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Here is a partial solution to the first question in the original post. Let's look at the equation \begin{equation}\label{1}\tag{1} \sum\nolimits_{i=1}^m (X+ \Theta_i)^{-1} = Q. \end{equation} Lemma …

The said claim follows from the following general result on elementary symmetric polynomials, denoted $e_k$ below. $\newcommand{\vx}{\mathbf{x}}\newcommand{\vy}{\mathbf{y}}$ Theorem A (S. 2018). $ …

The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational Math …

The following Matlab code suggests that the function fails to be concave in the regime of interest ($0\le T\le 0.4$, $E, W \ge 0$, $p \ge 20$, $W\ge E$. f = @(E,W,T,p) (p*(W+1.000000000*10^5*W^.7*(.1 …

I have not given full thought to whether the algorithm will converge under the hypotheses placed on the intermediate values. However, it is worth noting here (perhaps the OP is already aware of this) …

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 …

Here is a very low-brow answer to the original question. Consider the lower-triangular matrix \begin{equation*} V = [V_{ij}] = \left[\binom{i-1}{j-1}\right]\quad \text{for}\quad i \ge j. \end{equatio …

Yes, even verifying local optimality is hard. Check out this famous paper of Murty and Kabadi, which considers NP-completeness in nonlinear programming, and in particular discusses hardness of local …

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 …

From Theorem 9, of this article it follows that $A \mapsto \log\frac{M_i(A)}{\det(A)}$ is convex on the set of positive definite matrices. The alleged convexity in the OP is a simple consequence of th …

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 …

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 …

Your minimization problem is equivalent to \begin{equation*} \min_{R^TR=I}\quad\prod_{i=1}^p r_i^T\Sigma r_i, \end{equation*} and it can be shown (using Hadamard's determinant inequality and some more …

Here's an alternative way of proving (what Yoav already did), but using different notation. Let us first write the optimization problem in matrix form. First, define \begin{equation*} d = \begin{ …