Convex analysis is the study of properties of convex sets and convex functions. For questions about optimization of convex functions over convex sets, please use the (convex-optimization) tag.

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Is the geometric-to-arithmetic function convex or concave?

Consider a vector $\mathbf{x} \in \mathbb{R}_{++}^N$. Also consider two functions, $g(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, and $a(\mathbf{x}): \mathbb{R}^N \rightarrow \mathbb{R}$, ...
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3answers
118 views

Concave function divided by a convex function. What is the result?

Let us say that I have a function $f(x)$ that we know is a concave. And let us also say that we have another function $g(x)$ that is a convex. If I make a new function, $h(x) = \frac{f(x)}{g(x)}$, ...
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40 views

An extreme point that is not strongly exposed

I want to construct an example that a point is extreme but is not exposed. This example can be in the following : A compact convex subset $K‎\subset‎\mathbb{R}^{2}$ and a point $u\in K$ such that ...
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2answers
31 views

Convex set for a set of points in 2d plane

There are set of five points $A(0,0) ,B(1,1) ,C(2,0) ,D(2,2).E(0,2) F(1.5,1.5)$ $S=\{A,B,C,D,E,F\}$ Please tell me whether my understanding is correct or not! The points $A,C,D,E$ forms a convex ...
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1answer
59 views

On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
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2answers
44 views

computation of subdifferential

This question mainly deals with subdifferential of a convex function with respect to the cost function $c(x,y)=\frac{|x-y|^2}{2}$ I want to compute the cost-subdifferential $\partial^{c}\phi$ of the ...
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1answer
28 views

A characterization of differentiability of a convex function

Let $\phi : \mathbb R^n \to \mathbb R$ be a convex function. For all point $x\in \mathbb R^n$, define the subdifferential as $$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
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1answer
42 views

How to verify correctness of a Fenchel conjugate derivation

Suppose I derived Fenchel conjugate of a function. My goal is to check if my solution is right. Suppose the steps are not available any more and only the final solution is present. Is there any ...
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28 views

Show that the Rosenbrock function is strictly convex for a specific region

So we know that the Rosenbrock function is a test function of sorts, but can anyone prove that a specific region is strictly convex? Rosenbrock eqn: $ f(x_{1},x_{2}) = 100(x_{2} - x_{1}^{2})^{2} + ...
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What are the extreme points of the set of probability measures on $(\mathbb N, \mathcal{F})$?

Say, we have some $\sigma$-algebra on $\mathbb N$ and let $\mathbb P$ be the set of all probability measures on it. We know that $\mathbb P$ is convex, so I wonder how do the extreme points look like. ...
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1answer
84 views

Tangent Cone is a cone?

I first give two definitions. Def1: A set $S$ is a cone if $x \in S, \lambda \geq 0 \implies \lambda x \in S$. Def2: Let $S$ be any set (we may assume $\mathbb{R}^n$ with the usual Euclidean ...
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31 views

is convex hull of intersection equal to intersection of convex hull

is $convexhull(S_1\cap S_2)=convexhull(S_1)\cap convexhull(S_2)$ where $S_1$ and $S_2$ are finite sets.
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1answer
33 views

Proof of Projection property of convex set

suppose a set S$\subseteq$ $R^{m*n}$ is convex.Prove that T={$x_1$ $\in$$R^{m}$ :($x_1$,$x_2$) $\in$ S } is convex. This is projection property of convex.Can somebody tell how to prove it.
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Isomorphisms of Convex Cones

A convex cone $C$ is a subsets $C \subseteq V$ of a vector space which is closed under positive linear combinations, i.e. for $\lambda, \mu > 0$ and $u,v \in C$ it is $\lambda u + \mu v \in C$. An ...
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region between two concentric circles are not convex set in eucledian space of order 2.

i want a counter example to show that the region between two concentric circles in R^2 is not a convex set...i think we will find two points in the comon region of concentric circles and then will ...
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1answer
62 views

When is $f(X)$ convex?

Let $X$ be a Banach space, and let $f: X \to X$ be a nonlinear operator, $\mathrm{Dom}(f)=X$. When is $f(X)$ convex?
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Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
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1answer
130 views

Lipschitz Smoothness, Strong Convexity and the Hessian

I'm working with the following two concepts: Lipschitz Smoothness - a function $f$ is Lipschitz smooth with constant $L$ if its derivatives are Lipschitz continuous with constant $L$, in other words ...
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1answer
88 views

Can a Lipschitz continuous function be strictly convex?

Let $\varphi:\mathbb R^n\to\mathbb R$, and suppose for all $x,y\in\mathbb R^n$, $$\|\varphi(x)-\varphi(y)\|\leq L\|x-y\|$$ for Lipschitz constant $L$. Is it possible for such a function to satisfy ...
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1answer
46 views

Interior of the sum of two sets?

Is it always true for two subsets $A $ and $B$ of a real Hilbert space $H$ that $\operatorname{int}(A+B)=\operatorname{int}(A)+\operatorname{int}(B)$ ?
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1answer
56 views

to prove XY-plane a convex set??

we know that R and R^2 have convex subsets because any two points in them can be joined by a line segment..but how we will prove it mathematically that XY-plane is a convex set??
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80 views

How to prove the open interval $(1,5)$ is a convex set?

I want to prove the interval $(1,5)$ is a convex set. A convex set is a set having all the convex linear combinations of its point in it, where a convex linear combination is a linear combination of ...
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0answers
25 views

Dual of the mixed $\ell_1/\ell_2$ norm?

The mixed $\ell_1/\ell_2$ norm $\Omega_{12} $ is defined as $\Omega_{12}(x) = \sum_g ||x_g||_2$ where $x_g$ are disjoint subsets of the elements of the vector $x$. This is used in machine learning ...
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1answer
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affine combination of convex functions.

In a complete smooth simply connected Riemannian manifold of non-positive curvature, the square of the distance function $d^2(p,x)$ is a smooth strictly convex function of $x$. It follows that this is ...
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102 views

Tangent line of a convex function

Let $V : \mathbb{R}^n \rightarrow \mathbb{R}$ be (strictly) convex and continuously differentiable on $\mathbb{R}^n$. Show that for any $y \in \mathbb{R}^n$ and any $x \in \mathbb{R}^n$ $$ V(y) \geq ...
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1answer
75 views

Concave function of two variables restricted to one variable

Suppose $u(x,y)$ is a concave and strictly increasing $\mathcal{C}^2$ function (think of a utility function from economics). Define the one variable function $f(x)=u(x,e^r(K-x))$ for all $x\in ...
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1answer
65 views

How to prove that this function is convex?

I want to prove convexity the following function (a,x > 0): $$ f(x)=\frac{x}{\sqrt{e^{-ax} + ax -1}} $$ What I tried: Using the Taylor series of $e^{-ax}$ $$ f(x) = \frac{1}{\sum_{i=2}^\infty ...
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can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
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19 views

Implementing a projection with KL-divergence

I want to implement the following and I am looking for an easy/fast way to implement it(the programming language does not matter). Assume that $p(\mathbf{x})$ is a proper probability distribution and ...
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30 views

Boundary points of probability simplex

I have a very simple question for which I know the answer but I can not prove it! What are the boundary points of a probability simplex? I know every probability vector with one zero component lies ...
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1answer
79 views

Convexity of Binomial Term

I am reading a book on the probabilistic method, and the following claim was made: $\dbinom{y}{n}$ is convex. Why is this the case?
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1answer
37 views

Proof that the image of an Itō integral is convex if the driving Wiener process is in a metric ball

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A := \int_0^1 f(t)\,d W_t$ be the Itō integral of an $L_2([0,1])$ deterministic function $f$ with respect to the Wiener process $W$. ...
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1answer
51 views

Prove that $x \rightarrow \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt$ is convex

To put it bluntly I'm stuck proving proving the subsequent inequality $$ \forall x>0, \int_0^\infty \frac{e^{-tx}}{1+e^{-t}}dt \int_0^\infty \frac{t^2 e^{-tx}}{1+e^{-t}}dt \geq {\left ( ...
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0answers
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Litterature: Convex Geometry using differential forms

I've been looking for papers or books that discuss/use differential forms in convex geometry. I have been very unsuccessful in my search and was wondering if anybody here had come across such ...
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1answer
125 views

Finding the dual cone

Looking at the example here, I'm trying to understand how the author finds the dual cone $K^*$. The question asks to find the dual cone of $\{Ax | x \succeq 0\}$ where $A \in \mathbb R^{m\ ...
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Proof that the set $\{ x \in \mathbb{R}^n: \max\{w_1^Tx, \ldots, w_N^Tx\} \geq \gamma\}$ is not convex in general

Let $w_1, \ldots, w_m$ and $x$ be vectors in $\mathbb{R}^n$, and $\gamma$ be some constant in $\mathbb{R}$. How can I prove that the set $\{ x \in \mathbb{R}^n: \max\{w_1^Tx, \ldots, w_N^Tx\} \geq ...
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Stronger than strict convexity, bounded hessian?

I've encountered a condition similar, but slightly stronger, to that of a function being strictly convex. The condition is $\phi(\lambda x+(1-\lambda)y)\leq \lambda \phi(x)+(1-\lambda)\phi(y) - ...
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1answer
128 views

Proving that a Hessian Matrix is positive definite

I'm currently stuck on a problem for my Artificial Intelligence class. The assignment is provided at the following link: http://courses.engr.illinois.edu/cs440/HW1.pdf The problems that I'm stuck on ...
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33 views

Legendre transform for HJB PDE

I am trying to understand section 2.3 of the following article: http://www.princeton.edu/~sircar/Public/ARTICLES/montreal.pdf . I think I understand that $H_v(t,g(t,z)) =z$, because by definition of ...
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1answer
36 views

Intersection of half planes vs union?

Can someone explain to be why we are taking intersection instead of union? Because taking the union means we are also taking the union of ALL the $y$s in $S$ no?
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55 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
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38 views

Conjugate of a function

Yes this is homework. Given $f(x) = 1^{T}(x)_+$ where $(x)_+ = \max\{0,x\}$, what is $f^*$? I know that the conjugate of a function $f$ is $f^*(y) = \sup (y^Tx - f(x))$ but I do not know how to show ...
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1answer
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Determining if a function is convex

Yes this is homework. For which values of $a$ is the function $f(x)=e^{-a \sqrt x}$ with $\mathbf dom f = \mathbf R_+$ convex? The possible answers are: $a \le 0$ $a \ge 0$ $-1 \le a \le 1 $ ...
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Convexity of a complicated function

Let $\mathbb{S}$ be a 2-D convex set in the positive quadrant. Let us define \begin{align} y_L=\min_{(x,y)\in\mathbb{S} }y \\ y_R=\max_{(x,y)\in\mathbb{S} }y \end{align} For any positive number ...
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Convexity of product of two given functions

Let $f(x)$ be a convex function of $x$ in a given positive interval. Also assume $f(x)\geq 0$ everywhere in that interval. Is the function $g(x)=xf(x)$ convex in that interval?. Is it possible that ...
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1answer
106 views

Composition of convex function and affine function

Let $g: E^{m} \rightarrow E^{1}$ be a convex function, and let $h: E^{n} \rightarrow E^{m} $ be an affine function of the form $h(x)=Ax+b$, where $A$ is an $m \times n$ matrix and $b$ is an $m \times ...
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Why is the feasible set of utility values (in bargaining problem) convex?

Let $S := \{x \in \Bbb{R}^n \mid x \ge 0, \sum_{i=1}^n x_i = 1\}$ be the set of mixed strategies. For a bimatrix game with pay-off matrices $A$, $B$ we denote $C := \{ (u, v) \mid \exists (x,y)\in ...
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Convex functions - two questions

I have two questions regarding convex functions: First question: Let f be convex function on closed interval [a,b]. Prove that f has maximum in x=a or x=b. I understand that $\forall ...
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2answers
62 views

Convexity of $\frac{1}{f}$ over the set where the concave function $f$ is positive

$S \subset R^n,~~f : S \rightarrow R $ is a concave function. $S^{'}= \{ x \in S: f(x)>0 \}. $ Prove that $\frac{1}{f}$ is a convex function on $S^{'}.$
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1answer
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Prove or disprove the concavity of the function [closed]

Prove or disprove the concavity of $f$ over the following two domains. $$f(x_1,x_2)=10-2(x_2-x^{2}_{1})^{2}$$ defined either over $$S_1=\{(x_1,x_2) : -1\leq x_1 \leq 1, -1 \leq x_2 \leq ...