Search Results

Fix $a \in \mathbb R^n$ and let $\|\cdot\|$ be any norm on $\mathbb R$ (e.g $\ell_1$ norm). For any $a \in \mathbb R^n$, it is clear that the function $f_a(x) := \|x-a\|_2 + \|x\|$ is strictly convex …

Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$). Question. …

Let $\Delta_n := \{x \in \mathbb{R}^n | x \ge 0, \sum_{1 \le i \le n}x_i = 1\}$ be the $n$-simplex. For $a, b \in \mathbb R^n$, with $\Delta_n \not \ni a$, consider the problem of computing the follo …

Let $H$ be a Hilbert space (e.g Euclidean $\mathbb R^n$), and fix a proper convex function $f:H \to (-\infty,+\infty]$. Given any $t \ge 0$, let $P_t:H \to H$ be the proximal operator of $f$ at level …

Let $C$ be a convex compact subset of a finite-dimensional normed vector space and let $f:C \to \mathbb R$ be strictly convex and uniformly continuous function (e.g it is sufficient that $f$ be Lipsch …

Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set $$ \alpha := \sup_{(x,y) \in C} ax + b y. $$ Question. In …

Studying some problems arising in decision-making under model uncertainty, I'm led to consider the following problems. Let $\mathbb E_P$ and $\mathbb V_P$ denote the expectation and variance operator …

Let $\mu \in \mathbb R^n$ and let $\Sigma$ be a positive-definite matrix of order $n \ge 2$. Fix $t \ge 0$ and define $\alpha(\mu,\Sigma,t) > 0$ by $$ \alpha(\mu,\Sigma,t) := \sup_{\|w\| = 1}\frac{1}{ …

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 fun …

Let $E$ be a large sparse $l$-by-$n$ matrix ($l$ and $n$ can be in the billions...) with coefficients in $\{-1, 0, 1\}$: the first row of $E$ is the vector $(1,0,0,\ldots,0) \in \mathbb R^n$, and ever …

Let $Z$ be a random variable with finite mean $\mathbb E[Z]$ and let $\phi:(-\infty,\infty) \rightarrow (-\infty,\infty]$ be a convex l.s.c function which is differentiable at $z=1$ with gradient $\ph …

Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds. Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \mathbb R^m$ …

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) = …

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here). …

Let $A$, $B$, and $C$ be centrally-symmeric convex bodies in $\mathbb R^n$. Note that any such set can such set induces a norm $\|\cdot\|_C$ on $\mathbb R^n$ defined by $\|x\|_C := \sup_{c \in C}c^\to …