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 $f$.

The classic reference in this subject is R.T. Rockafellar, Convex Analysis.

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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Prove that every convex function is continuous

A function $f : (a,b) \to \Bbb R$ is said to be convex if $$f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$$ whenever $a < x, y < b$ and $0 < \lambda <1$. Prove that every convex function is continuous. Usually it uses the fact: If $a…
cowik
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Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it to solve problems, but in reality I still lack…
trembik
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Is the composition of $n$ convex functions itself a convex function?

Is a set of convex functions closed under composition? I don't necessarily need a proof, but a reference would be greatly appreciated.
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Midpoint-Convex and Continuous Implies Convex

Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks. Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an interval $(a,b)$.
Jack
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How do you prove that $\{ Ax \mid x \geq 0 \}$ is closed?

Let $A$ be a real $m \times n$ matrix. How do you prove that $\{ Ax \mid x \geq 0, x \in \mathbb R^n \}$ is closed (as in, contains all its limit points)? The inequality $x \geq 0$ is interpreted component-wise. This fact is used in some proofs…
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Geometric intuition of conjugate function

I am looking for a geometric and intuitive explanation of the conjugate function and how it maps to the below analytical formula. $$ f^*(y)= \sup_{x \in \operatorname{dom} f } (y^Tx-f(x))$$
Abhishek Bhatia
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cosh x inequality

While reading an article on Hoeffding's Inequality, I came across a curious inequality. Namely $$\cosh x \leq e^{x^2/2} \quad \forall x \in \mathbb{R}$$ I tried many ways to prove it and finally, the Taylor series approach worked: $$e^x = 1 + x +…
Gautam Shenoy
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Does local convexity imply global convexity?

Question: Under what circumstances does local convexity imply global convexity? Motivation: Classically, a twice differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ is convex if and only if the second derivative is nonnegative everywhere.…
Nick Alger
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How pathological can a convex function be?

Let $f : \mathbb R \to \mathbb R$ be convex. How weird can $f$ be? I know $f$ can easily be non-differentiable at finitely many points, for example $f(x) = \sum_{i=1}^n | x - c_i|$. Can it be non-differentiable at infinitely many points? Or on a set…
alfalfa
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Example of a function such that $\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$ but $\varphi$ is not convex

Rudin's Real and Complex Analysis Chapter 3 Exercise 4 is: Assume that $\varphi$ is a continuous real function on $(a,b)$ s.t. $$\varphi\left(\frac{x+y}{2}\right)\leq \frac{\varphi(x)+\varphi(y)}{2}$$ for all $x,y\in(a,b)$. Prove that $\varphi$ is…
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The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is $g(f)$ strictly convex. My attempt, since $f$ is…
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Convexity of the product of two functions in higher dimensions

Exercise 3.32 page 119 of Convex Optimization is concerned with the proof that if $f:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto f(x)$ and $g:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto g(x)$ are both convex, nondecreasing (or nonincreasing) and positive,…
Anthony Labarre
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Does there exist a space filling curve which sends every convex set to a convex set?

Does there exist a surjective continuous function $f:[0,1]\to [0,1]^2$ which maps every convex set to a convex set? Such a function could be considered an especially "regular" sort of space-filling curve. There are of course many well-known…
user456828
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What's the difference between interior and relative interior?

As defined in Convex Optimization written by Stephen Boyd & Lieven Vandenberghe, both interior and relative interior seems to describe a same thing: a set that peels away its boundary points. So, what on earth is the difference between these two…
BioCoder
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Convex surface on which any two points $a,b$ can be joined by a curve of length $(\pi/2-\epsilon)|a-b|$

I am trying to solve an exercise on page 13 of the book Metric structures on Riemannian and non-Riemannian spaces by Gromov. Construct a closed, convex surface $X$ in $\mathbb R^3$ such that any two points $a,b\in X$ can be joined by a curve…
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