Group Constraints for FMINCON

Hi all! I'm writing a model that needs to minimize 6 variables through FMINCON: o, a, b, w, kappa, gamma. The variable options is defined as follows:

optimset(optimset(@fmincon),'Diagnostics','off','Display','off','LargeScale','off','MaxSQPIter',1000,'TolFun',1e-6)

The variables are already subject to the following lower/upper boundaries:

• o, a, b, w between 0 and Inf;
• kappa, gamma strictly positive between (2 * options.TolCon) and Inf;

I need to ensure a few constraints are respected:

• o, a, b, w must approximately sum to 1;
• kappa * gamma must be approximately equal to 1;

If I didn't have to consider the last two variables, I would have probably used A and b parameters of FMINCON as follows:

A = [-eye(3); ones(1,3)];
b = [(zeros(3,1) + (2 * options.TolCon)); (1 - (2 * options.TolCon))];

But going for that approach with two distinct constraints (an additive one and a multiplicative one) is pretty weird and I really have no clue about how to set A and b.

It seems that the nonlcon parameter may be what I'm looking for, but it's unclear to me how to formulate it properly.

Thanks!

Answer 1

[John D'Errico]

Approximately equals is NOT an equality constraint.

But you can write it as TWO inequality constraints. So if you want kappa*gamm approximately equal to 1, then that means you want the product to be within some given tolerance of 1. That is:

kappa*gamm >= 1 - tol

kappa*gamm <= 1 + tol

You can then swap the inequality direction on the first constraint by multiplying by -1.

-kappa*gamm <= -(1 - tol)

Note my use of gamm as a variable name, instead of gamma. Since there is a function called gamma that is often quite useful, I strongly suggest not using a variable named gamma.

[Tommaso Belluzzo]

Thanks for your answer. May I kindly as you how I can implement this on the point of view of Matlab code?

I think I have to write a nonlcon function like this?

function [c,ceq] = my_nonlcon(x,options)
    kappa = x(5);
    gamm = x(6);
    kg = kappa * gamm;

    c1 = kg - (1 - (2 * options.TolCon));
    c2 = kg - (1 + (2 * options.TolCon));
    
    c = [c1; c2];
    ceq = [];
end

Together with A and b contraints formulated as follows:

A = [[-eye(3) zeros(3)]; [ones(1,4) zeros(1,2)]; zeros(2,6)];
b = [(zeros(3,1) + (2 * options.TolCon)); (1 - (2 * options.TolCon)); zeros(2,1)];

Am I right?