Searching for max-min in MATLAB

Question

I am writing a matlab code where i calculate the max-min. I am using matlab's "fminimax" to solve the following problem:

 ki=G(i,:);
 ki(i)=0;
 fs(i)=-((G(i,i)*pt(i)+sum(ki.*pt)+C1)-(C2*(sum(ki.*pt)+C1)));

G: is a system matrix. pt: is the optimization variable.

When the actual system matrix is used, the "fminimax" stops after one iteration and returns the initial value of "pt", no matter what the initial value for "pt", ie no solution is found. (the initial value is defined as X0 in the documentation). The system has the following parameters: G is in the order of e-11, pt is in the order of e-1, and c1 is in the order of e-14.
when i try a randomly generated test matrix and different parameters, the "fminimax" finds a solution for the problem, and everything works fine. G in order of e-2, pt in order of e-2, c1 is in the order of e-7.
I tried to scale the actual system: "fminimax" lasted more than one iteration, however, it still returned the initial value of pt, ie it couldn't find a solution.
I tried to change the tolerance of the "fminmax", using "options" [StepTolerance, OptimalityTolerance, ConstraintTolerance, and functiontolerance]. There were no impact at all. still no solution.

I thought that the problem might be that the precision of "fminimax" is not that high, or it is not suitable to solve the problem. i think it is also slow.
i downloaded CPLX, and i wanted to transform the max-min problem into linear programing, using a method i found in a book. However, when i tried my code on a simple minimax it didn't give the same solution.

I thought of using CVX for example, but the problem is not convex.

What might be the problem?
PS the system matrix, G, has different realizations, i tried some of them. However, the "fminimax" responds in the same way for all of them, ie it wasn't able to find an adequate solution.

Answer 1

I am not convinced that the optimization solvers are broken. If the problem is nonconvex, then there can be multiple local minimizers. Given the information you have provided, we have no way of knowing whether you started at an initial condition.

The first place you need to start is by getting more information from the optimization exit condition... Did it finish because it hit the iteration limit? (I hope not since it isn't doing many iterations)... Did it finish because a tolerance was hit (eg the function did not change by more than xxxx)? Or perhaps it could not find a feasible solution? (I don't know if you have any constraints that need to be met).

More than likely, I wold guess that you are starting at a local minimizer without realizing it. So you need to determine whether you are indeed at a local minimizer by looking at the jacobian of the function evaluated at your initial guess. Either calculate it analytically or use a finite step approximation....