Abstract
AbstractWe 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.
Highlights
We study the convergence of the Nelder-Mead simplex method [35] for the solution of the unconstrained minimization problem f (x) → min f : Rn → R, where f is continuous
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 investigated the spectra of the matrices TiP (i), using a simultaneous similarity reduction on T to block lower triangular matrices we proved a convergence result (Theorem 9) for the simplex sequence S(k) to rank-one matrices of the form xeT for some vector x
Summary
We study the convergence of the Nelder-Mead simplex method [35] for the solution of the unconstrained minimization problem f (x) → min f : Rn → R , where f is continuous. In 1998 McKinnon [29] constructed a strictly convex function f : R2 → R with continuous derivatives for which the Nelder-Mead simplex algorithm converges to a nonstationary point. – The function values at all simplex vertices in the standard Nelder-Mead algorithm converge to the same value. In 2012 Lagarias, Poonen, Wright [22] significantly improved the results of the earlier paper [23] for the restricted Nelder-Mead method, where expansion steps are not allowed. Let F be the class of twice-continuously differentiable functions R2 → R with bounded level sets and everywhere positive definite Hessian They proved that for any f ∈ F and any nondegenerate initial simplex S(0), the restricted NelderMead algorithm converges to the unique minimizer of f. This will be discussed at the end of Section 5
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