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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Does function that maps bounded convex sets (minus straight line segments) to bounded convex sets must be continuous everywhere?

This question in the title came to my mind while I was sitting with my granny in front of my house maybe about half an hour ago. Although it looks innocent I do not know at the moment some simple ...
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Convex integral inequality

I cannot prove that if $f(x)$ is convex on $[a,b]$ then $f\Big(\frac{a+b}2\Big) \le \frac1{b-a}\int_a^b f(x)\,dx \le \frac{f(a)+f(b)}2 .$
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linearize average success probability constraint

In my optimization problem, I've a constraint to calculate the average success probability of a path. $x_{i,j}$ is binary variable defined as: $$ \begin{align} \label{eq3:1} x_{i,j} = \begin{cases} ...
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34 views

Convex or non-convex function

I want to minimize the following function $$\frac{a}{bxy+cd}e^{\frac{a}{bxy+cd}}H+2-\Gamma(1,\frac{eaf}{b(1-x)},\frac{eagf}{bx(1-y)})$$ where $a,b,c,d,e,f,g,H$ are constants and greater than $0$. $\...
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How to prove the convexity of $f$ if the strict epigraph of $f$ is convex

I have trouble to prove the equivalence of the following two definitions of convex function. For convenience, I list them as follows: Def-A. Let $f : I\to \mathbb{R}$ be a function, where $I$ is (and ...
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Inequality for convex functions

Let $a,b:\mathbb{R}\to\mathbb{R}_+$ be two strictly convex and differentiable functions, such that $a\geq b$. Let $x\leq\alpha\leq y\leq\beta$, where $\alpha,\beta\in\mathbb{R}_+$ are two constants. ...
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47 views

When we can permute between the integral and convex hull?

Is there a relation between the following expressions? $$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right) $$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds $$ where $A$ ...
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38 views

Can a quasiconvex function be made convex by composition with a diffeomorphism?

Assume we are given a continuous quasiconvex function $f: \mathbb{R}^n \to \mathbb{R}$. Intuitively I feel that quasiconvexity means that there should exist a diffeomorphism $h: \mathbb{R}^n \to \...
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1answer
26 views

Finding minimizer from different order

Let a nonnegative function $f(x,y)$: $\mathbb R^2\to \mathbb R$ be second order continuous differentiable. We also know that $f$ is not convex in its two arguments, but only separately in each of them....
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Invertibility of a polylogarithmic map

Consider a map defined on $\Bbb R\times(0,+\infty)$ and given by $$M:(a,b)\to(\rho,E),$$ $$\rho = \int_{\Bbb R^n}\frac{dx}{1+\exp(a+b|x|^2)}\\E=\int_{\Bbb R^n}\frac{|x|^2dx}{1+\exp(a+b|x|^2)}.$$ I ...
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25 views

Exact line search in convex optimization

For a convex function $f$ what do we know about convexity of the exact line search problem? $$\min_{\alpha \ge 0} f(x+ \alpha p_k)$$ I think because the function is convex and is linear in variable, ...
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27 views

Showing that this set satisfies the closed criterion

Suppose we have the subset $S = \{ \lambda v \mid \lambda \geq 0 \} + K $, where $v$ is a vector in $\mathbb{R^3}$ and $K$ is a convex hull of six other vectors. How do I show that it satisfies the ...
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90 views

Why is the dual cone of $l^1$ is $l^\infty$?

I just noticed somewhere in Convex Optimization that the dual cone of $l^1$ is $l^\infty$! (A diamond in $\mathbb{R}^2$ for $l^1$ is a square in $\mathbb{R}^2$ for $l^\infty$.) In fact I cannot ...
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21 views

Upper bound for sum of powers

I have a sequence of positive real numbers $\{x_i\}_{i=1}^{N}$ and $k \in \mathbb{N}$. I was wondering if one can find an upper bound of the type $$ \sum_{i=1}^{N}{x_i^k} \leq f\left(\sum_{i=1}^{N}{...
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Algorithm to find the point in a convex polygon closest to an external point

Given a convex polygon $P$, and a point $q$ of the plane, external to $P$, what is the fastest algorithm to compute the closest point in $P$ to $q$? A linear algorithm of course works, computing the ...
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Is this implicit mapping convex?

I am interested in the convexity properties of the following mapping on the $n\times 1$ vector $x$: $$ x_{j}=y_{j}^{\beta}\left(\sum_{i=1}^{n}B_{ij}x_{i}\right)^{\alpha} $$ where $\beta>0$, $y_j\...
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63 views

Convex sets in infinite dimensional Banach spaces

I am reviewing functional analysis and getting stuck in this question. Let $X$ be an infinite dimensional Banach space. Show that there exist convex sets $K_1, K_2$ such that $K_1\cap K_2=\emptyset,...
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53 views

A convex set in $\mathbb{R}^n$ whose closure is $\mathbb{R}^n$

Let $S$ be a convex set in $\mathbb{R}^n$. Prove that if a closure of S is $\mathbb{R}^n$ then $S=\mathbb{R}^n$
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28 views

Is generalized mean convex / concave?

The generalized mean can be given using the following equation: $ M_p(x_1, \dots, x_n) = (\frac{1}{n}\sum_i x_i^p)^{1/p} $ Is it convex /concave when $p<1$ ?
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Conjugate of difference of convex functions

I am reading through this tutorial on DC programming and the author makes a startling claim without proof: If $g$ and $h$ are two lower semi-continuous convex function, then the conjugate function of ...
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1answer
44 views

For which values of $\gamma$ does this inequality hold?

Edited: Just realised my first post was somewhat misleading and not precise. Thanks to the two commetators that pointed it out. I am working on an article and ended up wondering for which values of $\...
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Looking for an entry level discussion on convex analysis

I have been studying for a qualifier and every so often I come across questions such as: Let $f_n:[a,b] \to \mathbb{R}$ be convex functions and suppose that $f(x) = \lim_{n \to \infty} f_n(x)$ exists ...
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52 views

Can we say a convex cone is a closed set without further proof?

There are some related problems: 1. dual cone is closed 2. Why is any subspace a convex cone? Consider a cone $\mathcal{C}(A)$: $$\mathcal{C}(A) = \{Ax: x\geq 0\}$$ This is a cone generated ...
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Proof of the Line Segment Principle for Convex Sets

I'm self studying a chapter on convex sets in preparation for a course on optimisation for economists however I'm having trouble understanding the proof of the line segment principle. I would ...
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1answer
44 views

Show that exactly one of the following two systems has a solution.

Let A be a $m \times n$ matrix, $\mathbf{c}$ an $n$-dimensional ector and $\mathbf{b} \ge \mathbf{0}$ an $m$-dimensional vector. Show that exactly one of the following two systems has a solution: $\...
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66 views

Support function of an ellipse

I defined the support function $h_A:R^n→R$ of a non-empty closed convex set $A\subseteq \mathbb{R}^n$ as $$h_A(x)= \sup\left\{x⋅a |\ a \in A\right\}$$ Everything I know about this topic I found it. I ...
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97 views

The Dual problem of a non constraints problem?

The primal problem is $min_{w\in R^d}: P(w)$ where $P(w)=\frac{1}{n}\sum_{i=1}^n\phi_i(w^Tx_i)+\frac{\lambda}{2}||w||^2$. The dual problem is $max_{\alpha\in R^n}: D(\alpha)$ where $D(\alpha)=\frac{...
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Convexity and Proof of one sided Derivative

Working on some real analysis work, I've been able to show that for a function $f$, which is convex on $[a,b]$, for $a\leq x_1< x_2< x_3\leq b$: $$\frac{f(x_2)-f(x_1)}{x_2-x_1} \leq \frac{f(x_3)...
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Dual cone and sum of closed cones

Picture below is from the 35 page of Schneider R.-Convex Bodies_ The Brunn-Minkowski Theory-Cambridge University Press (2013) , I think $C^o$ is always closed no matter $C$ is closed or not. Because ...
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Convexity in oriented matroid theory: proof on closure operator?

I would like to try to solve the following problems. Problem from the Oriented Matroids book by Bjorner, Las Vergnas, Sturmfels, White, and Ziegler. It is problem 3.9 on page 152. Attempt ...
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Minimal Ellipsoid in $R^{2}$; why is it the Ellipsoid 2 in the figure?

It is stated in the book Convex Optimization, Boyd in page 47 that the ellipsoid 2 is the minimal because no other ellipsoid (centered at the origin) contains the point (top point) and is contained in ...
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Cones of positive semidefinite matrices generated by matrices of rank $1$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive ...
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1answer
21 views

Is it possible to move vertices of a regular polygon to shape a given convex polygon?

can vertices of a regular polygon (n-gon) in the plane be moved (slide) one at a time to form a given convex polygon so that the polygons in between remain convex?
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Dual cone of a L1 norm cone?

I am listening to convex optimization lectures and I hear that dual cone of a $L1$ norm cone is a $L-\infty$ norm cone. Can anybody please explain how? I understand that every point in the dual cone ...
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convex conjugate

$X$ is a Banach space and $X^{*}$ denotes its dual. Let $f:X\rightarrow\mathbb{R}$ be an arbitrary convex function. The Fenchel conjugate of $f$ is the function $f^{*}:X^{*}\rightarrow\mathbb{R}$, ...
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$f:\mathbf{R}^n \to \mathbf{R}$'s derivative in each argument has the same sign everywhere. What is $f$'s shape?

We have a differentiable $f:\mathbf{R}^n \to \mathbf{R}$ with the property that each partial derivative has the same sign everywhere in its domain. Does this mean that the sublevel sets of $f$ (sets ...
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489 views

Does there exist regular planar pentagon?

How to prove or disprove that the boundary of any convex body in $\mathbb{R}^3$ (treated as a surface) includes 5 points which form a regular planar pentagon?
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A question of the proof of the duality mapping for convex bodies.

In picture below ,why the set $A$ is convex ? Below picture is from the A characterization of the duality mapping for convex bodies ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
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A question of the duality mapping for convex bodies

In picture below ,why the define of $\varphi$ is independent of the choice of sequence $(K_i)_{i\in N}$ ? Below picture is from the A characterization of the duality mapping for convex bodies ~~~...
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1answer
146 views

Chebyshev sets in finite dimension are closed and convex

Prove a finite-dimensional converse to the “best approximation theorem”: Let $K$ be a subset of a finite-dimensional Hilbert space $H$ which satisfies the following property: for each $x \in H$ there ...
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discrete convexity arising in a simple discrete optimization problem

Let $S$ be a fixed integer satisfying $S \ge 1$, let $a$ range over the integers between $1$ and $S$ inclusive, and for $i = 1, \dotsc, a$, let each $x_i$ range over the nonnegative integers, such ...
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Can Extragradient method be expressed with proximal steps?

As we know, for solving saddle point problems, forward-backward algorithm is generally not guaranteed to converge. But extragradient method converge Structured Prediction via the Extragradient Method ...
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How to Calculate the normal cone of a covex set at a point?

Let $C$ be a convex set of $\mathbb{R}^d$ and $\overline{x}\in C$ we define the normal cone of $C$ at $\overline{x}$ by \begin{equation} N_C(\overline{x}) = \{ y \in \mathbb{R}^d \ <y ,c-\...
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Proof of Convexity Using Quasiconvexity

Suppose $f_{1}, \ldots, f_{n},~n\geq 3$, are continuously differentiable functions defined on $\mathbb{R}$ such that for each $i\in \{1,\ldots,n\}$, we have: (i) $f_{i}(v_{i}^{0})=0$ and $f_{i}^{'}(...
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1answer
38 views

Proving an inequality involving a strictly convex function

Given, $f$ is a strictly convex function. Based on what assumptions on '$x$' and '$y$', can I say that the following inequality stands true : $$f(x) \; + f(y) \; > \; f(x + y) \; \; ?$$
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Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate?

Is $\frac{1}{2}\|x\|^2$ the only function that is equal to its convex conjugate? The convex conjugate is defined as $$ f^{*}(x) = \sup_y\{\langle x, y\rangle - f(y)\}. $$
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58 views

A term for a function $f$ such that $x f'(x)$ is decreasing

Consider a differentiable, monotonically increasing function $f$. If $f'(x)$ is increasing, then $f$ is convex. If $f'(x)$ is decreasing, then $f$ is concave. Is there a term that describes $f$ when ...
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22 views

Set of marginals is convex [closed]

Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let $S_{1,2}$ denote a convex closed ...
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30 views

The standard n-simplex is compact set

$ " Let \ K\subseteq \mathbb R^n \ be \ a \ compact \ set,\ then\ the\ convex \ hull \ of\ K\ is\ also\ compact \ set\ " $ In order to prove this we use that the standard n-simpex as defined ...
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459 views

Simplex - Help with a proof

I am trying to prove the following (basic) claim about simplex. If you check my proof and help me with the part where I stuck, I would appreciate it very much. Let $X$ be a finite set and define $\...