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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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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139 views

Can someone confirm this sketch of a proof for the existence of smooth partitions of unity on an unbounded convex open euclidean set?

EDIT!! The problem originally described (see below) has been reduced to the correctness of a simple extension of an argument from Rudin's PMA. Feel free to skip to the proposed solution, below. As ...
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2answers
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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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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29 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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48 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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111 views

Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
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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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21 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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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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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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22 views

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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10 views

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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14 views

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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56 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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28 views

interchange of convex hull operation and intersection

Let $A^{\epsilon}$ be a set. Let $\overline{co}(A)$ be the closed convex hull of $A$, i.e., the smallest convex set that contains $A$. My question is under what condition, the following is true ...
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24 views

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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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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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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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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2answers
18 views

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
16 views

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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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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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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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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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 ...
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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 ...
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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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67 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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12 views

Log-concave function changes when scalar is added

A function $f$ is log-concave if $\log(f)$ is concave. Intuitively, one might guess that adding a scalar to a function would not affect properties like concavity, log-concavity, quasi-concavity etc., ...
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18 views

Subgradient of function of two variables

i do not have any experience in convex analysis and I would be most grateful if you would help me with the concept of subgradient. I get the concept of subderivative (one dimension) but it is hard ...
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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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33 views

Prove that $tx+(1-t)x \ge x^ty^{1-t}$

Given conditions are $x>0$ $y>0$ and $0 \le t \le 1$ There is a hint given which says $Log$ is a concave increasing function. How do I apply this here? There is also a generalization of this ...
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Show $A=\{(x,y) \in \mathbb{R^2} \mid x^2+2y^2<2p\}$ is convex

Show $A=\{(x,y) \in \mathbb{R^2} \mid x^2+2y^2<2p\}$ is convex (i.e. $(a,b) \in A \implies ta+(1-t)b \in A\ \forall\ 0\leq t \leq 1$. I have $x_1^2+2y_1^2 <2p$ and $x_2^2+2y_2^2 <2p$ for ...
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If F is convex and lower semicontinuous in norm, then F is weakly lower semicontinuous

I have to prove the following statement: If $F$ is convex and lower semicontinuous in norm, then $F$ is weakly lower semicontinuous.
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37 views

Quasiconvexity of a function on the positive orthant using rays

It is seen in Boyd's book on Convex Optimization book that to show a function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ is quasiconvex, it is enough to show that $f$'s restriction to a line is ...
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Is a $k$-differentiable convex function $k$-continuously differentiable?

It is known that a differentiable convex function is continuously differentiable. Is a $k$-differentiable convex function $k$-continuously differentiable?
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22 views

convex hull of union of positive definite matrices

Is it true that any element of ${\rm co}\Big\{\bigcup_{x \in [a,b]} S(x) \Big\}$ is in $\mathbb{S}_{> 0}^n$ (cone of positive definite $n \times n$ matrices), given that $S(x) \in \mathbb{S}_{> ...
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Convexity of n $4u^4x^{2.5}y^{-5}$ over $x>0, y >0$ and $u>0$

I am trying to find if the function $4u^4x^{2.5}y^{-5}$ is convex over $u>0, y>0$ and $x>0$. The thing which comes to my mind immediately is to check the positive semi-definiteness of the ...
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10 views

Property of a $C^\infty$ convex function

Hey guys I need your help. Let $\Omega$ be a bounded, 2 or 3 dimensional domain with smooth boundary. Let $c\in H^2(\Omega)$ with Neumann boundary conditions. We define ...
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Supporting hyperplane of a polarity of a convex body.

Recently, I am studying in combinatorial convexity and related topics. I use the book "Combinatorial Convexity and Algebraic Geometry" (GTM 168) as my main reference. The book is very good, all the ...
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29 views

Are these two optimization problems equivalent?

I have two problems as follow. $min_x: ||x-y||_2^2 + \lambda_1 ||x|| \quad \ \ (1)$ and $min_x: ||x-y||_2^2 + \lambda_2 ||x||^2 \quad (2)$ Here $||\cdot||$ could be any norm and ...
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35 views

Explain why for an extreme point some inequalities in the constraints become equalities

Suppose we have a linear program $$ \max_x c^Tx\\ \text{subject to } Ax \leq b $$ where $\leq$ denotes pairwise inequality, i.e. $a^T_i x \leq b_i, i = 1,..., n$. If $y$ is an extreme point, why is ...
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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
21 views

Convexity of the Pareto front: formal definition

Does anyone have a reference to a formal definition of what convexity of a Pareto front in multiobjective optimisation means? All literature I've seen uses the term without defining it.
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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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27 views

Get the global minimum with functions convex in a subset of the domain; numerical methods.

I have a $C^\infty$ function $f:\mathbb{R}^n\to \mathbb{R}$ that is positive and known to have a zero and a global minimum in an unknown point $x$. Furthermore, $f$ is convex in the set $$ S = ...
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66 views

To show a closed convex set $S \subseteq R^n$ is bounded if and only if $S$ contains no rays.

I want to show that a closed convex set $S \subseteq R^n$ is bounded if and only if $S$ contains no rays. Where $r \in S$ is a ray of $S$ if $x \in S$ implies that $x+\mu r \in S$ for all $\mu \in ...