Coding and minimizing a fitness function using the Genetic Algorithm

This is a demonstration of how to create and minimize a fitness function using the Genetic Algorithm in the Genetic Algorithm and Direct Search Toolbox.

Contents

A simple fitness function

Here we want to minimize a simple function of two variables

   min f(x) = 100 * (x(1)^2 - x(2)) ^2 + (1 - x(1))^2;
    x

Coding the fitness function

We create an M-file named simple_fitness.m with the following code in it:

   function y = simple_fitness(x)
   y = 100 * (x(1)^2 - x(2)) ^2 + (1 - x(1))^2;

The Genetic Algorithm solver assumes the fitness function will take one input x where x is a row vector with as many elements as number of variables in the problem. The fitness function computes the value of the function and returns that scalar value in its one return argument y.

Minimizing using GA

To minimize our fitness function using the GA function, we need to pass in a function handle to the fitness function as well as specifying the number of variables in the problem. 

   FitnessFunction = @simple_fitness;
   numberOfVariables = 2;
   [x,fval] = ga(FitnessFunction,numberOfVariables)

Optimization terminated: stall generations limit exceeded.

x =

    0.7745    0.6098


fval =

    0.0610

The x returned by the solver is the best point in the final population computed by GA. The fval is the value of the function @simple_fitness evaluated at the point x.

A fitness function with additional arguments

Sometimes we want our fitness function to be parameterized by extra arguments that act as constants during the optimization. For example, in the previous fitness function, say we want to replace the constants 100 and 1 with parameters that we can change to create a family of objective functions. We can re-write the above function to take two additional parameters to give the new minimization problem 

   min f(x) = a * (x(1)^2 - x(2)) ^2 + (b - x(1))^2;
    x

a and b are parameters to the fitness function that act as constants during the optimization (they are not varied as part of the minimization). One can create an M-file called parameterized_fitness.m containing the following code: 

   function y = parameterized_fitness(x,a,b)
   y = a * (x(1)^2 - x(2)) ^2 + (b - x(1))^2;

Minimizing using additional arguments

Again, we need to pass in a function handle to the fitness function as well as the number of variables as the second argument.

GA will call our fitness function with just one argument 'x', but our fitness function has three arguments: x, a, b. We can use an anonymous function to capture the values of the additional arguments, the constants a and b. We create a function handle 'FitnessFunction' to an anonymous function that takes one input 'x', but calls 'parameterized_fitness' with x, a, and b. The variables a and b have values when the function handle 'FitnessFunction' is created, so these values are captured by the anonymous function. 

   a = 100; b = 1; % define constant values
   FitnessFunction = @(x) parameterized_fitness(x,a,b);
   numberOfVariables = 2;
   [x,fval] = ga(FitnessFunction,numberOfVariables)

Optimization terminated: maximum number of generations exceeded.

x =

    1.1599    1.3461


fval =

    0.0256

Vectorizing your fitness function

Consider the previous fitness function again:

   f(x) = a * (x(1)^2 - x(2)) ^2 + (b - x(1))^2;

By default, the GA solver only passes in one point at a time to the fitness function. However, sometimes speed up can be achieved if the fitness function is vectorized to take a set of points and return a set of function values.

For example if the solver wants to evaluate a set of five points in one call to this fitness function, then it will call the function with a matrix of size 5-by-2, i.e. , 5 rows and 2 columns (recall 2 is the number of variables).

Create an M-file called vectorized_fitness.m with the following code:

   function y = vectorized_fitness(x,a,b)
   y = zeros(size(x,1),1); %Pre-allocate y
   for i = 1:size(x,1)
     x1 = x(i,1);
     x2 = x(i,2);
     y(i) = a * (x1^2 - x2) ^2 + (b - x1)^2;
   end

This vectorized version of the fitness function takes a matrix x with an arbitrary number of points, the rows of x, and returns a column vector y of length the same as the number of rows of x.

We need to specify that the fitness function is vectorized using the options structure created using GAOPTIMSET. The options structure is passed in as the third argument. 

   FitnessFunction = @(x) vectorized_fitness(x,100,1);
   numberOfVariables = 2;
   options = gaoptimset('Vectorized','on');
   [x,fval] = ga(FitnessFunction,numberOfVariables,options)

Optimization terminated: maximum number of generations exceeded.

x =

    1.0580    1.1235


fval =

    0.0051

Copyright 2004 The MathWorks, Inc.