Solving Nonlinear Constrained Optimization Problems with Matlab

Hello everyone and welcome.

In this video, I’m going to show you how to solve nonlinear constrained optimization problems with Matlab.

This optimization solution method is based on an optimization solver named “fmincon solver” in Matlab. To enhance the robustness of the solver, I use a simple code for running the solver multiple times (say: 10 times), each with random initial solution, and the final solution is the best solution found in all the runs. Thereby, the chance to get the global optimal solution is significantly improved.

Here is the detail of the benchmark function.

Let’s see how it works.

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Matlab code

objective_function.m

function Output = objective_function(Input)
x = Input(1);
y = Input(2);
Output = sin(y)*exp((1-cos(x))^2) + cos(x)*exp((1-sin(y))^2) + (x-y)^2;

nonlinear_constraint.m

function [C Ceq] = nonlinear_constraint(Input)
x = Input(1);
y = Input(2);
C = (x+5)^2 + (y+5)^2 - 25;  % inequality constraint 
Ceq = [];  % no equality constraint

fminconsolver.m

function [x,fval,exitflag,output,lambda,grad,hessian] = fminconsolver(x0,lb,ub)
options = optimoptions('fmincon');
options = optimoptions(options,'Display', 'off');
options = optimoptions(options,'PlotFcn', {  @optimplotfval @optimplotconstrviolation });
[x,fval,exitflag,output,lambda,grad,hessian] = ...
fmincon(@objective_function,x0,[],[],[],[],lb,ub,@nonlinear_constraint,options);

main_code.m

clc
clear all
close all
%--------------------------------------------------------------------------
% customization: lower bound lb = [1xn], and upper bound lb = [1xn]
lb = [-10 -6.5];
ub = [0 0];
%--------------------------------------------------------------------------
n=10; % number of starts
[x y]=size(lb);
S=zeros(n,y);
R=zeros(n,1);
% multi-start optimization solver
for j=1:n
    for i = 1:y
        x0(i)=lb(i) + (ub(i)-lb(i))*rand; % generate initial solution
    end
    [x,fval,exitflag,output,lambda,grad,hessian] = fminconsolver(x0,lb,ub);
    S(j,:)=x;
    R(j,1)=fval;
end
[x1 y1]=find(R == min(R));
objective_function_value = R(min(x1),1)
optimal_solution = S(min(x1),:)

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