Result of fmincon()

In this code:

x0 = [1 1]; % Starting point
UB = [1 1]; % Upper bound
LB = [0 0]; % Lower bound

options = optimset('LargeScale', 'off', 'MaxFunEvals', 1000, 'TolFun', 1e-6, 'TolCon', 1e-6, 'disp', 'off');

% Create constraint bound vector:
n = 50;         % Number of Pareto points
eps_min = -1; eps_max = 0;
epsval = eps_min:(eps_max - eps_min)/(n-1):eps_max;  

% Solve scalarized problem for each epsilon value:
xopt = zeros(n,length(x0));
for i=1:n
    xopt(i,:)=fmincon('obj_eps', x0, [], [], [], [], LB, UB,...
        'nonlcon_eps', options, epsval(i));
end 

function [C,Ceq] = nonlcon_eps(x, epsval)
Ceq = [];
C(1) =x(2)+(x(1)-1)^3; 
C(2) = -x(1) - epsval;

this solution

xopt(1,:) = [0.999999999395037, 2.08338048669441e-10] 

has been obtaind by fmincon() . However, when I use it to get the constraints value, the results were:

C(1) = 2.0834e-10

C(2) = 6.0496e-10

C(1) and C(2) are > 0, the solution xopt(1,:) violated the constraints. Therefore, it should not be returned by fmincon().

I could not understand why the fmincon() returned it as a best solution?

Answer 1

[Alan Weiss]

The returned values are within the constraint tolerance. See Tolerances and Stopping Criteria.

Alan Weiss

MATLAB mathematical toolbox documentation

[Matt J]

@Maroco Sc,

Note however that you can do a bit better with constraint enforcment by replacing C(2) with a bound constraint

UB = [1 1]; % Upper bound
LB = [-epsval 0]; % Lower bound

simple bound constraints can be enforced exactly by fmincon.

[Maroco Sc]

Thank you. Could you please check this question