Convergence properties of the Nelder-Mead simplex method in low dimensions
Convergence properties of the Nelder-Mead simplex method in low dimensions
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DOI：
10.1137/s1052623496303470
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发表时间：
1998-12-21
影响因子：
3.1
通讯作者：
Wright, PE
中科院分区：
文献类型：
--
作者：
Lagarias, JC;Reeds, JA;Wright, PE
Nelder-Mead单纯形算法，首次发表于1965年，是一种非常流行的多维无约束最小化的直接搜索方法。尽管Nelder-Mead算法被广泛使用，但基本上没有理论结果被明确证明。本文给出了Nelder-Mead算法应用于1维和2维严格凸函数的收敛性质。我们证明收敛到一个极小1维，和各种有限的收敛结果2维。麦金农的一个反例给出了二维严格凸函数族和Nelder-Mead算法收敛于非极小值的一组初始条件。目前还不知道是否Nelder-Mead方法可以证明收敛到一个更专门的一类凸函数的二维极小。
The Nelder-Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder-Mead algorithm. This paper presents convergence properties of the Nelder-Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder-Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder-Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.