Incorrect solution for symmetric problems in fmincon

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

Answer 1

[Alan Weiss]

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]

Thanks for the quick reply!

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?

[Matt J]

It seems to me that it could conclude that from a comparison of the gradient of objective & linear constraint.

On the contrary, the gradient of the objective at your initial point is grad=[-1,-1]. Therefore in both feasible directions u= [1,-1] and u=[-1,1], the directional derivative is zero. So, the algorithm cannot see a feasible direction of increase or decrease.

[Sargondjani]

Thanks Matt. I understand now it's really a second order thing.