Improved Nelder Mead’s Simplex Method and Applications

Abstract

Derivative free optimization algorithms are often used when it is difficult to find function derivatives, or if finding such derivatives are time consuming. The Nelder Mead’s simplex method is one of the most popular derivative free optimization algorithms in the fields of engineering, statistics, and sciences. This algorithm is favored and widely used because of its fast convergence and simplicity. The simplex method converges really well with small scale problems of some variables. However, it does not have much success with large scale problems of multiple variables. This factor has reduced its popularity in optimization sciences significantly. Two solutions of quasi gradients are introduced to improve it in terms of the convergence rate and the convergence speed. The improved algorithm with higher success rate and faster convergence which still maintains the simplicity is the key feature of this paper. This algorithm will be compared on several benchmark functions with the original simplex method and other popular optimization algorithms such as the genetic algorithm, the differential evolution algorithm, and the particle swarm algorithm. Then the comparing results will be reported and discussed.