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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). $ …

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) …

One approach is to solve the optimization problem: \begin{equation*} \min_x\quad \|Ax-y\|_\infty. \end{equation*} This is a nonsmooth optimization problem, but is amenable to a variety of scalable opt …

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 …

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 …

For your problem, where the relations between objects are specified via a distance matrix, the formulation of Correlation Clustering, seems to be more appropriate. Here you do not need to pick $k$ in …

This function is not differentiable (consider $A=0$). If you are interested in learning about its subdifferential (and more on subdifferential of spectral functions), please refer to the excellent pap …

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 …

To expand on my comment (and given the update by the OP), it is clear that $R=UP^T$ (where $A=PDP^T$ and $B=ULU^T$) maximizes the trace. This follows because $\text{tr}(RAR^TB) \le \langle\lambda^\dow …

As far as I know, there is no "analytic" solution to your problem. Fortunately, this problem happens to be a special case of optimizing (over $\mathbb{C}^n$) a quadratic function subject to two quadra …

Here is one way to solve your problem numerically using CVX under Matlab: % assuming we have defined v and m above cvx_begin sdp variable d(2*m,2*m) diagonal maximize det_rootn(v*d*v') su …

Here is a simple argument. First, define $r_t = \|x_t - x^\ast\|$, where $x^\ast$ denotes an optimal point. Since $f$ is convex, we have \begin{equation*} f(x^\ast) \ge f(x_t) + \langle f'(x_t), x …

Here are a few references that should help you get started in the area of approximating determinants: Approximation of the determinant of large sparse symmetric positive definite matrices Determinan …

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 …