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This problem arises when minimizing the operator equation $X P X^* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$. The (reduced) task: Given $P$ and …

The following was resolved through personal communication, but I thought I'd post it here in case anyone has a similar problem in the future. I'm going to assume $P$ is nonsingular (strictly positi …

I'm looking specifically at the optimization problem $$ \begin{align*} \text{maximize: }& n - \frac{\|\lambda\|_1^2}{\|\lambda\|_2^2}\\ \text{subj. to: }& \lambda \succeq \epsilon\mathbf{1} \end{align …

$\newcommand\xx{\mathbf{x}}\newcommand\yy{\mathbf{y}}\newcommand\A{\mathbf{A}}\newcommand\aalpha{\boldsymbol{\alpha}}\newcommand\bbeta{\boldsymbol{\beta}}\newcommand\E{\mathbb{E}}\newcommand\inner[1]{ …

Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$ $$ …

In a quadratic program (QP), do linear equality constraints always reduce the norm of the minimizer? Specifically, let $P \succ 0$, $A \in \mathsf{M}_{m\times n}$ and $q\in\mathbb{R}^n$. Define $$x^* …

We can show more, namely that if $K$ is a closed convex set (such as $\ker A$) containing the origin and \begin{align*} x_c^* &= \operatorname*{argmin}_x \,\frac{1}{2}x^\mathsf{T} P x - q^\mathsf{T}x\ …

A more general version of the result Beenakker cites that allows for normal $B$ (not just symmetric):$\newcommand\tr{\operatorname{tr}}\newcommand\E{\mathbb{E}}$ Take $x\sim\mathcal{N}(0,I)$ and reca …