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14
votes
1answer
375 views

Minimize sum of $\ell_2$ norm and linear combination, on simplex

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 ...
7
votes
1answer
191 views

When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
7
votes
1answer
199 views

In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something. Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...
4
votes
1answer
74 views

Extremal Lipschitz convex functions

Let $B_d$ the unit ball in $\mathbb{R}^d$, and let $F_d$ be the set of convex functions with Lipschitz constant at most 1 from $B_d$ to $\mathbb{R}$. When $d=1$ (so the domain is the just the ...
1
vote
0answers
18 views

Concavity of maxima [closed]

Suppose we have the following optimization problem : $\min\limits_x kf(x) + g(x)$ where $f$ is a decreasing convex function in $x$ and $g$ is an increasing convex function. Can we say that $x^*$ is ...
0
votes
0answers
36 views

Bounds on the curvature of a sequence of convex functions

Let $\{f_n\}$ be a sequence of (real-valued) smooth convex functions on $[0,1]$, with $f_n(0) = f_n(1) = 0$ for all $n$. Let $t_n \in [0,1]$ be the minimizer of $f_n$ and assume that $M_n:= f_n(t_n) ...
2
votes
2answers
186 views

What is the dual of an semidefinitely representable (SDR) cone?

The Question Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product. Let $\mathcal K\subset V$ be a cone defined as $$ \mathcal K=\Big\{x\in V\ \...
3
votes
3answers
262 views

How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
6
votes
2answers
180 views

Gaussian and the convex hull of moment curves

Let $c_1,\dots, c_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c_1,\dots, c_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a ...
1
vote
0answers
43 views

Projecting on a convex compact polytope with special form

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 ...
1
vote
0answers
73 views

Projecting on a a special polyhedron

Let $X$ be an $n$-by-$p$ matrix and consider the closed convex polyhedron $$\mathcal P_X := \{y \in \mathbb R^n | \|X^Ty\|_\infty \le 1\}.$$ Notice that $\mathcal P_X$ is symmetric about the origin. ...
8
votes
2answers
199 views

Constructing an independent uniform random variable from two independent ones

Does there exist a continuous (differentiable) function $h:[0,1]\times [0,1] \to [0,1]$ such that if $\alpha,\beta\in [0,1]$ are independent and uniformly distributed on $[0,1]$, the random variable $...
1
vote
0answers
50 views

Derivatives of Minkowski function?

Let $A\subset \mathbb R^n$ and $M$ be the convex hull of the set $A$, e.g., $M:=Conv(A)$. The Minkowski function on $M$ is defined as follows \begin{align*} &f: \mathbb R^n \to \mathbb R\\ &f(...
0
votes
0answers
173 views

On the remarkable property of the concave function's level sets

Consider a smooth function $f(x) \colon \mathbb{R}^2_+ \to \mathbb{R}_+$ such that $f$ is concave and positively homogeneous of order one. Consider a linear transform $P$ given by matrix $$ P = \...
6
votes
0answers
93 views

orthogonal projector onto the set of convex functions

Let $\Omega\subset \mathbb R^d$ be an open, convex domain, and consider the Hilbert space $L^2(\Omega)$. Each sum of convex functions is convex, hence the subset $Conv(\Omega)$ of all convex functions ...
-1
votes
1answer
63 views

Convex Optimization in an Ellipsoid

Suppose we want to minimize a linear objective inside an ellipsoid that is, $\min _x l^Tx$ such that $(x - \mu)^TA(x - \mu) \leq \beta ^2$. Here, A is PSD and $\mu$ is a fixed vector. Can this be ...
5
votes
1answer
91 views

Two (new?) variants of convex functions

I find that the following two types of functions are useful to my research. (i) We know that a function $f: \mathbb{R}_+^m\rightarrow \mathbb{R}$ is called convex if for all ${\bf x,y}\in \mathbb{R}...
0
votes
0answers
65 views

Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that $\...
3
votes
1answer
126 views

Generalization of standard convex problem

In standard convex programming, the objective function and each of the constraint inequalities are convex. in such case, if the KKT condition hold for a point, and Slater condition is also hold for ...
1
vote
0answers
123 views

Proximal mapping of composition with linear operator

Let $A$ be an orthogonal matrix. Then the proximal mapping $prox_{f \circ A}(x)$ can be evaluated efficiently by $$ (I + \partial (f \circ A))^{-1}(x) = prox_{f \circ A}(x) = A^T prox_{f}(A x), $$ as ...
0
votes
0answers
53 views

A question about the approximation of convex cones

I have the following question which maybe is too naive. Let $K$ be a convex cone on $\mathbf{R}^n$. Can we approximate $K$ by a sequence of polyhedral convex cone $K_i$ such that for any compactly ...
3
votes
1answer
136 views

A family of convex bodies in Banach-Mazur position

Let $\{K_i\}$ be a family of smooth, origin-symmetric, strictly convex bodies such that $K_i$ converge in the Hausdorff distance (or you may assume $\partial K_i\to \partial K$ smoothly, in the sense ...
5
votes
1answer
165 views

Order between two completely monotone functions?

I am wondering if the following assertion is true: Let $f,g:\mathbb{R}_+\rightarrow [0,1]$ be completely monotone functions on $\mathbb{R}_+^*$, that is, $(-1)^n f^{(n)}(x)\geq 0$ and $(-1)^n g^{(n)}...
0
votes
1answer
56 views

How to compute the direction of slowest ascent from the minimum of a strongly convex function?

Consider a twice differentiable strongly convex function $f:\mathbb{R}^n \rightarrow \mathbb{R^+}$ that attains its minimum value at the point $x^*$. I am wondering if one can compute a direction of ...
1
vote
0answers
60 views

concavity of a vector function

I'm given a function $g:\mathbb{R}^n \mapsto \mathbb{R}$, $g(y) = \prod_{i\in[n]} (1+y_i\cdot c_i)$, where $c_i>0$. Let $e_a,e_b$ be two arbitrary standard vectors. It is easy to show that for any ...
2
votes
1answer
58 views

Bounding the difference in the value of a strongly convex function at its integer minimum and other integer points

I am currently working on a problem where I have to minimize a $m$-strongly convex function $$f ~: ~\mathbb{R}^n \rightarrow \mathbb{R}^+$$ over a bounded integer lattice, $$L = \mathbb{Z}^n \cap [-...
3
votes
0answers
211 views

Area defined with $\pm$ closedness

Denote $B_n\subset\Bbb R^n$ to be unit ball at origin. Denote $S\subset B_n$ to region of type $\mathsf I$ if it satisfies $$s\in S\iff\forall t\in S, s+t\in S\mbox{ or }s-t\in S$$ I am convinced $\...
1
vote
1answer
33 views

Finding an unfrustrated set of local linear constraints with given minimal value

Let $ \{F_{i}\} $ be a finite set of linear functionals on a convex compact subset $B\subset\mathbb{R}^{n}$ such that each $F_i$ is $k$-local (acts on $k\ll n$ variables only). Assume $F=\sum_{i}F_{i}$...
4
votes
0answers
104 views

Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. ...
4
votes
1answer
75 views

Self-concordant function for dual cone

I wonder if there is any existing result for self-concordant function in the literature about the following question. Suppose $f$ is a self-concordant barrier function of a proper cone $K$ (pointed, ...
0
votes
1answer
347 views

Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...
6
votes
3answers
508 views

Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
3
votes
2answers
163 views

Constructing a quasiconvex function [closed]

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, \...
4
votes
0answers
77 views

Linear projections of convex sets with unique preimages of boundary points

Fix a compact convex subset $C \subset \mathbb{R}^n$ with nonempty interior. For any subspace $S \subset \mathbb{R}^n$, let $P_S$ denote the orthogonal linear projection onto $S$. I'd like to claim ...
5
votes
0answers
71 views

Concavity of mixed volumes and mixed discriminants

For $n\times n$ symmetric matrices $A_1, \ldots, A_n$, the mixed discriminant $D(A_1, \ldots, A_n)$ can be defined as $1/n!$ times the coefficient of $t_1\ldots t_n$ in the homogeneous polynomial $\...
6
votes
1answer
300 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
1
vote
2answers
390 views

A question involving Mazur's Lemma

Consider the Mazur's Lemma (H. Brezis - "Functional analysis, ..."): "Assume $(x_n)$ converges weakly to $x$. Then there exists a sequence $(y_n)$ made up of convex combinations of the $x_n$'s that ...
3
votes
1answer
124 views

Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) d(u,...
3
votes
1answer
107 views

Gradient estimate of convex functions

Consider a special type of convex function $g(\cdot):\mathbb{R}^d \to \mathbb{R}_+\cup\{+\infty\}$ such that $g(x)=+\infty$ as $|x|\to \infty$. Then $g$ is differentiable almost everywhere within its ...
4
votes
1answer
222 views

monotone parabolic systems, convex variational structure and Legendre transform

The context: for my research I am currently looking at parabolic systems of the type $$ \left\{ \begin{array}{ll} \partial_t b(u)-\Delta u=0 \qquad & (t,x)\in \mathbb{R}^+\times\Omega\\ u=0 & ...
3
votes
1answer
133 views

generalized mean inequality extension

from generalized inequality, we now that for $p>q$, we have $M_p(\mathbf{x})\ge M_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ ...
1
vote
1answer
60 views

monotonicity alike functions

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$, under what condition, we have ${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow {f_2}\...
0
votes
0answers
86 views

Largest ball with fixed center in a a convex region

Let $x_0$ be a point contained inside a compact, convex set $C\subset\mathbb{R}^d$, which is of the form $C=\{x:f(x)\leq0\}$ for some explicit convex function $f$. Is there a computationally ...
1
vote
0answers
74 views

A version of isotone projection cones

We write $a \succeq b$, where both $a, b \in \mathbb{R}^n$, as a shorthand for $a_i \ge b_i$ for all $1 \le i \le n$. Let $C$ be a closed convex cone in the first orthant of $\mathbb{R}^n$ and denote ...
2
votes
0answers
64 views

A convex analysis theorem improvement

John's theorem states that to any full-dimensional symmetric convex set $K\subseteq R^n$ and any Ellipsoid $E\subseteq R^n$ that is centered at origin, there exists an invertible linear map $T$ so ...
1
vote
1answer
62 views

number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where $i_\alpha\in\{0,1,\...
0
votes
1answer
150 views

On the upper bound of Hermitian matrices

Suppose we are given a Hermitian matrix $A$, how to describe the following set of Hermitian $S=\{X:X\geq \pm A\}$, where $Y\geq B$ is $Y-B$ is semidefinite matrix. This is of course a convex set, and ...
1
vote
0answers
49 views

Uniqueness of (generalized) Moreau decomposition

Let $H$ be some Hilbert space, which we can take to be the usual finite-dimensional Euclidean space if needed. For a function $f : H \to \mathbb{R}$, let $f^* : H \to \mathbb{R}$ be its conjugate dual,...
7
votes
1answer
497 views

Uniform convergence of convex functions

It is a well-known result that if a sequence of convex function $f_n(\cdot)$ converges on a dense set $C'$ of an open set $C$, then the limit function $f$ exists on $C$, and the converge is uniform ...
1
vote
1answer
61 views

characterization of a certain closed convex cone

Consider $x_1,\cdots,x_n \in \mathbb{R}^d$, and the closed convex cone in $\mathbb{R}^n$ defined by $$\mathcal{K}(\underline{x}):=\{(\varphi(x_1),\cdots,\varphi(x_n)):\varphi \textrm{ convex on }\...