weighted sum of matrices optimization

Question

I'm beginner in optimization and welcome any guide in this field.

I have 15 matrices (i.e., Di of size (n*m)) and want to find best weights (i.e., wi) for weighted averaging them and make a better matrix that is more similar to one given matrix (i.e., Dt).

In fact my objective function is like it:

min [norm2(sum(wi * Di) - Dt) + norm2(W)]
for i=1 ... 15    s.t. sum(wi) = 1 , wi >= 0

How can I optimize this function in Matlab?

Answer 1

You are describing a simple Quadratic programming, that can be easily optimized using Matlab's quadprog.

Here how it goes:

You objective function is [norm2(sum(wi * Di) - Dt) + norm2(W)] subject to some linear constraints on w. Let's re-write it using some simplified notations. Let w be a 15-by-1 vector of unknowns. Let D be an n*m-by-15 matrix (each column is one of the Di matrices you have - written as a single column), and Dt is a n*m-by-1 vector (same as your Dt but written as a column vector). Now some linear algebra (using the fact that ||x||^2 = x'*x and that argmin x is equivalent to argmin x^2)

[norm2(sum(wi * Di) - Dt)^2 + norm2(W)^2] =
(D*w-Dt)'*(D*w-Dt) + w'*w =
w'D'Dw - 2w'D'Dt + Dt'Dt + w'w =
w'(D'D+I)w - 2w'D'Dt + Dt'Dt

The last term Dt'Dt is constant w.r.t w and therefore can be discarded during minimization, leaving you with

H = 2*(D'*D+eye(15));
f = -2*Dt'*D;

As for the constraint sum(w)=1, this can easily be defined by

Aeq = ones(1,15);
beq = 1;

And a lower bound lb = zeros(15,1) will ensure that all w_i>=0.

And the quadratic optimization:

w = quadprog( H, f, [], [], Aeq, beq, lb );

Should do the trick for you!