凸優化

定義

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

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

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.