Formulating minimization problem in matlab with three dimensional matrix?

Nonconvex
1 
    mf = 383;
numclus = 5;
clear cvx;
cvx_begin quiet
    variables U(mf,mf,numclus) lambda(1)
    expressions S(1)
    S = 0;
    for i = 1:mf-1
        for j = 1:mf-1
            for k = 1:numclus
                S = S + U(i,j,k)*E(i,j,k) + lambda*((U(i,j+1,k)-U(i,j,k))^2 + (U(i+1,j,k)-U(i,j,k))^2);
            end
        end
    end
    minimize(S)
    subject to
        sum(U,3) == 1
        U >= 0  
        lambda >= 0
cvx_end

Above is my attempt at a minimization problem. The first term in the sum is a distance minimizer with E being a matrix of values that I calculated prior to running cvx. The later term is like a “gradient” for U. The error that Matlab is spitting out is:

Disciplined convex programming error:
Cannot perform the operation: {real affine} .* {convex}

However, all the values I am multiplying are not vectors? I am not sure whats going on here. I know the code is not very efficient, I am still trying to figure out how to write everything in matrix notation with the use of for loops but this is the problem I have now. Please any feedback is appreciated! Am very desperate!

As a side note, these matrices are pretty big and I was wondering if there was a way for cvx to stop after a certain amount of time or to have it spit out how long its taking (using tic and toc?)

2

I believe the issue here is that lambda*((U(i,j+1,k)-U(i,j,k))^2 is not jointly convex in lambda and U. CVX is only for solving convex problems. See the CVX user guide for a list of acceptable expressions.

3

ah, good to know. I just removed that part from my response.