Multiplication of very large 2D and 3D matrices

I have an optimization problem with 2D variable matrix and 3D cost matrix. I need to minimize the multiplication of these two matrices*.

Variable matrix, V, is 1500x1500 and consists of only 0 and 1. Cost matrix, c, is 1500x1500x1M. V consists of origin-destination pairs. c consists of distances between pairs for every user (1M).

I have 2 problems:

1. c is extremely large. So, it is impossible to keep it in memory.
2. How can I multiply these two matrices in an efficient way?

*In brief, the variable matrix (1500x1500) should be constructed such that sum of the multiplication of it with every dimension of the cost matrix (1500x1500) is minimized.

[Mitchell Thurston]

This is what I would try:

result = zeros(1500);

for i = 1:1500
    for k = 1:1500
        if (V(i,k))
            result(i,k) = sum(C(k,i,:));
        end
    end
end

I'm not positive on your index notation though.