Global vs. Local Optimization Using ga

Searching for a Global Minimum

Sometimes the goal of an optimization is to find the global minimum or maximum of a function—a point where the function value is smaller or larger at any other point in the search space. However, optimization algorithms sometimes return a local minimum—a point where the function value is smaller than at nearby points, but possibly greater than at a distant point in the search space. The genetic algorithm can sometimes overcome this deficiency with the right settings.

As an example, consider the following function.

Plot the function.

t = -10:.1:103;
for ii = 1:length(t)
    y(ii) = two_min(t(ii));
end
plot(t,y)

The function has two local minima, one at $$x = 0$$, where the function value is –1, and the other at $$x = 101$$, where the function value is $$–1 – 1/e$$. Since the latter value is smaller, the global minimum occurs at $$x = 101$$.

Run ga Using Default Parameters

The code for the two_min helper function is at the end of this example. Run ga with default parameters to minimize the two_min function. Use the gaplot1drange helper function (included at the end of this example) to plot the range of the ga population at each iteration.

rng default % For reproducibility
options = optimoptions('ga','PlotFcn',@gaplot1drange);
[x,fval] = ga(@two_min,1,[],[],[],[],[],[],[],options)

ga stopped because the average change in the fitness value is less than options.FunctionTolerance.

x = 
-0.0688

fval = 
-1.0000

The genetic algorithm returns a point very close to the local minimum at $$x = 0$$. Note that all individuals lie between –60 and 60. The population never explores points near the global minimum at $$x = 101$$.

Increase Initial Range

One way to make the genetic algorithm explore a wider range of points—that is, to increase the diversity of the populations—is to increase the initial range. The initial range does not have to include the point x = 101, but it must be large enough so that the algorithm generates individuals near x = 101. Set the InitialPopulationRange option to [-10;90] and rerun the solver.

options.InitialPopulationRange = [-10;90];
[x,fval] = ga(@two_min,1,[],[],[],[],[],[],[],options)

ga stopped because it exceeded options.MaxGenerations.

x = 
100.9783

fval = 
-1.3674

This time, the custom plot shows a much wider range of individuals. There are individuals near 101 from early on, and the population mean begins to converge to 101.

Helper Functions

This code creates the two_min helper function.

function y = two_min(x)
if x <= 100
    y = -exp(-(x/100)^2);
else
    y = -exp(-1) + (x-100)*(x-102);
end
end

This code creates the gaplot1drange helper function.

function state = gaplot1drange(options,state,flag)
%gaplot1drange Plots the mean and the range of the population.
%   STATE = gaplot1drange(OPTIONS,STATE,FLAG) plots the mean and the range
%   (highest and the lowest) of individuals (1-D only).  
%
%   Example:
%   Create options that use gaplot1drange
%   as the plot function
%     options = optimoptions('ga','PlotFcn',@gaplot1drange);

%   Copyright 2012-2014 The MathWorks, Inc.

if isinf(options.MaxGenerations) || size(state.Population,2) > 1
    title('Plot Not Available','interp','none');
    return;
end
generation = state.Generation;
score = state.Population;
smean = mean(score);
Y = smean;
L = smean - min(score);
U = max(score) - smean;

switch flag

    case 'init'
        set(gca,'xlim',[1,options.MaxGenerations+1]);
        plotRange = errorbar(generation,Y,L,U);
        set(plotRange,'Tag','gaplot1drange');
        title('Range of Population, Mean','interp','none')
        xlabel('Generation','interp','none')
    case 'iter'
        plotRange = findobj(get(gca,'Children'),'Tag','gaplot1drange');
        newX = [get(plotRange,'Xdata') generation];
        newY = [get(plotRange,'Ydata') Y];
        newL = [get(plotRange,'Ldata') L];
        newU = [get(plotRange,'Udata') U];       
        set(plotRange,'Xdata',newX,'Ydata',newY,'Ldata',newL,'Udata',newU);
end
end