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 that this set is convex

I need a help with prooving that a given set is a convex set: $\{ x \in R^n | Ax \leq b, Cx = d \}$ I know the definition of convexity: $X \in R^n$ is a convex set if $\forall \alpha \in R, 0 ...
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Concavity near the boundary

I am trying to understand a paper, and have come across an "it is easy to see ..." but i don't find it easy at all to see. As far as I can understand, I see the problem thus: We have a convex bounded ...
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Every face of compact convex set is closed?

Well, this is my doubt: Let $\vec{E}$ be a n.v.s. and $K\subset \vec{E}$ a compact convex set. Then every face of $K$ is closed. Any hint in order to prove it is welcome. Thanks in advance!
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Equations for interior of Platonic solids

It is well-known that for Platonic solids: The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$. The interior of octahedron is $|x|+|y|+|z|<a$. But ...
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Is the sub-level set of a concave function convex?

Suppose I have a set defined as follows: \begin{align*} S=\{ x: f(x) \le c\} \end{align*} where $f(x)$ is continuous and concave function defined over some $x \in K$ where $K$ is compact and convex. ...
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Test Convex Hull of Vectors

My mathematical background is generally not so great so please pardon me if my question appears silly. I am trying to test the convex hull of 3 vectors for an intersection with coordinate axes as ...
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21 views

Convex, non-negative function starting from 0

Let $h(x)$ be defined for $x \ge 0$. Suppose that $h(x)$ is non-negative and convex with $h(0) = 0$. I need to show that for $x_2 \ge x_1$ $h(x_2)\ge h(x_1)$. I need to do this from the definition of ...
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Trying to show that $(c_0, \| \cdot \|_s)$ is strictly convex, where $\| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |$

I'm trying to show that $ (c_0, \| \cdot \|_s) $ is a strictly convex space, where $$ \| x \|_s = \underset{i = 1}{\overset{\infty}{\sum}} \frac{1}{2^i} | x_i |,$$ $ x = (x_1, x_2, ..., x_i, ...) \in ...
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Example for a Schur-convex function that is not convex

Let $x \succ y $ be the majorization pre-order on real vectors. (Wikipedia link) We say a function from real vectors to the reals is Schur convex if $x\succ y$ implies $f(x) ≥ f(y)$. With the result ...
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$f:\Bbb R\to\Bbb R$ increasing and convex $\Rightarrow f(x_0)\le f(x)-c(x-x_0)$

Let $f:\Bbb R\to\Bbb R$ such that $f',f''\ge0$ on the whole real line. Then for every $x_0$ fixed, $\exists\; c\in\Bbb R$ s.t. $$ f(x_0)\le f(x)-c(x-x_0)\;\;,\;\;\forall x\in\Bbb R. $$ Now ...
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Calculating the Convex hull of a specific set in $\mathbb{R}^3$

I have to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup \{(k,k^2/4-1,-k^2/4-k)|l\le -2\}$. I am aware with the Fenchel-Bunt theorem, so I just have to consider every (closed) ...
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Matrix convexity!

Given $M\in\Bbb Z_{\geq0,\leq b}^{n\times n}$, if $\mathsf{rank}(M-Q_i)=\mathsf{rank}(Q_i)$ where $i\in\{1,2\}$ with $Q_i\in\Bbb R_{\geq0}^{n\times n}$, then if $\forall ...
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On the weak* compactness of subdifferentials

Let $X$ be a normed vector space over $\mathbb R$ and $X'$ its dual space (the set of norm-continuous linear functionals on $X$). Let $f:X\to\mathbb R$ be a convex function. Consider the ...
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$f$ convex, $\lim_{x\to\infty}\frac{f(x)}{x}=0$, then $f$ is constant

Let $f$ be a convex function of $\Bbb R$ and suppose $\lim\limits_{x\to\pm\infty}\frac{f(x)}{x}=0$. How we can prove that $f$ is constant function?
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Convexity of a certain function connected to the norm

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex? It is certainly the case if the normed space has ...
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Convex Constraint on Sine Wave Simularity

So lets say you have a vector X = [x1 x2 x3 ..... xn] You want to optimize a cost function over X. However you want to constrain the vector X to look like a sine wave. Say you can parameterize a ...
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Is multiplication of monotonically decreasing convex functions convex?

I'm aware that if $h(x)$ and $f(x)$ are convex functions, $g(x) = h(x)f(x)$ may not necessarily be convex. I'm curious whether $g(x)$ is convex if both $h(x)$ and $f(x)$ are also monotonically ...
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Any example of strongly convex functions whose gradients are Lipschitz continuous in $\mathbb{R}^N$

Let $f:\mathbb{R}^N\to\mathbb{R}$ be strongly convex and its gradient is Lipschitz continuous, i.e., for some $l>0$ and $L>0$ we have $$f(y)\geq f(x)+\nabla f(x)^T(y-x)+\frac{l}{2}||y-x||^2,$$ ...
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What is the definition of convexity from $f : \mathbb{R}^2 \rightarrow \mathbb{R}$?

$f(\lambda x + (1-\lambda y) \leq \lambda f(x) + (1- \lambda) f(y)$. This is the definition of convexity I am used to. If $f$ is a convex function, then $f : \mathbb{R} \rightarrow \mathbb{R}$. What ...
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Infinite dimensional convex cone

For every infinite set $I$, the closed convex cone $S:=\{f\in \mathbb{R}^{(I)}:f\geq 0\}$ in $\mathbb{R}^{(I)}$, equipped with the finest locally convex topology, has empty interior. How do I ...
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Proving a function is convex

From the Defintion of convex: Theorem to be proven: If $f$ is differentiable and $f'$ is increasing, then $f$ is convex. Use Proof by Contradiction. Consider, $I = (a, b)$ with $a < x < ...
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How to proof that a straight line can split a convex to at most two regions?

I am self-studying the book "Concrete Mathematics". The authors state the statement: "A straight line can split a convex region into at most two new regions, which will also be convex" 1) How can one ...
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Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
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Convexity and equality in Jensen inequality

Theorem 3.3 from W. Rudin, Real and complex analysis, says: Let $\mu$ be a probabilistic measure on a $\sigma$-algebra of subsets of a given set $\Omega$. If a function $f:X \rightarrow \mathbb R$ ...
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38 views

Does having positive second derivative at a point imply convexity in some neighborhood?

Suppose that I have a real valued function of a single variable $f(x)$ which is twice differentiable in some open interval $I$. Then, I know from calculus that if $f''(x) >0 $ on $I$, then $f$ is ...
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is this a convex optimization problem?

Can someone clarify is this a convex optimization problem or not. $min \| X-UV\|_{F}\quad $ s.t $ \quad U \geq ,V\geq0$ . If not , what makes the problem non-convex?
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Is closure of convex hull of C equal to convex hull of closure.

If $C$ is a set in a topological vector space (or in particular a metric space), can we say that $\text{cl}(\text{conv}(C)) = \text{conv}(\text{cl}(C))$, where cl$(\cdot)$ represents closure and ...
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Non-linear analysis

1) I am looking for a book which would give the proof of the following theorem(see below). I didn't find any book who does it: in infinite dimension (in Rockafellar Convex analysis book we are in ...
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Elements of a Convex Set

Let $ S \subset \mathbb R^n $ be a convex set. Given $ \vec x, \vec y, \vec z \in S $ and three positive numbers such that $ a+b+c=1 $, show that $a\vec x+b\vec y+c\vec z$ is in $S$ also. Ok, so, I ...
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Example of maximal monotone operators in non-reflexive Banach spaces with applications in PDE

My question is about examples of maximal monotone operators that are defined in non-reflexive Banach spaces and have applications in PDEs, variational inequalities, etc (any application actually)? If ...
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Proof sketch for a convex function, help. [duplicate]

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
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Preimage of Legendre-Fenchel transform

Let $X$ be a Banach space with dual $X'$, and let $f : X'\to (-\infty,+\infty]$ be a convex lower semicontinuous function. Does there exist some characterization or some nontrivial results concerning ...
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When are the extreme points of a set the bondary?

Let $X$ be a convex compact set. When is the set of extreme points equal to the boundary of $X$? NOTE: by boundary I mean $\overline{X} \setminus \mbox{Int}(X)$.
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sup is bounded or not?

The sup is as following: $c_f = sup_{x,s\in D} \ f(y) - f(x) - (y-x)^Tb$ where $y=x+\alpha(s-x)$, $\alpha \in (0,1 )$ is constant and $b$ is a constant vector. $D$ is a convex compact set and ...
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Proof sketch for a convex function, help.

Assume that $f(x)$ has two derivatives in $(0,2)$ and $0<a<b<a+b<2$. Prove that if $f(a)\ge f(a+b)$ and $f″(x)\le 0$ $\forall x \in (0, 2)$, then: $$\frac{af(a)+bf(b)}{a+b} \ge f(a+b) ...
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Proving a function of matrix is convex

I have a function of a matrix and a vector $f(A,b)=y^\top (I-A)^{-1} b$ and I want to know the conditions under which it is convex. For functions of a vector, the positive definiteness of the Hessian ...
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28 views

Steepest Descent Sequence

How can I compute the first three iterates for the steepest descent sequence $f(x_1,x_2) = \frac{(x_1^2+3x_2^2)}{2}$ beginning at $x_0 = (\frac{\sqrt{3}}{2}, \frac{1}{2 \sqrt{3}})^T$ $\nabla ...
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Convexity proof - can I get some pointers?

Prove that $C \subset \mathbb{R}^n$ is convex iff $\forall m \in \mathbb{N}$ and every set of $m$ points $\{x_1,...,x_m\} \subset C$ we have that $\sum_{i=1}^m \lambda_i x_i \in C$ Where ...
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Show the following statements are equivalent - convexity

Let $C \subset \mathbb{R}^n$ be a set. Show the following are equivalent: (a) The set $C$ is convex. (b) The function $\delta_C : \mathbb{R}^n \to \mathbb{R} \cup \infty$ defined as: ...
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Convexity Proof with constraints on the gradient

Consider a minimization problem $(P)$ : minimize $f(x)$ subject to $\delta_C(x) \leq 0$ Now assume that $\emptyset \neq C \subset \mathbb{R}^n$ is convex and let $f: \mathbb{R}^n \to \mathbb{R}$ be ...
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Projection on Epigraph of a convex function

Given a convex function $h:\mathbb{R}^n \mapsto \mathbb{R}$, and a point $(x,\alpha) \in \mathbb{R}^n \times \mathbb{R}$, how can I find a closed formula to compute the projection of $(x,\alpha)$ in ...
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Term for a Convex Function whose derivative is also convex

Let $f(x)$ be a monotone non-decreasing convex function such that its derivative $\frac{d}{dx}f(x) = f'(x)$ is also a convex function. Is there a term in literature that is used to refer to such ...
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Confusion on convex functions

I got a problem while solving a problem regarding convex functions on an interval $(a,b)$. What I had to show is if $f$ is convex then $f'$ exists except possibly at countably many points and is ...
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Disjoint Convex Sets That Are Not Strictly Separated

Question 2.23 out of Boyd and Vanderberghe: Give an example of two closed convex sets that are disjoint but cannot be strictly separated. The obvious idea is to take something like unbounded sets ...
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Prove that the following two definitions of the convex hull are equivalent.

I was wondering if a topology expert could help me solve this proof, as I have no idea but want to understand these concepts. This is not for homework. Let X be a point set, not necessarily finite, ...
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uniform convergences of convex functions

Let $f_n(\cdot)$ be a sequence of continuous and convex function on $\mathbb{R}^d$, and be supported on a full dimensional compact convex set $D$. If $f_n(\cdot)$ converges point-wise to $f$ in the ...
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How to derive the solution in quadratic optimization

I'm reading the book "Convex Analysis and Optimization" written by Prof. Bertsekas. In Example 2.2.1, there are the following description: I don't know how to ...
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40 views

Prove that the Cartesian Product of two Convex Sets is a Convex Subset

Here's the problem: Suppose that $S\subset \mathbb R^m$ is a convex set and $T\subset \mathbb R^n$ is a convex set. Show that the set $$S \times T = \{ (x_1 ,...,x_{m+n}\in \mathbb R^{m+n}):(x_1 ...
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Proving Convexity of an Open Disk

I need to prove that the following set is convex: $$ \{(x,y):x^2 +y^2 \lt 2\} $$ Obviously, this an open disk of radius $\sqrt2$. My intuition is to use triangle inequality for this proof because a ...
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77 views

Is the norm of a convex function convex?

I know that the norm of $x\in R^n$, $(\sum\limits_{i=1}^n|x_i|^2)^{0.5}$ is a convex function. Also, not any composition of two convex functions is convex. So my question is: Lets say we have a real ...