A Sequential Simplex Program for Solving Minimization Problems

TLDR

Nelder and Mead have developed a simple, robust direct-search procedure for finding the minimum of a function. The method evaluates a function at n + 1 simplex vertices, then iteratively reflects the worst vertex through the centroid, applying extension or contraction as needed, until convergence criteria are satisfied.

Abstract

Nelder and Mead [2] have developed a simple, robust direct-search procedure for finding the minimum of a function. It consists of evaluating a function of n variables at the (n + 1) vertices of a general simplex. The simplex is then moved away from the largest function value by replacing the vertex having this value with one located by reflection through the centroid of the other vertices. Extension or contraction is then applied depending on the contours of the response surface. This continues until either the specified number of trials has been used, the function values differ among themselves by less than a specified amount, or the coordinates of the function are changing by less than a specified amount.