E04LYF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

NAG Library Routine Document

E04LYF

+− 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

E04LYF is an easy-to-use modified-Newton algorithm for finding a minimum of a function,

F (x_{1}, x_{2}, \dots, x_{n})

subject to fixed upper and lower bounds on the independent variables,

x_{1}, x_{2}, \dots, x_{n}

when first and second 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 E04LYF (

N, IBOUND, FUNCT2, HESS2, BL, BU, X, F, G, IW, LIW, W, LW, IUSER, RUSER, IFAIL)

INTEGER	N, IBOUND, IW(LIW), LIW, LW, IUSER(*), IFAIL
REAL (KIND=nag_wp)	BL(N), BU(N), X(N), F, G(N), W(LW), RUSER(*)
EXTERNAL	FUNCT2, HESS2

3 Description

E04LYF is applicable to problems of the form:

Minimize F (x_{1}, x_{2}, \dots, x_{n}) subject to l_{j} \leq x_{j} \leq u_{j}, j = 1, 2, \dots, n

when first and second derivatives of

F (x)

are available.

Special provision is made for problems which actually have no bounds on the

x_{j}

, problems which have only non-negativity bounds and problems in which

l_{1} = l_{2} = \dots = l_{n}

and

u_{1} = u_{2} = \dots = u_{n}

. You must supply a subroutine to calculate the values of

F (x)

and its first derivatives at any point

x

and a subroutine to calculate the second derivatives.

From a starting point you supplied there is generated, on the basis of estimates of the curvature of

F (x)

, 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 \geq 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 $l_{j}$ and $u_{j}$ individually.
$IBOUND = 1$: If there are no bounds on any $x_{j}$ .
$IBOUND = 2$: If all the bounds are of the form $0 \leq x_{j}$ .
$IBOUND = 3$: If $l_{1} = l_{2} = \dots = l_{n}$ and $u_{1} = u_{2} = \dots = u_{n}$ .

Constraint:

0 \leq IBOUND \leq 3

.

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

You must supply this routine to calculate the values of the function

F (x)

and its first derivatives

\frac{\partial F}{\partial x_{j}}

at any point

x

. It should be tested separately before being used in conjunction with E04LYF (see the E04 Chapter Introduction).

The specification of FUNCT2 is:

SUBROUTINE FUNCT2 (

N, XC, FC, GC, IUSER, RUSER)

INTEGER	N, IUSER(*)
REAL (KIND=nag_wp)	XC(N), FC, GC(N), RUSER(*)

1: N – INTEGERInput: On entry: the number $n$ of variables.
2: XC(N) – REAL (KIND=nag_wp) arrayInput: On entry: the point $x$ at which the function and its derivatives are required.
3: FC – REAL (KIND=nag_wp)Output: On exit: the value of the function $F$ at the current point $x$ .
4: GC(N) – REAL (KIND=nag_wp) arrayOutput: On exit: $GC (j)$ must be set to the value of the first derivative $\frac{\partial F}{\partial x_{j}}$ at the point $x$ , for $j = 1, 2, \dots, n$ .
5: IUSER( $*$ ) – INTEGER arrayUser Workspace
6: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: FUNCT2 is called with the parameters IUSER and RUSER as supplied to E04LYF. You are free to use the arrays IUSER and RUSER to supply information to FUNCT2 as an alternative to using COMMON global variables.

FUNCT2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04LYF is called. Parameters denoted as Input must not be changed by this procedure.

4: HESS2 – SUBROUTINE, supplied by the user.External Procedure

You must supply this routine to evaluate the elements

H_{i j} = \frac{\partial^{2} F}{\partial x_{i} \partial x_{j}}

of the matrix of second derivatives of

F (x)

at any point

x

. It should be tested separately before being used in conjunction with E04LYF (see the E04 Chapter Introduction).

The specification of HESS2 is:

SUBROUTINE HESS2 (

N, XC, HESLC, LH, HESDC, IUSER, RUSER)

INTEGER	N, LH, IUSER(*)
REAL (KIND=nag_wp)	XC(N), HESLC(LH), HESDC(N), RUSER(*)

1: N – INTEGERInput: On entry: the number $n$ of variables.
2: XC(N) – REAL (KIND=nag_wp) arrayInput: On entry: the point $x$ at which the derivatives are required.
3: HESLC(LH) – REAL (KIND=nag_wp) arrayOutput: On exit: HESS2 must place the strict lower triangle of the second derivative matrix $H$ in HESLC, stored by rows, i.e., set $HESLC ((i - 1) (i - 2) / 2 + j) = \frac{\partial^{2} F}{\partial x_{i} \partial x_{j}}$ , for $i = 2, 3, \dots, n$ and $j = 1, 2, \dots, i - 1$ . (The upper triangle is not required because the matrix is symmetric.)
4: LH – INTEGERInput: On entry: the length of the array HESLC.
5: HESDC(N) – REAL (KIND=nag_wp) arrayOutput: On exit: must contain the diagonal elements of the second derivative matrix, i.e., set $HESDC (j) = \frac{\partial^{2} F}{\partial x_{j}^{2}}$ , for $j = 1, 2, \dots, n$ .
6: IUSER( $*$ ) – INTEGER arrayUser Workspace
7: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace: HESS2 is called with the parameters IUSER and RUSER as supplied to E04LYF. You are free to use the arrays IUSER and RUSER to supply information to HESS2 as an alternative to using COMMON global variables.

HESS2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which E04LYF is called. Parameters denoted as Input must not be changed by this procedure.

5: BL(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the lower bounds

l_{j}

.

If IBOUND is set to

0

,

BL (j)

must be set to

l_{j}

, for

j = 1, 2, \dots, n

. (If a lower bound is not specified for any

x_{j}

, the corresponding

BL (j)

should be set to

- 10^{6}

.)

If IBOUND is set to

3

, you must set

BL (1)

to

l_{1}

; E04LYF will then set the remaining elements of BL equal to

BL (1)

.

On exit: the lower bounds actually used by E04LYF.

6: BU(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: the upper bounds

u_{j}

.

If IBOUND is set to

0

,

BU (j)

must be set to

u_{j}

, for

j = 1, 2, \dots, n

. (If an upper bound is not specified for any

x_{j}

the corresponding

BU (j)

should be set to

10^{6}

.)

If IBOUND is set to

3

, you must set

BU (1)

to

u_{1}

; E04LYF will then set the remaining elements of BU equal to

BU (1)

.

On exit: the upper bounds actually used by E04LYF.

7: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

X (j)

must be set to a guess at the

j

th component of the position of the minimum, for

j = 1, 2, \dots, n

. The routine checks the gradient and the Hessian matrix at the starting point, and is more likely to detect any error in your programming if the initial

X (j)

are nonzero and mutually distinct.

On exit: the lowest point found during the calculations. Thus, if

IFAIL = 0

on exit,

X (j)

is the

j

th component of the position of the minimum.

8: F – REAL (KIND=nag_wp)Output

On exit: the value of

F (x)

corresponding to the final point stored in X.

9: G(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the value of

\frac{\partial F}{\partial x_{j}}

corresponding to the final point stored in X, for

j = 1, 2, \dots, n

; the value of

G (j)

for variables not on a bound should normally be close to zero.

10: IW(LIW) – INTEGER arrayWorkspace

11: LIW – INTEGERInput

On entry: the dimension of the array IW as declared in the (sub)program from which E04LYF is called.

Constraint:

LIW \geq N + 2

.

12: W(LW) – REAL (KIND=nag_wp) arrayWorkspace

13: LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which E04LYF is called.

Constraint:

LW \geq \max (N \times (N + 7), 10)

.

14: IUSER( $*$ ) – INTEGER arrayUser Workspace

15: RUSER( $*$ ) – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by E04LYF, but are passed directly to FUNCT2 and HESS2 and may be used to pass information to these routines as an alternative to using COMMON global variables.

16: 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 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.

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

or

- 1

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

Note: E04LYF 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 $BL (j) > BU (j)$ for some $j$ ,
or	$IBOUND = 3$ and $BL (1) > BU (1)$ ,
or	$LIW < N + 2$ ,
or	$LW < \max (10, N \times (N + 7))$ .

$IFAIL = 2$: There have been $50 \times N$ 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 $F (x)$ 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 so as 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 E04LYF 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 $(\sim 10^{6})$ . This indicates that there is a mistake in user-supplied subroutines FUNCT2 or HESS2, 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.

$IFAIL = 11$: It is very likely that you have made an error in forming the second derivatives.

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.

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

F (x)

and the distance of the starting point from the solution. The number of operations performed in an iteration of E04LYF is roughly proportional to

n^{3} + O (n^{2})

. In addition, each iteration makes one call of HESS2 and at least one call of FUNCT2. So, unless

F (x)

, the gradient vector and the matrix of second derivatives can be evaluated very quickly, the run time will be dominated by the time spent in user-supplied subroutines FUNCT2 and HESS2.

Ideally the problem should be scaled so that at the solution the value of

F (x)

and the corresponding values of

x_{1}, x_{2}, \dots x_{n}

are each 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 E04LYF will take less computer time.

9 Example

A program to minimize

F = {(x_{1} + 10 x_{2})}^{2} + 5 {(x_{3} - x_{4})}^{2} + {(x_{2} - 2 x_{3})}^{4} + 10 {(x_{1} - x_{4})}^{4}

subject to

\begin{aligned} 1 & \leq & x_{1} & \leq & 3 \\ - 2 & \leq & x_{2} & \leq & 0 \\ 1 & \leq & x_{4} & \leq & 3 . \end{aligned}

starting from the initial guess

(3, - 1, 0, 1)

. (In practice, it is worth trying to make user-supplied subroutines FUNCT2 and HESS2 as efficient as possible. This has not been done in the example program for reasons of clarity.)

9.1 Program Text

Program Text (e04lyfe.f90)

9.2 Program Data

None.

9.3 Program Results

Program Results (e04lyfe.r)

E04LYF (PDF version)

E04 Chapter Contents

E04 Chapter Introduction

NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012