SJE000112 - SIAM J. OPTIM. Vol. 9, No. 1, pp. 112147 c 1998...

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CONVERGENCE PROPERTIES OF THE NELDER–MEAD SIMPLEX METHOD IN LOW DIMENSIONS * JEFFREY C. LAGARIAS , JAMES A. REEDS , MARGARET H. WRIGHT § , AND PAUL E. WRIGHT SIAM J. O PTIM . c ± 1998 Society for Industrial and Applied Mathematics Vol. 9, No. 1, pp. 112–147 Abstract. The Nelder–Mead simplex algorithm, first published in 1965, is an enormously pop- ular 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 algo- rithm 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. Key words. direct search methods, Nelder–Mead simplex methods, nonderivative optimization AMS subject classifications. 49D30, 65K05 PII. S1052623496303470 1. Introduction. Since its publication in 1965, the Nelder–Mead “simplex” al- gorithm [6] has become one of the most widely used methods for nonlinear uncon- strained optimization. The Nelder–Mead algorithm should not be confused with the (probably) more famous simplex algorithm of Dantzig for linear programming; both algorithms employ a sequence of simplices but are otherwise completely different and unrelated—in particular, the Nelder–Mead method is intended for unconstrained op- timization. The Nelder–Mead algorithm is especially popular in the fields of chemistry, chem- ical engineering, and medicine. The recent book [16], which contains a bibliography with thousands of references, is devoted entirely to the Nelder–Mead method and vari- ations. Two measures of the ubiquity of the Nelder–Mead method are that it appears in the best-selling handbook Numerical Recipes [7], where it is called the “amoeba algorithm,” and in Matlab [4]. The Nelder–Mead method attempts to minimize a scalar-valued nonlinear func- tion of n real variables using only function values, without any derivative information (explicit or implicit). The Nelder–Mead method thus falls in the general class of di- rect search methods ; for a discussion of these methods, see, for example, [13, 18]. A large subclass of direct search methods, including the Nelder–Mead method, maintain at each step a nondegenerate simplex , a geometric figure in n dimensions of nonzero volume that is the convex hull of n + 1 vertices.
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SJE000112 - SIAM J. OPTIM. Vol. 9, No. 1, pp. 112147 c 1998...

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