NAG Toolbox: nag_opt_uncon_simplex (e04cb)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

A guess at the position of the minimum. Note that the problem should be scaled so that the values of the $\mathbf{x}\left(i\right)$ are of order unity.

2:     $\mathrm{tolf}$ – double scalar

The error tolerable in the function values, in the following sense. If $f_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n+1$, 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_uncon_simplex (e04cb) should terminate if

$$\sqrt{\frac{1}{n+1}\sum _{i=1}^{n+1}{\left(f_i-f_m\right)}^2}<\mathbf{tolf}\text{.}$$

(1)

You may specify $\mathbf{tolf}=0$ if you wish to use only the termination criterion (2) on the spatial values: see the description of tolx.

Constraint: $\mathbf{tolf}$ 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.

3:     $\mathrm{tolx}$ – double scalar

The error tolerable in the spatial values, in the following sense. If $LV$ denotes the ‘linearized’ volume of the current simplex, and if ${LV}_{\mathrm{init}}$ denotes the ‘linearized’ volume of the initial simplex, then you can request that nag_opt_uncon_simplex (e04cb) should terminate if

$$\frac{LV}{{LV}_{\mathrm{init}}}<\mathbf{tolx}\text{.}$$

(2)

You may specify $\mathbf{tolx}=0$ if you wish to use only the termination criterion (1) on function values: see the description of tolf.

Constraint: $\mathbf{tolx}$ 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.

4:     $\mathrm{funct}$ – function handle or string containing name of m-file

funct must evaluate the function $F$ at a specified point. It should be tested separately before being used in conjunction with nag_opt_uncon_simplex (e04cb).

[fc, user] = funct(n, xc, user)

Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of variables.

2:     $\mathrm{xc}\left(\mathbf{n}\right)$ – double array

The point at which the function value is required.

3:     $\mathrm{user}$ – Any MATLAB object

funct is called from nag_opt_uncon_simplex (e04cb) with the object supplied to nag_opt_uncon_simplex (e04cb).

Output Parameters

1:     $\mathrm{fc}$ – double scalar

The value of the function $F$ at the current point $\mathbf{x}$.

2:     $\mathrm{user}$ – Any MATLAB object

5:     $\mathrm{monit}$ – function handle or string containing name of m-file

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

If no monitoring is required, monit may be string nag_opt_uncon_simplex_dummy_monit (e04cbk)

[user] = monit(fmin, fmax, sim, n, ncall, serror, vratio, user)

Input Parameters

1:     $\mathrm{fmin}$ – double scalar

The smallest function value in the current simplex.

2:     $\mathrm{fmax}$ – double scalar

The largest function value in the current simplex.

3:     $\mathrm{sim}\left(\mathbf{n}+1,\mathbf{n}\right)$ – double array

The $n+1$ position vectors of the current simplex.

4:     $\mathrm{n}$ – int64int32nag_int scalar

$n$, the number of variables.

5:     $\mathrm{ncall}$ – int64int32nag_int scalar

The number of times that funct has been called so far.

6:     $\mathrm{serror}$ – double scalar

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

7:     $\mathrm{vratio}$ – double scalar

The current value of the linearized volume ratio used in termination test (2).

8:     $\mathrm{user}$ – Any MATLAB object

monit is called from nag_opt_uncon_simplex (e04cb) with the object supplied to nag_opt_uncon_simplex (e04cb).

Output Parameters

1:     $\mathrm{user}$ – Any MATLAB object

6:     $\mathrm{maxcal}$ – int64int32nag_int scalar

The maximum number of function evaluations to be allowed.

Constraint: $\mathbf{maxcal}\ge 1$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the dimension of the array x.

$n$, the number of variables.

Constraint: $\mathbf{n}\ge 1$.

2:     $\mathrm{user}$ – Any MATLAB object

user is not used by nag_opt_uncon_simplex (e04cb), but is passed to funct and monit. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Output Parameters

1:     $\mathrm{x}\left(\mathbf{n}\right)$ – double array

The value of $\mathbf{x}$ corresponding to the function value in f.

2:     $\mathrm{f}$ – double scalar

The lowest function value found.

3:     $\mathrm{user}$ – Any MATLAB object

4:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

Constraint: if $\mathbf{tolf}\ne 0.0$ and $\mathbf{tolx}\ne 0.0$ then both should be greater than or equal to the machine precision.

Constraint: if $\mathbf{tolf}=0.0$ then $\mathbf{tolx}$ is greater than or equal to the machine precision.

Constraint: if $\mathbf{tolx}=0.0$ then $\mathbf{tolf}$ is greater than or equal to the machine precision.

Constraint: $\mathbf{maxcal}\ge 1$.

Constraint: $\mathbf{n}\ge 1$.

   $\mathbf{ifail}=2$

maxcal function evaluations have been completed without any other termination test passing. Check the coding of funct before increasing the value of maxcal.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: e04cb_example

function e04cb_example


fprintf('e04cb example results\n\n');

global print_info
print_info = false;

x0     = [-1; 1];
tolf   = sqrt(x02aj);
tolx   = sqrt(tolf);
maxcal = int64(100);
[x, f, user, ifail] = e04cb( ...
                             x0, tolf, tolx, @funct, @monit, maxcal);

fprintf('Minimum function value = %12.3e\n',f);
fprintf('Minumum location,    x = (%7.3f,%7.3f)\n',x);

fig1 = figure;
[xx,yy] = meshgrid([-1.5:0.1:1.5],[-2:0.1:2]);
z = exp(xx).*(4*xx.*(xx+yy)+2*yy.*(yy+1)+1);
[~,h] = contour(xx,yy,z);
h.LevelList = [0:0.1:1,1.3,1.6,2,2.6,3.2,4,5,6.5,8,10, ...
		13,16,20,25,32,40,50,65,80,100];
colormap(lines);
text(x(1),x(2),'*');
text(x0(1),x0(2),'X');
title('Contours of f: X - Initial Point, * - Local Minimum');



function [fc, user] = funct(n, xc, user)
  fc = exp(xc(1))*(4*xc(1)*(xc(1)+xc(2))+2*xc(2)*(xc(2) +1)+1);

function [user] = monit(fmin, fmax, sim, n, ncall, serror, vratio, user)
global print_info
  if (print_info)
    fprintf('\nNumber of function calls             = %10d\n', ncall);
    fprintf('Smallest function value              = %10.6f\n', fmin);
    fprintf('The simplex is\n');
    disp(sim);
    fprintf('Standard deviation in f(vertices)    = %10.6f\n', serror);
    fprintf('Current/initial simplex volume ratio = %10.6f\n', vratio);
  end

e04cb example results

Minimum function value =    1.789e-08
Minumum location,    x = (  0.500, -1.000)