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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 …

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 …

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 \\ …

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 …

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 …

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 …

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 …

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).

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 …

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 …

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 …

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 …