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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Multiplicative version of convex hull

The convex hull of a finite set of points, $(x_i,y_i) \in \mathbb{R_+}^2$ ($i=1,...,n$), is defined as: $$\left\{(\sum_{i=1}^{n} \alpha_i x_i,\sum_{i=1}^{n} \alpha_i y_i) \mathrel{\Bigg|} (\forall i: ...
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1answer
15 views

Faster Algorithms for Convex Hulls

I was interested in the following: Given two polyhedra $P_1, P_2$ specified in the form: $$ P_1 = {x : A_1x \le b_1 } $$ $$ P_2 = {x : A_2x \le b_2 } $$ Whereas $ x \in R^n$ and $b_1, b_2$ are ...
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1answer
22 views

How to show that $f(x,y,z) = (1-x^{2})^{2}+z^{2}+y^{2}+yz$ is a convex function on $S =\{(x,y,z) \in \mathbb{R}^3|\frac{1}{\sqrt 3} < x\}$?

Information: In the previous problem I had to find stationary points and the Hessian matrix and I found out that in the stationary points $(-1,0,0) $ and $ (1,0,0)$ were local minimums, and in the ...
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26 views

Concavity of function implies convex upper contour

Today I saw a theorem in class that stated the following: $f$ is concave $ \Rightarrow\{z \in \mathbb R^n : f(z) \ge c\}$ is convex. The proof is relatively straight forward and I understand. ...
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1answer
20 views

Barrier cone of a convex set. Why it is a cone?

Barier cone $L$ of a convex set C is defined as $\{x^*|<x, x^*> \le \beta, x\in C\}$ for some $\beta \in \mathbb{R}$. However, consider an scenario when $x_1\in L$, $\beta>0$ and $<x, ...
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1answer
45 views

Where does this definition of convex set come from?

My textbook says the following. We call a set $E\subset R^k$ convex if $$\lambda x+(1-\lambda ) y\in E$$ whenever $x\in E, y\in E, 0<\lambda <1.$ I'm totally at a loss as to what this ...
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19 views

Submodular function, square of which is also submodular?

A Submodular function $ f:2^E \rightarrow R $ is a function that satisfies the following two equivalent definitions: for every $ S,T\subseteq E: f(S) + f(T) \geq f(S\cup T)+f(S\cap T) $ for every $ ...
2
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1answer
23 views

Find projection of a point to a ball and box?

Find the projection of a point $y$ on a closed ball (center $x_0$, radius $r$) and hyperbox $H=\{x| a\leq x \leq b\}, a,b \in R^n$. I have no idea on how to proceed the proof. Please show me some ...
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7 views

Find the distance and projection convex analysis

Let $$C:= \{(x_1,x_2,x_3)| x_1^2+x_2^2+x_3^2 \leq 1, x_1^2-x_2\leq 0\},$$ and $y = (1,2,3)$. Find the distance $d_C(y)$ from $y$ to $C$ and projection $\pi(y)$ of $y$ on $C$. I have no idea ...
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1answer
34 views

Upper bound for a convex fractional function

Consider the following convex fractional function $$f\left( {\bf{x}} \right) = \frac{1}{{a - {{\bf{b}}^T}{\bf{x}}}}$$ where ${a - {{\bf{b}}^T}{\bf{x}}} \ge 0$. Is it possible to obtain a linear or ...
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1answer
37 views

Minimum of a convex function w.r.t. a subset of its domain

Let $X$ be a convex set and $f:X\mapsto \mathbb R$ be a continously differentiable convex function and $x_0$ be an element of $X$ such that $$\forall x\in X:f(x_0)\le f(x).$$ Let $y_0\in Y\subset X$ ...
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23 views

How to prove $\int_{\Omega} \sum_{i=1}^{N} f_i(x)dx$ is equivilant with $\sum_{i=1}^{N} \int_{\Omega} f_i(x)u_i(x)dx$

I have a 2D image in $\Omega$ space. Assume that the space can be separated into $N$ sub-regions $\Omega_i$ such that $\Omega_i \cap\Omega_j=\emptyset$; $\Omega_i \cup \Omega_j=\Omega, \forall ...
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1answer
18 views

Partitioning in convex problem (variables in two subsets)

Consider the following problem from textbook Convex Optimization Algorithm p.10: \begin{equation} \begin{aligned} &{\text{min}} & & F(x)+G(y)\\ ...
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4answers
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If points in a convex set $C$ escape to infinity roughly in direction $v$ then an infinite ray in that direction exists

Let $C\subseteq \mathbb R^n$ a convex set. Assume there is a sequence $\{c_k\}_{k\in\mathbb N}$ with $c_k\in C$, $|c_k|\to\infty$ such that $v:=\lim \frac 1{|c_k|}c_k$ exists. Does this imply that ...
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78 views

Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt. I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
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3answers
763 views

How can it be proved that the geometric mean function is concave?

A function $f: \mathbb R ^n \rightarrow \mathbb R $ is said to be concave if $\forall x,y \in \mathbb{R}^n, \forall \lambda \in [0,1]$ we have $ \lambda f(x) + (1-\lambda)f(y) \le f( \lambda x + ...
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2answers
50 views

Proof that Right hand and Left hand derivatives always exist for convex functions.

Definition A function $f$ is convex on an interval if for $a,x, \text{and} \;b$ in the interval with $a\lt x\lt b$, we have $$\frac{f(x)-f(a)}{x-a}\lt \frac{f(b)-f(a)}{b-a}.$$ While reading the ...
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Is the minimum of a parametric convex function convex again?

Let $I$ and $J$ be compact intervals. Let $f:I\times J\to\mathbb R$ be differentiable and strictly convex. Is the function $g:I\to\mathbb R$ defined by $$ g(x) = \min_{y\in J} f(x,y) $$ ...
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2answers
84 views

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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1answer
29 views

Compact set has supporting hyperplane parallel to any hyperplane $H$?

Let $E$ be a non-empty compact set and $H$ is any hyperplane. Show that $E$ has a supporting hyperplane parallel to $H$. I have no idea to proceed the proof. Can anyone give me some hints? Thanks ...
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1answer
76 views

Example of sum of log-concave is not always log-concave

I know that sum of log-concave is not always log-concave. Could anyone provides me with an example to prove this? Like probability distribution fn (pdf) of normal distribution is log-concave; on ...
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2answers
28 views

Convexity of space of integrals of homotopic paths

Let $B$ be a Banach space. Consider a set $Q \subset B$, a path $\gamma_0:[0,1] \to Q$ and the set of paths $\Gamma_0$ of paths $[0,1] \to Q$ homotopic (with ends fixed) to the path $\gamma_0$. Let ...
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21 views

Interpolating a convex conjugates

Let $\|\cdot\|_2$ denote the 2 norm on $\mathbb R^N$. i.e. $\|x\|_2=\sqrt{x_1^2+x_2^2+\cdots+x_N^2}$. Then for any $\gamma \in (0,+\infty)$, we define $$ |x|_\gamma:= \begin{cases} ...
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3answers
28 views

strictly convex functions and limits

Suppose I have a strictly convex function $f(x)$ for $x\geq 0$, with $f(0) = 0$, $f'(0) =0$ and $f''(0) >0$. Is it obvious that $f$ must be superlinear as $x\to +\infty$? Alternatively, how can I ...
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Show that the set $\{y_1a_1+y_2a_2: -1\le y_1,y_2\le 1\}$ is a polyhedron

Show that the set is a polyhedron and express it in the form: $S = \{Ax\leq b, Fx = g\}$, $S=\{y_1a_1+y_2a_2 | -1\leq y_1\leq 1, -1\leq y_2\leq 1\}$ where $a_1,a_2\in\mathbb{R}^n$ My attempt: A ...
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250 views

Subsets of $\mathbb{R}^2$ that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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1answer
51 views

Partitioning $\mathbb{R}^d$ with two convex sets

The problem/puzzle is: Find two convex sets in Euclidean space, $A, B\subseteq\mathbb{R}^d$, such that the number of connected components of $\mathbb{R}^d\setminus (A\cup B)$ is the maximum ...
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1answer
23 views

Why is this transformation convex?

Let $f:\mathbb{R}^n \to \mathbb{R}\cup\{ \infty \}$ be convex. It's claimed that this implies $g:\mathbb{R}^n \times (0,\infty) \to \mathbb{R}\cup\{ \infty \},(x,y)\mapsto yf(\frac{x}{y})$ is convex. ...
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sum of fractional functions optimization problem

Consider the following sum of fractional functions optimization problem $$ \begin{array}{l} \mathop {\min }\limits_{\bf{x}} \,\,\,\sum\limits_{i = 1}^p {\frac{1}{{{\bf{a}}_i^T{\bf{x}} + {b_i}}}} \\ ...
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1answer
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Polyhedral cone as conic hull of a finite set

I am reading notes on optimization and it was claimed that all polyhedral cones in $K\subseteq \mathbb{R}^n$ can be written Cone(R) where $R\subseteq \mathbb{R}^n$ is a finite set. That is, if K is a ...
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Convexity of Log Determinant of Function

Given a function $g(x): \mathbb{R} \to \mathbb{R}^{N x N}$, under what circumstances is $f(x) = - \log \det g(x)$ a convex function? Assume that each of the $N^2$ entries in $g(x)$ is convex over ...
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27 views

Biconjugate of a nonconvex function

Is biconjugate of a non-convex function, the tightest lower bound on that function? If yes why?
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Inner construction of polyhedral set: cone part uniquely determined

The "inner construction" of a polyhedral set is: given $V,R\subseteq \mathbb{R}^n$, $|V|,|R|\in \mathbb{Z}^+$ (nonempty, finite), put $S:=Co(V)+Cone(R)$. It was claimed that Cone(R) is uniquely ...
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1answer
24 views

Where can I find the nowhere subdifferentiable example of rockafellar?

I'm told that Rockafellar gave an example of a real extended function defined on a locally convex space, whose subdifferential is empty at each point of its domain. The function is proper, lower ...
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Joint Convexity Proof

Let $x$ be $n \times 1$ vector and $Y$ be $n \times n$ matrix. Prove that $f(x,Y) = x'Y^{-1}x$ is jointly convex in $x$ and $Y$ when $Y \succ 0$.
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1answer
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How to define a “line” and “symmetry w.r.t. a line” in $L_2(\lambda)$ space

For any $x,y\in L_2([0,1],\lambda)$, define the inner product $\langle. , . \rangle$ by \begin{equation} \langle x, y \rangle=\int_{[0,1]} x(t) y(t) \lambda (dt) \end{equation} Is it proper to ...
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Prove that $f(x,y,z)=\ln x-\alpha\ln y-(1-\alpha)\ln z$ is quasiconvex

I am trying to show that $f(x,y,z)=\ln x-\alpha\ln y-(1-\alpha)\ln z$ defined for $x,y,z\in\mathbb{R}^+$ and $\alpha\in(0,1)$ is quasiconvex. This is equivalent to showing that the set ...
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1answer
483 views

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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409 views

Is second derivative of a convex function convex?

If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
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Is mutual information convex in the joint distribution?

Assume some joint distribution $P(X,Y) = P(Y|X)P(X)$. It is well know that, for fixed $P(Y|X)$, mutual information is a concave function of $P(X)$ and, for fixed $P(X)$, a convex function of $P(Y|X)$ ...
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invex functions and their usefulness?

An invex function $f$ is a differentiable function from $\Bbb R^n$ to $\Bbb R$ that for some function $\eta : \Bbb R^n \times \Bbb R^n \to \Bbb R^n$ satisfies for all $x, u$, $f(x) - f(u) \geq \eta(x, ...
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1answer
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Continuity of convex functions that have continuous restrictions to closed subspaces

Let $X$ be an infinite-dimensional normed vector space , let $U\subset X$ be an infinite-dimensional closed subspace, and let $f:X\to[0,\infty)$ be convex. Question: If the restriction $f|_U$ is ...
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48 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
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46 views

Practical application of Gauss-Lucas theorem

Let $z_1,z_2,z_3 \in \mathbb C$ pairwise distinct be the affix of points $A, B$ and $C$. Let $P(x)=(x-z_1)(x-z_2)(x-z_3)$. Let $z_4$ and $z_5$ be the roots of $P'$ (with the possibilty that ...
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44 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
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28 views

Upper bound using a convex function

Let $g, f: K\times S \to \Bbb R$ be convex and continuous functions on compact and convex sets $K,S \subset \Bbb R^n$. Does there exist a differentiable strongly convex function $h$ on $S$ (w.r.t. ...
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0answers
33 views

is $\ell^1$-norm a strongly convex function? [closed]

Is $\ell^1$-norm defined by $|x|_1:=\sum |x_i|$ where $x \in \Bbb R^N$ a strongly (or strictly) convex function? I think, as "strong convexity" is defined, the $L^1$-norm has $f''=0$ (second ...
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1answer
34 views

Complex analysis question regarding Cauchy's integral formula and holomorphic functions

Let $U \subseteq \Bbb{C} $ be an open convex subset of $\Bbb{C}$ We also assume that $\partial U$ is smooth. We fix a point $z_0 \in U.$ Using Cauchy's integral formula, show that $$\lvert ...
4
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1answer
35 views

Area of the lattice generated from $(n, n\sqrt{2} \mod 1)$

I plotted $\Big\{ (n, n \sqrt{2} \, \mathrm{mod} \,1) \;\Big| -50 \leq n \leq 50 \Big\}$ and even though the $n \sqrt{2}$ is a line, the pattern that emerges is a lattice. What is the basis of this ...
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1answer
28 views

Finding KKT conditions for nonlinear optimization problem.

I have an optimization like below: $\text{ minimize } \sum_k - \log_2 x_k $ $\text{subject to: } x_k \leq q , k =1,2, \cdots, N .$ I can form the Lagrange of the problem as below: $L(x, ...