Thank everybody in advance. I’d like to solve an optimization problem for a matrix function $ f(C)$ . However, the matrix pseudo-inverse constraint gives me big troubles.

Vectors $ V_{n\times 1}$ , $ F_{m\times 1}$ and matrix $ B_{m\times n}$ are known. Matrix $ C_{n\times m}$ is unknown. How to numerically solve: $ $ \min\limits_{E}f(C)=V^T CF $ $ $ $ {\rm subject\,\, to}: BC=I_{m\times m} \ m<n $ $ $ I_{m\times m}$ is an $ m\times m$ Identity matrix.

I know this problem can be (sort of, because non-square matrices) formulated into linear matrix inequality (LMI). Similar to: optimization of inverse matrix with constraint on matrix elements. But my processor does not allow such numerical method (LMI) to be used. Plus I do not know how to non-square LMI. So I am looking for a different approach to solve this problem.

I have an ugly gradient descent approach for this problem. But it is complicated and I do not like it. I will not post it for now to limit people’s thoughts. I will post it 4 days after the question.

In addition, I’m not sure if adding following constraint would make the problem easier: $ $ {\rm subject\,\,to}: -I_{n\times n}<(CF)^T I_{n\times n}<I_{n\times n} $ $

*Appendix* – Formulate problem to (sort of) LMI: **(May not be correct, since matrix is not sqaure)** Define invertible matrix $ E_{n\times n}$ , $ $ BC = BEE^{-1}C=(BE)(E^{-1}C)=I \ E^{-1}C = pinv(BE) \ C=E\,\,pinv(BE) $ $ Introducing $ E$ is to represent all possible $ C$ – parameterize $ C$ . Then the problem $ \min\limits_{E}f(C)=V^T CF$ is equivalent of finding the $ \min$ of scalar $ \alpha$ , that below inequality is feasible (i.e. some $ E$ exists) $ $ V^T CF < \alpha \ V^T E\,\,pinv(BE) F < \alpha $ $ With Schur complement, it is equivalent to following LMI $ $ \left[\begin{array}{cc} BE & F\ V^T E & \alpha \end{array} \right]<0 $ $