Questions tagged [convex-analysis]

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Smooth approximation of nonnegative, nondecreasing, concave functions

Let $f\colon [0, \infty)\to\mathbb{R}$ be nonnegative, nondecreasing, and concave. Prove the following claim or give a counter example: There is a sequence of functions $f_n\colon [0, \infty)\to\...
Froomfondel's user avatar
10 votes
1 answer
318 views

Convex functions in convex sets

Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in ...
Stefan Steinerberger's user avatar
3 votes
2 answers
82 views

Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$. If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
Aimar's user avatar
  • 33
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0 answers
100 views

Characterization of a "complex" hull?

This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides ...
M.G.'s user avatar
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1 vote
1 answer
55 views

Viscosity characterization of convex functions

Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then $$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$ I was looking for a similar ...
Luca.b's user avatar
  • 113
6 votes
2 answers
512 views

Conditions for including cones

Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
dzk's user avatar
  • 61
2 votes
1 answer
99 views

Tangent cone of a closed convex cone

Let $K \subset \mathbb{R}^n$ be a closed convex set. Given a point $u \in K$, the tangent cone of $K$ at $u$ is defined as (or characterized by) $$ T_K(u) := \mathrm{cl}(\left\{ t (v - u) \mid v \in K,...
aest's user avatar
  • 133
4 votes
2 answers
117 views

An upper bound of gradient norm for convex functions near minimizer

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ ...
Jean Legall's user avatar
1 vote
1 answer
62 views

Legendre transformation of vector valued function

Good afternoon. Is there any generalisation of Legendre--Fenchel transformation to the vector-valued functions $f: \mathbb{R}^n \to \mathbb{R}^n$?
Dmitry Vilensky's user avatar
5 votes
1 answer
171 views

Large ideally convex sets

Let $E$ be a Banach space. A set $C \subseteq E$ is called ideally convex if for every bounded sequence $(x_n)$ in $C$ and for every sequence $(\lambda_n)$ in $[0,1]$ that sums up to $1$ the vector $\...
Jochen Glueck's user avatar
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On convexity of special fractals in the plane

Let $X \subset \mathbb{R}^2$ be a subset of the plane with Hausdorff dimension $1<dim_H(X)<2$. For a subset $Y \subset \mathbb{R}^n$ we define $Y$ to be convex if for every $y_1,y_2 \in Y$ the ...
gigi's user avatar
  • 1,303
0 votes
0 answers
80 views

Decomposing convex functions as (simple, useful, convex) + (convex)

Let $f$ be a nice convex function, let $X = \{ x_i : i \in [N] \}$ be a collection of points in the domain of $f$, and define the 'bundle approximation' $$ f(x; X) = \max \{ f(x_i) + \langle \nabla f (...
πr8's user avatar
  • 559
7 votes
1 answer
478 views

Asking for an English version of Aleksandrov's famous 1939 paper in Convex Geometry

I have difficulty even in finding a Russian version of the next paper: "Aleksandrov, A. D., Almost everywhere existence of the second differential of a convex function and some properties of ...
Wenqing Ouyang's user avatar
0 votes
0 answers
35 views

Solving constrained convex minimization problems by reduction to a scalar equation

Let $f,g:\mathbb R^d \to \mathbb R$ be functions such that $g$ is convex and there exists $x_0 \in \mathbb R^n$ such that $f(x_0) \le 0$. $f$ is strictly convex. Consider the problem $$ \tag{*} \...
dohmatob's user avatar
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0 answers
73 views

The study of directional derivatives for functions that are minimums of convex functions

Has research been conducted on the topic of directional derivatives of functions that are minimums of convex functions? It would be greatly appreciated if you could provide the appropriate reference. ...
Samira Fallah's user avatar
2 votes
0 answers
51 views

Whether $d_x(t) := \|P_t(x)-x\|_H$ is increasing in $t$ where $P_t:H \to H$ is the proximal operator of a convex function

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 $...
dohmatob's user avatar
  • 6,338
0 votes
0 answers
55 views

Support function of the intersection of a hyper-ellipsoid and a Euclidean ball

Le $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_d$ be positive numbers. For any $x \in \mathbb R^d$ and $r \ge 0$, define $\gamma(x,r) := \sup_{z \in E(r)}x^\top z$, where $$ E(r) := E \cap B_2^d(r)...
dohmatob's user avatar
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0 votes
0 answers
78 views

Bound for the laplacian of a strictly convex function from above by the gradient of it

Let $V \in C^2(\mathbb{R}^d; \mathbb{R})$ a (strictly) convex function with $ \int_{\mathbb{R}^d} \mathrm{e}^{-V(x)} \, dx = 1.$ I am trying to show that $$ \int_{\mathbb{R}^d} |\nabla_x V(x) |^2\...
kumquat's user avatar
  • 43
3 votes
0 answers
80 views

Convex minorants to convex functions, given partial Taylor expansion and smoothness estimate

Let $V$ denote a strictly convex function (in arbitrary dimension) whose Hessian is $L$-Lipschitz. Given only this knowledge, and the values of $\left\{ V \left( x \right), \nabla V \left( x \right), \...
πr8's user avatar
  • 559
1 vote
1 answer
74 views

Strict convexity of the cost function is enough to ensure the existence and uniqueness of the optimal transport map

Let $X=Y = \mathbb R^d$ and $c:X \times Y \to [0, \infty)$ be Borel measurable. Let $\mu, \nu$ be Borel probability measures on $X,Y$ respectively. Let $\mathcal T$ be the set of all Borel measurable ...
Akira's user avatar
  • 713
1 vote
1 answer
129 views

Is a Lipschitz continuous gradient equivalent to this condition?

I know if a function $f: \mathbb{R}^n \to \mathbb{R}$ is $L$-smooth, i.e. its gradient $\nabla f$ is $L$-Lipschitz continuous, then it satisfies the following inequality for any $x, x_0 \in \mathbb{R}^...
aest's user avatar
  • 133
5 votes
1 answer
382 views

Is the Legendre transform as an operator Lipschitz?

Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is: $$ f^*(y) := \...
gdavtor's user avatar
  • 161
4 votes
0 answers
125 views

A Lipschitzian's condition for the measure of nonconvexity

I'm actually working on the measure of nonconvexity and its application. Especially, the Eisenfeld–Lakshmikantham MNC defined - in a Banach space - by: $$\alpha(A)=\sup_{b\in\overline{\operatorname{...
Motaka's user avatar
  • 291
7 votes
0 answers
323 views

Are there any characterizations of $C^2$ convex functions?

There are several characterizations of convex functions with the Lipschitz continuous gradient. If we already know that the function is of class $C^1$, then we have the following equivalent conditions:...
Piotr Hajlasz's user avatar
0 votes
1 answer
57 views

Strict inclusion for recession cone of closure of a convex set

Let $C$ be a nonempty closed convex subset of $\mathbb{R}^n$. The recession cone of $C$ is given by $$R_C=\left\lbrace d\in\mathbb{R}^n:x+td\in C, \forall t>0, \forall x\in C\right\rbrace.$$ It is ...
Chivul's user avatar
  • 29
1 vote
0 answers
127 views

The image of zero-measure set under normal mapping is Lebesgue measurable

Let $u$ be a convex function defined on a bounded open set $\Omega$ in $\mathbb{R}^n$. Then $u$ is twice differentiable a.e. Let $E_u$ be the set on which $u$ is not twice differentiable. Then $E_u$ ...
User1999's user avatar
2 votes
1 answer
184 views

On the infimal convolution of two norms on $\mathbb R^n$

$\newcommand{\R}{\mathbb R}$For natural $n$, $a\in\R^n$, and real $t>0$, let \begin{equation*} K:=K_{n,t}(a):=\inf_{x\in\R^n}(\|a-x\|_2+t\|x\|_1), \end{equation*} \begin{equation*} M:=M_{n,...
Iosif Pinelis's user avatar
0 votes
1 answer
60 views

Is the mapping $F(a):= \arg\min_{x \in \mathbb R^n} \|x-a\|_2 + \|x\|_1$ non-expansive?

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 ...
dohmatob's user avatar
  • 6,338
0 votes
1 answer
90 views

Orthogonal projection of a point centrally-symmetric closed convex subset of $\mathbb R^n$ never expands the coordinates of the point

Let $C$ be a closed convex subset of $\mathbb R^n$ which is symmetric about the standard coordinate axes. For example, think of $C$ as the unit-ball for an $\ell_p$-norm, for some $p \in [1,\infty]$. ...
dohmatob's user avatar
  • 6,338
1 vote
0 answers
22 views

Minimax statistical estimation of proximal transform $\mbox{prox}_g(\theta_0)$, from linear model data $y_i := x_i^\top \theta_0 + \epsilon_i$

tl;dr: My question pertains the subject of minimax estimation theory (mathematical statistics), in the context of linear regression. Given a vector $\theta_0 \in \mathbb R^d$, consider the linear ...
dohmatob's user avatar
  • 6,338
3 votes
0 answers
128 views

Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?

I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
P. Quinton's user avatar
6 votes
2 answers
201 views

For most directions does the supporting hyperplane meeting a bounded convex set meet it in one point?

Let $C\subseteq \mathbb R^n$ be non-empty, convex and compact. For $v\in S^{n-1}$, let $H_v$ be the supporting hyperplane in the direction of $v$ (i.e., $H_v$ is the boundary of the smallest closed ...
Alexander Pruss's user avatar
2 votes
1 answer
106 views

Analytic value of $\alpha := \sup_{(x,y) \in C} ax+by$, where $C := \{(x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$

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. ...
dohmatob's user avatar
  • 6,338
0 votes
1 answer
61 views

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
  • 6,338
18 votes
3 answers
657 views

Convergence of convex functions

I can prove the following result. Theorem 1. Let $f_n:\mathbb{R}^n\to \mathbb{R}$ be a sequence of convex functions that converges almost everywhere to a function $f:\mathbb{R}^n\to\mathbb{R}$. Then ...
Piotr Hajlasz's user avatar
3 votes
1 answer
144 views

Probabilistic Taylor theorem for concave functions

This paper proves a probabilistic version of Taylor's theorem \begin{equation*} \mathbb{E}g(X) = \sum_{k=0}^{n-1} \frac{g^{(k)}(0)}{k!} \mathbb{E}X^k + \frac{\mathbb{E}X^n}{n!} \mathbb{E} g^{(n)}(X_{(...
Dejan Evisal's user avatar
2 votes
0 answers
59 views

Convex ordering of measures that are obtained by different push-forwards of a same measure

Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...
theouscidda6's user avatar
0 votes
0 answers
60 views

Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$

Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...
dohmatob's user avatar
  • 6,338
2 votes
1 answer
99 views

Can we use the solution to two optimisation problems to solve a third, bigger, one?

Background Say we have an optimization problem $$\min_x f(x) = g(x) + h(x)$$ where $g$ is differentiable and convex, and $h$ are convex but not necessarily differentiable. If $g$ is the mean squared ...
user19904's user avatar
4 votes
2 answers
519 views

Unit ball of the sum space

Let $V$ be a vector space and $\|\cdot \|_1$ and $\|\cdot\|_2$ two norms on $V$. Let $\|\cdot\|_+$ be given by $$ \|v\|_+ := \inf_{v = v_1 + v_2} \|v_1\|_1 + \|v_2\|_2 $$ It is well-known that $\|\...
Willie Wong's user avatar
  • 33.9k
0 votes
0 answers
45 views

Intersection of the simplex with a space vector

The sets are defined in $\mathbb{R}_+^n$ $(n\geq 1)$. The relative interior of a convex set $C$ will be denoted $\mathring C$. The codimension of a submanifold $M$ of a manifold $N$ will be denoted ...
G. Panel's user avatar
  • 513
5 votes
3 answers
465 views

How to prove this (corollary of) hyperplane separation theorem?

$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$. The theorem is as follows. If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...
Ypbor's user avatar
  • 159
0 votes
0 answers
62 views

solution of equivalent problem Kantorovich for case squared distance function

We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to $$ \inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
Giovanni Arquimedes Wences Naj's user avatar
5 votes
1 answer
210 views

Proving the set $\lbrace \frac{(x + y)^2}{\sqrt{y}} \leq x - y + 5, y > 0 \rbrace$ is convex

I have recently picked up a course on Convex Analysis in my spare time, but feel I'm not quite up to speed with the 'tricks' for proving a set is convex. I have managed to prove this by moving all ...
AlwaysLost123's user avatar
0 votes
2 answers
325 views

Smooth approximation for non differentiable function

Let $f(t) = \min(\frac{1}{\lvert t\rvert}, 1)$. I would like to find a smooth approximating function $g$ such that $f(t) \leq g(t)$ for all real $t$. Is there a nice function $g$ out there? Any ...
Johnny T.'s user avatar
  • 3,373
0 votes
0 answers
48 views

Minimal value of $f(x) := \|x-x_0\|_S^2 + a\|x-x_0\|_S\|x\| + b\|x\|^2$

Fix $x_0 \in \mathbb R^n$, $a,b \ge 0$ and an $n \times n$ positive-definite matrix $S$. For any $x \in \mathbb R^n$, define $$ f(x) := \|x-x_0\|_S^2 + a\|x-x_0\|_S\|x\| + b\|x\|^2, $$ where $\|z\|_S :...
dohmatob's user avatar
  • 6,338
3 votes
2 answers
194 views

Hausdorff dimension of the non-differentiability set a convex function

Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and $$ E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}. $$ Then we have the following result which is Theorem: If $X= \...
Akira's user avatar
  • 713
-4 votes
1 answer
73 views

Convex combination of $\frac{1}{x}$ inequality [closed]

Let $0 < x_1 \leq ... \leq x_n$ and $\sum \alpha_i = 1, \alpha_i \geq 0$. Show $\sum \frac{\alpha_i}{x_i} \leq \frac{x_1 + x_n - \sum \alpha_i x_i}{x_1 x_n} $. Since the left side looks like a ...
Maximilian's user avatar
3 votes
1 answer
263 views

Derivative of distance function to a convex set in CAT(0) space

Let $(X,d)$ be a complete CAT(0) space. We denote by $T_x X$ the tangent cone at a point $x\in X$ and by $d_x$ its associated distance. So $(T_x X,d_x)$ is also a complete CAT(0) space. In CAT(0) ...
Othmane J's user avatar
4 votes
0 answers
233 views

Generalized Jensen's inequality for positively homogeneous functions

The function $f:V \to \hat{\mathbb{R}}$ is said to be positively homogeneous iff $f(\alpha v) = \alpha f(v)$ for every $\alpha \in \mathbb{R}_{++}$. Here $V$ is a real vector space and $\hat{\mathbb{R}...
Nik Pronko's user avatar

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