Learn more about Genetic Algorithm and Direct Search Toolbox

Performing a Pattern Search Using the Optimization Tool GUI

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Example — A Linearly Constrained Problem Displaying Plots Example — Working with a Custom Plot Function

Example — A Linearly Constrained Problem

This section presents an example of performing a pattern search on a constrained minimization problem. The example minimizes the function

where

subject to the constraints

where

Performing a Pattern Search on the Example

To perform a pattern search on the example, first enter

optimtool('patternsearch')

to open the Optimization Tool, or enter optimtool and then choose patternsearch from the Solver menu. Then type the following function in the Objective function field:

@lincontest7

This is an M-file included in Genetic Algorithm and Direct Search Toolbox software that computes the objective function for the example. Because the matrices and vectors defining the starting point and constraints are large, it is more convenient to set their values as variables in the MATLAB workspace first and then enter the variable names in the Optimization Tool. To do so, enter

x0 = [2 1 0 9 1 0];
Aineq = [-8 7 3 -4 9 0];
bineq = [7];
Aeq = [7 1 8 3 3 3; 5 0 5 1 5 8; 2 6 7 1 1 8; 1 0 0 0 0 0];
beq = [84 62 65 1];

Then, enter the following in the Optimization Tool:

• Set Start point to x0.

• Set the following Linear inequalities:

  • Set A to Aineq.

  • Set b to bineq.

  • Set Aeq to Aeq.

  • Set beq to beq.

The following figure shows these settings in the Optimization Tool.

Then click Start to run the pattern search. When the search is finished, the results are displayed in Run solver and view results pane, as shown in the following figure.

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Displaying Plots

The Plot functions pane, shown in the following figure, enables you to display various plots of the results of a pattern search.

Select the check boxes next to the plots you want to display. For example, if you select Best function value and Mesh size, and run the example described in Example — Finding the Minimum of a Function Using the GPS Algorithm, the tool displays the plots shown in the following figure.

The upper plot displays the objective function value at each iteration. The lower plot displays the mesh size at each iteration.

Note   When you display more than one plot, clicking on any plot while the pattern search is running or after the solver has completed opens a larger version of the plot in a separate window.

Plot Options describes the types of plots you can create.

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Example — Working with a Custom Plot Function

To use a plot function other than those included with the software, you can write your own custom plot function that is called at each iteration of the pattern search to create the plot. This example shows how to create a plot function that displays the logarithmic change in the best objective function value from the previous iteration to the current iteration.

This section covers the following topics:

• Creating the Custom Plot Function

• Using the Custom Plot Function

• How the Plot Function Works

Creating the Custom Plot Function

To create the plot function for this example, copy and paste the following code into a new M-file in the MATLAB Editor.

function stop = psplotchange(optimvalues, flag)
% PSPLOTCHANGE Plots the change in the best objective function 
% value from the previous iteration.
  
% Best objective function value in the previous iteration
persistent last_best
 
stop = false;
if(strcmp(flag,'init')) 
        set(gca,'Yscale','log'); %Set up the plot
        hold on;
        xlabel('Iteration'); 
        ylabel('Log Change in Values');
        title(['Change in Best Function Value']);
end
 
% Best objective function value in the current iteration
best = min(optimvalues.fval);  
 
 % Set last_best to best
if optimvalues.iteration == 0
last_best = best;
        
else
        %Change in objective function value
			 change = last_best - best; 
        plot(optimvalues.iteration, change, '.r');
end

Then save the M-file as psplotchange.m in a folder on the MATLAB path.

Using the Custom Plot Function

To use the custom plot function, select Custom function in the Plot functions pane and enter @psplotchange in the field to the right. To compare the custom plot with the best function value plot, also select Best function value. Now, when you run the example described in Example — A Linearly Constrained Problem, the pattern search tool displays the plots shown in the following figure.

Note that because the scale of the y-axis in the lower custom plot is logarithmic, the plot will only show changes that are greater than 0. The logarithmic scale shows small changes in the objective function that the upper plot does not reveal.

How the Plot Function Works

The plot function uses information contained in the following structures, which the Optimization Tool passes to the function as input arguments:

• optimvalues — Structure containing the current state of the solver

• flag — String indicating the current status of the algorithm

The most important statements of the custom plot function, psplotchange.m, are summarized in the following table.

Custom Plot Function Statements

M-File Statement	Description
persistent last_best	Creates the persistent variable last_best, the best objective function value in the previous generation. Persistent variables are preserved over multiple calls to the plot function.
set(gca,'Yscale','log')	Sets up the plot before the algorithm starts.
best = min(optimvalues.fval)	Sets best equal to the minimum objective function value. The field optimvalues.fval contains the objective function value in the current iteration. The variable best is the minimum objective function value. For a complete description of the fields of the structure optimvalues, see Structure of the Plot Functions.
change = last_best - best	Sets the variable change to the best objective function value at the previous iteration minus the best objective function value in the current iteration.
plot(optimvalues.iteration, change, '.r')	Plots the variable change at the current objective function value, for the current iteration contained in optimvalues.iteration.

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Using Direct Search

Performing a Pattern Search from the Command Line

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