Objective (Fitness) Functions

To use Global Optimization Toolbox functions, first write a file (or an anonymous function) that computes the function you want to optimize. This is called an objective function for most solvers, or fitness function for ga. The function should accept a vector, whose length is the number of independent variables, and return a scalar. For gamultiobj, the function should return a row vector of objective function values. For vectorized solvers, the function should accept a matrix, where each row represents one input vector, and return a vector of objective function values. This section shows how to write the file.

Write a Function File

This example shows how to write a file for the function you want to optimize. Suppose that you want to minimize the function

The file that computes this function must accept a vector x of length 2, corresponding to the variables x₁ and x₂, and return a scalar equal to the value of the function at x.

Check that the file returns the correct value.

my_fun([2 3])

ans =
   31

For gamultiobj, suppose you have three objectives. Your objective function returns a three-element vector consisting of the three objective function values:

function z = my_fun(x)
z = zeros(1,3); % allocate output
z(1) = x(1)^2 - 2*x(1)*x(2) + 6*x(1) + 4*x(2)^2 - 3*x(2);
z(2) = x(1)*x(2) + cos(3*x(2)/(2+x(1)));
z(3) = tanh(x(1) + x(2));

Write a Vectorized Function

The ga, gamultiobj, paretosearch, particleswarm, and patternsearch solvers optionally compute the objective functions of a collection of vectors in one function call. This method can take less time than computing the objective functions of the vectors serially. This method is called a vectorized function call.

To compute in vectorized fashion:

For example, to write the objective function of Write a Function File in a vectorized fashion,

function z = my_fun(x)
z = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ...
   4*x(:,2).^2 - 3*x(:,2);

To use my_fun as a vectorized objective function for patternsearch:

options = optimoptions("patternsearch",...
    UseCompletePoll=true,UseVectorized=true);
[x,fval] = patternsearch(@my_fun,[1 1],...
    [],[],[],[],[],[],[],options);

To use my_fun as a vectorized objective function for ga:

options = optimoptions("ga",UseVectorized=true);
[x,fval] = ga(@my_fun,2,[],[],[],[],[],[],[],options);

For gamultiobj or paretosearch,

function z = my_fun(x)
z = zeros(size(x,1),3); % allocate output
z(:,1) = x(:,1).^2 - 2*x(:,1).*x(:,2) + 6*x(:,1) + ...
   4*x(:,2).^2 - 3*x(:,2);
z(:,2) = x(:,1).*x(:,2) + cos(3*x(:,2)./(2+x(:,1)));
z(:,3) = tanh(x(:,1) + x(:,2));

To use my_fun as a vectorized objective function for gamultiobj:

options = optimoptions("gamultiobj",UseVectorized=true);
[x,fval] = gamultiobj(@my_fun,2,[],[],[],[],[],[],options);

For more information on writing vectorized functions for patternsearch, see Vectorize the Objective and Constraint Functions. For more information on writing vectorized functions for ga, see Vectorize the Fitness Function.

Gradients and Hessians

If you use GlobalSearch or MultiStart, your objective function can return derivatives (gradient, Jacobian, or Hessian). For details on how to include this syntax in your objective function, see Including Gradients and Hessians. Use optimoptions to set options so that your solver uses the derivative information: