Weighted Centroids in Adaptive Nelder–Mead Simplex: With heat source locator and multiple myeloma predictor applications

Abstract

This paper introduces a novel approach for enhancing the Nelder–Mead Simplex method by utilizing the weighted mean of simplex vertices to efficiently determine the search direction towards optima. The Nelder–Mead algorithm is commonly used for solving unconstrained optimization problems through iterative direct search. While the Nelder–Mead algorithm excels in low-dimensional scenarios, its challenges in higher dimensions are addressed in our study.We reveal that the Nelder–Mead Simplex algorithm’s operations rely not only on the problem dimension but also on adaptive descending parameters for calculating the weighted centroid of vertices. The adaptive weights are determined based on the concept that generating a descending sequence depends on the fitness values of vertices. Consequently, the search direction tends to prioritize vertices with superior fitness values over others.Through extensive testing on renowned high-dimensional benchmark functions, the proposed algorithm outperforms both the standard and adaptive Nelder–Mead simplex algorithms in terms of performance demonstrating superior consistency and reliability. Furthermore, it is applied to an inverse heat source location problem formulated as a linear elliptic partial differential equation, serving as a large-scale real-world application. Additionally, the algorithm successfully addresses a neural network model solving a classification problem concerning the prediction of multiple myeloma, a type of bone marrow cancer, in patients, achieving an average accuracy of 92.3% with a minimal standard deviation of 0.13%. When the results obtained from eight different methods used in the study are compared in terms of performance in general, the proposed method is among the top three optimization algorithms. These real-world applications showcase the method’s versatility.