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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Generalization of Brouwer’s fixed-point theorem

Perhaps the most widely known version of Brouwer’s famous fixed-point theorem reads as follows: For any $n\in\mathbb N$, let $A\subseteq\mathbb R^n$ be a compact (with respect to the Euclidean ...
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1answer
41 views

Open ball around a point in the hyperplane in a locally convex space

Let $f$ be a continuous linear functional in a locally convex space $X$ and let $z\in X$ so that $f(z)=\alpha$. Let $V_z$ be an open neighborhood around $z$. It is my intuition that $V_z$ should ...
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17 views

Probability that a point lies in an uncertain convex hull

Given $n+1$ independent random vectors $X_i \sim N(\mu_i,\Sigma_i)$, where each $\mu_i \in \mathbb{R}^n,$ let $C$ denote the random region formed by taking the convex hull of a realization of the set ...
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28 views

Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question ...
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1answer
54 views

Minkowski sum and vectors

Problem: Given two convex polygons A, B, we can define Minkowski sum, as A + B = {a + b: a $\in$ A, b $\in$ B}, where a + b vector sum. Prove that: for every external perpendicular u to an edge of ...
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119 views

Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
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1answer
32 views

A convexity argument

Let $(\alpha_n)$ be a sequence of positive real numbers s.t. $\sum \alpha_n=1.$ Consider a sequence of complex numbers $(\beta_n)$ s.t $|\beta_n|=const$ for all $ n \in \mathbb{N}.$ Suppose that ...
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53 views

convexity of a nonlinear function

I tried to search similar answers to my problem here, but unfortunately I'm a little bit lost on this subject, originally from a physical problem, but can be stated as: Let $A\subset R^3$ be a ...
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1answer
36 views

Functions mapping convex sets on convex sets

A function $f:\Bbb R^n\to\Bbb R^m$ is said that maps convex sets on convex sets if it satisfies: (1) the image $f(A)$ of any convex set $A$ is a convex set; (2) the preimage $f^{-1}(B)$ of any convex ...
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3answers
38 views

Convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant 1$

I would like to ask the convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant1$. We know that $y\geqslant\frac{1}{x}$ is convex for $x>0$. But if we transform the inequality into ...
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1answer
37 views

How to prove conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u)

I don't know how to prove that if $ M \subseteq R^n, \forall u \in R^n $ then conv(M)+u=conv(M+u) and Aff(M)+u=Aff(M+u). conv is convex hull and Aff is affine hull. Yes it is a homework question, ...
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31 views

If we know the convex conjugate of $f(x,y)$, what can we say about the conjugate of $f$ in $x$?

Say $f^*(x,y)$ is the convex conjugate of $f(x,y)$. Now take $g_{y_0}(x) := f(x, y_0)$. Is there any relationship between $g^*_{y_0}(x)$ and $f^*(x, y_0)$?
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32 views

Subset relation between convex cone and its dual

Let $C$ be a convex cone and $C^*$ its dual cone. It seems for me that either $C\subseteq C^*$ or $C^* \subseteq C$ at least in 2 dimension. Is it correct? if so, is it the case also for higher ...
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56 views

Convex dense subset of $\Bbb{R}^n$ is the entire space

Say we have a convex dense set $X\subset\Bbb{R}^n$, does it follow that $X=\Bbb{R}^n$ ? For $n=1$ it's true because convex set of real numbers are intervals, and if it's dense then it's $\Bbb{R}$. ...
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1answer
17 views

Optimality condition

I was looking at a few results of convex optimization and I'm stuck with a part of a proof. Consider the following minimization problem: \begin{align} \text{minimize} \quad &\Phi(x) \\ ...
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17 views

linear inequalities and reduction

Let $X\subset \mathbb R^m$ be a convex compact subset. Let $E=\{-1,0,1\}^m$. What would be the necessary or sufficient conditions on $X$ such that for any $f\in [-1,1]^m$, there exists at least ...
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15 views

Find maximum value of ${a}_{0}=A$ ${a}_{n}=B$

We have numbers $A$ and $B$, sequence ${a}_{0},\cdots,{a}_{n}$ such that: ${a}_{0}=A$ ${a}_{n}=B$. All greater than 0. Find maximum value of $\prod_{i=1}^{n}\frac{{a}_{i}}{{a}_{i-1}+{a}_{i}}$ Hint: ...
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1answer
106 views

closure of a convex set in a normed linear space is convex ?

Is it true that if $A$ is a convex set in a normed linear space $V$ , then the closure of $A$ is also convex ? (I know that the interior is convex )
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29 views

Convexity of distance function

For any $n \in \mathbb{N}$, $a \in \mathbb{R}$ with $a > 1$ and $k_i > 0$ for $i = 1,\ldots,n$ define the following function: $$f: \mathbb{R}_{>0}^n \to \mathbb{R}, x \mapsto \sum_{i=1}^n ...
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47 views

general definition of concavity, mean-preserving spread and concavity

The usual definition of concavity is: for any $x_1$ and $x_2$ and any $t\in[0,1]$, $$f(tx_1+(1-t)x_2)\geqslant tf(x_1)+(1-t)f(x_2).$$ I am wondering how to generalize this definition to more than 2 ...
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1answer
28 views

Convexity, Hessian matrix, and positive semidefinite matrix

I would like to ask whether my understanding of convexity, Hessian matrix, and positive semidefinite matrix is correct. For a twice differentiable function $f$, it is convex iff its Hessian $H$ is ...
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1answer
30 views

Convexity of $f(x,y)=\frac{x}{y^2}$

I would like to ask the convexity of function $$f(x,y)=\frac{x}{y^2}$$ where $x\geqslant0, y>0$. Since $f(x,y)$ is differentiable but not twice differentiable, I used the first order condition ...
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20 views

How to perform a quasiconvex optimization

I have a quasiconvex objective function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ which I would like to minimize over a simplex $S\subseteq \mathbf{R}^n$. I have looked pretty hard but have been unable ...
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3answers
66 views

Convexity implies absolute continuity?

The following is taken from an exam: $f:[a,b]\rightarrow\mathbb{R}$ is convex implies $f$ is absolutely continuous (recall $f'$ exists a.e.) One has local Lipschitz-ness by convexity, but how to ...
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21 views

function that is higher at constant sequence than random sequence is concave

I have a T-sequence $x=(x_1,x_2,...,x_T)$ where each $x_i$ random, but they have same expectation $c$, i.e., $E[x_i]=c$, for all $i$. Another T-sequence that is constant, i.e., $x_i=c$, for all $i$. ...
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1answer
51 views

What is the closed form for this norm $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$?

I read a paper, which has the equation above $\|Z\|=\sup_{\|u\|=1,\|v\|=1} u^TZv$. Is this a well known norm and what is the closed form for this norm. Also what does this norm describe? Thanks in ...
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1answer
20 views

A lower semi-continuous convex function being not continuous on its domain

Let $f : \mathbb{R}^N \longrightarrow \mathbb{R} \cup \{+\infty \}$ be a lower semi-continuous convex proper function. Let $dom f$ be the domain of $f$, i.e. $dom f:= \{ x \in \mathbb{R}^N \ | \ f(x) ...
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Is the subdifferential always convex and closed set?

Two properties of the subdifferential set are stated as follows: Given a function $f : \mathbb{R}^n → \mathbb{R}$, (i) the subdifferential set $\partial f(x)$ is always convex and closed, even if ...
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Whether it is jointly convex or not?

I know that $f(x, Y)=x^HY^{-1}x$ is a convex function on $x$ and $Y$ jointly, where Y is positive definite. Now, if $Y=\Sigma_{n=1}^{N}A_nSA_n^H+I$, where $S$ is positive semidefinite and $I$ is the ...
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1answer
40 views

Strong duality of SDPs

On pp. 654 of Boyd's book, it is claimed that strong duality holds between the SDPs B.2 and B.3 (at the bottom of this page). Does it require additional assumption that one of them is strictly ...
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Existence of affine hull of set S

"Obviously, the intersection of an arbitrary collection of affine sets is again affine. Therefore, given any $S \subset R^{n}$ there exists a unique smallest affine set containing $S$ (namely, the ...
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1answer
55 views

Lipschitz-constant gradient implies bounded eigenvalues on Hessian

I've read in a few places that if we have a Lipschitz gradient $$\|\nabla f(x) - \nabla f(y)\|\leq L\|x-y\|,\, \forall x,y, $$ we can equivalently say $\nabla^2f\preceq LI.$ But I'm having a hard ...
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Pricing Function is convex

I am now reading Alternative Characterization of American Put Options by Carr et all (available at http://www.math.nyu.edu/research/carrp/papers/pdf/amerput7.pdf). There is a theorem called 'Main ...
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Extreme points of the closed unit sphere in $\big(C_K([0,1]),\|\;\|_\infty\big)$

Lets take $(C_K([0,1]),\|\|_\infty)$, which is a normed vector space. $$ C_K([0,1])=\{x:[0,1]\to K\;\;|\;x\text{ is continuous } \}\\ \|x\|_\infty=\sup_{t\in[0,1]}|x(t)| $$ So, since the closed unit ...
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1answer
58 views

Minkowski functional being homogeneous

Let $(X,\|\|)$ a normed vector space over $K$ and $E\subset X$ convex and absorbing. Let $p_E(x)=\inf_{x\in tE}\{t>0\}$, $E_1=\{x\in X:p_E(x)<1\}$ and $E_2=\{x\in X:p_E(x)\le 1\}$. I want to ...
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24 views

Convexity of a complex function

I have a problem in which I need to find the minimum of the function $f:\mathbf{C}\rightarrow\mathbf{R}$ given by $$f(s) = v^*M(s)^*M(s)v$$ with $v \in \mathbf{C}^{n+1}$ and so that $|v_i|$ ...
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1answer
23 views

Three Minkowski functionals resulting the same

Let $(X,\|\|)$ a normed vector space over $K$ and $E\subset X$ convex and absorbing. Let $p_E(x)=\inf_{x\in tE}\{t>0\}$, $E_1=\{x\in X:p_E(x)<1\}$ and $E_2=\{x\in X:p_E(x)\le 1\}$. I want to ...
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10 views

Two characterizations for the interior and closure of a convex and absorbing set when the Minkowski functional is continuous

Let $(X,\|\;\|)$ be a normed vector space over $K$ and E$\subset X$ be convex and absorbing. Lets define $E_1=\{x\in X:p_E(x)<1\}$ and $E_2=\{x\in X: p_E(x)\le 1\}$. I want to prove that: $$ ...
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Minkowski functional characterization for convex and absorbing sets

Let $(X,\|\;\|)$ be a normed vector space over $K$. Let $E\subset X$ be convex and absorbing. And let $E_1=\{x\in X:p_E(x)<1\}$, $E_2=\{x\in X: p_E(x)\le 1\}$; where $p_E(x)=\inf_{x\in ...
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1answer
22 views

Boundary of a convex body

I need your help in defining a boundary for a given convex body, $P$ without knowing the shape of $P$ meaning that I can't say that $P$ is a polygon or any shape which is convex. Meaning that by ...
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49 views

Prove a linear combination of a convex set is convex

Suppose $S$ is a convex subset of $\mathbb R^n$ , and suppose $T: \mathbb R^n \rightarrow \mathbb R^m $ is any linear transformation. Prove that the set $\,\{T(x)\,|\,x \in S \}$ is also convex. ...
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Caracterization of a convex set

Let $X$ be a vector space over $K$. I want to prove that: $$ E\subset X\text{ is convex } \Leftrightarrow (s+t)E=sE+tE\;\;\forall s,t\ge 0 $$ I'm trying the $(\Rightarrow)$ part and I've already ...
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1answer
61 views

How to prove that for a nonempty convex subset $S \subset X$ ($X$ is normed vector space) it is true that $\partial \overline{S} = \partial S$?

I am having trouble with the concept of convexity. This is the statement I'm trying to prove. Let $X$ be normed vector space. If $S \subset X$ is convex and $S^\circ \neq \emptyset$ then $\partial ...
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1answer
25 views

Factor space norm calculation when the subspace is finite-dimensional

Let $(X,\|\;\|_X)$ be a normed vector space and let $M$ be a closed finite-dimensional subspace of $X$. I want to prove that: $$ \forall x\in X\;\;\exists\;m\in M\text{ s.t. } ...
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68 views

Is the intersection of 2 convex hulls a convex hull?

Given two finite sets of points, $X$ and $Y$, in $\mathbb R^d$ and assuming that $\text{conv}(X)\cap\text{conv}(Y)\neq\varnothing$ I would guess that the intersection is a convex hull of some other ...
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1answer
15 views

Can we say anything about the minimum of a perspective function compared to that of the original function?

Given convex function $f(x)$, its perspective function is $g(x,t) = tf(x/t), t>0$ is also convex. Is the minimum of $g$ over $(x,t)$ always less than (or larger than) the minimum of $f(x)$? Note ...
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1answer
23 views

Subgradient inequality for strongly convex functions

I need some help to follow the argument made here which says that $$ f(x_t) - f(x^*) \geq \frac{\alpha}{2}\|x_t-x^*\|^2 $$ if $f$ is $\alpha$ strongly convex and $x^*$ the minimizer of $f$. From the ...
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2answers
44 views

Is the function $f(x,y) = \frac{ax+by}{1+cx+dy}; a,b,c,d>0 $ convex?

$$f(x,y) = \frac{ax+by}{1+cx+dy}; a,b,c,d>0$$ Also please suggest an easy way to determine the convexity of such functions? I would also appreciate if I can numerically verify it quickly (instead ...
2
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1answer
66 views

An inequality regarding the derivative of two concave functions

Let $f:\mathbb{R}^+\to \mathbb{R}^+$ and $g:\mathbb{R}^+\to \mathbb{R}^+$ be two strictly increasing, strictly concave and twice differentiable functions with $f(0)=g(0)=0$ and $f'(0)=g'(0)>0$. We ...
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21 views

Formal definition of a face of a polyhedron

Given an $n$-dimensional convex polyhedron, an $(n-1)$-dimensional face of it can be defined as an intersection of the polyhedron with a supporting hyperplane. What is the formal definition in the ...