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As noted in the comments, this problem is not really a research level problem. Afaik, versions of it were originally solved in the 50s. Here is an entire survey that discusses efficient algorithms (i …

One approach is to solve the optimization problem: \begin{equation*} \min_x\quad \|Ax-y\|_\infty. \end{equation*} This is a nonsmooth optimization problem, but is amenable to a variety of scalable opt …

Your problem is a special case of a Fractional Linear Program, so as such following the recipe provided on Wikipedia you should be able to solve it by using a reformulation to an equivalent linear pro …

This is indeed a classic problem. Recall the more general problem of computing the prox operator of an lsc convex function $f$, i.e., \begin{equation*} \text{prox}_f(y) := \operatorname{argmin}\quad \ …

The paper cited in my answer here provides a detailed proof of the two-block case of alternating minimization (block coordinate descent). In particular, as mentioned in my comment, the convergence fol …

Building on Nesterov's work, in his Ph.D thesis, Peter Richtarik considers first-order methods with relative error of approximation guarantees. I haven't looked in too closely, but I am sure that a la …

Unless I'm mistaken, the following argument provides a solution. Since the Frobenius norm is orthogonally invariant we can assume without loss of generality that $S$ is diagonal. I'll write $Q$ inste …

A nonempty set $S$ in a normed linear space $X$ is called a Chebyshev set if for each $u \in X$ there is exactly one nearest point in $S$ to $u$ (i.e., for a Chebyshev set, nearest points always exist …

Although not monotone at the operator level (as suggested by C. Mooney's proof), the monotonicity of prox-residual norms is known (probably you are already aware of it). Let $P_\eta^g$ denote the pr …

To some extent. Here's some relevant material where group theoretic objects show up in optimization (though a lot of it is convex algebraic geometry). Orbitopes Group majorization and a host of maj …

You are trying to solve what is known as a best approximation problem. von Neumann's alternating projections does not work here (as might have been perhaps suggested above) You can use Dykstra's pr …

If $C$ is positive semidefinite, then so is $\begin{bmatrix} C & C\\ C & C\end{bmatrix}$ for the simple reason that it is nothing but the Kronecker product of $\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatri …

One reference that I could locate is the following Minimum norm ellipsoids as a measure in high-cycle fatigue criteria by Nestor Zouain, presented at a conference in 2005 (see $\S5$ of that pdf). Howe …

A somewhat more general theorem is available (though not entirely suitable for consumption by a beginner, but maybe it's ok). See Theorem 15.4 in Convex Analysis, by R. T. Rockafellar. Thm.15.4 (R …

Maybe this is overkill, but I would recommend that you read the masterful paper: Optimizing the spectral radius, by Yurii Nesterov and Vladimir Protasov, 2012. Unless I'm mistaken, your problem is a …