NAG Library Routine Document

H02CBF

Note: this routine uses optional parameters to define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 9 of this document. If, however, you wish to reset some or all of the settings please refer to Section 10 for a detailed description of the algorithm, to Section 11 for a detailed description of the specification of the optional parameters and to Section 12 for a detailed description of the monitoring information produced by the routine.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

+− 8 Further Comments

8.1 Scaling

8.2 Description of the Printed Output

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

+− 10 Algorithmic Details

10.1 Overview

10.2 Definition of the Search Direction

10.3 The Main Iteration

10.4 Choosing the Initial Working Set

+− 11 Optional Parameters

11.1 Description of the Optional Parameters

12 Description of Monitoring Information

1 Purpose

H02CBF solves general quadratic programming problems with integer constraints on the variables. It is not intended for large sparse problems.

2 Specification

SUBROUTINE H02CBF (

N, NCLIN, A, LDA, BL, BU, CVEC, H, LDH, QPHESS, INTVAR, LINTVR, MDEPTH, ISTATE, XS, OBJ, AX, CLAMDA, STRTGY, IWRK, LIWRK, WRK, LWRK, MONIT, IFAIL)

INTEGER	N, NCLIN, LDA, LDH, INTVAR(LINTVR), LINTVR, MDEPTH, ISTATE(N+NCLIN), STRTGY, IWRK(LIWRK), LIWRK, LWRK, IFAIL
REAL (KIND=nag_wp)	A(LDA,), BL(N+NCLIN), BU(N+NCLIN), CVEC(), H(LDH,*), XS(N), OBJ, AX(max(1,NCLIN)), CLAMDA(N+NCLIN), WRK(LWRK)
EXTERNAL	QPHESS, MONIT

3 Description

H02CBF uses a ‘Branch and Bound’ algorithm in conjunction with E04NFF to try and determine integer solutions to a general quadratic programming problem. The problem is 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,

where

A

is an

m_{L}

n

matrix and

f (x)

may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for

f (x)

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

c

is an

n

-element vector.

Problem type	$f (x)$	Matrix $H$
FP	Not applicable	Not applicable
LP	$c^{T} x$	Not applicable
QP1	$\frac{1}{2} x^{T} H x$	symmetric
QP2	$c^{T} x + \frac{1}{2} x^{T} H x$	symmetric
QP3	$\frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal
QP4	$c^{T} x + \frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal

Table 1

Only when the problem is linear or the matrix

H

is positive definite can the technique be guaranteed to work; but often useful results can be obtained for a wider class of problems.

The default problem type is QP2 and other objective functions are selected by using the optional parameter Problem Type. For problems of type FP, the objective function is omitted and H02CBF attempts to find a feasible point for the set of constraints.

Branch and bound consists firstly of obtaining a solution without any of the variables

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

constrained to be integer. Suppose

x_{1}

ought to be integer, but at the optimal value just computed

x_{1} = 2.4

. A constraint

x_{1} \leq 2

is added to the system and the second problem solved. A constraint

x_{1} \geq 3

gives rise to a third sub-problem. In a similar manner a whole series of sub-problems may be generated, corresponding to integer constraints on the variables. The sub-problems are all solved using E04NFF.

In practice the routine tries to compute an integer solution as quickly as possible using a depth-first approach, since this helps determine a realistic cut-off value. If we have a cut-off value, say the value of the function at this first integer solution, and any sub-problem,

W

say, has a solution value greater than this cut-off value, then subsequent sub-problems of

W

must have solutions greater than the value of the solution at

W

and therefore need not be computed. Thus a knowledge of a good cut-off value can result in fewer sub-problems being solved and thus speed up the operation of the routine. (See the description of MONIT in Section 5 for details of how you can supply your own cut-off value.)

4 References

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

Gill P E, Murray W, Saunders M A and Wright M H (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298

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

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

Pardalos P M and Schnitger G (1988) Checking local optimality in constrained quadratic programming is NP-hard Operations Research Letters 7 33–35

5 Parameters

1: N – INTEGERInput

On entry:

n

, the number of variables.

Constraint:

N > 0

2: NCLIN – INTEGERInput

On entry:

m_{L}

, the number of general linear constraints.

Constraint:

NCLIN \geq 0

3: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array A must be at least

N

NCLIN > 0

and at least

1

NCLIN = 0

On entry: the

i

th row of A must contain the coefficients of the

i

th general linear constraint, for

i = 1, 2, \dots, m_{L}

NCLIN = 0

then the array A is not referenced.

4: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which H02CBF is called.

Constraint:

LDA \geq \max (1, NCLIN)

5: BL( $N + NCLIN$ ) – REAL (KIND=nag_wp) arrayInput

6: BU( $N + NCLIN$ ) – REAL (KIND=nag_wp) arrayInput

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_{L}

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

l_{j} = - \infty

), set

BL (j) \leq - bigbnd

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

u_{j} = + \infty

), set

BU (j) \geq bigbnd

; the default value of

bigbnd

10^{20}

, but this may be changed by the Infinite Bound Size. To specify the

j

th constraint as an equality, set

BL (j) = BU (j) = β

, say, where

|β| < bigbnd

Constraints:

$BL (j) \leq BU (j)$ , for $j = 1, 2, \dots, N + NCLIN$ ;
if $BL (j) = BU (j) = β$ , $|β| < bigbnd$ .

7: CVEC( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array CVEC must be at least

N

if the problem is of type LP, QP2 (the default) or QP4, and at least

1

otherwise.

On entry: the coefficients of the explicit linear term of the objective function when the problem is of type LP, QP2 (the default) and QP4.

If the problem is of type FP, QP1, or QP3, CVEC is not referenced.

8: H(LDH, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array H must be at least

N

if it is to be used to store

H

explicitly, and at least

1

otherwise.

On entry: may be used to store the quadratic term

H

of the QP objective function if desired. In some cases, you need not use H to store

H

explicitly (see the specification of QPHESS). The elements of H are referenced only by QPHESS. The number of rows of

H

is denoted by

m

, whose default value is

n

. (The Hessian Rows may be used to specify a value of

m < n

If the default version of QPHESS is used and the problem is of type QP1 or QP2 (the default), the first

m

rows and columns of H must contain the leading

m

m

rows and columns of the symmetric Hessian matrix

H

. Only the diagonal and upper triangular elements of the leading

m

rows and columns of H are referenced. The remaining elements need not be assigned.

If the default version of QPHESS is used and the problem is of type QP3 or QP4, the first

m

rows of H must contain an

m

n

upper trapezoidal factor of the symmetric Hessian matrix

H^{T} H

. The factor need not be of full rank, i.e., some of the diagonal elements may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of H, the fewer iterations will be required. Elements outside the upper trapezoidal part of the first

m

rows of H need not be assigned.

If a nondefault version of QPHESS is supplied, then in some cases it may be desirable to use a one-dimensional array to transmit data to QPHESS. (This is illustrated in the example program in Section 9 in H02CCF.) H is then declared as a vector with dimension (LDH), where

LDH \geq N \times (N + 1) / 2

In other situations, it may be desirable to compute

H x

H^{T} H x

without accessing H – for example, if

H

H^{T} H

is sparse or has special structure. The parameters H and LDH may then refer to any convenient array.

If the problem is of type FP or LP, H is not referenced.

9: LDH – INTEGERInput

On entry: the first dimension of the array H as declared in the (sub)program from which H02CBF is called.

Constraints:

if the problem is of type QP1, QP2 (the default), QP3 or QP4, $LDH \geq N$ or at least the value of the optional parameter Hessian Rows ( $default value = n$ );
if the problem is of type FP or LP, $LDH \geq 1$ .

10: QPHESS – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

In general, you need not provide a version of QPHESS, because a ‘default’ subroutine with name E04NFU is included in the Library. However, the algorithm of H02CBF requires only the product of

H

H^{T} H

and a vector

x

; and in some cases you may obtain increased efficiency by providing a version of QPHESS that avoids the need to define the elements of the matrices

H

H^{T} H

explicitly. QPHESS is not referenced if the problem is of type FP or LP, in which case QPHESS may be the routine E04NFU.

The specification of QPHESS is:

SUBROUTINE QPHESS (

N, JTHCOL, H, LDH, X, HX)

INTEGER	N, JTHCOL, LDH
REAL (KIND=nag_wp)	H(LDH,*), X(N), HX(N)

1: N – INTEGERInput

On entry: this is the same parameter N as supplied to H02CBF.

2: JTHCOL – INTEGERInput

On entry: specifies whether or not the vector

x

is a column of the identity matrix.

$JTHCOL = j > 0$: The vector $x$ is the $j$ th column of the identity matrix, and hence $H x$ or $H^{T} H x$ is the $j$ th column of $H$ or $H^{T} H$ , respectively, which may in some cases require very little computation and QPHESS may be coded to take advantage of this. However special code is not necessary because $x$ is always stored explicitly in the array X.
$JTHCOL = 0$: $x$ has no special form.

3: H(LDH, $*$ ) – REAL (KIND=nag_wp) arrayInput

On entry: this is the same parameter H as supplied to H02CBF.

4: LDH – INTEGERInput

On entry: this is the same parameter LDH as supplied to H02CBF.

5: X(N) – REAL (KIND=nag_wp) arrayInput

On entry: the vector

x

6: HX(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the product

H x

if the problem is of type QP1 or QP2 (the default), or the product

H^{T} H x

if the problem is of type QP3 or QP4.

QPHESS must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which H02CBF is called. Parameters denoted as Input must not be changed by this procedure.

11: INTVAR(LINTVR) – INTEGER arrayInput

On entry:

INTVAR (i)

must contain the index of the solution vector

x

which is required to be integer. For example, if

x_{1}

and

x_{3}

are constrained to take integer values then

INTVAR (1)

might be set to

1

and

INTVAR (2)

3

. The order in which the indices are specified is important, since this determines the order in which the sub-problems are generated. As a rule-of-thumb, the important variables should always be specified first. Thus, in the above example, if

x_{3}

relates to a more important quantity than

x_{1}

, then it might be advantageous to set

INTVAR (1) = 3

and

INTVAR (2) = 1

. If

k

is the smallest integer such that

INTVAR (k)

is less than or equal to zero then H02CBF assumes that

k - 1

variables are constrained to be integer; components

INTVAR (k + 1)

\dots

INTVAR (LINTVR)

are not referenced.

12: LINTVR – INTEGERInput

On entry: the dimension of the array INTVAR as declared in the (sub)program from which H02CBF is called. Often LINTVR is the number of variables that are constrained to be integer.

Constraint:

LINTVR > 0

13: MDEPTH – INTEGERInput

On entry: the maximum depth (i.e., number of extra constraints) that H02CBF may insert before admitting failure.

Suggested value:

MDEPTH = 3 \times N / 2

Constraint:

MDEPTH \geq 1

14: ISTATE( $N + NCLIN$ ) – INTEGER arrayInput/Output

On entry: need not be set if the (default) optional parameter Cold Start is used.

If the optional parameter Warm Start has been chosen, ISTATE specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first

n

elements of ISTATE refer to the upper and lower bounds on the variables, and the next

m_{L}

elements refer to the general linear constraints (if any). Possible values for

ISTATE (j)

are as follows:

$ISTATE (j)$	Meaning
0	The corresponding constraint should not be in the initial working set.
1	The constraint should be in the initial working set at its lower bound.
2	The constraint should be in the initial working set at its upper bound.
3	The constraint should be in the initial working set as an equality. This value must not be specified unless $BL (j) = BU (j)$ .

The values

- 2

- 1

and

4

are also acceptable but will be reset to zero by the routine. If H02CBF has been called previously with the same values of N and NCLIN, ISTATE already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The routine also adjusts (if necessary) the values supplied in XS to be consistent with ISTATE.

Constraint:

- 2 \leq ISTATE (j) \leq 4

, for

j = 1, 2, \dots, N + NCLIN

On exit: the status of the constraints in the working set at the point returned in XS. The significance of each possible value of

ISTATE (j)

is as follows:

$ISTATE (j)$	Meaning
$- 2$	The constraint violates its lower bound by more than the feasibility tolerance.
$- 1$	The constraint violates its upper bound by more than the feasibility tolerance.
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
$1$	This inequality constraint is included in the working set at its lower bound.
$2$	This inequality constraint is included in the working set at its upper bound.
$3$	This constraint is included in the working set as an equality. This value of ISTATE can occur only when $BL (j) = BU (j)$ .
$4$	This corresponds to optimality being declared with $XS (j)$ being temporarily fixed at its current value. This value of ISTATE can occur only when $IFAIL = 1$ on exit.

15: XS(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an initial estimate of the solution.

On exit: the point at which H02CBF terminated. If

IFAIL = 0

1

3

, XS contains an estimate of the solution.

16: OBJ – REAL (KIND=nag_wp)Output

On exit: the value of the objective function at

x

x

is feasible, or the sum of infeasibilities at

x

otherwise. If the problem is of type FP and

x

is feasible, OBJ is set to zero.

17: AX( $\max (1, NCLIN)$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the final values of the linear constraints

A x

NCLIN = 0

, AX is not referenced.

18: CLAMDA( $N + NCLIN$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the values of the Lagrange-multipliers for each constraint with respect to the current working set. The first

n

elements contain the multipliers for the bound constraints on the variables, and the next

m_{L}

elements contain the multipliers for the general linear constraints (if any). If

ISTATE (j) = 0

(i.e., constraint

j

is not in the working set),

CLAMDA (j)

is zero. If

x

is optimal,

CLAMDA (j)

should be non-negative if

ISTATE (j) = 1

, non-positive if

ISTATE (j) = 2

and zero if

ISTATE (j) = 4

19: STRTGY – INTEGERInput

On entry: determines a branching strategy to be used throughout the computation, as follows:

$STRTGY$	Meaning
$0$	Always left branch first, i.e., impose an upper bound constraint on the variable first.
$1$	Always right branch first, i.e., impose a lower bound constraint on the variable first.
$2$	Branch towards the nearest integer, i.e., if $x_{k} = 2.4$ then impose an upper bound constraint $x_{k} \leq 2$ , whereas if $x_{k} = 2.6$ then impose the lower bound constraint $x_{k} \geq 3.0$ .
$3$	A random choice is made between a left-hand and a right-hand branch.

Constraint:

STRTGY = 0

1

2

3

20: IWRK(LIWRK) – INTEGER arrayWorkspace

21: LIWRK – INTEGERInput

On entry: the dimension of the array IWRK as declared in the (sub)program from which H02CBF is called.

Constraint:

LIWRK \geq 2 \times N + 3 + 2 \times MDEPTH

22: WRK(LWRK) – REAL (KIND=nag_wp) arrayWorkspace

23: LWRK – INTEGERInput

On entry: the dimension of the array WRK as declared in the (sub)program from which H02CBF is called.

Constraints:

if the problem type is QP2 (the default) or QP4,
- if $NCLIN > 0$ , $LWRK \geq 2 \times N^{2} + 9 \times N + 5 \times NCLIN + 4 \times MDEPTH$ ;
- if $NCLIN = 0$ , $LWRK \geq N^{2} + 9 \times N + 4 \times MDEPTH$ ;
if the problem type is QP1 or QP3,
- if $NCLIN > 0$ , $LWRK \geq 2 \times N^{2} + 8 \times N + 5 \times NCLIN + 4 \times MDEPTH$ ;
- if $NCLIN = 0$ , $LWRK \geq N^{2} + 8 \times N + 4 \times MDEPTH$ ;
if the problem type is LP,
- if $NCLIN = 0$ , $LWRK \geq 9 \times N + 1 + 4 \times MDEPTH$ ;
- if $NCLIN \geq N$ , $LWRK \geq 2 \times N^{2} + 9 \times N + 5 \times NCLIN + 4 \times MDEPTH$ ;
- otherwise $LWRK \geq 2 \times {(NCLIN + 1)}^{2} + 9 \times N + 5 \times NCLIN + 4 \times MDEPTH$ ;
if the problem type is FP,
- if $NCLIN = 0$ , $LWRK \geq 8 \times N + 1 + 4 \times MDEPTH$ ;
- if $NCLIN \geq N$ , $LWRK \geq 2 \times N^{2} + 8 \times N + 5 \times NCLIN + 4 \times MDEPTH$ ;
- otherwise $LWRK \geq 2 \times {(NCLIN + 1)}^{2} + 8 \times N + 5 \times NCLIN + 4 \times MDEPTH$ .

24: MONIT – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

MONIT may be used to print out intermediate output and to affect the course of the computation. Specifically, it allows you to specify a realistic value for the cut-off value (see Section 3) and to terminate the algorithm. If you do not require any intermediate output, have no estimate of the cut-off value and require an exhaustive tree search then MONIT may be the dummy routine H02CBU.

The specification of MONIT is:

SUBROUTINE MONIT (

INTFND, NODES, DEPTH, OBJ, X, BSTVAL, BSTSOL, BL, BU, N, HALT, COUNT)

INTEGER	INTFND, NODES, DEPTH, N, COUNT
REAL (KIND=nag_wp)	OBJ, X(N), BSTVAL, BSTSOL(N), BL(N), BU(N)
LOGICAL	HALT

1: INTFND – INTEGERInput: On entry: specifies the number of integer solutions obtained so far.
2: NODES – INTEGERInput: On entry: specifies the number of nodes (sub-problems) solved so far.
3: DEPTH – INTEGERInput: On entry: specifies the depth in the tree of sub-problems the algorithm has now reached.
4: OBJ – REAL (KIND=nag_wp)Input: On entry: specifies the value of the objective function of the end of the latest sub-problem.
5: X(N) – REAL (KIND=nag_wp) arrayInput: On entry: specifies the values of the independent variables at the end of the latest sub-problem.
6: BSTVAL – REAL (KIND=nag_wp)Input/Output: On entry: normally specifies the value of the best integer solution found so far.

On exit: may be set a cut-off value if you are an experienced user as follows. Before an integer solution has been found BSTVAL will be set by H02CBF to the largest machine representable number (see X02ALF). If you know that the solution being sought is a much smaller number, then BSTVAL may be set to this number as a cut-off value (see Section 3). Beware of setting BSTVAL too small, since then no integer solutions will be discovered. Also make sure that BSTVAL is set using a statement of the form
IF (INTFND.EQ.0) $BSTVAL =$ cut-off value

on entry to MONIT. This statement will not prevent the normal operation of the algorithm when subsequent integer solutions are found. It would be a grievous mistake to unconditionally set BSTVAL and if you have any doubts whatsoever about the correct use of this parameter then you are strongly recommended to leave it unchanged.
7: BSTSOL(N) – REAL (KIND=nag_wp) arrayInput: On entry: specifies the solution vector which gives rise to the best integer solution value so far discovered.
8: BL(N) – REAL (KIND=nag_wp) arrayInput: On entry: $BL (i)$ specifies the current lower bounds on the variable $x_{i}$ .
9: BU(N) – REAL (KIND=nag_wp) arrayInput: On entry: $BU (i)$ specifies the current upper bounds on the variable $x_{i}$ .
10: N – INTEGERInput: On entry: specifies the number of variables.
11: HALT – LOGICALInput/Output: On entry: will have the value .FALSE..

On exit: if HALT is set to .TRUE., E04NFF/E04NFA will be brought to a halt with $IFAIL = - 1$ . This facility may be useful if you are content with any integer solution, or with any integer solution that fits certain criteria. Under these circumstances setting $HALT = .TRUE.$ can save considerable unnecessary computation.
12: COUNT – INTEGERInput/Output: On entry: unchanged from previous call.

On exit: may be used by you to save the last value of INTFND. If a subsequent call of MONIT has a value of INTFND which is greater than COUNT, then you know that a new integer solution has been found at this node.

MONIT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which H02CBF is called. Parameters denoted as Input must not be changed by this procedure.

25: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = - 1$: Algorithm terminated at your request ( $HALT = .TRUE.$ ).

$IFAIL = 1$: Input parameter error immediately detected.

$IFAIL = 2$: No integer solution found. (Check that BSTVAL has not been set too small.)

$IFAIL = 3$: MDEPTH is too small. Increase the value of MDEPTH and re-enter H02CBF.

$IFAIL = 4$: The basic problem (without integer constraints) is unbounded.

$IFAIL = 5$: The basic problem is infeasible.

$IFAIL = 6$: The basic problem requires too many iterations.

$IFAIL = 7$: The basic problem has a reduced Hessian which exceeds its assigned dimension.

$IFAIL = 8$: The basic problem has an invalid parameter setting.

$IFAIL = 9$: The basic problem, as defined, is not standard.

$IFAIL = 10$: LIWRK is too small.

$IFAIL = 11$: LWRK is too small.

$IFAIL = 12$: An internal error has occurred within the routine. Please contact NAG with details of the call to H02CBF.

7 Accuracy

H02CBF 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 Further Comments

This section contains some comments on scaling and a description of the printed output.

8.1 Scaling

Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See Chapter E04 and Gill et al. (1981) for further information and advice.

8.2 Description of the Printed Output

This section describes the (default) intermediate printout and final printout produced by H02CBF. The intermediate printout is a subset of the monitoring information produced by the routine at every iteration (see Section 12). You can control the level of printed output (see the description of the Print Level in Section 11.1). Note that the intermediate printout and final printout are produced only if

Print Level \geq 10

(the default).

The following line of summary output (

< 80

characters) 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.
Step	is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.
Ninf	is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective	is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the 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 nonincreasing. 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. Once optimal multipliers are obtained, the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Norm Gz	is $‖Z_{R}^{T} g_{FR}‖$ , the Euclidean norm of the reduced gradient with respect to $Z_{R}$ (see Sections 10.2 and 10.4). During the optimality phase, this norm will be approximately zero after a unit step.

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.

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

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 (

default value = \sqrt{ε}

, where

ε

is the machine precision; see Section 11.1), State will be ++ or -- respectively.

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, 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 iterate.

Lower Bound

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

BL (j) \leq - bigbnd

Upper Bound

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

BU (j) \geq bigbnd

Slack

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

BL (j)

and

BU (j)

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

BL (j) \leq - bigbnd

and

BU (j) \geq bigbnd

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

BL (j)

and

BU (j)

are replaced by

BL (n + j)

and

BU (n + j)

respectively, and with the following change in the heading.

L Con

gives the name (L) and index

j

, for

j = 1, 2, \dots, m

, of the 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.

9 Example

This example minimizes the quadratic function

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

, where

c = {(- 0.02, - 0.2, - 0.2, - 0.2, - 0.2, 0.04, 0.04)}^{T}

H = (\begin{array}{r} 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{array})

subject to the bounds

\begin{array}{l} - 0.01 \leq x_{1} \leq 0.01 \\ - 0.1 \leq x_{2} \leq 0.15 \\ - 0.01 \leq x_{3} \leq 0.03 \\ - 0.04 \leq x_{4} \leq 0.02 \\ - 0.1 \leq x_{5} \leq 0.05 \\ - 0.01 \leq x_{6} \\ - 0.01 \leq x_{7} \end{array}

to the general constraints

\begin{array}{r} x_{1} & + & x_{2} & + & x_{3} & + & x_{4} & + & x_{5} & + & x_{6} & + & x_{7} & = & - 0.13 \\ 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 & - 0.0049 \\ 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 & - 0.0064 \\ 0.02 x_{1} & + & 0.04 x_{2} & + & 0.01 x_{3} & + & 0.02 x_{4} & + & 0.02 x_{5} & \leq & - 0.0037 \\ 0.02 x_{1} & + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & - 0.0012 \\ - 0.0992 & \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} \\ - 0.003 & \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 & 0.002 \end{array}

and the variable

x_{4}

is constrained to be integer.

The initial point, which is infeasible, is

x_{0} = {(- 0.01, - 0.03, 0.0, - 0.01, - 0.1, 0.02, 0.01)}^{T} .

The optimal solution (to five figures) is

x^{*} = {(- 0.01, - 0.073328, - 0.00025809, 0.0, - 0.063354, 0.014109, 0.0028312)}^{T} .

The document for H02CCF includes an example program to solve the same problem using some of the optional parameters described in Section 11.

Note: the remainder of this document is intended for more advanced users. Section 10 contains a detailed description of the algorithm which may be needed in order to understand Sections 11 and 12. Section 11 describes the optional parameters which may be set by calls to H02CCF and/or H02CDF. Section 12 describes the quantities which can be requested to monitor the course of the computation.

10 Algorithmic Details

H02CBF implements a basic branch and bound algorithm (see Section 3) using E04NFF as its basic sub-problem solver. See below for details of its algorithm.

10.1 Overview

H02CBF is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is based on 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 parameters of H02CBF or appear in the printed output. H02CBF has two phases:

(i)	finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and
(ii)	minimizing the quadratic objective function within the feasible region (the optimality phase).

The computations in both phases are performed by the same subroutines. 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. The feasibility phase does not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when

m_{L} \leq n

. Once any iterate is feasible, all subsequent iterates remain feasible.

H02CBF has been designed to be efficient when used to solve a sequence of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., E04WDF). In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the Warm Start in Section 11.1.

In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate

\bar{x}

is defined by

\bar{x} = x + α p

(1)

where the step length

α

is a non-negative scalar, and

p

is called the search direction.

At each point

x

, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the Feasibility Tolerance; see Section 11.1). The working set is the current prediction of the constraints that hold with equality at the solution of a linearly constrained QP problem. The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of

x

into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of

A

corresponding to fixed variables may be ignored.

Let

m_{W}

denote the number of general constraints in the working set and let

n_{FX}

denote the number of variables fixed at one of their bounds (

m_{W}

and

n_{FX}

are the quantities Lin and Bnd in the monitoring file output from H02CBF; see Section 12). Similarly, let

n_{FR}

(

n_{FR} = n - n_{FX}

) denote the number of free variables. At every iteration, the variables are reordered so that the last

n_{FX}

variables are fixed, with all other relevant vectors and matrices ordered accordingly.

10.2 Definition of the Search Direction

Let

A_{FR}

denote the

m_{W}

n_{FR}

sub-matrix of general constraints in the working set corresponding to the free variables, and let

p_{FR}

denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along

p

A_{FR} p_{FR} = 0 .

(2)

In order to compute

p_{FR}

, the

T Q

factorization of

A_{FR}

is used:

A_{FR} Q_{FR} = (0 T),

(3)

where

T

is a nonsingular

m_{W}

m_{W}

upper triangular matrix (i.e.,

t_{i j} = 0

i > j

), and the nonsingular

n_{FR}

n_{FR}

matrix

Q_{FR}

is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of

Q_{FR}

are partitioned so that

Q_{FR} = (Z Y),

where

Y

n_{FR}

m_{W}

, then the

n_{Z}

(n_{Z} = n_{FR} - m_{W})

columns of

Z

form a basis for the null space of

A_{FR}

. Let

n_{R}

be an integer such that

0 \leq n_{R} \leq n_{Z}

, and let

Z_{R}

denote a matrix whose

n_{R}

columns are a subset of the columns of

Z

. (The integer

n_{R}

is the quantity Zr in the monitoring output from H02CBF. In many cases,

Z_{R}

will include all the columns of

Z

.) The direction

p_{FR}

will satisfy (2) if

p_{FR} = Z_{R} p_{R},

(4)

where

p_{R}

is any

n_{R}

-vector.

Let

Q

denote the

n

n

matrix

Q = (\begin{array}{l} Q_{FR} \\ I_{FX} \end{array}),

where

I_{FX}

is the identity matrix of order

n_{FX}

. Let

H_{Q}

and

g_{Q}

denote the

n

n

transformed Hessian and transformed gradient

H_{Q} = Q^{T} H Q and g_{Q} = Q^{T} (c + H x)

and let the matrix of first

n_{R}

rows and columns of

H_{Q}

be denoted by

H_{R}

and the vector of the first

n_{R}

elements of

g_{Q}

be denoted by

g_{R}

. The quantities

H_{R}

and

g_{R}

are known as the reduced Hessian and reduced gradient of

f (x)

, respectively. Roughly speaking,

g_{R}

and

H_{R}

describe the first and second derivatives of an unconstrained problem for the calculation of

p_{R}

At each iteration, a triangular factorization of

H_{R}

is available. If

H_{R}

is positive definite,

H_{R} = R^{T} R

, where

R

is the upper triangular Cholesky factor of

H_{R}

. If

H_{R}

is not positive definite,

H_{R} = R^{T} D R

, where

D = diag (1, 1, \dots, 1, μ)

, with

μ \leq 0

The computation is arranged so that the reduced-gradient vector is a multiple of

e_{R}

, a vector of all zeros except in the last (i.e.,

n_{R}

th) position. This allows the vector

p_{R}

in (4) to be computed from a single back-substitution

R p_{R} = γ e_{R}

(5)

where

γ

is a scalar that depends on whether or not the reduced Hessian is positive definite at

x

. In the positive definite case,

x + p

is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. If

H_{R}

is not positive definite,

p_{R}

satisfies the conditions

p_{R}^{T} H_{R} p_{R} < 0 and g_{R}^{T} p_{R} \leq 0,

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

x + α p

10.3 The Main Iteration

If the reduced gradient is zero,

x

is a constrained stationary point in the subspace defined by

Z

. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that

x

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

λ_{C}

and

λ_{B}

for the general and bound constraints are defined from the equations

A_{FR}^{T} λ_{C} = g_{FR} and λ_{B} = g_{FX} - A_{FX}^{T} λ_{C} .

(6)

Given a positive constant

δ

of the order of the machine precision, a Lagrange-multiplier

λ_{j}

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

λ_{j} \leq δ

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

λ_{j} \geq - δ

when the associated constraint is at its lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section 12) from the working set.

If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and you can force H02CBF to continue until the minimum value of the sum of infeasibilities has been found; see the discussion of the Minimum Sum of Infeasibilities in Section 11.1. At such a point, the Lagrange-multiplier

λ_{j}

corresponding to an inequality constraint in the working set will be such that

- (1 + δ) \leq λ_{j} \leq δ

when the associated constraint is at its upper bound, and

- δ \leq λ_{j} \leq (1 + δ)

when the associated constraint is at its lower bound. Lagrange-multipliers for equality constraints will satisfy

|λ_{j}| \leq 1 + δ

If the reduced gradient is not zero, Lagrange-multipliers need not be computed and the nonzero elements of the search direction

p

are given by

Z_{R} p_{R}

(see (4) and (5)). The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints. If

H_{R}

is positive definite and

x + p

is feasible,

α

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

\bar{x}

will be zero, and Lagrange-multipliers are computed. Otherwise,

α

is set to

α_{M}

, the step to the ‘nearest’ constraint (with index Jadd; see Section 12), which is added to the working set at the next iteration.

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

A_{FR}

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

A_{FR}

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

A_{FR}

changes. Explicit representations are recurred of the matrices

T

Q_{FR}

and

R

; and of vectors

Q^{T} g

, and

Q^{T} c

. The triangular factor

R

associated with the reduced Hessian is only updated during the optimality phase.

One of the most important features of H02CBF is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonal elements of the

T Q

factor

T

(the printed value Cond T; see Section 12). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T.

H02CBF includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see Gill et al. (1989)). The main feature of the anti-cycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a choice of constraints to be added to the working set. Let

α_{M}

denote the maximum step at which

x + α_{M} p

does not violate any constraint by more than its feasibility tolerance. All constraints at a distance

α

(

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.

10.4 Choosing the Initial Working Set

At the start of the optimality phase, a positive definite

H_{R}

can be defined if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive definite by definition, corresponding to the case when

A_{FR}

contains

n_{FR}

constraints.) The idea is to include as many general constraints as necessary to ensure that the reduced Hessian is positive definite.

Let

H_{Z}

denote the matrix of the first

n_{Z}

rows and columns of the matrix

H_{Q} = Q^{T} H Q

at the beginning of the optimality phase. A partial Cholesky factorization is used to find an upper triangular matrix

R

that is the factor of the largest positive definite leading sub-matrix of

H_{Z}

. The use of interchanges during the factorization of

H_{Z}

tends to maximize the dimension of

R

. (The condition of

R

may be controlled using the Rank Tolerance. Let

Z_{R}

denote the columns of

Z

corresponding to

R

, and let

Z

be partitioned as

Z = (\begin{matrix} Z_{R} & Z_{A} \end{matrix})

. A working set for which

Z_{R}

defines the null space can be obtained by including the rows of

Z_{A}^{T}

as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by

Z_{R}

, as described in Section 10.2.

The artificially augmented working set is given by

{\bar{A}}_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}),

(7)

so that

p_{FR}

will satisfy

A_{FR} p_{FR} = 0

and

Z_{A}^{T} p_{FR} = 0

. By definition of the

T Q

factorization,

{\bar{A}}_{FR}

automatically satisfies the following:

{\bar{A}}_{FR} Q_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}) Q_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}) (\begin{matrix} Z_{R} & Z_{A} & Y \end{matrix}) = (\begin{matrix} 0 & \bar{T} \end{matrix}),

where

\bar{T} = (\begin{array}{l} I & 0 \\ 0 & T \end{array}),

and hence the

T Q

factorization of (7) is available trivially from

T

and

Q_{FR}

without additional expense.

The matrix

Z_{A}

is not kept fixed, since its role is purely to define an appropriate null space; the

T Q

factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with

Z_{A}

when

Z_{R}^{T} g_{FR} = 0

, since this simply involves repartitioning

Q_{FR}

. The ‘artificial’ multiplier vector associated with the rows of

Z_{A}^{T}

is equal to

Z_{A}^{T} g_{FR}

, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an A appears alongside the entry in the Jdel column of the monitoring file output (see Section 12).

The number of columns in

Z_{A}

and

Z_{R}

, the Euclidean norm of

Z_{R}^{T} g_{FR}

, and the condition estimator of

R

appear in the monitoring file output as Art, Zr, Norm Gz and Cond Rz respectively (see Section 12).

Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of H02CBF does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e.,

m_{L} \leq n

). (Use of the simplex method in this situation leads to savings in storage.) At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. If a temporary bound is ‘deleted’ from the working set, an F (for ‘Fixed’) appears alongside the entry in the Jdel column of the monitoring file output (see Section 12).

11 Optional Parameters

Several optional parameters in H02CBF define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of H02CBF 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. A full description of each optional parameter is provided in Section 11.1.

Optional parameters may be specified by calling one, or both, of the routines H02CCF and H02CDF prior to a call to H02CBF.

H02CCF reads options from an external options file, with Begin and End as the first and last lines respectively and each intermediate line defining a single optional parameter. For example,

Begin
  Print Level = 5
End

The call

CALL H02CCF (IOPTNS, INFORM)

can then be used to read the file on unit IOPTNS. INFORM will be zero on successful exit. H02CCF should be consulted for a full description of this method of supplying optional parameters.

H02CDF can be called to supply options directly, one call being necessary for each optional parameter. For example,

CALL H02CDF ('Print Level = 5')

H02CDF should be consulted for a full description of this method of supplying optional parameters.

All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by H02CBF (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.

11.1 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 X02AJF).

Keywords and character values are case and white space insensitive.

Check Frequency

i

Default

= 50

Every

i

th iteration, a numerical test is made to see if the current solution

x

satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is refactorized and the variables are recomputed to satisfy the constraints more accurately. If

i \leq 0

, the default value is used.

Cold Start

Default

Warm Start

This option specifies how the initial working set is chosen. With a Cold Start, H02CBF chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within Crash Tolerance).

With a Warm Start, you must provide a valid definition of every element of the array ISTATE (see Section 5 for the definition of this array). H02CBF will override your specification of ISTATE if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of ISTATE which are set to

- 2

- 1 ​ or ​ 4

will be reset to zero, as will any elements which are set to

3

when the corresponding elements of BL and BU are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when H02CBF is called repeatedly to solve related problems.

Crash Tolerance

r

Default

= 0.01

This value is used in conjunction with the optional parameter Cold Start (the default value) when H02CBF 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 may be used to reset all optional parameters to their default values.

Expand Frequency

i

Default

= 5

This option is part of an anti-cycling procedure designed to guarantee progress even on highly degenerate 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 Feasibility Tolerance is

δ

. Over a period of

i

iterations, the feasibility tolerance actually used by H02CBF (i.e., the working feasibility tolerance) increases from

0.5 δ

δ

(in steps of

0.5 δ / i

At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the working feasibility tolerance is reinitialized to

0.5 δ

If a problem requires more than

i

iterations, the resetting procedure is invoked and a new cycle of

i

iterations is started with

i

incremented by

10

. (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 H02CBF reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.

i \leq 0

, the default value is used. If

i \geq 9999999

, no anti-cycling procedure is invoked.

Feasibility Phase Iteration Limit

i_{1}

Default

= \max (50, 5 (n + m_{L}))

Optimality Phase Iteration Limit

i_{2}

Default

= \max (50, 5 (n + m_{L}))

The scalars

i_{1}

and

i_{2}

specify the maximum number of iterations allowed in the feasibility and optimality phases. Optimality Phase Iteration Limit is equivalent to Iteration Limit. Setting

i_{1} = 0

and

Print Level > 0

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

i_{1} < 0

i_{2} < 0

, the default value is used.

Feasibility Tolerance

r

Default

= \sqrt{ε}

r \geq ε

r

defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about

6

decimal digits, it would be appropriate to specify

r

10^{- 6}

. If

0 \leq r < ε

, the default value is used.

H02CBF attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the Minimum Sum of Infeasibilities can be used to find the minimum value of the sum. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise

r

by a factor of

10

100

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

Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance

r

Hessian Rows

i

Default

= n

Note that this option does not apply to problems of type FP or LP.

This specifies

m

, the number of rows of the Hessian matrix

H

. The default value of

m

n

, the number of variables of the problem.

If the problem is of type QP,

m

will usually be

n

, the number of variables. However, a value of

m

less than

n

is appropriate for QP3 or QP4 if

H

is an upper trapezoidal matrix with

m

rows. Similarly,

m

may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2, in which case the last

n - m

rows and columns of

H

are assumed to be zero. In the QP case,

m

should not be greater than

n

; if it is, the last

m - n

rows of

H

are ignored.

i < 0

i > n

, the default value is used.

Infinite Bound Size

r

Default

= 10^{20}

r > 0

r

defines the ‘infinite’ bound

bigbnd

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

bigbnd

will be regarded as

+ \infty

(and similarly any lower bound less than or equal to

- bigbnd

will be regarded as

- \infty

). If

r \leq 0

, the default value is used.

Infinite Step Size

r

Default

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

r > 0

r

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

r

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

r \leq 0

, the default value is used.

Iteration Limit

i

Default

= \max (50, 5 (n + m_{L}))

Iters

Itns

See optional parameter Feasibility Phase Iteration Limit.

List

Default

Nolist

Normally each optional parameter specification is printed as it is supplied. Optional parameter Nolist may be used to suppress the printing and optional parameter List may be used to restore printing.

Maximum Degrees of Freedom

i

Default

= n

Note that this option does not apply to problems of type FP or LP.

This places a limit on the storage allocated for the triangular factor

R

of the reduced Hessian

H_{R}

. Ideally,

i

should be set slightly larger than the value of

n_{R}

expected at the solution. It need not be larger than

m_{N} + 1

, where

m_{N}

is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than

m_{N}

For quadratic problems, a minimizer may lie on any number of constraints, so that

n_{R}

may vary between

1

and

n

. The default value of

i

is therefore the number of variables

n

. If Hessian Rows

m

is specified, the default value of

i

is the same number,

m

Minimum Sum of Infeasibilities

Default

= NO

If no feasible point exists for the constraints, this option is used to control whether or not H02CBF will calculate a point that minimizes the constraint violations. If

Minimum Sum of Infeasibilities = N O

, H02CBF will terminate as soon as it is evident that no feasible point exists for the constraints. The final point will generally not be the point at which the sum of infeasibilities is minimized. If

Minimum Sum of Infeasibilities = Y ES

, H02CBF will continue until the sum of infeasibilities is minimized.

Monitoring File

i

Default

= - 1

i \geq 0

and

Print Level \geq 5

, monitoring information produced by H02CBF at every iteration is sent to a file with logical unit number

i

. If

i < 0

and/or

Print Level < 5

, no monitoring information is produced.

Optimality Tolerance

r

Default

= ε^{0.8}

r \geq ε

r

defines the tolerance used to determine if the bounds and general constraints have the right ‘sign’ for the solution to be judged to be optimal.

0 \leq r < ε

, the default value is used.

Print Level

i

Default

= 10

The value of

i

controls the amount of printout produced by H02CBF, as indicated below. A detailed description of the printed output is given in Section 8.2 (summary output at each iteration and the final solution) and Section 12 (monitoring information at each iteration). If

i < 0

, the default value is used.

The following printout is sent to the current advisory message unit (as defined by X04ABF):

$i$	Output
$0$	No output.
$1$	The final solution only.
$5$	One line of summary output ( $< 80$ characters; see Section 8.2) for each iteration (no printout of the final solution).
$\geq 10$	The final solution and one line of summary output for each iteration.

The following printout is sent to the logical unit number defined by the Monitoring File:

$i$	Output
$< 5$	No output.
$\geq 5$	One long line of output ( $> 80$ characters; see Section 12) for each iteration (no printout of the final solution).
$\geq 20$	At each iteration, the Lagrange-multipliers, the variables $x$ , the constraint values $A x$ and the constraint status.
$\geq 30$	At each iteration, the diagonal elements of the upper triangular matrix $T$ associated with the $T Q$ factorization (3) (see Section 10.2) of the working set, and the diagonal elements of the upper triangular matrix $R$ .

Print Level \geq 5

and the unit number defined by Monitoring File is the same as that defined by X04ABF, then the summary output is suppressed.

Problem Type

a

Default

=

QP2

This option specifies the type of objective function to be minimized during the optimality phase. The following are the five optional keywords and the dimensions of the arrays that must be specified in order to define the objective function:

LP	H not referenced, $CVEC (N)$ required;
QP1	$H (LDH *)$ symmetric, CVEC not referenced;
QP2	$H (LDH *)$ symmetric, $CVEC (N)$ required;
QP3	$H (LDH *)$ upper trapezoidal, CVEC not referenced;
QP4	$H (LDH *)$ upper trapezoidal, $CVEC (N)$ required.

For problems of type FP, the objective function is omitted and neither H nor CVEC are referenced.

The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.

$a$	Option
Quadratic	QP2
Linear	LP
Feasible	FP

In addition, the keyword QP is equivalent to the default option QP2.

H = 0

, i.e., the objective function is purely linear, the efficiency of H02CBF may be increased by specifying

a

as LP.

Rank Tolerance

r

Default

= 100 ε

Note that this option does not apply to problems of type FP or LP.

This parameter enables you to control the condition number of the triangular factor

R

(see Section 10). If

ρ_{i}

denotes the function

ρ_{i} = \max \{|R_{11}|, |R_{22}|, \dots, |R_{i i}|\}

, the dimension of

R

is defined to be smallest index

i

such that

|R_{i + 1, i + 1}| \leq \sqrt{r} |ρ_{i + 1}|

. If

r \leq 0

, the default value is used.

12 Description of Monitoring Information

This section describes the long line of output (

> 80

characters) which forms part of the monitoring information produced by H02CBF. (See also the description of the optional parameters Monitoring File and Print Level in Section 11.1.) You can control the level of printed output.

To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices

1

through

n

refer to the bounds on the variables, and indices

n + 1

through

n + m_{L}

refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).

When

Print Level \geq 5

and

Monitoring File \geq 0

, the following line of output is produced at every iteration on the unit number specified by Monitoring File. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Itn	is the iteration count.
Jdel	is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd	is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step	is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.
Ninf	is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective	is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the 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 nonincreasing. 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. Once optimal multipliers are obtained, the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Bnd	is the number of simple bound constraints in the current working set.
Lin	is the number of general linear constraints in the current working set.
Art	is the number of artificial constraints in the working set, i.e., the number of columns of $Z_{A}$ (see Section 10.4).
Zr	is the number of columns of $Z_{R}$ (see Section 10.2). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $Zr = n - (Bnd + Lin + Art)$ . The value of $n_{Z}$ , the number of columns of $Z$ (see Section 10.2) can be calculated as $n_{Z} = n - (Bnd + Lin)$ . A zero value of $n_{Z}$ implies that $x$ lies at a vertex of the feasible region.
Norm Gz	is $‖Z_{R}^{T} g_{FR}‖$ , the Euclidean norm of the reduced gradient with respect to $Z_{R}$ (see Sections 10.2 and 10.4). During the optimality phase, this norm will be approximately zero after a unit step.
NOpt	is the number of nonoptimal Lagrange-multipliers at the current point. NOpt is not printed if the current $x$ is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
Min Lm	is the value of the Lagrange-multiplier associated with the deleted constraint. If Min Lm is negative, a lower bound constraint has been deleted, if Min Lm is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration, Min Lm will be zero.
Cond T	is a lower bound on the condition number of the working set.
Cond Rz	is a lower bound on the condition number of the triangular factor $R$ (the Cholesky factor of the current reduced Hessian; see Section 10.2). If the problem is specified to be of type LP, Cond Rz is not printed.
Rzz	is the last diagonal element $μ$ of the matrix $D$ associated with the $R^{T} D R$ factorization of the reduced Hessian $H_{R}$ (see Section 10.2). Rzz is only printed if $H_{R}$ is not positive definite (in which case $μ \neq 1$ ). If the printed value of Rzz is small in absolute value, then $H_{R}$ is approximately singular. A negative value of Rzz implies that the objective function has negative curvature on the current working set.

H02CBF (PDF version)

H Chapter Contents

H Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentH02CBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Scaling

8.2 Description of the Printed Output

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

10 Algorithmic Details

10.1 Overview

10.2 Definition of the Search Direction

10.3 The Main Iteration

10.4 Choosing the Initial Working Set

11 Optional Parameters

11.1 Description of the Optional Parameters

12 Description of Monitoring Information

NAG Library Routine Document

H02CBF