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Analytic formula for minimizer of $f(x) := \sqrt{(x-a)^\top S(x-a)}+ r \|x\|_2$

Let $S$ be a positive-definite $n \times n$ matrix and define $\|z\|_S := \sqrt{x^\top S x}$ for any $x \in \mathbb R^n$. Let $a$ be a fixed vector in $\mathbb R^n$ and $r \ge 0$, and consider the ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
189 views

Analytic expression for the Moreau envelope of $x \mapsto \|Ax\|$

Given an $m \times n$ matrix $A$ and a vector $c \in \mathbb R^n$, define $\eta(A,c) \ge 0$, $$ \eta(A,c) := \sup_{u \in \mathbb R^n} u^\top c - \frac{1}{2}\|u\|^2 - \|Au\|. $$ Note that $\eta(A,c) = \...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
1k views

$\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by $$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$ whereas the nuclear norm is ...
Santiago Armstrong's user avatar
1 vote
0 answers
139 views

Subgradient chain rule

Suppose $$F:\mathbb{R}^n \to \mathbb{R},\; F(x)=\mathrm{max}_\mathrm{eig}(C-\mbox{diag}(x)).$$ I am trying to find a subgradient of $F$ at $x_0$. A subgradient of $\mathrm{max}_\mathrm{eig}$ is given ...
Stephen T.'s user avatar
-1 votes
1 answer
137 views

Does a half plane contain intersection of some other half planes? [closed]

I'm doing research in Optimization and I have found this obstacle in the way. If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
Nothing's user avatar
  • 19
1 vote
1 answer
291 views

Norm of solution of quadratic program

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^* ...
Conner DiPaolo's user avatar
4 votes
2 answers
604 views

A certain type of quadratic constrained quadratic program (QCQP)

Let $P_1$, $P_2$ be two Hermitian matrices. Can anyone comment on the following QCQP? $$\begin{array}{ll} \text{minimize} & z^{H} z\\ \text{subject to} & z^{H} P_1 z +1 \leq 0\\ & z^{H} ...
dineshdileep's user avatar
  • 1,421
2 votes
2 answers
2k views

Hessian of function of covariance matrices

Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$, $$ \mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...
liubenyuan's user avatar
2 votes
1 answer
1k views

Subgradient of Minimum Eigenvalue

Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function \begin{align} f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2) \end{align} where $\lambda_{\text{min}}$ ...
dineshdileep's user avatar
  • 1,421
1 vote
2 answers
229 views

Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints

Let $C_1$,$C_2$,...$C_N$ be $M \times M$ indefinite hermitian matrices. What can we say about the following quadratic constriants \begin{align} w^{H}C_1w>0 \\\ w^{H}C_2w>0 \\\ ...~~~~~~~~~~ \\\ ....
dineshdileep's user avatar
  • 1,421
-1 votes
1 answer
175 views

Regularized Gradient with respect to a matrix (with a specific structure)

Suppose we have a typical logdet function $\mathcal{L}$ $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q}, $$ where $\...
liubenyuan's user avatar