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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.

Convex analysis is the study of properties of convex sets and convex functions.

Formally, a set $S$ is convex if, for all $x, y \in S$ and $t \in [0,1]$, $tx + (1-t)y \in S$. Intuitively, this says that for any two points in $S$, the line segment connecting those points is also in $S$.

Formally, a function $f: S \to \mathbb{R}$ defined on a convex set $S$ is convex, if for all $x,y \in S$ and $t \in [0,1]$, $f(tx + (1-t)y) \leq t f(x) + (1-t) f(y)$. Intuitively, this says that, for any two points on the graph of $f$, the line segment connecting those points lies above the graph of $f$.

Some of the more important results in convex analysis include Carathéodory's Theorem, Jensen's Inequality, Minkowski's Theorem, and the supporting hyperplane theorem.

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Can a polynomial function limited by x axis values be convex?

Consider the given set: $$ S = x ∈ E^2: x_2 − x_1^2 = 0, −1 \le x1 \le 1 $$ When I draw the set I find that it is a polynomial where the $$x_1$$ axis is cut at -1 and 1. The max value for $$x_2$$ is 1. It seems like a convex set to me but the book…
moli
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convex and compact set

Hello guys I am struggling to solve this exercise could someone help me please! The question is the following: Let $S \subseteq \mathbb{R}^k$ be a convex and compact set. For $i=1,\ldots,n$, let $f_i: S \to S$ be continuous functions. Show that…
Ani
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Concave functions generating convex set

If $g_i$ are concave functions where $i = 1,2,\ldots,m$ and $b_i$ are constants where $i = 1,2,\ldots,m$. Why is the set : $$ S = {x : g_i(x) \ge b_i i = 1,2,...,m} $$ a convex set? I know by definition if $f$ is a concave function then $$ f( tx_1 +…
moli
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Differentiability and convex function

If $E\subseteq \mathbb R^d$ is convex and $f:E\to\mathbb R$ is a convex function, is it true that : For all $x\in E$, if $f$ has partial derivatives at $x$ along all directions $u$ such that $x+tu\in E$ for small enough $t>0$ (i.e., $f$ is…
TrivialPursuit
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Any example of known strong convexity constant?

A continuously differentiable function $f(x)$ is strongly convex on $\mathbb{R}^{n}$ if there exists a positive constant $\mu$ such that for any $x, y \ \in \mathbb{R}^{n}$, \begin{align} f(y)\geq f(x) + \langle \nabla f(x), y-x\rangle +…
Ogiad
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Why is the bound in Radon's Theorem optimal?

I have read about Radon's Theorem on Wikipedia. The theorem states that Any set of $d + 2$ points in $\mathbb{R}^d$ can be partitioned into two sets whose convex hulls intersect. I strongly suspect that the bound $d+2$ is even optimal, i.e. the…
3nondatur
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Proving a square set is convex

Assume there is a set $S$ in $\mathbb{R}^2 $ that is a square with $x \in [-1 ,1]$ and $y\in [-1,1]$. I need to prove that this set is convex. Hence, I thought of the following: Suppose $a, b \in S$ such that $a = (x_1, y_1)$, $b = (x_2, y_2)$ and…
mathguy69
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Let $f:\mathbb{R}^n\to\mathbb{R}$ .Prove $f$ is convex iff for any $x\in\mathbb{R}^n$ and $d\neq 0$ $g_{x,d}(t)=f(x+td)$ is convex

Let $f:\mathbb{R}^n\to\mathbb{R}$ .Prove $f$ is convex iff for any $x\in\mathbb{R}^n$ and $d\neq 0$ $g_{x,d}(t)=f(x+td)$ is convex(where g is one-dimensional). The way that we assume that $f$ is convex and proving $g$ is I have managed to do but…
convxy
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Suppose that $f$ is not strictly convex on $C$. Prove that there exist $x ,y \in !n(x \not= y)$ such that $f$ is affine over the segment $[x, y]$

Let $f$ be a convex function defined on a convex set $C$. Suppose that $f$ is not strictly convex on $C$. Prove that there exist $x, y \in \mathbb{R}^n,(x \not= y)$ such that $f$ is affine over the segment $[x, y]$. My idea was assuming by…
convxy
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Proving convexity of the set$ \{ (x,y): 25y^2 - x^2 \geq 9 \} $

I need to show that the following set is convex: $$\{(x,y): \{||(x,3)|| \leq 5y, \: y \leq 1\} $$ So I rewrote the first constraint to $ 25y^2 - x^2 \geq 9$ and the minimum of this constraint is at $(x,y) = (0, 0.6)$ So I came up with the…
mathguy69
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How to prove that f is convex?

This is from Bela Bollobas's book on functional analysis. Given: $f: (a,b) \to (c,b) \text{ and } \phi: (c,b) \to \mathbb{R} \quad \phi^{-1}f, \phi \text{ are both convex }$ To show: $f$ is convex What I've tried: Looked at the definitions a lot of…
user1752323
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Show that the set ${Ax, x \in C}$ is convex

I'm doing the exercise below and I'm stuck on how to show what is being asked for. Given an $m \times n$ matrix $A$ and a convex set $ C \in \mathbb{R}^n$, show that the set $\{Ax |x \in C\}$ is convex. In this case, I also want to show or give a…
Danilo Perl
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Is the function convex or concave?

I am trying to solve question below: Is function $$f(x, y, z) = x+y +z + \ln(xyz)$$ strongly convex or strongly concave in the $A = \{ (x, y, z) | x > 0, y < 0, z < 0 \}$? Also, determine the (possible) local and global extremes values?
Aada Soikkeli
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Equivalence of two definitions of the Legendre transform

Let $I$ be an open interval and let $f:I\to\mathbb R\:$ be a convex function with an invertible derivative $(f')^{-1}:=\phi$. Then $$f^*(y):=\text{sup}\ \{xy-f(x):x\in I\}=\phi(y)y-f(\phi(y))$$ for all $y\in f'(\mathbb R)$. How do you prove this or…
Filippo
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How to prove $f(U,V)=||A-UVU^T||_F^2$ is convex?

I think one way is to verify if both $\frac{\partial^2 f}{\partial U^2}$ and $\frac{\partial^2 f}{\partial V^2}$ are postive semi-definite, is there other way to prove it?
Pei
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