Implementing the Nelder-Mead simplex algorithm with adaptive parameters

Authors:
Fuchang Gao;Lixing Han
Affiliations:
Department of Mathematics, University of Idaho, Moscow, USA 83844;Department of Mathematics, University of Michigan-Flint, Flint, USA 48502
Venue:
Computational Optimization and Applications
Year:
2012

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Abstract

In this paper, we first prove that the expansion and contraction steps of the Nelder-Mead simplex algorithm possess a descent property when the objective function is uniformly convex. This property provides some new insights on why the standard Nelder-Mead algorithm becomes inefficient in high dimensions. We then propose an implementation of the Nelder-Mead method in which the expansion, contraction, and shrink parameters depend on the dimension of the optimization problem. Our numerical experiments show that the new implementation outperforms the standard Nelder-Mead method for high dimensional problems.