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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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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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Midpoint-convexity and continuity implies convexity

Assume that function $f$ is continuous on an interval $(a,b)$. Given that $$ f \left( \frac{x+y}{2} \right) \leqslant \frac{f(x) + f(y)}{2} \,,$$ how can I show that $f$ is convex?
Jack
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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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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.
Phillip Cloud
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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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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…
littleO
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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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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…
bobdylan
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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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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…
Jiajun Wu
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Unit Ball with p-norm

I am having trouble understanding the definition of p-norm unit ball. What I know is that for infinity (maximum norm), then it will shape as a square. I need a "click" to understand this, can someone be so kind to explain this to me in simple…
dresden
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Why is every $p$-norm convex?

I know that $p$-norm of $x\in\Bbb{R}^n$ is defined as, for all $p\ge1$,$$\Vert{x}\Vert_p=\left(\sum_{i=1}^{n} \vert{x_i}\vert^p\right)^{1/p}.$$ The textbook refers to "Every norm is convex" for an example of convex functions. I failed to prove…
Danny_Kim
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The measurability of convex sets

How to prove the measurability of convex sets in $R^n$? I have seen a proof, but too long and not very intuitive. If you have seen any, please post it here.
cjr
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A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, for example $f(x)=\int\lfloor x\rfloor\,dx$. I just…
Mario Carneiro
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