NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

e04 Chapter Contents

e04 Chapter Introduction

NAG C Library Function Document

nag_opt_simplex_easy (e04cbc)

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NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

e04 Chapter Contents

e04 Chapter Introduction

▸▿ Contents

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

© The Numerical Algorithms Group Ltd. 2018

1

Purpose

nag_opt_simplex_easy (e04cbc) minimizes a general function

F (x)

of

n

independent variables

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

by the Nelder and Mead simplex method (see Nelder and Mead (1965)). Derivatives of the function need not be supplied.

2

Specification

#include <nag.h>

#include <nage04.h>

void

nag_opt_simplex_easy (Integer n, double x[], double *f, double tolf, double tolx,

void	(funct)(Integer n, const double xc[], double fc, Nag_Comm *comm),

void	(monit)(double fmin, double fmax, const double sim[], Integer n, Integer ncall, double serror, double vratio, Nag_Comm comm),

Integer maxcal, Nag_Comm *comm, NagError *fail)

3

Description

nag_opt_simplex_easy (e04cbc) finds an approximation to a minimum of a function

F

of

n

variables. You must supply a function to calculate the value of

F

for any set of values of the variables.

The method is iterative. A simplex of

n + 1

points is set up in the

n

-dimensional space of the variables (for example, in

2

dimensions the simplex is a triangle) under the assumption that the problem has been scaled so that the values of the independent variables at the minimum are of order unity. The starting point you have provided is the first vertex of the simplex, the remaining

n

vertices are generated by nag_opt_simplex_easy (e04cbc). The vertex of the simplex with the largest function value is reflected in the centre of gravity of the remaining vertices and the function value at this new point is compared with the remaining function values. Depending on the outcome of this test the new point is accepted or rejected, a further expansion move may be made, or a contraction may be carried out. See Nelder and Mead (1965) and Parkinson and Hutchinson (1972) for more details. When no further progress can be made the sides of the simplex are reduced in length and the method is repeated.

The method can be slow, but computational bottlenecks have been reduced following Singer and Singer (2004). However, nag_opt_simplex_easy (e04cbc) is robust, and therefore very useful for functions that are subject to inaccuracies.

There are the following options for successful termination of the method: based only on the function values at the vertices of the current simplex (see (1)); based only on a volume ratio between the current simplex and the initial one (see (2)); or based on which one of the previous two tests passes first. The volume test may be useful if

F

is discontinuous, while the function-value test should be sufficient on its own if

F

is continuous.

4

References

Nelder J A and Mead R (1965) A simplex method for function minimization Comput. J. 7 308–313

Parkinson J M and Hutchinson D (1972) An investigation into the efficiency of variants of the simplex method Numerical Methods for Nonlinear Optimization (ed F A Lootsma) Academic Press

Singer S and Singer S (2004) Efficient implementation of the Nelder–Mead search algorithm Appl. Num. Anal. Comp. Math. 1(3) 524–534

5

Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of variables.

Constraint:

n \geq 1

.

2: $x [n]$ – doubleInput/Output

On entry: a guess at the position of the minimum. Note that the problem should be scaled so that the values of the

x [i - 1]

are of order unity.

On exit: the value of

x

corresponding to the function value in f.

3: $f$ – double *Output

On exit: the lowest function value found.

4: $tolf$ – doubleInput

On entry: the error tolerable in the function values, in the following sense. If

f_{i}

, for

i = 1, 2, \dots, n + 1

, are the individual function values at the vertices of the current simplex, and if

f_{m}

is the mean of these values, then you can request that nag_opt_simplex_easy (e04cbc) should terminate if

\sqrt{\frac{1}{n + 1} \sum_{i = 1}^{n + 1} {(f_{i} - f_{m})}^{2}} < tolf .

(1)

You may specify

tolf = 0

if you wish to use only the termination criterion (2) on the spatial values: see the description of tolx.

Constraint:

tolf

must be greater than or equal to the machine precision (see Chapter x02), or if tolf equals zero then tolx must be greater than or equal to the machine precision.

5: $tolx$ – doubleInput

On entry: the error tolerable in the spatial values, in the following sense. If

L V

denotes the ‘linearized’ volume of the current simplex, and if

{L V}_{init}

denotes the ‘linearized’ volume of the initial simplex, then you can request that nag_opt_simplex_easy (e04cbc) should terminate if

\frac{L V}{{L V}_{init}} < tolx .

(2)

You may specify

tolx = 0

if you wish to use only the termination criterion (1) on function values: see the description of tolf.

Constraint:

tolx

must be greater than or equal to the machine precision (see Chapter x02), or if tolx equals zero then tolf must be greater than or equal to the machine precision.

6: $funct$ – function, supplied by the userExternal Function

funct must evaluate the function

F

at a specified point. It should be tested separately before being used in conjunction with nag_opt_simplex_easy (e04cbc).

The specification of funct is:

void	funct (Integer n, const double xc[], double fc, Nag_Comm comm)

1: $n$ – IntegerInput

On entry:

n

, the number of variables.

2: $xc [n]$ – const doubleInput

On entry: the point at which the function value is required.

3: $fc$ – double *Output

On exit: the value of the function

F

at the current point

x

.

4: $comm$ – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_opt_simplex_easy (e04cbc) you may allocate memory and initialize these pointers with various quantities for use by funct when called from nag_opt_simplex_easy (e04cbc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

Note: funct should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_opt_simplex_easy (e04cbc). If your code inadvertently does return any NaNs or infinities, nag_opt_simplex_easy (e04cbc) is likely to produce unexpected results.

7: $monit$ – function, supplied by the userExternal Function

monit may be used to monitor the optimization process. It is invoked once every iteration.

If no monitoring is required, monit may be specified as NULLFN.

The specification of monit is:

void	monit (double fmin, double fmax, const double sim[], Integer n, Integer ncall, double serror, double vratio, Nag_Comm *comm)

1: $fmin$ – doubleInput

On entry: the smallest function value in the current simplex.

2: $fmax$ – doubleInput

On entry: the largest function value in the current simplex.

3: $sim [(n + 1) \times n]$ – const doubleInput

On entry: the

n + 1

position vectors of the current simplex, where

sim [(j - 1) \times (n + 1) + i - 1]

is the

j

th coordinate of the

i

th position vector.

4: $n$ – IntegerInput

On entry:

n

, the number of variables.

5: $ncall$ – IntegerInput

On entry: the number of times that funct has been called so far.

6: $serror$ – doubleInput

On entry: the current value of the standard deviation in function values used in termination test (1).

7: $vratio$ – doubleInput

On entry: the current value of the linearized volume ratio used in termination test (2).

8: $comm$ – Nag_Comm *

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

user – double *
iuser – Integer *
p – Pointer: The type Pointer will be void *. Before calling nag_opt_simplex_easy (e04cbc) you may allocate memory and initialize these pointers with various quantities for use by monit when called from nag_opt_simplex_easy (e04cbc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

8: $maxcal$ – IntegerInput

On entry: the maximum number of function evaluations to be allowed.

Constraint:

maxcal \geq 1

.

9: $comm$ – Nag_Comm *

The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).

10: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $maxcal = 〈value〉$ .
Constraint: $maxcal \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $tolf = 0.0$ and $tolx = 〈value〉$ .
Constraint: if $tolf = 0.0$ then $tolx$ is greater than or equal to the machine precision.

On entry, $tolx = 0.0$ and $tolf = 〈value〉$ .
Constraint: if $tolx = 0.0$ then $tolf$ is greater than or equal to the machine precision.
NE_REAL_2: On entry, $tolf = 〈value〉$ and $tolx = 〈value〉$ .
Constraint: if $tolf \neq 0.0$ and $tolx \neq 0.0$ then both should be greater than or equal to the machine precision.
NW_TOO_MANY_FEVALS: maxcal function evaluations have been completed without any other termination test passing. Check the coding of funct before increasing the value of maxcal.

7

Accuracy

On a successful exit the accuracy will be as defined by tolf or tolx, depending on which criterion was satisfied first.

8

Parallelism and Performance

nag_opt_simplex_easy (e04cbc) 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

Local workspace arrays of fixed lengths (depending on n) are allocated internally by nag_opt_simplex_easy (e04cbc). The total size of these arrays amounts to

n^{2} + 6 n + 2

double elements.

The time taken by nag_opt_simplex_easy (e04cbc) depends on the number of variables, the behaviour of the function and the distance of the starting point from the minimum. Each iteration consists of

1

or

2

function evaluations unless the size of the simplex is reduced, in which case

n + 1

function evaluations are required.

10

Example

This example finds a minimum of the function

F (x_{1}, x_{2}) = e^{x_{1}} (4 x_{1}^{2} + 2 x_{2}^{2} + 4 x_{1} x_{2} + 2 x_{2} + 1) .

This example uses the initial point

(- 1, 1)

(see Section 10.3), and we expect to reach the minimum at

(0.5, - 1)

.

10.1

Program Text

Program Text (e04cbce.c)

10.2

Program Data

None.

10.3

Program Results

Program Results (e04cbce.r)

NAG C Library Manual, Mark 26

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

e04 Chapter Contents

e04 Chapter Introduction

© The Numerical Algorithms Group Ltd. 2018