# Selective Reflections for the Nelder-Mead Algorithm

## Abstract

The Nelder-Mead Algorithm is an established algorithm to search for a minimum to a function. Since its first description, the algorithm has had various improvements made to it. With this project, I searched for improvements to the algorithm through alterations to its reflective direction, and geometric structure. I found in testing that using reflective directions that maintain the geometry of the algorithm’s search structure, a simplex, tend to perform better than those that do not. The algorithm’s speed can also be increased in higher test dimensions by increasing the number of points in the simplex by one. With these discoveries, the Nelder-Mead algorithm can find solutions to problems it previously could not, and do so with fewer function evaluations.

Selective Reflections for the Nelder-Mead Algorithm

The Nelder-Mead Algorithm is an established algorithm to search for a minimum to a function. Since its first description, the algorithm has had various improvements made to it. With this project, I searched for improvements to the algorithm through alterations to its reflective direction, and geometric structure. I found in testing that using reflective directions that maintain the geometry of the algorithm’s search structure, a simplex, tend to perform better than those that do not. The algorithm’s speed can also be increased in higher test dimensions by increasing the number of points in the simplex by one. With these discoveries, the Nelder-Mead algorithm can find solutions to problems it previously could not, and do so with fewer function evaluations.