The Nelder--Mead Method in High Dimensions: An Exact Moment-of-Inertia Identity and Dimension-Free almost sure Convergence

Abstract Gal´antai (2023) proved that the vertices of the Nelder–Mead simplices in Rn converge almost surely to a common limit in a random model, but his criterion requires all |T |ℓ products of transformation matrices, |T | = 3n + 3 + (n + 1)!, and was verified only for n 6. We derive an exact identity for the moment of inertia Φk of the simplex. This yields a deterministic convergence criterion. For probabilistic models of the iteration, we show that Gal´antai’s theorem holds in every dimension with blocks of length one, but for n 5 its uniform model consists almost entirely of shrinks, which never occur for strictly convex functions. If the replaced vertex has uniformly distributed rank, the vertices converge almost surely to a common limit in every dimension whenever contractions are more than four times as frequent as expansions (standard coefficients), with expected rate 1 c/n per iteration. For Gal´antai’s insertion model, almost sure convergence follows from a spectral-radius condition, which we prove in every dimension with the moment of inertia as a Lyapunov function. The deterministic criterion and the result for uniformly distributed ranks extend to the original, unordered method. We combine the reflection and contraction coefficients of Puhan and Bu˝rmen (2026), an expansion coefficient adapted to the share of Φk carried by the replaced vertex, and Kelley’s sufficient decrease safeguard with its constant divided by (n + 1)2. We give a sufficient condition for the convergence of the resulting algorithm to a critical point. In tests up to n = 300, it required fewer function evaluations than the method of Gao and Han in all cases except a test example with a neural network that is motivated by machine learning, and fewer than the coefficients of Puhan and Bu˝rmen in eleven of the fifteen other cases. MSC: 65K10; 90C56; 60F15