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Questions tagged [convex-analysis]

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1
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0answers
19 views

Second order necessary and sufficient conditions for convex nonsmooth optimization

For convex smooth optimization, first and second order necessary and sufficient conditions are well known. Does such standard second order necessary and sufficient conditions exist for convex ...
2
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1answer
79 views

Reference for Minkowski functional when 0 is not in the interior

The Minkowski functional on a normed linear space $E$ is usually defined for convex (or sometimes even non convex) subsets $C$ of $E$ such that $0 \in \operatorname{int}(C)$. Is there any standard ...
3
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2answers
98 views

Product of concave functions and harmonic mean

I discovered something interesting, and I would like to know whether it is a known result or not. Say that a function $f: \Omega \subset \mathbb{R} \rightarrow \mathbb{R_+^*}$ is $\alpha$-concave if $...
2
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0answers
53 views

Smooth dependence of convex functions on Monge-Ampère measure

Let $\Omega\subset\mathbb{R}^2$ be a bounded convex domain, $f$ be a positive smooth function on $\Omega$ and $\phi:\partial\Omega\rightarrow\mathbb{R}$ be a continuous function. It is known that as ...
1
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1answer
50 views

Reference on vector-valued convex conjugate

The following definition of convex conjugate is taken from Wiki: Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot ,...
1
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1answer
169 views

Boundary regularity for the Monge-Ampère equation $\det D^2u=1$

Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying $$ \det D^2u=1,\quad u|_{\partial\Delta}=0. $$ Classical ...
1
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0answers
142 views

Is this set in $\mathbb{R}^d$ closed? [closed]

Let $X$ be a convex set (in my case $X$ is infinite dimensional too) and for each $ i \in \{1,2,\dots,d \} $, let $f_{i}: X \rightarrow \mathbb{R} $ be convex functions where $C_{i} := \{ f_{i}(x) : x ...
1
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2answers
55 views

Alternative characterization of epi-convergence

I am struggling with the proof of a property of epi-convergence. We need the following definitions: For a sequence of sets $(C^\nu)_\nu$ in $\mathbb R^n$, the outer limit is the set $\limsup_\nu C^\...
1
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0answers
115 views

The perturbation of a convex function can also be convex?

$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly increasing on both dimensions (i.e. if $x_1>x_2$ then $f(x_1,y)>f(x_2,y)$), lipschitz continuous function defined on a ...
3
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0answers
55 views

Legendre transform on signed measure space

Let $X$ be an open set in $\mathbb{R}^n$ and $M(X)$ be the space of finite signed measures defined on $X$. $L(p)$ is a lower-semicontinuous convex functional defined on $M(X)$. My question is: (1) ...
0
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0answers
43 views

Convex functions and infinite convex combinations

Let $f: D\rightarrow \mathbb R$ be a convex function on a convex subset in $\mathbb R^n$. Let $t_i>0$ with $\sum_{i=1}^\infty t_i=1$ and $x_\in D$ be such that the series $\sum_{i=1}^\infty t_i x_i ...
2
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1answer
51 views

Given the following two assumptions, How to conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?

Please refer attached 6-page short paper for details. Let $M(y)=y+2r\nabla g_r(y)$. Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper) and $ \ \lim_{k ...
1
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0answers
46 views

Convexity on large scales

Has the following concept ever been studied/have a standard name? Let $f:\mathbb{R}\to \mathbb{R}$ be continuous. We say $f$ is (mid-point) convex on large scales provided $$f\left( \frac{x+y}{2}\...
1
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0answers
58 views

Solve simple stochastic variational inequality

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 $\...
1
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0answers
25 views

Dual representation of problems involving $f$-divergences

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 ...
1
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1answer
119 views

Characterization of state spaces of Boolean algebras

A state space of a Boolean algebra is a Choquet simplex but not all Choquet simplices can be viewed as state spaces of Boolean algebras. Is it known which Choquet simplices are precisely state spaces ...
16
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1answer
497 views

Extreme points of convex compact sets

Preparing to a lecture on Krein--Milman theorem I read in W. Rudin's Functional analysis textbook (1973) that it is unknown whether any convex compact set in any topological vector space has an ...
1
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1answer
66 views

A property on some unbounded metric spaces

Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property: $\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\...
1
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1answer
51 views

Reference Request: Differentiability of Moreau Envelope

I've recently come across many results discussing the differentiation of the Moreau envelope defined by $$ e(f)(x)\triangleq \min_{h \in H} \|h-x\|^2 + f(z) , $$ where $f$ is a convex functional on a ...
4
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1answer
98 views

a compact set with nonempty convex sections

Let $X = [0,1]^d$ be the unit cube in the $d$-dimensional Euclidean space. For every $x \in X$ and every coordinate $i=1,2,\ldots,d$ denote by $x_{-i} := (x_j)_{j \neq i}$. Given a set $Y \subseteq X$ ...
2
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1answer
77 views

Why the convexity condition on the definition of a face of a convex set?

A face of a closed convex set $X\subseteq\mathbb{R}^n$ is defined to be a set $F\subseteq X$ such that: $F$ is convex. Every line segment from $X$ whose interior meets $F$ is contained in $F$. Is ...
2
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1answer
92 views

small perturbation of BV function

consider an interesting real analysis question: define average operator on $[0,1]$: $A_{\epsilon} f (x) = \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f(y) dy , f \in BV[0,1] $ ( may clarify ...
4
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1answer
142 views

Is the preimage of a face under an affine map a face?

Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $...
4
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2answers
126 views

A kind of exponential concavity for polynomials?

Is there $C > 0$ such that the inequality $$ \prod_{n\in\mathbb{N}} p(n)^{a_n} \leq p\left(C\prod_{n\in\mathbb{N}} n^{a_n}\right) $$ holds for all finitely supported sequences $(a_n)$ with $a_n\geq ...
9
votes
1answer
425 views

Is KL divergence $D(P||Q)$ strongly convex over $P$ in infinite dimension

By KL divergence I mean $D(P||Q) = \int dP \log(\frac{dP}{dQ})$. I am looking for the conditions under which this strong convexity is true and possible references. I could not find an answer for ...
3
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1answer
189 views

concavity of $\log [ (1+\frac{x_0}{1+x_0+x_1/2+x_2/4}) (1+\frac{x_1}{1+x_0/4+x_1+x_2/2}) (1+\frac{x_2}{1+x_0/2+x_1/4+x_2}) ]$

Let $f:~ [0,1]^3 \rightarrow \mathbb{R}$ be $$ f(x_0,x_1,x_2)= \log \left[ \left(1+\frac{x_0}{1+x_0+\frac{x_1}{2}+\frac{x_2}{4}}\right) \left(1+\frac{x_1}{1+x_1+\frac{x_2}{2}+\frac{x_0}{4}}\right) \...
2
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0answers
51 views

Star-convex curve and Fourier series

Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
4
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3answers
142 views

Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$

I want to prove that function \begin{equation} f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt \end{equation} is quasi-concave. One approach is to obtain the closed form ...
1
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1answer
84 views

Every closed and convex subset of a uniformly convex metric space is Chebyshev?

I came across the statement ``Every closed and convex subset of a uniformly convex b-metric space is Chebyshev'' in [1]. Here, the term `convex' is in the sense of Takahashi. I tried looking up for ...
1
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0answers
50 views

Are the sets whose convex hull surface admits multiple representations a shy set of sets?

Consider a compact subset $A$ of $R^n$. Let me call $A$ special if any point $x$ that is on the boundary of $\textsf{conv.hull}(A)$ admits a unique representation as a convex combination of points in $...
0
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0answers
73 views

Volume of parametric integral of convex set

Given $t,\mu>0$. I am interested in computing the volume of the $n$-dimesnional set $$\int_{0}^{t}e^{A(t-\tau)}\begin{pmatrix}0\\ 0\\ \vdots\\ 0\\1\end{pmatrix}U\:{\rm{d}}\tau,$$ where the set $U$ ...
2
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1answer
110 views

Quasi-concavity of $f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$ on $[0~1000]$

I want to prove that function $f:[0~1000]\rightarrow R$, $$f(x)=(1-\frac{x}{1000})\log_2(1+2^x)$$ is quasi-concave. Any idea how to do the proof? I already tried to prove that any super-level set is ...
4
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2answers
140 views

Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix

Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that $\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that $$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
0
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0answers
45 views

Optimization of generalized Fisher information over space of probability measures

Assume we already have $p(\theta)$ apriori desnsity function about unknown parameter $\theta$, and we consider an optimization problem as follows: \begin{equation} \max_{P \in \mathcal{M}} \int_{\...
0
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1answer
87 views

1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$ \begin{split} \mathbb ...
0
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0answers
114 views

Convexity of the Frobenius norm of the product of matrices

I have a question similar to Convexity of the Frobenius norm of the product of two matrices. I am not able to comment on that question as I don't have enough reputation, and that is why I have asked a ...
-1
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1answer
85 views

subgradient calculus [closed]

I have to calculate a subgradient of the following function $$ p(x) = \max\{e(Cy-Cx)\ :\ Cy \geq Cx, y \in X\}, $$ where $C$ is a $p\times n$ matrix and $X$ is a convex polyhedral set. It is a non-...
5
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1answer
222 views

On faces of convex sets of positive semidefinite matrices

A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
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0answers
40 views

In convex optimization we know that the optimum solution is on which hyper plane

We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
-1
votes
1answer
97 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 ...
1
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0answers
34 views

How is the minimax oracle used to find the oracle complexity of projected subgradient?

I am going through a set of blog posts on the complexity of projected gradient method. https://blogs.princeton.edu/imabandit/2013/03/15/orf523-oracle-complexity-large-scale-optimization/ defines the ...
3
votes
0answers
108 views

Weak to weak$^*$ continuity of the duality mapping

Let $X$ be a uniformly convex and uniformly smooth Banach space. We consider the duality mapping $J_p^X$ defined as the sub-gradient $\partial (\frac1p\|\cdot\|^p)$. Is there a characterisation of the ...
1
vote
1answer
68 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^* ...
4
votes
2answers
248 views

Is this function always bounded below?

Is there any global constant $C$ such that $$C<\frac{\sum_{i=1}^{n}x_{i}\log x_{i}-x_{i}+(1-x_{i})\log(1-x_{i})}{\sum_{i=1}^{n}x_{i}}+\log(\sum_{i=1}^{n}x_{i})-\log(\sum_{i=1}^{n}x_{i}^{2})$$ for ...
6
votes
2answers
138 views

Inequality on permutation polytope

Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n), c = (c_1, \cdots, c_n) \in \mathbb{R}^n$ with $a_1 \geq \cdots \geq a_n, b_1 \geq \cdots \geq b_n, 0 < c_1 \leq \cdots \leq c_n$. In addition, ...
3
votes
2answers
204 views

Mixtures of log-convex functions are log-convex: a reference

A referee of a submitted paper requested details on the statement that $\int_0^a e^{-tx^2}\,dx$ is log-convex in real $t$, for each $a>0$. While there are a number of ways to prove this statement, ...
2
votes
1answer
194 views

example of convexity

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $(S_1,S_2)\in \mathcal{B}(F)^2$. We define $$W(S_1,S_2)=\{(\langle S_1 y\; ,\;y\rangle,\langle S_2 ...
4
votes
0answers
129 views

Sharp constant for inequality with convex functions

This is a follow up to this question, where the optimal constant was left open. Let $P \subset \mathbb{R}^n$ be bounded, convex, and open. Let \begin{equation} \mathcal{H} := \{f : P \rightarrow \...
2
votes
0answers
32 views

Do internal stable sets contain big manifolds?

Given two strictly concave functions $u_{i}$ with continuous derivatives in $\mathbb{R}^{k}$. We define their upper levels at a point $x$ of these functions as the set of points y such that $u_i(y)>...
3
votes
0answers
53 views

Behavior of vector field that is the arg min of a convex function

Say $f(x,y)$ is a real valued function that is strongly convex and twice differentiable in both $x$ and $y$. Here $x$ and $y$ are real vectors of different dimensions. We define $y^\star(x)= \arg \...