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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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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31 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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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
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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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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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1answer
24 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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62 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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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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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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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 ...
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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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68 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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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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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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27 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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36 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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17 views

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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30 views

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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23 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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28 views

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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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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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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70 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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22 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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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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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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68 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 ...
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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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38 views

A question about Hilbert Spaces and convex sets

I am struggling with this and could really do with some help: Let $H$ be a Hilbert space over $\mathbb{R}$, $\{v_n\}$ be a sequence of vectors in $H$, and $C$ be a convex subset of $H$ containing ...
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Jensen's inequality; what's the need for the probability measure?

Jensen's inequality states that if: $\mu$ is a probability measure on $\Omega$, $f$ is integrable function (on $\Omega$) and $\phi$ is convex on the range of f then: $\phi \left( ...
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Is $f(x)$ convex if $\log f(x)$ is convex?

One of the convex composition rules states that $h(g(x))$ is convex if $h(x)$ is convex and non-decreasing, and $g(x)$ is convex. Now I want to go the other way - I know that $\log(f(x))$ is convex ...
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66 views

Convex and continuous function on compact set implies Lipschitz

Let the function $f: C \rightarrow \mathbb{R}$ be convex and continuous, where $C \subset \mathbb{R}^n$ is a compact set. Prove or disprove that $f$ is Lipschitz continuous on $C$. Comments: If $f$ ...
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52 views

what is the closed form solution for $\min_x ||y-x||^2_2+\lambda ||x||_2$

$y$ and $x$ are vectors. $\|\cdot\|_2$ is Euclidean norm. In one paper I read, they say the closed form solution is $x=\max\{y-\lambda \frac{y}{\|y\|_2}, 0\}$. I don't know why $x$ need to be ...
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55 views

$f'$ decreasing everywhere but not defined in one point. Is $f$ concave?

Small issue: Suppose that $f:[a,b] \rightarrow \mathbb{R}$ is a continuous function, differentiable except on a finite set of points, let say in one point $y$. For $x<y$ and $x>y$ we have ...
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1answer
43 views

Is the set of probability density functions convex?

Given is the set of probability density functions defined as $P:=\left \{ p(x)\mid p(x)\, is\ a\ probability \ density \ function \right \}$ Is $P$ a convex set? I am not sure that here i have to ...
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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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Sufficient condition for self-concordance?

I've revised a previous question that was ill-formed. Consider the following two definitions. Def'n 1 (Lipschitz continuity of Hessian): A function $f:\mathbb{R}^n\to\mathbb{R}$ is said to have a ...
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37 views

Log convexity and the gamma function

I am writing an essay on the gamma function. I have learnt and understood convex theory and how the log-convex nature of the gamma function makes it a unique extension of the factorials ...
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93 views

Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
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1answer
32 views

Boundedness of sublevel sets of an integral function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an increasing function, i.e., $x<y \Rightarrow f(x) < f(y)$ for all $x,y$. Assume that $\lim_{|x| \rightarrow \infty} |f(x)| = \infty$ Define ...
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53 views

convex polygon triangulation

Suppose we have given a convex polygon on $n$ vertices $P= \{ a_1, \cdots , a_n \}$ in the plane (arranged clockwise). How can we prove that there exist atleast two indices $i$ such that circle ...
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1answer
52 views

Is this $\underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y) $ correct?

$\underset{x<1,y<1}{\sup}f(x,y) = \underset{x<1}{\sup} \underset{y<1}{\sup}\ f(x,y).$ Intuitively, I think the above equation holds for all $f(x,y)$. Am I right?