Scaled function for optimization

clear

if true
    % Unscaled formulation:
    % min (x^3 - y)^2 - x
    % subject to
    % x^2 + 3150 - y <= 0
    % -15 <= x <= 15
    % -4000 <= y <= 4000
    obj = @(x) (x(1)^3 - x(2))^2 - x(1);
    nonlcon = @(x) deal(x(1)^2 + 3150 - x(2), []);
    lb = [-15, -4000];
    ub = [15, 4000];

    % Solve the problem a few times
    fprintf("Unscaled optimisation results:\n");
    x_unscaled = zeros(5,2);
    fval_unscaled = zeros(5,1);
    for i = 1:5
        options = optimoptions('ga','Display','none','ConstraintTolerance',1e-6,'PlotFcn', @gaplotbestf);
        [x_unscaled(i,:), fval_unscaled(i)] = ga(obj, 2, [], [], [], [], lb, ub, nonlcon, options);
        fprintf("Objective value: %0.3f, x = %0.3f, y = %0.3f\n",fval_unscaled(i), x_unscaled(i,1), x_unscaled(i,2));
    end
end

% Scaled formulation:
% Substitute w = x/15 and z = y/4000
% and multiply the objective and constraints by reasonable factors
% (so that all of their terms are somewhere around the order of 1)
% to obtain...
% min 0.01*(((15*w)^3 - (4000*z))^2 - 15*w)
% subject to
% 0.001*((15*w)^2 + 3150 - (4000*z)) <= 0
% -1 <= w <= 1
% -1 <= z <= 1
if true
    obj = @(x) 0.01*(((15*x(1))^3 - (4000*x(2)))^2 - (15*x(1)));
    nonlcon = @(x) deal(0.001*((15*x(1))^2 + 3150 - (4000*x(2))), []);
    lb = [-1, -1];
    ub = [1, 1];

    % Solve the problem a few times
    fprintf("Scaled optimisation results:\n");
    x_scaled = zeros(5,2);
    fval_scaled = zeros(5,1);
    for i = 1:5
        options = optimoptions('ga','Display','none','ConstraintTolerance',1e-6,'PlotFcn', @gaplotbestf);
        [x_scaled(i,:), fval_scaled(i)] = ga(obj, 2, [], [], [], [], lb, ub, nonlcon, options);
        % Rescale for x and y afterwards
        fval_scaled(i,1) = fval_scaled(i,1) * 100;
        x_scaled(i,1) = x_scaled(i,1) * 15;
        x_scaled(i,2) = x_scaled(i,2) * 4000;
        fprintf("Objective value: %0.3f, x = %0.3f, y = %0.3f\n",fval_scaled(i), x_scaled(i,1), x_scaled(i,2));
    end
end

From the run, I got this solution. But, I do not understand the coding, especially the 0.01 and 0.001 which multiply in the scaled objective function and constraint. How does it come from?

Answer 1

[Torsten]

From the run, I got this solution. But, I do not understand the coding, especially the 0.01 and 0.001 which multiply in the scaled objective function and constraint. How does it come from?

Did you read the program documentation ?

% Substitute w = x/15 and z = y/4000

% and multiply the objective and constraints by reasonable factors

% (so that all of their terms are somewhere around the order of 1)