Optimizing noisy CNLS problems by using Nelder-Mead algorithm: new method to compute simplex step efficiency

Nelder-Mead simplex algorithm (NMA) is a derivative-free algorithm that can be used to solve complex nonlinear least-squared (CNLS) problems. NMA can fit equivalent electrical circuit models to noisy electrochemical impedance spectroscopy (EIS) data. The ability of NMA simplex to adapt itself onto the local landscape of mathematical functions is governed by the simplex steps (reflection, expansion, inside and outside contraction and shrink steps). However, according to EIS literature, a method to compute the simplex step efficiency has not been reported yet. Herein, we provide and recommend a new method to compute the simplex step efficiency. The new method was used to evaluate the adaptive and standard NMA modifications (SNMA and ANMA). Be advised that by using the new method we detected an unknown property of a more successful ANMA (vs. SNMA), i.e. the application of adaptive parameters decreased only the inside contraction step efficiency. However, the aforementioned drawback was resolved by modifying and boosting the existing ANMA, which was a process that demonstrated the usefulness of the new method. Thereby, SNMA, ANMA, and the modified ANMA fitting engines were embedded in the free (MIT licensed) software by using Python programming language. (C) 2019 Elsevier B.V. All rights reserved.