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The application of proximal gradient to this exact problem is considered in (equations (2) and (35) in) the Uncertainty in Artificial Intelligence (UAI) 2019 paper Fast Proximal Gradient Descent for A …

The problem is convex and is a Second Order Cone problem (SOCP) for $\beta \le 0.5$. The statement that the problem is convex for $\beta > 0.5$ is incorrect. There is no closed-form solution. If $\bet …

@Arian has supplied the solution for $k = 1$. It is a known result that for $k = n$, the desired convex hull is the set of all $n$ by $n$ doubly stochastic matrices. I offer this conjecture for the …

f(x) is increasing from 0 1o 1, where it reaches its global maximum, and decreasing from 1 to $\infty$. It is concave from 0 to 2.00841388..., and convex beyond that (note that the 2nd derivative of f …

The derivation of this dual is provided in example 8.8 "Sum of singular values revisited" of section 8.6 "Semidefinite duality and LMIs" of Mosek Modeling Cookbook 3.2.1.

This is a partial answer, which addresses the practicalities of and workarounds for solving convex optimization problems not satisfying Slater's condition. It does not address the existence of a polyn …

If $\nabla g$ and $\nabla h$ are linearly independent at a given point, then the Linear Independence Constraint Qualification (LICQ) is satisfied, and presuming that $f$, $g$, and $h$ are all continuo …

Yes, this is convex because the objective function and all constraints are convex. The objective function is affine (linear), which is convex. The semidefinjite constraint on X is convex. The trace e …

Solve the unconstrained least squares problem in "one-shot", for example by QR or SVD (if not too big), if you consider that to be "one-shot". Then if the optimal $x$ to the unconstrained least squar …

Unlike textbook examples, most optimization problems don't have closed form solutions for the KKT conditions. I believe the inequality constraint renders this such a case. The projection you seek can …

The matrix variable $X$ only appears linearly (affinely), so this can be formulated and solved as a (convex) Linear Semidefinite optimization feasibility problem. Unless the solver runs into numerical …

$f(x)$ is not convex. Here is a counterexample to its convexity in MATLAB notation. C = [2 1;1 2] x1 = [1 2]' x2 = [2 1]' x3 = 0.5*(x1 + x2) Then f(x1) = f(x2) = 8 f(x3) = 9 > 0.5*(f(x1) + f(x2))

Minimizing a concave function subject to convex constraints is Concave Programming. If the constraints of a Concave Programming problem are compact, as in your example, there must be a global optimum …

The Extended Monotropic Programming problem deals with a more general extension to all n variables at a time than the two variables at a time extension you are interested in, except that the literatur …

This is easy to formulate and solve (presuming it's not too large or otherwise unpleasant) in CVX or YALMIP. CVX: cvx_begin variable X(length(a),length(a)) semidefinite minimize(norm(a*X-b)) cvx_end …