Convergence of the Restricted Nelder--Mead Algorithm in Two Dimensions

• Main Authors: Lagarias, Jeffrey C. (Author), Poonen, Bjorn (Contributor), Wright, Margaret H. (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Society for Industrial and Applied Mathematics, 2012-11-26T19:10:17Z.
• Subjects: Article
• Online Access: Get fulltext

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.
National Science Foundation (U.S.) (Grant DMS-0841321)
National Science Foundation (U.S.) (Grant DMS-1069236)