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.

  • $\begingroup$ 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. $\endgroup$ – Mark L. Stone Mar 25 '20 at 1:46
  • $\begingroup$ @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. $\endgroup$ – ARandomName Mar 25 '20 at 2:06
  • $\begingroup$ 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? $\endgroup$ – Mark L. Stone Mar 25 '20 at 12:04
  • $\begingroup$ 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. $\endgroup$ – Mark L. Stone Mar 25 '20 at 14:21
  • $\begingroup$ I want $B$ to be close to $A^2$ and $C$ be close to $A^4$. $\endgroup$ – ARandomName Mar 26 '20 at 5:21

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