One of the famous problem in SDP is the matrix norm minimization (see S. Boyd, *Convex Optimization*, p. 170).

Consider:

\begin{equation}\label{eq:Lasse} \begin{aligned} &\min_{\mathbf{x}} & & \|A(x)-M\|_2 \\ & & & A(x)=-A(x)^T \end{aligned} \end{equation}

Here

- $x\in \mathbb{R}^n$
- $A(x)=x_1A_1+\cdots+x_nA_n$, with $A_i\in \mathbb{R}^{n\times n}$ and $A_i=-A_i^T$. So we consider $A_i$ are skew-symmetric. We also assume each column of $A(x)$, $A_i(x)$, $\|A_i\|_2=1$.
- $M\in \mathbb{R}^{n\times n}$ is given.
- Here we consider the spectral norm of matrices.

So this SDP finds the optimal solution $x$ to minimize a particular metric between $A(x)$ and $M$ and this metric is quantified by $\|A(x)-M\|$. Here we suppose $x^*$ is the optimal solution.

My question is that is $x^*$ the optimization solution of the following problem?

\begin{equation} \begin{aligned} &\max_{\mathbf{x}} & & \langle A(x), M\rangle \\ &\text{ s.t.} & & A(x)=-A(x)^T \end{aligned} \end{equation}

The motivation for asking this problem is from the fact that in the vector case (assume $\|c\|_2=1$)

\begin{equation} \begin{aligned} &\min_{\mathbf{x}, \|x\|_2=1} & & \|x-c\|_2 \end{aligned} \end{equation}

the optimal solution is $x^*=c/\|c\|_2$. And $x^*$ is also the solution of the following problem

\begin{equation} \begin{aligned} &\max_{\mathbf{x}, \|x\|_2=1} & & \langle x, c\rangle. \end{aligned} \end{equation}

So not sure if it also fits to the case of matrices. Any reference or papers are welcome.

Also if it is not the same for spectral norm, will it be the same for Frobenius norm? why and why not?

Sincerely appreciate your help.