How do I use the max() function on an optimization expression when using the genetic algorithm?

I am using the genetic algorithm to optimize a 5 by 5 planar antenna array in the aim of achieving a predetermined sidelobe level (SLL_imp in the following code) with as few antenna elements as possible. Moreover, the optimization variable is a 5 by 5 binary matrix (A_mn in the following code) which represents the state (either included or eliminated) of the individual antenna elements. However, I am getting an 'Invalid data type' error when I use max() on AF (an optimization expression in the following code that will be ultimately used to construct the objective/error function). How can I get around this problem?

% Sparse array optimization using GA
clear, clc

% Create binary optimization variables
A_mn = optimvar('A_mn',5,5,'Type','integer','LowerBound',0,'UpperBound',1);

% Create an optimization problem with the objective function
prob = optimproblem("Objective",error_fcn(A_mn));

options = optimoptions("ga","PlotFcn","gaplotbestf");

rng default % For reproducibility
[sol,fval,exitflag] = solve(prob,"Solver","ga","Options",options);

function y = error_fcn(A_mn)
    % Define some variables to be used in AF (array factor) computation
    j = sqrt(-1);
    M = 5; N = 5;           
    f = 5.4e+9;             
    c = (3e+8)/sqrt(1);     
    lambda = c/f;           

    k = (2*pi)/lambda;
    d_x = 0.5*lambda;       
    d_y = 0.5*lambda;       

    theta = linspace(-180, 180, 1001);
    phi = 0;

    theta_0 = 0;
    phi_0 = 0;
    beta_x = -k*d_x*sind(theta_0)*cosd(phi_0);      
    beta_y = -k*d_y*sind(theta_0)*sind(phi_0);  
    psi_x = k*d_x*sind(theta).*cosd(phi) + beta_x;
    psi_y = k*d_y*sind(theta).*sind(phi) + beta_y;

    % Compute the AF (array factor)
    AF = 0;
    for n = 1:N
        for m = 1:M
            AF = AF + (exp(j*((m-1)*psi_x + (n-1)*psi_y)));
        end
    end

    AF_mag = abs(AF);
    AF = 0;

    for n = 1:N
        for m = 1:M
            AF = AF + (A_mn(m,n)*AF_mag);
        end
    end
    
    % Compute the normalized AF (array factor) in decibel
    AF_max = max(AF);       % ERROR OCCURS ON THIS LINE    
    AF_dB = 20*log10(AF/AF_max);

    % Find the sidelobes of the AF (array factor)
    lobes = findpeaks(AF_dB);
    % Identify the maximum sidelobe level
    SLL = lobes(floor(0.5*length(lobes)));
    SLL_imp = -12.0425;     % the imposed sidelobe level that we want to optimize for
    
    y = (SLL - SLL_imp)^2;
end

Answer 1

[Alan Weiss]

If you check the Supported Operations for Optimization Variables and Expressions, you see that max is not supported. (Why? I believe it is because max is not differentiable.) Therefore you must use fcn2optimexpr to work around the issue of unsupported functions. So instead of the line

prob = optimproblem("Objective",error_fcn(A_mn));

use

prob = optimproblem("Objective",fcn2optimexpr(@error_fcn,A_mn));

Alan Weiss

MATLAB mathematical toolbox documentation

[Saeed Raffoul]

Thank you, fcn2optimexpr solves the issue of the unsupported max function. However, I am getting an error, concerning the solve function, which I am failing to understand:

Error using optim.problemdef.OptimizationProblem/solve

Array indices must be positive integers or logical values.

Error in ArrayThinning_GA (line 13)

[sol,fval,exitflag] = solve(prob,"Solver","ga","Options",options);

Caused by:

Failure in user-supplied fitness function evaluation. GA cannot continue.

Which array is not being indexed with positive integers or logical values? I can confirm that the array, lobes, is only indexed by postive integers.

[Alan Weiss]

Without seeing the entire function call and the functions they call, I am not sure. The problem might relate to some supposedly integer values being slightly different than integers. Sometimes, you can fix this issue with a call to round on the appropriate variables just before the function gives its outputs.

However, the issue might lie elsewhere. For more helpful answers, please provide more details.