Non-linear constraints with several input variables in fmincon

Hello everyone,

I am trying to solve the following optimization problem with fmincon:

To do this, I have made the following code:

fun = @(x) x'*L*x;

nonlcon = @(x) consfun(A,B,x);

x = fmincon(fun,x0,[],[],[],[],[],[],nonlcon);

Where ''consfun'' is the following function:

function [c,ceq] = consfun(A,B,x)
    c = (norm(A*x,inf)/b)-D);
    ceq = [];
end

However, the final solution, x, does not satisfy the non-linear constraint so I wonder if the code is correct in relation to the optimisation problem posed.

Can anyone help me?

Thank you very much for your time!

[Alan Weiss]

Did fmincon claim to give a feasible solution? If so, then in what way was the constraint violated? I mean, was consfun(A,B,x) > 0? If not, then you may need to search for a feasible solution. See Converged to an Infeasible Point.

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 1

[Matt J]

Your nonlinear constraints are not differentiable. That doesn't always spell disaster, but it breaks the assumptions of fmincon. Also, since your problem can be reformulated as a quadratic program, it would be better to use quadprog.

[m,n]=size(A);
e=ones(m,1);

Aineq=[A;-A];
bineq=[(B+b*D).*e ; -(B-b*D).*e];

x=quadprog(L,zeros(n,1),Aineq,bineq);

[Perry Mason]

That’s a nice reformulation, but Does it really hold with the inf norm?

[Matt J]

@Perry Mason you mean you don't see the equivalence of the linear constraints with the inf norm constraints?

[Perry Mason]

I meant that the constraint in the original problem of javm6 is defined in terms of the infinity norm.

I reckon your reformulation works if the constraint is defined using euclidean norm, but will this reformulation still work even if the infinity norm is used?

Cheers

[Matt J]

The reformulation is equivalent to the original, posted problem with the inf-norm constraints. The inf-norm constraint in the can be re-written as the linear system,

-b*D <= A*x-B <= b*D

or

 A*x <=  B+b*D
-A*x <= -(B-b*D)