MATLAB: Optimization Problem: How to minimize a sum of matrices multipled by a vector

Question

I have m NxN matrices called A1, A2, …, Am. Their entries are known.

I also have an Nx1 vector called X. Its entries are known.

I have an Nx1 vector called Y. Its entries are known.

I want to solve the following minimization problem for the scalars a1, a2, …, am:

MINIMIZE: (a1.A1 + a2.A2 + a3.A3 + …. + am.Am)X – Y

this equals: minimize (TX-Y) where T is an NxN matrix, X is Nx1, and Y is Nx1

How do I set up this problem to solve it in matlab? I tried looking in the optimization toolbox, but I failed to set up lsqlin in a way that solves this.

Answer 1

So the question is, how to construct T? It should be as simple as,

T=reshape(  vertcat(A1,A2,..,Am)*X  , N , [] );

Now you can just go and plug into LSQLIN.