Questions tagged [convex-analysis]

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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18 views

Convex optimization - comparing optimal values (any counter example?)

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1 $ and $ \Omega_2$...
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1answer
25 views

If a product of a Lipschitz function and another bounded function is Lipschitz, is the other function Lipschitz?

Let $x=[0,1]^n,\;p: [0,1]^n \rightarrow [0,1]^n$. Define $V(x) = p(x).x$, i.e. the inner product of $p(x)$ and $x$. We are given $V(.)$ is convex. Can we say $p(x)$ is Lipschitz? My approach: $V(.)$ ...
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How to solve the optimization problem? (any counter example?)

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1 $ and $ \Omega_2$...
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how to prove chance constraint with two random parameters is a convex set?

Consider the following chance constraint: $$P(A_{\xi}x \leq b_{\xi}) \geq \eta, $$ where $\eta \in (0, 1)$ and $\xi$ is a random vector following a discrete uniform distribution. $A_{\xi} \in R^{m\...
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1answer
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Proving $u = \text{prox}_f(x) \iff x - u \in \partial f(u)$

I want to show that $u = \text{prox}_f(x) \iff x - u \in \partial f(u)$, where $f$ is a proper, convex and lower semicontinuous function where $\partial$ is the subdifferential and $$ \text{prox}_f(x) ...
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3answers
49 views

Show $M=\{(x,y):y\geq x^2\}$ is a convex set

Show that the following set is convex. $$M = \left\{ (x,y) : y \geq x^2 \right\}$$ First I take two arbitrary points $(x_1,y_1)$ and $(x_2,y_2) \in M$. Then I need to show that $$\lambda(x_1,y_1)+[1-\...
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Does a closed convex function with open domain tend to infinity as it approaches the boundary?

If we have a closed differentiable convex function $f$ with open domain, given a sequence $\{x_{k}\}$ such that: $$ \{x_{k}\} \subset \text{dom}f: \: x_{k} \rightarrow \bar{x} $$ where $\bar{x} \in \...
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11 views

Proof convexity of multi-stage dynamic program

Optimization problem A paper describes the maximization of an objective function $$R = \sum_{i=1}^n \pi_i(p_i, x_i, y_i)$$ by deciding on $p_i, x_i \forall i = 1, ..., n$. $y_i$ is defined as $y_i = ...
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40 views

Is $2x + y \geq k$ a convex set?

I'm asked to prove that $2x+y \geq k$ is a convex set. My teacher told me to take two points $(x_1,y_1)$ and $(x_2,y_2)$ within the set. Then he showed two inequalities: $2 x_1 + y_1 \geq k$ $2 x_2 + ...
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21 views

Convexity condition for smooth submanifold-with-boundary in $\mathbb{R}^n$

One can prove that a differentiable function $f\colon \mathbb{R}\to \mathbb{R}$ is strictly convex if and only if for each $a\in\mathbb{R}$ we have $$f'(a)(x-a)+f(a)\leq f(x)\qquad\text{for all }x\in\...
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Faces of the Cartesian product of two non-polyhedral facially exposed convex cones?

Let $C_1$ and $C_2$ be convex cones in $\mathbb{R}^n$, such that every face of $C_i$ has the form $H\cap C_i$, where $H$ is a supporting hyperplane to $C$. My question is: Is it possible to ...
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1answer
22 views

How to check convexity of the following constraint $w_1^2 + w_2^2 - w_2^2\alpha + 1 \ge \eta$?

Usually for function of two variable checking the convexity of a constraint is quite simple since we just need to compute the Hessian. In this case, I am running into a problem where my constraint is ...
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1answer
14 views

Construct certain pairs of $n$-dimensional convex sets.

In $n$-dimensions, construct a convex set $A$--not a hypersphere--having an inscribed convex set $B$ of volume $p$ times that of $A$, for $n=9$, $p =\frac{29}{64}$, $n=15$, $p=\frac{8}{33}$, $n=27$, $...
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1answer
18 views

Integral representation of a convex function

I have a doubt about a step used in the proof of the following theorem (Rao and Ren, Theory of Orlicz Spaces): Let $\phi:(a,b)\to\mathbb{R}$ a function. $\phi$ is convex if and only if for each closed ...
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2answers
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Sum of convex function and increasing function [duplicate]

I have a sum of increasing function and a convex function over some domain. Can I say that the sum is also a convex function ? Or when can i say that sum of convex function with increasing function is ...
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1answer
36 views

Do a set of vertices uniquely determine a polytope?

The question in title arises because I am trying to prove that a polytope is the convex hull of its vertices, i.e., $\mathcal{P}=conv(V)$. Here is how far I have got. Convex hull of a finite set of ...
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8 views

the convexity about a special discrete function

Assume that $f:\Re\rightarrow \Re$ is convex, and random variables $X_1,X_2,\dots,$ are nonnegative and independently and identically distributed. Prove that $g(n)=E[f(\sum_{i=1}^n X_i)]$ is convex on ...
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1answer
29 views

$f(x)$ is convex if the perspective of $f(x)$ is convex?

In pages 89 of Boyd & Vandenberghe's Convex Optimization, It says if $f: \mathbb{R}^n\rightarrow \mathbb{R}$ is convex, then its perspective $g: \mathbb{R}^{n+1}\rightarrow \mathbb{R}$ defined by $...
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1answer
14 views

What is the dual norm of infinity norm ball with a non-unity radius?

The dual norm of an infinity norm ball with a unit radius can be expressed as $$\|x\|_1 = \underset{y \in \mathbb{R}^n,\|y\|_{\infty} \le {\color{blue}1}}{\sup }x^Ty$$ What is a dual norm of a non-...
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1answer
60 views

Does a strictly convex extended-real-valued function attain a minimum on a compact convex set?

The question says it all. I'd be most interested in hearing the answer for an arbitrary topological vector space, though functions on $\mathbb R ^n$ are also of interest. I know that if a strictly ...
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12 views

Dual cone's dual cone is the closure of primal cone's convex hull

Assume $K$ is a cone and its dual cone is $K^* = \{y:x^Ty \geq 0,\, \forall x \in K\}$. Then we have $K^{**} = \text{cl}(\text{conv}\ K)$, where cl means closure, conv means convel hull. How to prove ...
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1answer
44 views

The other side of Jensen's inequality

For a convex function $f: X \to \mathbf{R}$, there is the famous Jensen's inequality $$f\left(\frac{\sum_{i=1}^n x_i}{n}\right) \leq \frac{\sum_{i=1}^n f(x_i)}{n}$$ Is there a lower bound to $f\left(\...
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1answer
15 views

Convexity bound in Lieb and Loss.

I'm currently reading through Lieb and Loss's Analysis text. At the end of the proof of theorem 1.9 the authors prove the inequality $$ \left( |a|+|b|\right)^{p}\leq(1-\lambda)^{1-p}|a|^{p}+\lambda^{1-...
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1answer
19 views

Area under the graph of a convex function

Consider the following problem: Suppose that $f$ is a twice differentiable real function such that $f''(x)>0$ for all $x\in[a,b]$. Find all numbers $c\in[a,b]$ at which the area between the graph $...
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2answers
57 views

What are the subdifferentials $\partial f(0)$ and $\partial f(1)$?

Let $ f: \mathbb{R} \to \mathbb{R} $ given by \begin{equation*} f(x) = \left\{ \begin{array}{rl} x \log x -x & \text{if } x \geq 0\\ \infty & \text{if else}\\ \end{array} \right. \end{equation*...
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1answer
23 views

Volume of convex body as an integral of its radial function

Let $C$ be a compact convex set in $\mathbb{R}^d$. Let the origin $O$ by in the internal of $C$. The gauge function $\gamma_C(.) : \mathbb{R}^d \to [0, \infty]$ of $C$ is defined as $$ \gamma_C(x) = \...
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1answer
47 views

convex sets in convex optimization

How to prove that the following set is not convex? $$M = \left\{ \mathbb{R}^{3}: x_{1}x_{2}x_{3}\le 1,x_{1}+x_{3}\ge 2,x_{1} \ge 0 \right\}$$ Thanks for any help. I tried to write it down as ...
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1answer
44 views

The function $f(x)=|x|^p,$ $x\in \mathbb{R}^{n}$ is strictly convex for $p>1$?

Let $p>1$. In the paper [1] below, it says that The function $f(x)=|x|^p,$ $x\in \mathbb{R}^{n}$ is strictly convex. I would like to prove that. By definition, We need to show that the Hessian ...
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1answer
24 views

Infinite-dimensional Bregman divergence

Let $C$ be a convex subset of $\mathbb R^n$ with nonempty interior. Let $f: C \to \mathbb R$ be a strictly convex function, differentiable in the interior of $C$, whose gradient $\nabla f$ extends to ...
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1answer
37 views

Simple Concave Function Question

Suppose we have a function $f:[0,\infty)\longrightarrow [0,\infty)$ that is concave and $a > b>0$. Then given a constant $c>0$ I claim that $f(a+c) - f(a) \le f(b+c) - f(b)$. If I draw a ...
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1answer
23 views

Clarification about the definition of subgradient

I have a basic clarificatory question about the definition of a subgradient from the book Infinite Dimensional Analysis by Aliprantis and Border. Here is the definition. Given a dual pair $\langle X,...
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1answer
48 views

Proving an inequality involving a concave function

I have encountered the following inequality, but I'm struggling to prove it: Suppose $f:\mathbb{R} \to \mathbb{R}$ is a concave function and $\theta > 1$. Then $$ \theta^k f(x) \geq f(\theta^k x)$$ ...
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1answer
14 views

Prove that the supremum of an affine function is concave

Suppose I have a function such that $F(\theta x+(1-\theta)y,z)=\theta F(x,z)+(1-\theta)F(y,z)$ for $\theta\in(0,1)$. I want to show that $F(x,z)$ is concave in the first argument when taking a ...
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2answers
53 views

Convex hulls of a set and its subsets

Suppose that $P$ is a set of $k>3$ points in $\mathbb{R}^2$. Let $\mathrm{Conv}(P)$ be the convex hull of $P$. I think that the following claim is true (I know how to prove it geometrically) : $\...
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1answer
23 views

Prove this infimum function is convex

Let $X \subset \mathbb{R}^n$ be convex, as well as the functions $f:X \to \mathbb{R}$ and $g:X \to \mathbb{R}$. Prove $h(b)=\inf \{f(x): g(x) \le b, x \in X \}$ defined on $\mathbb{R}$ is convex also. ...
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1answer
19 views

Example of a monotone Lipschitz operator that is not cocoercive?

Let $\mathcal{X}$ be a real Hilbert space, let $x,y\in\mathcal{X}$, and let $L\in\left]0,+\infty\right[$. I am looking for an operator $T\colon\mathcal{X}\to\mathcal{X}$ which is \begin{align} \text{...
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1answer
36 views

Convexity of constrained profit function [closed]

I have to show that the following constrained profit function, $\pi$, is convex. A firm choses non-negative quantity, $x$, of inputs with price $p$ and non-negative quantity, $y$, of outputs with ...
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46 views

How do I prove that a theoretical set is Convex?

I am trying to get my head around convex analysis proofs and I am not sure how to begin. I think the definition I should use for proofs is that a set C is convex if for any $u,v\in C$, the point $tu+(...
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1answer
36 views

Topological properties of convex sets

I was trying to prove some 'facts' regarding convex sets which seem natural, but I seem to stuck, so I was wondering whether tbey are indeed true, which I'll describe below. Let $X$ be a topological ...
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1answer
29 views

What are some conditions that are sufficient for the following projection property?

Let $X$ be a locally convex topological vector space, and let $f: X \times X \to [0,\infty]$. Say that $f$ has the projection property if the following holds: For all compact, convex $C \subseteq X$ ...
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1answer
39 views

Example of a strongly convex function where the Lipschitz constant $L$ is equal to the strong convexity parameter $u$

I am trying to come up with like three strongly convex function $f\colon\mathbb{R}\to\mathbb{R}$ where the Lipschitz constant $L$ is equal to the strong convexity parameter $u$, i.e. for every $x,y\in\...
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2answers
44 views

If every $x$ with $f(x) = f(\bar x)$ is local minimum, then $\bar x$ is global minimum.

I am writing to ask for your help guys. The question I am having trouble with is: Let $f: \mathbb{R}^n \to \mathbb{R}$ be a contionuous function and $\bar x \in \mathbb{R}$. Show that the following ...
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1answer
31 views

Convex hull and supporting hyperplanes

Let $S \subseteq \mathbb{R}^n$ be a nonempty set and $p \in S$ be a point. A (closed) halfspace $H\subseteq \mathbb{R}^n$ is said to support $S$ at $p$ if $S \subseteq H$ and $p \in \partial H$. For ...
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1answer
76 views

How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?

Assume $A \in \textbf{S}^n_{++}$, an ellipsoid centered at the origin given by $$\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$ Then we have $\mathcal{E}_A \subseteq \mathcal{E}_B $ if and only if $B-A ...
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1answer
117 views

Convex quadratic program plus one non-convex term

How would you approach a standard convex quadratic program with convex constraints but one non-convex term? Say $|x|^{0.4}$. $$\begin{array}{ll} \underset{x}{\text{minimize}} & \frac12 x^{T} Q x + ...
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32 views

Are these two separation results equivalent?

Let $X$ be a real topological vector space, and let $A$ and $B$ be nonempty, disjoint, convex subsets of $X$. S1. If $A$ is open, then there exists $c \in \mathbb R$ and a nonzero, linear, continuous, ...
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2answers
70 views

Normal cone of the complement of an ellipsoid

On page 68, exercise 2.38 in Boyd & Vanderberghe's book, Convex Optimization, a normal cone of a boundary point $x$ of a set $C$ is defined as $$\mathcal N_C(x) := \left\{ g \mid g^T x \ge g^T y, \...
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1answer
38 views

Expression for the Clarke subdifferential of a weakly convex function

Let $\gamma\in\left]0,+\infty\right[$, let $f$ be a proper, convex, lower semicontinuous function from a real Hilbert space $\mathcal{X}$ to $\left]-\infty,+\infty\right]$, and set $g=f-\frac{\gamma}{...
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0answers
17 views

The family of distributions with log-convex Mills' ratio/inverse hazard function

Consider a twice-differentiable PDF $f(x)$. If $f(x)$ is log-concave, then so is its CDF, $F(x)$. As a consequence, the Mills' ratio or inverse hazard function $[1-F(x)]/f(x)$ is monotonic ...
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1answer
17 views

Rewriting the definition of Strong Convexity

Description I've come across the following transition in a textbook of Convex Optimisation. I am struggling in figuring out how to transform in the equation below so that I'd appreciate if anyone hits ...

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