How to express a product Ax where A and x are both cvx affine expressions?

Nonconvex
1

Dear CVX community,

This is my optimization program:
$$ \underset{C,\alpha}{\text{min}}|{C}|_1 + |{\alpha}|_1 \quad s.t. \begin{array}{l}
\quad |y-HCW\alpha|_2 < \delta \ \quad diag(C) = 0 \end{array} $$

I have written a function to compute HCW\alpha. This function first computes W\alpha to obtain a vector, say x_{est}, then is performed the product Cx_{est} to finally multiply H by Cx_{est}. Due the nature of my model is necessary to perform those transformations in that order.

When the function is about to perform the product Cx_{est}, where both C (a 256x256 matrix) and x_{est} (a 256x1 vector) are cvx real affine expressions, the following message is delivered:

??? Error using ==> cvx.mtimes at 126

Disciplined convex programming error:
Only scalar quadratic forms can be specified in CVX

I do not figure out how to rewrite this product as a valid convex expression. Any hint would be a great help. Thanks.

2

Why don’t you lay out more clearly what your optimization model is? What are your optimization variables? What is the mathematical constraint (?) you’re trying to specify?

3

It is done Mark, I hope that in this way my question is going to be clearer. Thanks.

4

CWα is not convex (unless W is the zero matrix). In the 1 by 1 case (for C and α), the Hessian can readily be seen to be indefinite. So rewriting will not get your model to be accepted by cvx, unless the model itself is changed.

5

Did you prove that your model is convex before even attempting this? There is a reason CVX does not allow most variable-variable products: they are rarely convex. If you do prove it, then you will likely show yourself how to represent it in CVX: because CVX’s rules mirror common proof steps!