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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General Techniques for Proving Convexity with Simple Example

I would like an exhaustive list of techniques to determine (or disprove) the convexity of a function. I think it is fair to assume the obvious things, sum of convex functions is convex and so on. ...
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116 views

Computing bounding box of polytope (system of linear inequalities)

Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints. I think ...
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125 views

convexity and lower semi-continuity for weak convergence

My question is a general one, whose answer can probably be found in any decent convex analysis book. I unfortunately don't have any at hand right now, so here it is: Let's consider a "reasonable" ...
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56 views

Question about the legendre transform in convex analysis

I'm work through a script about convex analysis, especially with dual problems. There is one step in a proof, which is not entirely clear to me. Let $U$ be a function, which is strictly concave, ...
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154 views

Taking a convex hull does not increase a supremum of a linear function

Let $X$ be a topological vector space, let $f:X\to\Bbb R$ be a continuous linear function and let $P(X)$ denote the set of all Borel probability measures on $X$. For any $M\subseteq X$ we define the ...
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33 views

$K$ cone, $x\in K$ and $y\in E$. Does $x+\lambda y\in K$ for all $\lambda\geq 0 $ implies $y\in K$?

Suppose that $K\subset E$, where $E$ is a Banach space and $K$ is a closed convex cone. Fix $x\in K$ and $y\in E$. Assume that $x+\lambda y\in K$ for all $\lambda\geq 0$. Can we conclude that $y\in ...
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$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^\infty\lambda_i\cdot a_i:a_i\in A, \lambda_i\ge0,\sum_{i=1}^\infty\lambda_i=1\right\}$ is superconvex

Let $X$ be a Banach space and $A\subset X$ a subset bounded. Denote by $\operatorname{sconv}(A)$ the superconvex hull of $A$: $$\operatorname{sconv}(A)=\left\{ \sum_{i=1}^{\infty}\lambda_i\cdot ...
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326 views

convex hull of orthogonal matrices

Where can I find the proof of the fact that the convex hull of the set of orthogonal matrices is the set of matrices with norm non greater than one? It is easy to show that a convex combination of ...
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603 views

sum of concave and convex function

Suppose $f$ is the sum of a concave and convex function, i.e. $$f=f_1+f_2$$ where $f_1$ is a concave function and $f_2$ is a convex function. I wonder if $f$ can be written as the following: ...
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290 views

An Orlicz norm is a norm

I had asked a question pertaining to Orlicz norms here. However, in the book I was reading, it said (and I paraphrase) "It's not difficult to show it is a norm on the space of integrable random ...
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242 views

Lovasz Extension Intuition

I am confused by the definition of Lovasz extension. The problem is I don't get the intuition behind the definition. In addition, Lovasz extension can be defined in different ways I don't see that ...
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149 views

Find conjugate indicator function

I'm doubt with this problem. Let $C=\left\{(x,y)\in \mathbb{R}^2|x+\frac{y^2}{2}\le 0\right\}$. I have to find $I_C^{*}(Y)$ defined by $I_C^{*}(Y)=\sup_{X \in \mathbb{R}^2} \left\{\langle ...
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123 views

Convex conjugate of absolute affine function?

Let $f:\mathbb{R}^n \rightarrow \mathbb{R} \cup \{ \infty \}$ be a convex function. The convex conjugate of $f$, which we call $f^*$ is defined as $f^*(y)=\sup \, \left \{ \langle y,x \rangle -f(x) ...
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75 views

Convex analysis problem

I have the following problem. Let $f:[a,b]\to \mathbb{R}$ be continuously convex. I have to prove that there exists $c\in (a,b)$ such that $$\frac{f(a)-f(b)}{b-a}\in \partial f(c)$$ Firstly, I'm ...
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1k views

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?
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Example of weakly discontinuous contraction

Can somebody give an example of a projector $P_c$ on the convex closed set C which is not weakly-continuous?
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113 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
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224 views

interior points and convexity

Question: Let If $P\subseteq \mathbb{R}^n$ be a convex set. show that $int(P)$ is a convex set. I know that a point $x$ is said to be the interior point of the set $P$ if there is an open ball ...
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158 views

Euclidean Metric and Convexity

Question: Consider the Euclidean metric space $(\mathbb{R}^n , \Vert\cdot \Vert)$. Let $X\subset \mathbb{R}^n$ and $f\colon X \to \mathbb{R}$. $X$ is said to be a convex set if for every $x,y \in X$ ...
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56 views

Generalizing the Definition of Convexity

The definition of convexity can be given as: Definition: Call a subset of $\mathbb{R} ^ k$, which will be denoted $E$, convex if given two elements of $E$, $\boldsymbol{x}$ and $\boldsymbol{y}$ and ...
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235 views

Jensen integral inequality in multidimensional case

The classical Jensen integral inequality says: Let $\mu$ be a probabilistic measure defined on some $\sigma$-algebra subsets of $\Omega$. If $f\colon \Omega \rightarrow \mathbb R$ be an integrable ...
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204 views

Concavity and Convexity

A set $X \subseteq \mathbb{R}^n$ is said to be convex if $tx + (1-t)y \in X$ for all $x,y \in X$ and $t \in (0,1)$. Given a convex set $X \subseteq \mathbb{R}^n$, a function $f: X \to \mathbb R$ is ...
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110 views

Why are close points in different regions close to the boundary?

I have a problem and it is bugging the heck out of me. It seems very obvious but now has become a major and frustrating stumbling block to a more productive line of thought. I have used a regular ...
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Need a way of computing the vertices of intersection of two simplices

I have two simplices $\Delta_1, \Delta_2$ defined as: The first simplex, $\Delta_1$, is the set of points defined as follows: $$\Delta_1 = \left\{\sum\theta_iu_i, \theta_i >= 0, ...
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184 views

Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.

If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$$ Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I ...
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148 views

Show that $\frac{x}{1-x}\cdot\frac{y}{1-y}\cdot\frac{z}{1-z} \ge 8$.

If $x,y,z$ are positive proper fractions satisfying $x+y+z=2$, prove that $$\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\ge 8$$ Applying $GM \ge HM$, I get ...
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194 views

Show that $2^n>1+n\sqrt{2^{n-1}}$

If $n$ be a positive integer greater than $1$, then prove that $$2^n>1+n\sqrt{2^{n-1}}$$ I found this problem under $AM \ge GM$ chapter. Help me to solve this problem using $AM \ge GM$. ...
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Prove that $\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$

If $a,b,c$ are positive , show that $$\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{c+a}+\dfrac{a^2+b^2}{a+b} \ge a+b+c$$ Trial: Here I proceed in this way ...
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36 views

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere?

Does every convex subset of $\mathbb{R}^n$ have normal almost everywhere? I want to know if I have any convex subset $M$ of $\mathbb{R}^n$ is it meaningful to talk about integral of second kind ...
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1answer
130 views

optimizing a logdet function with respect to a scalar and the Hessian matrix

Given a logdet function $\mathcal{L}(\gamma)$, $$ \mathcal{L}(\gamma) = \log\vert \mathbf{I} + \gamma\mathbf{S} \vert - \mathbf{q}^T(\gamma^{-1}\mathbf{I} + \mathbf{S})^{-1} \mathbf{q}, $$ where ...
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Jensen-like Inequality

I have the following question: Suppose we have a function $g:\mathbb{Z}_+ \cup \{0\} \rightarrow \mathbb{R}_+$ with the property, $g(\lfloor \frac{x+y}{2} \rfloor)$ + $g(\lceil \frac{x+y}{2} \rceil) ...
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236 views

I don't understand this proof of the AM-GM inequality?

The proof uses this lemma which I understand: $\mathbf {Lemma}$: Suppose $x$ and $y$ are positive real numbers such that $x>y$. If we decrease $x$ and increase $y$ by some positive quantity $E$ ...
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107 views

Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
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47 views

Distance between convex set and non-convex set?

So in http://en.m.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma there is some talk about distance between a mintowksi sum and a convex set. But I couldn't get how distance is being defined. Can ...
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How to show that $ Ax \le b$ is convex?

For $$ A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^m, c \in \mathbb{R} $$ one has to show that $$ K:= \{ x \in \mathbb{R}^n: Ax \le b \}$$ is convex. Now I'm aware that by definition, a set ...
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536 views

Two cubes in unit cube

A cube of side one contains two cubes of sides $a$ and $b$ having non-overlapping interiors. How to prove the inequality $a+b \le 1?$ The same question in higher dimensions.
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proving that this function does not define a norm on $\mathbb R^2$ since the convexity

This problema use the previous part to conclude something, so I write all the parts. First I have to prove that every norm in $\mathbb R^n$ is a convex function, I did it, it only requires the ...
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137 views

Equivalent definitions of uniform convexity.

I think I'm dumb, but I can't follow a simple proof from "Functional analysis and infinite dimensional geometry". They show that two different definitions of modulus of convexity of a norm are the ...
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108 views

A convex function that is bounded on a neighborhood is Lipschitz

Let consider a normed vector space $V$. I want to prove that If $f:V\to \mathbb R$ is a convex function and if for some $x_0 \in V$ the function is bounded on a neighborhood $W$ of $x_0$, then ...
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54 views

A reference to learn about duality

I am interested in learning about duality in convex optimization. I am looking for something to read which is: Reasonably short. Fairly self-contained (if it is a chapter in a textbook, I would ...
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64 views

convex function, inequality

If $f: R^n\rightarrow R$ is convex and $f(\alpha x)=\alpha f(x), \alpha \geq 0$, show that: a) $f(x+y)\leq f(x)+f(y)$ for all $x,y\in R^n$ b) $f(0)\geq 0$ c) $f(-x)\geq -f(x)$ for all $x\in R^n$ d) ...
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52 views

Proof of convexity of a function

I have to prove that function $J(x)=e^{x^3+x^2+1}$ is convex on $[0,\infty]$. I used a Theorem which says: **$U\subset R^n$ is convex set with non-empty interior and $J\in C^2(U)$. Function $J$ is ...
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35 views

Is the following function of several variables concave?

Suppose that $w_1,\dots,w_p\in(0,1)$ satisfy the condition $\sum_i w_i=1$, and let $$F(w_1,\dots,w_p) = \frac{1-\sum_i w_i^2}{\sum_i(1-w_i)^2 \left[\sum_i \frac{w_i}{1-w_i}\right]^2}.$$ Is function ...
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1answer
926 views

Is $f(x,y) = x^2y + x y^2$ (quasi-) concave or convex?

I should analyze whether the function $$f(x,y) = x^2y + x y^2 \text{ where } x,y > 0$$ is (quasi-) concave or convex. Thus, as usual, I set up the Hessian as $$ D^2f(x,y) = \left( ...
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183 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
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70 views

Convex function Inequality (3 - point version)

I was reading this article on inequalities (which some of you may find useful) here. On page 7, I came across this question by Titu Andreescu, which I shall reproduce here: Question: Let f be a ...
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130 views

Bound on Expectation of a convex function of a Random variable

My friend asked me the following question, which I at first thought was simple and straightforward: If $X$ is an integrable random variable and $g$ is a convex function(all real valued), then is it ...
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79 views

Is $f(X)=-tr(AXBX^T)$ convex?

Given $A,B \in \mathbb{R}^{N \times N}$ and they are non-negative matrix. Is $f(X)=-tr(AXBX^T)$ convex when $X$ is also non-negative? If yes, how can I show that?
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40 views

convexity of two linear spaces connected by a convex equality constraint

If there are two sets of linear constraints in different variables, Ax <= b with x_l <= x <= x_u and Cy <= d with y_l <= y <= y_u, and a set of equality constraints of a specific ...
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45 views

Determining the value of m for an m-convex set that is also non-convex

I'm looking within my PhD at atm at decomposing a random non-convex subset of the Euclidean Plane into a union of n convex sets, particularly hoping that the these sets (that from the overall ...