Abstract

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.

Keywords

MathematicsDimension (graph theory)Convergence (economics)CounterexampleSimplex algorithmSimplexMathematical optimizationRegular polygonConvex optimizationLinear programmingCombinatorics

Affiliated Institutions

Related Publications

Publication Info

Year
1998
Type
article
Volume
9
Issue
1
Pages
112-147
Citations
7269
Access
Closed

External Links

View on DOI.org

Social Impact

Altmetric
PlumX Metrics

Social media, news, blog, policy document mentions

Citation Metrics

7269
OpenAlex

Cite This

Jeffrey C. Lagarias, James A. Reeds, Margaret H. Wright et al. (1998). Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions. SIAM Journal on Optimization , 9 (1) , 112-147. https://doi.org/10.1137/s1052623496303470

Identifiers

DOI
10.1137/s1052623496303470