3. 凸函数(1)

Convex functions(1)

0. 前言

• 主要介绍了凸函数及其性质

1. Definition

• $f:\mathcal{R^n}\rightarrow \mathcal{R}$ is convex if $\textbf{dom}\ f$ is convex set and $$f[\theta x+(1-\theta)y]\leq \theta f(x)+(1-\theta)f(y)$$ for all $x, y\in\textbf{dom}\ f, 0\leq\theta\leq 1$

2. Convex Examples

• on $\mathcal{R}$
  • affine
  • exponential
  • powers
  • powers of absolute value
  • negative entropy
• on $\mathcal{R}^n$
  • affine
  • norms
• on $\mathcal{R}^{m\times n}$
  • affine
  • spectral norm

3. Restriction of a convex function to a line

$f:R^n\rightarrow R$ is convex if and only if the function $g:R\rightarrow R$

$$g(t)=f(x+tv), \textbf{dom}\ g=\{t\mid x+tv\in \textbf{dom}\ f\}$$

is convex for any $x\in\textbf{dom}\ f, v\in \mathcal{R}^n$

把 $f$ 定义域限定在任意空间直线上是凸的，那么 $f$ 是凸的。注意 $x+tv$ 中 $x,v\in\mathcal{R^n}$ 只有 $t\in\mathcal{R}$ ，$x,v$ 可变并控制是哪条直线

4. Extended-value extension

5. First-order condition

6. Second-order condition

一阶条件和二阶条件，总的来说和一元函数差不多

Epigraph

Jensen’s inequality