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You have a bilevel optimization problem. Usually, these problems are quite hard. Even if the lower level problem and the upper level problems are convex, the full problem may be NP hard. However, you …

A lot of things are known for the convergence of alternating projections for these convex feasibility problems. I suggest to start with H.H. Bauschke and J.M. Borwein: On projection algorithms for …

I think the claim is true under some additional assumption: Let $\bar x$ be another solution of $$\min_x f(x) ~~ s.t.\\ g_{t^*}(x)\leq0 ~ $$ for which some constraints are not fulfilled, i.e. $g_{\bar …

First things first: Be aware that global minima may be out of reach. Here are two possibilities that come to mind: If you have differentiability, you may use projected gradient descent: Start with an …

There is also Inverse problems in spacetime I: Inverse problems for Einstein equations - Extended preprint version, by Yaroslav Kurylev, Matti Lassas, Gunther Uhlmann and related papers. Sound …

I deleted a previous wrong and misleading answer. The Moreau-Yoshida envelope is a special case of th infimal convolution of two convex functions $f$ and $g$ which is defined as $$ f\Box g(x) = \inf_{ …

A bit too long for a comment: Let's consider a optimization problem of the form $$ \min_x f(x)\quad \text{s.t.}\quad x\in C. $$ If we now consider "symmetry" a bit more abstract by saying that you ha …

I am not sure if there is a standard name for these type of problems. I would call problems of this type sparse approximation problems because you want to solve a linear equation approximately (assumi …

This work like this: The $\infty$-norm constraints are straigtforward. In the first problem you write $$ -1 \leq x_i \leq 1 $$ or, more explicitely $$ x_i\leq 1\\-x_i\leq 1. $$ One could even just wr …

I would guess that the method is going to converge (weakly), even with constant stepsizes. Off the top of my head I don't know a precise reference. The method is close in spirit to the "hybrid project …

Yes, this is true and can found, for example in Chapter 7 of the "Handbook of Discrete and Computational Geometry" edited by Csaba D. Toth, Joseph O'Rourke, Jacob E. Goodman (chapter by Alexander Barv …

I second Czenek's recommendation. Moreover, it could be helpful to know this cute little characterization of invexity (due to Craven and Grower, see also here): Theorem: A differentiable function $f$ …

If you consider general linear programming problems and the solution in dependence on changes in the right hand side, you want to look for sensitivity analysis in linear programming and more specifica …

As a shameless plug for my own work: As you want to use the $\infty$-norm as a stopping criterion, you may be interested in homotopy methods. For the sake of completeness, assume that you want to find …

The specific problem you gave has the property that for every given data $(A,b,y)$ there is exactly one $x$ that solves the problem. Hence, there a solution map $(A,b,y) \mapsto x$ mapping $\mathbb{R} …