Need help for a nonlinear constraint optimization problem

Dear all,

I am going to solve the following nonlinear constraint optimization problem using MATLAB. I was wondering if you could please help me how to do that,

minimize -trace(W^-1)
subject to
          (A-B1*k)W+W*(A-B1*k)'=-B2*B2';
          sqrt(trace(C'*W*C)) <= 1;
          max(eig(A-B1*k)) <= -1;

A, B1 and B2 are known matrices and k is the unknown 2*2 matrix. (W is also a 2*2 matrix).

I wrote the following code but got error,

function [c,ceq] = constr(k)

c(1) = (A-B1*k)*W+W*(A-B1*k)'+B2*B2';
c(2) = sqrt(trace(C'*W*C))-1;
c(3) = max(eig(A - B1*k))+1;
ceq = [];
end

clc
clear
close all

A = [-1 0
     0 -2];
B1 = eye(2);
B2 = eye(2);
C = eye(2);
g2 = 3; g3 = -1;

x0=[1;1];
A=[];
b=[];
Aeq=[];
beq=[];
lb=[];
ub=[];

nonlcon = @constr;
fun = @(k)(-trace(inv(W)));
k = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

Thanks a lot.

[John D'Errico]

Nowhere in this code do you show how W is a function of k. Yes, you have written this:

fun = @(k)(-trace(inv(W)));

But that just uses some fixed matrix W, and ignores the value of k.

You cannot minimize a constant. Well you can try, but it will be rather boring.

[Mohammad]

Actually I am going to find k such that -trace(inv(W)) which is a function of k minimized.

[Torsten]

You will have to define 8 unknown variables (assuming W and k are arbitrary 2x2 matrices).

Let the first 4 of them form the matrix W and the last 4 of them form the matrix k:

W=[x(1) x(2);x(3) x(4)];
k=[x(5) x(6);x(7) x(8)];

Now your code has to work on this 8-element vector x, e.g.

function [c,ceq] = constr(x)

W=[x(1) x(2);x(3) x(4)];
k=[x(5) x(6);x(7) x(8)];
mat = (A-B1*k)*W+W*(A-B1*k)'+B2*B2';
ceq(1) = mat(1,1);
ceq(2) = mat(2,1);
ceq(3) = mat(1,2);
ceq(4) = mat(2,2);

c(1) = sqrt(trace(C'*W*C))-1;
c(2) = max(eig(A - B1*k))+1;

end

... although I ask myself how you want to prevent A-B1*k to have complex eigenvalues.

Best wishes

Torsten.

[Mohammad]

Thanks!

but I am getting this error,

Index exceeds matrix dimensions.

Error in constr (line 6) W = [x(5) x(6);x(7) x(8)];

[Torsten]

Your x0-vector contains 8 elements now ?

Best wishes

Torsten.

[Mohammad]

Got it. Thanks!