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If you throw in positive homogeneity, then the first class of functions is what is called sublinear, see for instance Proposition 1.1.4 ("Fundamentals of Convex Analysis"; Hiriart-Urruty, Claude Lema …

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 …

Here is a partial answer to one of the questions. I may expand on the others if I find time (or hopefully someone else provides an answer). Without additional restrictions, the circle, sphere etc., t …

To my knowledge, this is the first place where I saw the result being claimed: C. O. Kiselman. Regularity classes for operations in convexity theory. Kodai Math. J. 15. 1992. In particular on the firs …

Unfortunately, the conjectured function is not concave. Here is a simple simpler counterexample. \begin{equation*} B = \begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix},\quad A = \begin{bmatrix} 2 & 0 \\ …

I don't see any reason why the two problems should have the same optimal solution. Consider both SDPs with $A(X)=X$, $B=[25\ \ {-10};\ {-10}\ \ 20]$, and $C=1$, and $m=1$. Let the matrices be $2\ti …

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 …

The function that you have is convex for unitarily invariant norms, but for the (basis dependent) elementwise absolute value, it can clearly break as a trivial counterexample below shows. \begin{equa …

Ok, after reading your comments, and some thinking, here is one way to tackle what seems to be going on: You have a nondifferentiable loss function. You wish to compute a subgradient of the loss, bu …

This is a standard result in convex analysis. See for example, $\S$3.2.5 of Convex Optimization by Boyd and Vandenberghe (just slightly modify their proof to conclude strictness).

A slightly more precise result is noted below. You may like it given that you tried Schur complements, and convexity arguments. Observation. $\DeclareMathOperator{tr}{tr}$ \begin{equation*} \m …

Yes, a lot can be said for your special case. Given the notation, I presume you are optimizing over $\mathbb{C}^n$. In this case, see Section 2 in the paper Strong Duality in Nonconvex Quadratic Optim …