# 凸优化知识

Definition1.

$(P_J): min_{\substack x} J(x)\quad subject\quad to \quad b=Ax$的可行解集（feasible set）为凸。

Definition2

A function $J(x):\Omega\rightarrow R$ is convex if $\forall x_{1},x_{2}\in \Omega$ if and only if $J(x_{2}) \ge J(x_{1})+\bigtriangledown J(x_{1})^{T}(x_{2}-x_{1})$. or alternatively if and only if $\bigtriangledown^{2} J(x_{1})$ (Hessian matrix) is positive-definite.

The squared $l2$-norm is strictly convex(严格凸)因为其Hessian matrix 严格正定 for all $x$。（$\bigtriangledown^{2} \lVert x \rVert_{2}^{2}=2I \ge 0$

$l_{p}$范数:为x向量各个元素绝对值p次方和的1/p次方.

$\lVert x \rVert_{p}^{p}=\sum_{i}\lvert x_{i} \rvert ^{p}$