How to rewrite the matrix optimization problem to a convex form?

Asked Feb 21 '21 at 03:25

Active Feb 21 '21 at 16:02

Viewed 120 times

Consider the following optimization problem:

\begin{equation} \begin{aligned} p^*=&\min_{\Omega} & & \|\Omega Q + Q\Omega^T+2M\| \\ &\text{ s.t.} & & \Omega = -\Omega^T \\ \end{aligned} \end{equation}

Note

$Q = qq^T$, where $\|q\|=1$ and $q\in\mathbb{R}^4$. So $Q\succeq0$.
$M\in\mathcal{S}^4$, i.e., it is symmetric.
$Q$, $M$ are given.
$\|\cdot\|$ can be spectral norm, or Frobenius norm.

So the matrix variable is $\Omega$, which is skew-symmetric.

This optimization problem is obvious not convex since $\Omega$ is not psd or pd. Is there any way to rewrite it as a convex optimization problem?

Thanks

Note:

This problem is from the Sylvester equation $$AX+XB=C,$$ with $X=\Omega$. However, here $2M\neq C$ and it does not meet the condition which guarantees the unique solution. So I want to find the best $\Omega$ such that $\|\cdot\|$ is minimized. Also there is a constraint on $\Omega$.

edited Feb 21 '21 at 16:02

asked Feb 21 '21 at 03:25

Denny

3

Why not? Objective is convex, constraint is affine. – A.Γ. Feb 21 '21 at 06:41
@A.Γ. Should the variable $\Omega$ be positive semidefinite s.t. the optimization becomes convex? – Denny Feb 21 '21 at 15:43
@RodrigodeAzevedo Sure, just edit it. – Denny Feb 21 '21 at 16:03
1

Have you considered using a Lagrange multiplier (matrix)? Take a look at this. – Rodrigo de Azevedo Feb 21 '21 at 17:34
2

No. Skew-symmetric matrices build a linear subspace as they satisfy $\Omega+\Omega^T=0$. You may prove straightforward that if $\Omega_1$ and $\Omega_2$ are skew-symmetric then its linear combination is skew-symmetric as well. – A.Γ. Feb 21 '21 at 19:25
Did you make progress? – Rodrigo de Azevedo Feb 24 '21 at 13:35

How to rewrite the matrix optimization problem to a convex form?

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