfied if its violation does not exceed

r

. If

r < ε

r \geq 1

, the default value is used. Using this keyword sets both optional parameters

Linear Feasibility Tolerance

and

Nonlinear Feasibility Tolerance

r

, if

ε \leq r < 1

. (Additional details are given under the descriptions of these optional parameters.)

Function Precision

r

Default

= ε^{0.9}

This parameter defines

ε_{r}

, which is intended to be a measure of the accuracy with which the problem functions

F (x)

and

c (x)

can be computed. If

r < ε

r \geq 1

, the default value is used.

The value of

ε_{r}

should reflect the relative precision of

1 + |F (x)|

; i.e.,

ε_{r}

acts as a relative precision when

|F|

is large, and as an absolute precision when

|F|

is small. For example, if

F (x)

is typically of order

1000

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

ε_{r}

would be

10^{- 6}

. In contrast, if

F (x)

is typically of order

10^{- 4}

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

ε_{r}

would be

10^{- 10}

. The choice of

ε_{r}

can be quite complicated for badly scaled problems; see Chapter 8 of Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy. However, when the accuracy of the computed function values is known to be significantly worse than full precision, the value of

ε_{r}

should be large enough so that e05ucc will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.

Hessian

a

Default

= NO

This option controls the contents of the upper triangular matrix

R

(see Section 5). e05ucc works exclusively with the transformed and reordered Hessian

H_{Q}

(6), and hence extra computation is required to form the Hessian itself. If

Hessian = NO

, r contains the Cholesky factor of the transformed and reordered Hessian. If

Hessian = YES

, the Cholesky factor of the approximate Hessian itself is formed and stored in r.

Infinite Bound Size

r

Default

= 10^{20}

This defines the ‘infinite’ bound

infbnd

in the definition of the problem constraints. Any upper bound greater than or equal to

infbnd

will be regarded as

\infty

(and similarly any lower bound less than or equal to

- infbnd

will be regarded as

- \infty

Constraint:

r_{\max}^{\frac{1}{4}} \leq infbnd \leq r_{\max}^{\frac{1}{2}}

Infinite Step Size

r

Default

= \max (bigbnd, 10^{20})

r > 0

r

specifies the magnitude of the change in variables that is treated as a step to an unbounded solution. If the change in

x

during an iteration would exceed the value of

r

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

r \leq 0

, the default value is used.

Line Search Tolerance

r

Default

= 0.9

The value

r

(

0 \leq r < 1

) controls the accuracy with which the step

α

taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of

r

, the more accurate the linesearch). The default value

r = 0.9

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

If there are no nonlinear constraints, a more accurate search may be appropriate when it is desirable to reduce the number of major iterations – for example, if the objective function is cheap to evaluate, or if a substantial number of derivatives are unspecified. If

r < 0

r \geq 1

, the default value is used.

Linear Feasibility Tolerance

r_{1}

Default

= \sqrt{ε}

Nonlinear Feasibility Tolerance

r_{2}

Default

= ε^{0.33}

\sqrt{ε}

The default value of

r_{2}

ε^{0.33}

Derivative Level = 0

1

, and

\sqrt{ε}

otherwise.

The scalars

r_{1}

and

r_{2}

define the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed

r_{1}

, and similarly for a nonlinear constraint and

r_{2}

. If

r_{m} < ε

r_{m} \geq 1

, the default value is used, for

m = 1, 2

On entry to the local optimizer an iterative procedure is executed in order to find a point that satisfies the linear constraints and bounds on the variables to within the tolerance

r_{1}

. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless

r_{1}

is comparable to the finite difference interval).

For nonlinear constraints, the feasibility tolerance

r_{2}

defines the largest constraint violation that is acceptable at an optimal point. Since nonlinear constraints are generally not satisfied until the final iterate, the value of optional parameter

Nonlinear Feasibility Tolerance

acts as a partial termination criterion for the iterative sequence generated by the local minimizer (see the discussion of optional parameter

Optimality Tolerance

These tolerances should reflect the precision of the corresponding constraints. 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

r_{1}

10^{- 6}

List

Nolist

Default

Optional parameter

List

enables printing of each optional parameter specification as it is supplied.

Nolist

suppresses this printing.

Major Iteration Limit

i

Default

= \max (50, 3 (n + n_{L}) + 10 n_{N})

Iteration Limit

i

Iters

i

Itns

i

The value of

i

specifies the maximum number of major iterations allowed before termination of each local subproblem. Setting

i = 0

and

Major Print Level > 0

means that the workspace needed by each local minimization will be computed and printed, but no iterations will be performed. If

i < 0

, the default value is used.

Major Print Level

i

Default

= 0

Print Level

i

The value of

i

controls the amount of printout produced by the major iterations of e05ucc, as indicated below. A detailed description of the printed output is given in Section 9.2 (summary output at each major iteration and the final solution) and Section 13 (monitoring information at each major iteration). (See also the description of the optional parameter

Minor Print Level

The following printout is sent to stdout:

$i$	Output
$0$	No output.

For the other values described below, the parameters used by the local minimizer are displayed in addition to intermediate and final output.

$i$	Output
$1$	The final solution only.
$5$	One line of summary output ( $< 80$ characters; see Section 9.2) for each major iteration (no printout of the final solution).
$\geq 10$	The final solution and one line of summary output for each major iteration.

The following printout is sent to the file associated with the FileID defined by the optional parameter

Monitoring File

$i$	Output
$< 5$	No output.
$\geq 5$	One long line of output ( $> 80$ characters; see Section 13) for each major iteration (no printout of the final solution).
$\geq 20$	At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector $c$ ), the values of the linear constraints (the vector $A_{L} x$ ), and the current values of the variables (the vector $x$ ).
$\geq 30$	At each major iteration, the diagonal elements of the matrix $T$ associated with the $T Q$ factorization (5) (see Section 11.1) of the QP working set, and the diagonal elements of $R$ , the triangular factor of the transformed and reordered Hessian (6) (see Section 11.1).

Minor Iteration Limit

i

Default

= \max (50, 3 (n + n_{L} + n_{N}))

The value of

i

specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of

i

also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If

i \leq 0

, the default value is used.

Minor Print Level

i

Default

= 0

The value of

i

controls the amount of printout produced by the minor iterations of e05ucc (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in Section 9.2 (summary output at each minor iteration and the final QP solution) and Section 13 (monitoring information at each minor iteration). (See also the description of the optional parameter

Major Print Level

.) The following printout is sent to stdout:

$i$	Output
$0$	No output.
$1$	The final QP solution only.
$5$	One line of summary output ( $< 80$ characters; see Section 9.2) for each minor iteration (no printout of the final QP solution).
$\geq 10$	The final QP solution and one line of summary output for each minor iteration.

The following printout is sent to the file associated with the FileID defined by the optional parameter

Monitoring File

$i$	Output
$< 5$	No output.
$\geq 5$	One long line of output ( $> 80$ characters; see Section 9.2) for each minor iteration (no printout of the final QP solution).
$\geq 20$	At each minor iteration, the current estimates of the QP multipliers, the current estimate of the QP search direction, the QP constraint values, and the status of each QP constraint.
$\geq 30$	At each minor iteration, the diagonal elements of the matrix $T$ associated with the $T Q$ factorization (5) (see Section 11.1) of the QP working set, and the diagonal elements of the Cholesky factor $R$ of the transformed Hessian (6) (see Section 11.1).

Monitoring File

Default

= - 1

(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)

i

is of the type Nag_FileID and is obtained by a call to x04acc.

i \geq 0

and

Major Print Level \geq 5

i \geq 0

and

Minor Print Level \geq 5

, monitoring information produced by e05ucc at every iteration is sent to a file with ID

i

. If

i < 0

and/or

Major Print Level < 5

and

Minor Print Level < 5

, no monitoring information is produced.

Optimality Tolerance

r

Default

= ε_{R}^{0.8}

The parameter

r

(

ε_{R} \leq r < 1

) specifies the accuracy to which you wish the final iterate to approximate a solution of each local problem. Broadly speaking,

r

indicates the number of correct figures desired in the objective function at the solution. For example, if

r

10^{- 6}

and a local minimization terminates successfully, the final value of

F

should have approximately six correct figures. If

r < ε_{r}

r \geq 1

, the default value is used.

The local optimizer will terminate successfully if the iterative sequence of

x

values is judged to have converged and the final point satisfies the first-order Kuhn–Tucker conditions (see Section 11.1). The sequence of iterates is considered to have converged at

x

α ‖p‖ \leq \sqrt{r} (1 + ‖x‖),

(16)

where

p

is the search direction and

α

the step length from (3). An iterate is considered to satisfy the first-order conditions for a minimum if

‖Z^{T} g_{FR}‖ \leq \sqrt{r} (1 + \max (1 + |F (x)|, ‖g_{FR}‖))

(17)

and

|{res}_{j}| \leq ftol for all j,

(18)

where

Z^{T} g_{FR}

is the projected gradient (see Section 11.1),

g_{FR}

is the gradient of

F (x)

with respect to the free variables,

{res}_{j}

is the violation of the

j

th active nonlinear constraint, and

ftol

is the

Nonlinear Feasibility Tolerance

Out_Level

i

Default

= 0

This option defines the amount of extra information to be sent to the Fortran unit number defined by

Punch Unit

. The possible choices for

i

are the following:

$i$	Meaning
0	No extra output.
1	Updated solutions only. This is useful during long runs to observe progress.
2	Successful start points only. This is useful to save the starting points that gave rise to the final solution.
3	Both updated solutions and successful start points.

Punch Unit

i

Default

= 6

This option allows you to send information arising from an appropriate setting of

Out_Level

to be sent to the Fortran unit number defined by

Punch Unit

. If you wish this file to be different to the standard output unit where other output is displayed then this file should be attached by calling x04acc prior to calling e05ucc.

Start Objective Check At Variable

i_{1}

Default

= 1

Stop Objective Check At Variable

i_{2}

Default

= n

Start Constraint Check At Variable

i_{3}

Default

= 1

Stop Constraint Check At Variable

i_{4}

Default

= n

These keywords take effect only if

Verify Level > 0

. They may be used to control the verification of gradient elements computed by objfun and/or Jacobian elements computed by confun. For example, if the first

30

elements of the objective gradient appeared to be correct in an earlier run, so that only element

31

remains questionable, it is reasonable to specify

Start Objective Check At Variable = 31

. If the first

30

variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.

i_{2 m - 1} \leq 0

i_{2 m - 1} > \min (n, i_{2 m})

, the default value is used, for

m = 1, 2

. If

i_{2 m} \leq 0

i_{2 m} > n

, the default value is used, for

m = 1, 2

Step Limit

r

Default

= 2.0

r > 0, r

specifies the maximum change in variables at the first step of the linesearch. In some cases, such as

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

F (x) = a x^{b}

, even a moderate change in the elements of

x

can lead to floating-point overflow. The parameter

r

is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate

x

, the first point

\tilde{x}

at which

F

and

c

are evaluated during the linesearch is restricted so that

{‖\tilde{x} - x‖}_{2} \leq r (1 + {‖x‖}_{2}) .

The linesearch may go on and evaluate

F

and

c

at points further from

x

if this will result in a lower value of the merit function (indicated by L at the end of each line of output produced by the major iterations; see Section 9.2). If L is printed for most of the iterations,

r

should be set to a larger value.

Wherever possible, upper and lower bounds on

x

should be used to prevent evaluation of nonlinear functions at wild values. The default value

Step Limit = 2.0

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

0.1

0.01

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

Step Limit

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

r \leq 0

, the default value is used.

Verify Level

i

Default

= 0

Verify

Verify Constraint Gradients

Verify Gradients

Verify Objective Gradients

These keywords refer to finite difference checks on the gradient elements computed by objfun and confun. The possible choices for

i

are as follows:

$i$	Meaning
$- 1$	No checks are performed.
$0$	Only a ‘cheap’ test will be performed.
$\geq 1$	Individual gradient elements will also be checked using a reliable (but more expensive) test.

It is possible to specify

Verify Level = 0

3

in several ways. For example, the nonlinear objective gradient (if any) will be verified if either

Verify Objective Gradients

Verify Level = 1

is specified. The constraint gradients will be verified if

Verify = YES

Verify Level = 2

Verify

is specified. Similarly, the objective and the constraint gradients will be verified if

Verify = YES

Verify Level = 3

Verify

is specified.

0 \leq i \leq 3

, gradients will be verified at the first point that satisfies the linear constraints and bounds.

i = 0

, only a ‘cheap’ test will be performed, requiring one call to objfun and (if appropriate) one call to confun.

1 \leq i \leq 3

, a more reliable (but more expensive) check will be made on individual gradient elements, within the ranges specified by the

Start Constraint Check At Variable

and

Stop Constraint Check At Variable

keywords. A result of the form OK or BAD? is printed by e05ucc to indicate whether or not each element appears to be correct. If a gradient element is determined to be extremely poor (i.e., if it appears to have no significant digits of accuracy at all) then e05ucc will also exit with an error indicator in argument fail.

10 \leq i \leq 13

, the action is the same as for

i - 10

, except that it will take place at the user-specified initial value of

x

i < - 1

4 \leq i \leq 9

i > 13

, the default value is used.

We suggest that

Verify Level = 3

be used whenever a new function function is being developed.

13 Description of Monitoring Information

This section describes the long line of output (

> 80

characters) which forms part of the monitoring information produced by e05ucc. (See also the description of the optional parameters

Major Print Level

Minor Print Level

and

Monitoring File

.) You can control the level of printed output.

When

Major Print Level \geq 5

and

Monitoring File \geq 0

, the following line of output is produced at every major iteration of e05ucc on the file specified by

Monitoring File

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

Maj	is the major iteration count.
Mnr	is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be $1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 11). Note that Mnr may be greater than the optional parameter $Minor Iteration Limit$ if some iterations are required for the feasibility phase.
Step	is the step $α_{k}$ taken along the computed search direction. On reasonably well-behaved local problems, the unit step (i.e., $α_{k} = 1$ ) will be taken as the solution is approached.
Nfun	is the cumulative number of evaluations of the objective function needed for the linesearch. Evaluations needed for the estimation of the gradients by finite differences are not included. Nfun is printed as a guide to the amount of work required for the linesearch.
Merit Function	is the value of the augmented Lagrangian merit function (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section 11.3). As the solution is approached, Merit Function will converge to the value of the objective function at the solution. If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or the local optimizer terminates. Repeated failures will prevent a feasible point being found for the nonlinear constraints. If there are no nonlinear constraints present (i.e., $ncnln = 0$ ) then this entry contains Objective, the value of the objective function $F (x)$ . The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.
Norm Gz	is $‖Z^{T} g_{FR}‖$ , the Euclidean norm of the projected gradient (see Section 11.2). Norm Gz will be approximately zero in the neighbourhood of a solution.
Violtn	is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.
Nz	is the number of columns of $Z$ (see Section 11.2). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., $Nz = n - (Bnd + Lin + Nln)$ .
Bnd	is the number of simple bound constraints in the predicted active set.
Lin	is the number of general linear constraints in the predicted working set.
Nln	is the number of nonlinear constraints in the predicted active set (not printed if ncnln is zero).
Penalty	is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian merit function (not printed if ncnln is zero).
Cond H	is a lower bound on the condition number of the Hessian approximation $H$ .
Cond Hz	is a lower bound on the condition number of the projected Hessian approximation $H_{Z}$ ( $H_{Z} = Z^{T} H_{FR} Z = R_{Z}^{T} R_{Z}$ ; see (6)). The larger this number, the more difficult the local problem.
Cond T	is a lower bound on the condition number of the matrix of predicted active constraints.
Conv	is a three-letter indication of the status of the three convergence tests (16)–(18) defined in the description of the optional parameter $Optimality Tolerance$ . Each letter is T if the test is satisfied and F otherwise. The three tests indicate whether: (i)the sequence of iterates has converged; (ii)the projected gradient (Norm Gz) is sufficiently small; and (iii)the norm of the residuals of constraints in the predicted active set (Violtn) is small enough. If any of these indicators is F for a successful local minimization you should check the solution carefully.
M	is printed if the quasi-Newton update has been modified to ensure that the Hessian approximation is positive definite (see Section 11.4).
I	is printed if the QP subproblem has no feasible point.
C	is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that $x$ is close to a Kuhn–Tucker point (see Section 11.1).
L	is printed if the linesearch has produced a relative change in $x$ greater than the value defined by the optional parameter $Step Limit$ . If this output occurs frequently during later iterations of the run, optional parameter $Step Limit$ should be set to a larger value.
R	is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of $R$ indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, $R$ is modified so that its diagonal condition estimator is bounded.