# 凸優化

## 定義

${\displaystyle {\mathcal {X}}\subset \mathbb {R} ^{n}}$為一凸集，且${\displaystyle f:{\mathcal {X}}\to \mathbb {R} }$為一凸函數。凸優化就是要找出一點${\displaystyle x^{\ast }\in {\mathcal {X}}}$，使得每一${\displaystyle x\in {\mathcal {X}}}$滿足${\displaystyle f(x^{\ast })\leq f(x)}$[1][2]在最佳化理論中，${\displaystyle {\mathcal {X}}}$稱為可行域${\displaystyle f}$稱為目標函數${\displaystyle x^{\ast }}$稱為全局最優值，或全域最佳解[3]

{\displaystyle {\begin{aligned}&\operatorname {min} &&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\end{aligned}}}

## 腳註

1. ^ Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude. Convex analysis and minimization algorithms: Fundamentals. 1996: 291.
2. ^ Ben-Tal, Aharon; Nemirovskiĭ, Arkadiĭ Semenovich. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. 2001: 335–336.
3. ^ 凸優化──凸函數的最小化. 線代啟示錄. 2013-08-28 [2013-09-25].
4. ^ Boyd/Vandenberghe, p. 7
5. ^ For methods for convex minimization, see the volumes by Hiriart-Urruty and Lemaréchal (bundle) and the textbooks by Ruszczyński and Boyd and Vandenberghe (interior point).

## 參考資料

• Ruszczyński, Andrzej. Nonlinear Optimization. Princeton University Press. 2006.