NAG Library Routine Document

E04KZF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E04KZF is an easy-to-use modified Newton algorithm for finding a minimum of a function Fx1,x2,…,xn, subject to fixed upper and lower bounds on the independent variables x1,x2,…,xn, when first derivatives of F are available. It is intended for functions which are continuous and which have continuous first and second derivatives (although it will usually work even if the derivatives have occasional discontinuities).

2 Specification

SUBROUTINE E04KZF (	N, IBOUND, FUNCT2, BL, BU, X, F, G, IW, LIW, W, LW, IUSER, RUSER, IFAIL)
INTEGER	N, IBOUND, IW(LIW), LIW, LW, IUSER(*), IFAIL
*double precision*	BL(N), BU(N), X(N), F, G(N), W(LW), RUSER(*)
EXTERNAL	FUNCT2

3 Description

E04KZF is applicable to problems of the form:

Minimize⁡Fx1,x2,…,xn subject to lj≤xj≤uj, j=1,2,…,n

when first derivatives are known.

Special provision is made for problems which actually have no bounds on the xj, problems which have only non-negativity bounds, and problems in which l1=l2=⋯=ln and u1=u2=⋯=un. You must supply a subroutine to calculate the values of Fx and its first derivatives at any point x.

From a starting point you supplied there is generated, on the basis of estimates of the gradient of the curvature of Fx, a sequence of feasible points which is intended to converge to a local minimum of the constrained function.

4 References

Gill P E and Murray W (1976) Minimization subject to bounds on the variables NPL Report NAC 72 National Physical Laboratory

5 Parameters

1: N – INTEGERInput

On entry: the number n of independent variables.

Constraint: N≥1.

2: IBOUND – INTEGERInput

On entry: indicates whether the facility for dealing with bounds of special forms is to be used. It must be set to one of the following values:

IBOUND=0: If you are supplying all the lj and uj individually.
IBOUND=1: If there are no bounds on any xj.
IBOUND=2: If all the bounds are of the form 0≤xj.
IBOUND=3: If l1=l2=⋯=ln and u1=u2=⋯=un.

Constraint: 0≤IBOUND≤3.

3: FUNCT2 – SUBROUTINE, supplied by the user.External Procedure

You must supply this routine to calculate the values of the function Fx and its first derivatives ∂F ∂xj at any point x. It should be tested separately before being used in conjunction with E04KZF (see Chapter E04).

The specification of FUNCT2 is:

SUBROUTINE FUNCT2 (	N, XC, FC, GC, IUSER, RUSER)
INTEGER	N, IUSER(*)
*double precision*	XC(N), FC, GC(N), RUSER(*)

1: N – INTEGERInput: On entry: the number n of variables.
2: XC(N) – double precision arrayInput: On entry: the point x at which the function and derivatives are required.
3: FC – double precisionOutput: On exit: the value of the function F at the current point x,
4: GC(N) – double precision arrayOutput: On exit: GCj must be set to the value of the first derivative ∂F ∂xj at the point x, for j=1,2,…,n.
5: IUSER(*) – INTEGER arrayUser Workspace
6: RUSER(*) – double precision arrayUser Workspace: FUNCT2 is called from E04KZF with the parameters IUSER and RUSER as supplied to E04KZF. You are free to use the arrays IUSER and RUSER to supply information to FUNCT2 as an alternative to using COMMON.

FUNCT2 must be declared as EXTERNAL in the (sub)program from which E04KZF is called. Parameters denoted as Input must not be changed by this procedure.

4: BL(N) – double precision arrayInput/Output

On entry: the lower bounds lj.

If IBOUND is set to 0, you must set BLj to lj, for j=1,2,…,n. (If a lower bound is not specified for a particular xj, the corresponding BLj should be set to -106.)

If IBOUND is set to 3, you must set BL1 to l1; E04KZF will then set the remaining elements of BL equal to BL1.

On exit: the lower bounds actually used by E04KZF.

5: BU(N) – double precision arrayInput/Output

On entry: the upper bounds uj.

If IBOUND is set to 0, you must set BUj to uj, for j=1,2,…,n. (If an upper bound is not specified for a particular xj, the corresponding BUj should be set to 106.)

If IBOUND is set to 3, you must set BU1 to u1; E04KZF will then set the remaining elements of BU equal to BU1.

On exit: the upper bounds actually used by E04KZF.

6: X(N) – double precision arrayInput/Output

On entry: Xj must be set to a guess at the jth component of the position of the minimum, for j=1,2,…,n. The routine checks the gradient at the starting point, and is more likely to detect any error in your programming if the initial Xj are nonzero and mutually distinct.

On exit: the lowest point found during the calculations of the position of the minimum.

7: F – double precisionOutput

On exit: the value of Fx corresponding to the final point stored in X.

8: G(N) – double precision arrayOutput

On exit: the value of ∂F ∂xj corresponding to the final point stored in X, for j=1,2,…,n; the value of Gj for variables not on a bound should normally be close to zero.

9: IW(LIW) – INTEGER arrayWorkspace

10: LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which E04KZF is called.

Constraint: LIW≥N+2.

11: W(LW) – double precision arrayWorkspace

12: LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which E04KZF is called.

Constraint: LW≥maxN×N+7,10.

13: IUSER(*) – INTEGER arrayUser Workspace

Note: the dimension of the array IUSER must be at least 1.

IUSER is not used by E04KZF, but is passed directly to FUNCT2 and may be used to pass information to that routine.

14: RUSER(*) – double precision arrayUser Workspace

Note: the dimension of the array RUSER must be at least 1.

RUSER is not used by E04KZF, but is passed directly to FUNCT2 and may be used to pass information to that routine.

15: 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.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

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, because for this routine the values of the output parameters may be useful even if IFAIL≠0 on exit, the recommended value is -1. When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.

6 Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: E04KZF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

IFAIL=1: On entry, N<1,
or IBOUND<0,
or IBOUND>3,
or IBOUND=0 and BLj>BUj for some j,
or IBOUND=3 and BL1>BU1,
or LIW<N+2,
or LW<max10,N×N+7.

IFAIL=2: There has been a large number of function evaluations, yet the algorithm does not seem to be converging. The calculations can be restarted from the final point held in X. The error may also indicate that Fx has no minimum.

IFAIL=3: The conditions for a minimum have not all been met but a lower point could not be found and the algorithm has failed.

IFAIL=4: Not used. (This value of the parameter is included to make the significance of IFAIL=5 etc. consistent in the easy-to-use routines.)

IFAIL=5
IFAIL=6
IFAIL=7
IFAIL=8: There is some doubt about whether the point x found by E04KZF is a minimum. The degree of confidence in the result decreases as IFAIL increases. Thus, when IFAIL=5 it is probable that the final x gives a good estimate of the position of a minimum, but when IFAIL=8 it is very unlikely that the routine has found a minimum.

IFAIL=9: In the search for a minimum, the modulus of one of the variables has become very large ∼106. This indicates that there is a mistake in FUNCT2, that your problem has no finite solution, or that the problem needs rescaling (see Section 8).

IFAIL=10: It is very likely that you have made an error in forming the gradient.

If you are dissatisfied with the result (e.g., because IFAIL=5, 6, 7 or 8), it is worth restarting the calculations from a different starting point (not the point at which the failure occurred) in order to avoid the region which caused the failure. If persistent trouble occurs and it is possible to calculate second derivatives it may be advisable to change to a routine which uses second derivatives (see the E04 Chapter Introduction).

7 Accuracy

When a successful exit is made then, for a computer with a mantissa of t decimals, one would expect to get about t/2-1 decimals accuracy in x and about t-1 decimals accuracy in F, provided the problem is reasonably well scaled.

8 Further Comments

The number of iterations required depends on the number of variables, the behaviour of Fx and the distance of the starting point from the solution. The number of operations performed in an iteration of E04KZF is roughly proportional to n3+On2. In addition, each iteration makes at least m+1 calls of FUNCT2 where m is the number of variables not fixed on bounds. So unless Fx and the gradient vector can be evaluated very quickly, the run time will be dominated by the time spent in FUNCT2.

Ideally the problem should be scaled so that at the solution the value of Fx and the corresponding values of x1,x2,…,xn are in the range -1,+1, and so that at points a unit distance away from the solution, F is approximately a unit value greater than at the minimum. It is unlikely that you will be able to follow these recommendations very closely, but it is worth trying (by guesswork), as sensible scaling will reduce the difficulty of the minimization problem, so that E04KZF will take less computer time.