Convergence of the Nelder-Mead method

We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of po...

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Bibliographic Details
Published inNumerical algorithms Vol. 90; no. 3; pp. 1043 - 1072
Main Author Galántai, Aurél
Format Journal Article
LanguageEnglish
Published New York Springer US 01.07.2022
Springer Nature B.V
Subjects
Algebra
Algorithms
Computer Science
Convergence
Convex analysis
Failure modes
Numeric Computing
Numerical Analysis
Original Paper
Simplex method
Theorems
Theory of Computation
Derivative-free minimization
MSC 65K10
Nelder-Mead simplex method
MSC 90C56
Convergence
Online AccessGet full text
ISSN1017-1398
1572-9265
1572-9265
DOI10.1007/s11075-021-01221-7

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Summary:We develop a matrix form of the Nelder-Mead simplex method and show that its convergence is related to the convergence of infinite matrix products. We then characterize the spectra of the involved matrices necessary for the study of convergence. Using these results, we discuss several examples of possible convergence or failure modes. Then, we prove a general convergence theorem for the simplex sequences generated by the method. The key assumption of the convergence theorem is proved in low-dimensional spaces up to 8 dimensions.
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ISSN:1017-1398
1572-9265
1572-9265
DOI:10.1007/s11075-021-01221-7