Convergence of the Nelder-Mead method
Aurél Galántai
Abstract
Abstract 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.
Topics & Concepts
Convergence (economics)Compact convergenceModes of convergence (annotated index)MathematicsNormal convergenceConvergence testsSimplexTheory of computationMatrix (chemical analysis)Dominated convergence theoremApplied mathematicsLocal convergenceMathematical optimizationKey (lock)Pure mathematicsRate of convergenceAlgorithmComputer scienceIterative methodCombinatoricsMaterials scienceIsolated pointEconomicsTopological spaceEconomic growthTopological vector spaceComputer securityComposite materialMatrix Theory and AlgorithmsAdvanced Optimization Algorithms ResearchIterative Methods for Nonlinear Equations