How can I optimize complex functions within the Genetic Algorithm and Direct Search Toolbox?

A function of a complex variable "z" is called a complex function "f(z)". I have a complex function in the form:

 f(z) = z^2 - z + 1

and I would like to determine the optimal complex value of this complex function.

Answer 1

[MathWorks Support Team]

This bug has been fixed for Release 14 (R14). For previous releases, read below for any possible workarounds:

In general, complex functions "f(z)" evaluate to a complex value. In this situation, you must define a measure of optimality of these complex results that evaluate to a real value (e.g., "|f(z)|").

The real and imaginary parts must be considered separate variables within the objective function. Therefore, the input argument to the objective function will be a 1-by-2 vector including the real and imaginary part. For this example, the objective function is in the form:

function f = objfun(X)

% Create complex value from real and imaginary parts:

z = X(:, 1) + i*X(:, 2); 

f = z.^2 - z + 1; % Evaluate complex function

f = abs(f); %Generate optimality measure

Then, using the initial guess "z = 1 + 2i", the function can be optimized by the following command:

[z f] = ga(@objfun, 2)

which produces the results:

 z =

     0.5000     0.8661

 f =

   6.2523e-005

Therefore, the complex value that produces a local minimum of "abs(f(z))" is found to be:

 z = 0.5 + i*0.8661

Note that this method is the same for the PATTERNSEARCH function, except a 1-by-2 initial guess must be specified in the second input argument.

For information on how to optimize complex functions using the Optimization Toolbox, see the Related Solution.