# Incorrect solution for symmetric problems in fmincon

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Sargondjani on 22 Apr 2019
Commented: Sargondjani on 22 Apr 2019
If I maximize XX(1)^2+XX(2)^2 subject to x1 + x2 <=1 and use starting value X0=[0.5,0.5] I get as solution X=[0.5,0.5], although the two optima are X=[1,0] and X=[0,1].
Any clue how to prevent this from happening? (Other than using an asymmetric starting value). I already tried changing algorithm to sqp but that doesn't help.
See code:
function [XX,VAL] = test_con_opt()
clc;
close all;
dbstop if error;
sum_x = 1;
AA = [1,1];
bb = sum_x; %Inequality constraint: x1 + x2 <= sum_x
lb = [0,0];
pwr = 2;
%X0 = [0.25,0.75];
%X0 = [0.75,0.025];
X0 = [0.5,0.5];
[XX,mVAL] = fmincon(@(XX)obj_fun(pwr,XX(1),XX(2)),X0,AA,bb,[],[],lb);
VAL = - mVAL;
end
function [mVAL] = obj_fun(pwr,x1,x2)
mVAL = - (x1^pwr + x2^pwr);
end

Alan Weiss on 22 Apr 2019
fmincon is a gradient-based algorithm. When your initial point is [0.5,0.5], the gradient is zero, and fmincon stops, since it is at a stationary point.
In general, you can take random initial points, which are unlikely to be exact stationary points (assuming that stationary points are isolated).
Alan Weiss
MATLAB mathematical toolbox documentation
Sargondjani on 22 Apr 2019
What I don't understand then: if the algorithm compared the change in the objective function and the linear constraint at the stationary point it should conclude that moving away from the stationary point increases the value of the objective function. It seems to me that it could conclude that from a comparison of the gradient of objective & linear constraint.
And are there any algothims that actually achieve this?