Jeffrey C. Lagarias

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular 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 algorithm 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.

Two-scale difference equations II. local regularity, infinite products of matrices and fractals

This paper studies solutions of the functional equation \[ f(x) = \sum_{n = 0}^N {c_n f(kx - n),} \] where

Sets of Matrices All Infinite Products of Which Converge

k \geqq 2

Two-scale difference equations I: existence and global regularity of solutions

is an integer, and

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

\sum\nolimits_{n = 0}^N {c_n = k}

The finiteness conjecture for the generalized spectral radius of a set of matrices

. Part I showed that equations of this type have at most one

The computational complexity of simultaneous diophantine approximation problems

L^1

Tiling with Polyominoes and Combinatorial Group Theory

-solution up to a multiplicative constant, which necessarily has compact support in

Apollonian circle packings: Number theory

[0,{N / {k - 1}}]

Reconstructing truncated integer variables satisfying linear congruences

. This paper gives a time-domain representation for such a function