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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 …

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 …

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 …

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 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 …

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 …

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 …

Rather than compute "the" Lipschitz constant, it is much easier to do a backtracking line search that roughly upper-bounds the Lipschitz constant. To get some ideas, on how one might do a simple line …

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 …

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 …

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 …

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 …

Here is a crude idea that might work (haven't thought too carefully about it). Let $a=\lambda_{\min}(B_1^{-1}A_1)$ and $b=\lambda_{\min}(B_2^{-1}A_2)$. Then, for there to be a feasible solution to th …

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{ …