There is a famous problem Optimization of Frobenius Norm and Nuclear Norm; however, this is not I want to ask (about proximal operator).
Suppose I have an easy optimization problem:
$$\min_Q \|Q-Q_N\|_F$$
where $\|\cdot\|_F$ is the Frobenius norm: $\|X\|_F = (\operatorname{tr}(X^TX))^{\frac{1}{2}}$.
We know we can consider the following:
$$\mathcal{A}(Q,t) = \begin{bmatrix}I & Q-Q_N\\ (Q-Q_N)^T & tI \end{bmatrix}\succeq 0$$
By Schur complement we have the following $$tI-(Q-Q_N)^T(Q-Q_N)\succeq 0$$
If $Q\in \mathbb{R}$, the above becomes $$t-\|Q-Q_N\|^2\geq 0 \Rightarrow t\geq \|Q-Q_N\|^2 \geq 0$$ So the original problem is equivalent to
\begin{align}
&\min_{t,Q} & &t \\
& s.t. & & \|Q-Q_N\|^2 \geq 0
\end{align} or
\begin{align}
&\min_{t,Q} & &t \\
& s.t. & & \mathcal{A}(t,Q)\succeq 0
\end{align}
The second formulation is a SDP
My question is: if $Q\in \mathbb{R}^{n\times n}$ how to obtain the above convex optimization problem (minimize over $t$) with $\|\cdot\|^2$ replaced by $\|\cdot\|_F^2$
Is there any method except the vectorization of matrices?
Best Answer
Are you asking for the problem
$$\text{minimize trace(X) subj. to . } \begin{bmatrix}I & Q-Q_N\\ (Q-Q_N)^T & X \end{bmatrix}\succeq 0$$
which minimizes the Frobenius norm of $Q-Q_N$?