Questions tagged [convex-analysis]

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4
votes
1answer
112 views

Is there a non-convex function with non-decreasing average rate of change?

$\newcommand{\R}{\mathbb R}$ Let $f$ be a function from $\R$ to $\R$. It is said that $f$ is midpoint-convex if for any real $x$ and $y$ we have $f(x-y)+f(x+y)\ge2f(x)$ or, equivalently, $f(x+y)-f(x)\...
2
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0answers
41 views

Separability of Minkowski Sum of well-behaved sets

Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
1
vote
1answer
63 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 ...
2
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0answers
36 views

A complete metric space with some convex-type property

Let $(X,d)$ be a complete metric space with this property: for each $x∈X$, $r>0$ and $y∈X$ with $d(x,y)<r$, there exists $z∈X$ such that $d(x,y)+d(y,z)=d(x,z)=r$. I want to know if the family ...
0
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0answers
16 views

Derivation of dual for infinite linear program

I'm reading the section on Linear Programming in Barbu and Precupanu's Convexity and Optimization in Banach Spaces, and had a couple of questions concerning their derivation of the dual problem for ...
2
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0answers
41 views

improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
0
votes
1answer
50 views

Concavity of a function along a path

Suppose that $f(x,y)$ is a continuously differentiable function and $g(x,y) =xy-f(x,y)$. I know that $g$ is concave if and only if $(-f_{xx})(-f_{yy}) -(1-f_{xy}) ^{2}>0$ and $f_{xx}>0$. Now ...
5
votes
1answer
85 views

When is the log-permanent concave?

Let $\operatorname{PSD}_n$ be the cone of $n\times n$ semidefinite positive matrices. For any $X\in \operatorname{PSD}_n$, define $$f(X)=\log(\det(X)).$$ Then $f$ is a concave function on $\...
1
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0answers
81 views

Given a finite set of points, does there exist a linear function pass through a point and strictly below the other points for all the points?

I guess my question is a follow up question of this one: usul, Existence of a strictly convex function interpolating given gradients and values, version: 2019-04-13. In usul's question, the answer ...
0
votes
0answers
27 views

Extended-value subgradients

(I am not an expert in convex analysis, as may become clear.) Let $R$ be the extended reals, $R = \mathbb{R} \cup \{\pm \infty\}$. Standard texts define the subgradient of a convex function $f: \...
3
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0answers
70 views

Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
0
votes
1answer
126 views

Finding the conjugate of a function

I know that the Fenchel conjugate of a function is $$f^*(x^*) = \sup_x\{\langle x, x^*\rangle - f(x)\}.$$ However, how do I find the Fenchel conjugate of the function $$f(x) = \frac{1}{p}\sum\limits_{...
2
votes
1answer
69 views

Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R} $. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
2
votes
1answer
242 views

When does $\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$ hold for a real function $f(x,y)$?

Let $f(x,y)$ be a real function of the variables $x,y$ (which can be real vectors). Under what conditions do we have the following equality: $$\min_x \max_yf(x,y) = \min_y \max_x f(x,y)$$ For ...
5
votes
1answer
176 views

Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{...
1
vote
0answers
38 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 ...
2
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0answers
45 views

Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...
11
votes
1answer
330 views

Aleksandrov's proof of the second order differentiability of convex functions

Aleksandrov [A], proved a remarkable property of convex functions. Theorem. If $f:\mathbb{R}^n\to\mathbb{R}$ is convex, then for almost every $x\in\mathbb{R}^n$ there is $Df(x)\in\mathbb{R}^n$ and ...
9
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0answers
226 views

Second order differentiability of convex functions

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a convex function. Then $f$ is locally Lipschitz and hence differentiable a.e. (Rademacher). Let $E\subset\mathbb{R}^n$ be the set of points where $f$ is ...
2
votes
1answer
53 views

Inequalities for upper semi-continuous affine functions on compact sets by using extreme points

Suppose $f_1\colon K\to [0,\infty)$ and $f_2\colon K\to[0,\infty)$ are two upper semi-continuous affine functions, $$ f_i(\lambda x+(1-\lambda)y)=\lambda f_i(x)+(1-\lambda)f_i(y)\ \mbox{ for all }\ x,...
1
vote
1answer
63 views

Extreme points of an intersection of convex set with countably many linear spaces

Let $V$ be some `nice' vector space and let $T: V\to \mathbb{R}$ be a linear functional over $V$. Define \begin{align} M= K \cap \bigcup_{i \in \mathbb{N} } \{ v \in V: T(v)=c_i \} \end{align} ...
1
vote
1answer
133 views

Log-concavity of function

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ My goal is to show that $$ G(y):=\frac{(f_2*f_0)(y)}{(f_0*f_0)(y)}- \left(\frac{(f_1*f_0)(y) }{(f_0*f_0)(y)}\right)^2$$ is log-concave. Let us ...
1
vote
2answers
98 views

Monotonicity of maximum of convex combination of two scaled concave functions

Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \...
3
votes
0answers
54 views

subgradient in a predual under weak* continuity

Let $X$ be a Banach space. Suppose $f:X^*\to\mathbb R\cup\{\infty\}$ is convex, has closed and bounded (and so weak*-compact) effective domain, and is weak*-continuous on its effective domain. In ...
0
votes
0answers
94 views

How are the $L^2$ and $\sup$ norms related on the space of strongly convex functions?

Given a convex compact set $X \subset \mathbb R^d$ with interior containing the orign let $V$ be the space of all smooth functions $f: X \to \mathbb R$ with the properties: The function is strongly ...
0
votes
0answers
30 views

Characterization of global subdifferentiability

Let $X$ be a locally convex space, $D \subseteq X$ a nonempty compact convex set, and $f: D\to\mathbb R$ a continuous convex function. Question: Is there any known, interesting, alternative ...
1
vote
0answers
193 views

Continuity of the Legendre transform of a Lipschitz function

Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{...
2
votes
0answers
52 views

Prove the equicontinuity of a maximizing sequence

Let $X$ be a compact subset of $\mathbb{R}$ and $c(x_1,x_2,x_3,x_4)$ be a fixed bounded continuous functions on $X^4$. Assume $\mu,\nu$ are probability measures on $X^2$, and $\mu\otimes\nu$ is the ...
7
votes
1answer
278 views

Does midpoint-convex imply rationally convex?

Say that a function $f$ defined on $\mathbb{Q}^n$ is midpoint convex if $f((x+y)/2) \le (f(x) + f(y))/2$. Say that it is rationally convex if, for $\lambda \in \mathbb{Q} \cap [0,1]$ and $\bar \lambda ...
1
vote
1answer
166 views

Most general form of Jensen's inequality

What is the most general form of Jensen's inequality? Wikipedia gives for example this more general form, which holds in every topological vector space. Are there even more general forms, for ...
0
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0answers
53 views

Proximity operator composition

Suppose that $f,g$ are convex and lower-semi continuous functions from $\mathbb{R}^d$ to $\mathbb{R}$ such that $f\circ g$ is lsc and convex. I know that the proximity operators $\operatorname{Prox}...
5
votes
0answers
153 views

Generic shadows of convex bodies

If $K \subset \mathbb{R}^m$ is a convex body (i.e., a compact convex set) and $T \mathbin\colon \mathbb{R}^m \to \mathbb{R}^n$ is a linear map then $TK \subset \mathbb{R}^n$ is a convex body as well. ...
3
votes
1answer
102 views

What is a non-trivial example of an unbounded subdifferential?

Let $f: X \to [ -\infty, \infty]$ be some function, Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$, $$\partial f(x)$$ is "unbounded"? (trivial examples ...
0
votes
1answer
217 views

Expectations, double integrals and Jensen's inequality

$\def\anonfunc#1{#1(\cdot)}$Consider two random variables distributed $v\sim \anonfunc G$ and $c \sim \anonfunc F$ with pdfs $\anonfunc g$ and $\anonfunc f$. Let the supports of $c$ and $v$ be $[x,y]$....
3
votes
0answers
66 views

Sufficient condition for convex conjugate to be second-order differentiable

Let $f:\mathbb{R}^n\to (-\infty,\infty]$ be a convex lower-semicontinuous function, we then define its conjugate by $$ f^*(y)=\sup_{x\in \mathbb{R}^n}\{x^Ty-f(x)\}. $$ Then there exist well-known ...
18
votes
2answers
1k views

A strange variant of the Gaussian log-Sobolev inequality

Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
3
votes
1answer
68 views

On the area-perimeter ratio of a convex limited set

(Previously asked on MSE) Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as $$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$ Where $d(v,C)$ is the distance ...
0
votes
2answers
192 views

Goldowsky-Tonelli theorem for upper semi continuous function

Let $f:(0, \infty) \rightarrow \mathbb{R}$ be a convex and continuous function. We know that $\partial^{e} f(. )$(either right or left derivative) is non-decreasing and upper semi-continuous function. ...
1
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0answers
47 views

How does one translate from convex hull to a set of facets (inequalities)? [duplicate]

Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...
1
vote
1answer
62 views

Separation in $l^1$ (Kreps-YanTheorem)

I have a question about the hypotheses of the Kreps-Yan Separation Theorem. I use the notation $l^p_+$ for the subspace of vectors all of whose coordinates are non-negative and define $l^p_- = -l^p_+$....
2
votes
0answers
98 views

Maximization of an integral functional over a closed convex set

I want to maximize $$F(w):=\sum_{1\le i,\:j\le2}\int\lambda^{\otimes2}({\rm d}(x,y))\left(w_i(x)f_j(x,y)\wedge w_j(y)f_i(y,x)\right)g_{ij}(x,y)$$ over the closed convex set $$S:=\left\{w\in{\mathcal L^...
1
vote
0answers
61 views

Gradient formula for Clarke's generalized gradient on a general Banach space

In Theorem 10.27 of the book Functional Analysis, Calculus of Variations and Optimal Control, there is the following gradient formula: ($\operatorname{co}$ deotes the convex hull). Is there an ...
1
vote
1answer
121 views

How can we calculate the generalized gradient of $L^2\ni x\mapsto a\min(x(s),by(t))$?

Let $(T,\mathcal T,\tau)$ be a measure space, $a,b\ge0$, $s,t\in T$ and $$f(x):=a\min(x(s),bx(t))\;\;\;\text{for }x\in L^2(\tau).$$ How can we calculate the generalized gradient $\partial_Cf(x)$ of ...
0
votes
3answers
98 views

Intersection of Subharmonic and Harmonic Functions (also: Extension of harmonic functions)

Let $\Omega \subset \mathbb{R}^n$ be open, convex and bounded with smooth boundary. Define $$\mathcal{A}(\Omega) = \left\{ C \subset \Omega \ \big\vert \begin{array}{l} \text{for any subharmonic ...
3
votes
1answer
256 views

When do convexity and lower semicontinuity imply continuity?

Let $X$ be a nonempty compact convex subset of a locally convex space. Say $X$ has the convex function property if every convex, lower semicontinuous $f:X\to\mathbb R$ is also continuous. Question: ...
4
votes
1answer
223 views

Given convex l.s.c. function $f$, find decreasing convex function $\phi$ such that $f(x) \equiv \sup_y x\phi(y)-\phi(-y)$

Let $f: \mathbb R \rightarrow (-\infty,+\infty]$ be a lower-semicontinuous convex function. Question Under what futher conditions does there exists a convex decreasing function $\phi: \mathbb R \...
1
vote
0answers
65 views

Extreme points of set of measures with given barycenter

Let $X$ be a convex compact metrizable subspace of a locally convex Hausdorff topological vector space, $x_0\in X$, and $P$ be the space of all Borel probability measures on $X$ with barycenter $x_0$. ...
2
votes
1answer
216 views

Does the Legendre-Fenchel transform/convex conjugate of strongly convex functions have any desirable properties?

It is well known in convex analysis that when a closed, proper, function $f$ is Legendre-type, that is, essentially strictly convex and essentially smooth, the Legendre transform yields a dual ...
0
votes
0answers
36 views

Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question. We consider the ...
0
votes
0answers
24 views

Edit: Minimizing sequence of a multi-valued function using proximal point algorithm

The following is the multi-valued function I want to minimize: $f_1:\mathbb{R}^2\rightarrow \mathbb{R}$ defined by: $$f_1(x_1,x_2)=100x_1^2,\tag{1}$$ for all $x=(x_1,x_2)\in \mathbb{R}^2.$ The ...

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