Understanding Nelder–Mead simplex dynamics via Chebyshev regularization and its state-based feedback control

The Nelder–Mead simplex algorithm is widely used for derivative-free optimization but performs poorly in high-dimensional settings because of persistent reflection dominance and distorted simplex dynamics. Earlier work introduced Chebyshev-based parameter regularization, producing static crude and refined regimes that balance reflection suppression and convergence efficiency. This article develops a dynamical perspective by introducing reflection- and contraction-dominance metrics to characterize simplex transformations over time. These diagnostics explain the gains of Chebyshev-regularized schemes and the efficiency gap between crude and refined regimes. Building on this insight, this article proposes a deterministic feedback mechanism that adaptively regulates parameters within the Chebyshev-defined admissible space according to the evolving simplex state. Sensitivity analysis confirms robustness across tested parameter values. Numerical experiments on quadratic problems show reduced function evaluations while controlling reflection activity, and tests on the extended Rosenbrock function confirm the validity of the diagnostics on non-convex landscapes. The framework is applicable to arbitrary Nelder–Mead parameter regimes.