Convergence of the restricted Nelder-Mead algorithm in two dimensions

Jeffrey C. Lagarias, Bjorn Poonen, Margaret H. Wright

Research output: Contribution to journalArticle

Abstract

The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function f of n real variables using only function values, without any derivative information. Each Nelder-Mead iteration is associated with a nondegenerate simplex defined by n + 1 vertices and their function values; a typical iteration produces a new simplex by replacing the worst vertex by a new point. Despite the method's widespread use, theoretical results have been limited: for strictly convex objective functions of one variable with bounded level sets, the algorithm always converges to the minimizer; for such functions of two variables, the diameter of the simplex converges to zero but examples constructed by McKinnon show that the algorithm may converge to a nonminimizing point. This paper considers the restricted Nelder-Mead algorithm, a variant that does not allow expansion steps. In two dimensions we show that for any nondegenerate starting simplex and any twice-continuously differentiable function with positive definite Hessian and bounded level sets, the algorithm always converges to the minimizer. The proof is based on treating the method as a discrete dynamical system and relies on several techniques that are nonstandard in convergence proofs for unconstrained optimization.

Original languageEnglish (US)
Pages (from-to)501-532
Number of pages32
JournalSIAM Journal on Optimization
Volume22
Issue number2
DOIs
StatePublished - 2012

Keywords

  • Derivative-free optimization
  • Direct search methods
  • Nelder-Mead method
  • Nonderivative optimization

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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