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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Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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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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rconvex image under nonlinear function

Let $X\subset R^3$ be a compact and convex set, and let $f: X\rightarrow R^3$ be a nonlinear function, with $f\in C^k$. What are the tools to investigate if the image $K=f(X)$ is also convex, in the ...
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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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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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122 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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29 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 convex-...
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57 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 A,...
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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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39 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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Required conditions for product of two submodular functions to be submodular?

3.32 of Boyd's Convex Optimization book says: Products and ratios of convex functions. In general the product or ratio of two convex functions is not convex. However, there are some results that ...
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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 $$xy\geqslant1(x,...
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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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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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42 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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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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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) \\ \text{...
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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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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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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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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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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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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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117 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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38 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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25 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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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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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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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 $f$...
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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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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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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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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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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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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: $$ \text{...
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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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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 tE}\{t>0\}...
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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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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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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. } \|[x]_M\|_{X/M}=\|x+m\|...
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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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45 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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22 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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69 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., ...