Name
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. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1137/110830150 | SIAM JOURNAL ON OPTIMIZATION |
Keywords | DocType | Volume |
direct search methods,nonderivative optimization,derivative-free optimization,Nelder-Mead method | Journal | 22 |
Issue | ISSN | Citations |
2 | 1052-6234 | 5 |
PageRank | References | Authors |
0.74 | 9 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Jeffrey C. Lagarias | 1 | 1049 | 139.02 |
| Bjorn Poonen | 2 | 77 | 16.89 |
| Margaret H. Wright | 3 | 1233 | 182.31 |