# Help with constrained or regularized optimization problem involving variable matrices and powers of matrices (or perhaps matrix logarithms)

I am attempting to solve the following optimization problem:

$$\small\min_{A,B,C} \| Y_A - AX_A \|_F + \| Y_B - BX_B \|_F + \| Y_C - CX_C \|_F + \lambda_1 \|B - A^2\|_F + \lambda_2 \|C - A^4\|_F$$

where $$Y_i, X_i$$ are $$n \times m$$ known data matrices, $$A, B, C$$ are $$n \times n$$ unknown matrices (decision variables), and $$\| \cdot \|_F$$ denotes the Frobenius norm. I've attempted to use cvxpy to solve this program, however it seems like the operation $$\|B - A^2\|_F$$ does not follow disciplined convex programming (DCP) rules. Is anyone able to help out with how I might solve this optimization problem? I'm kind of assuming that taking matrix powers is not the right approach, but I don't know how else to formulate the problem. If anyone can help me solve the problem as is, that would be much appreciated.

• I believe your formulation is non-convex, hence it can not be reformulated per DCP rules. What are you trying to accomplish with the penalty terms involving $A^2$ and $A^4$? Are you trying to keep $C$ and $B$ close to $A$? If so, why the powers on $A$? The title mentions matrix logarithm. If you want to use matrix logarithms for the penalty terms, and want to make a computationally intensive, but perhaps "satisfying": formulation, you could use symmetrized quantum relative entropy for the penalty terms, which can be easily formulated using CVXQUAD with CVX under MATLAB. Mar 25 '20 at 1:46
• @MarkL.Stone, does taking matrix powers somehow make this non-convex? That doesn't seem to be the case after looking at the expression $A^2$ in cvxpy, but I don't know why it's non-convex. Also, yes I am trying to enforce that $B$ be close to $A^2$ and similarly for $C$. I guess it's also not clear to me how matrix logarithms work and if it will remain convex after doing that. But yes, I was thinking I could take the matrix logarithm and then I would avoid explicitly taking matrix powers. Mar 25 '20 at 2:06
• Yes the matrix powers "somehow" make it non-convex. You're still too vague on what you're really trying to do. Are penalty terms involving C-A and B-A not adequate? Mar 25 '20 at 12:04
• As for matrix logarithms, quantum relative entropy, etc., in order to keep problem convex, would need argument sot be positive semidefinite. I presume that would not be the case. Mar 25 '20 at 14:21
• I want $B$ to be close to $A^2$ and $C$ be close to $A^4$. Mar 26 '20 at 5:21