Less Is More: Simplified Nelder-Mead Method for Large Unconstrained Optimization

Kayo Gonçalves-E-Silva, Daniel Aloise, Samuel Xavier-De-Souza, Nenad Mladenović

Nelder-Mead method (NM) for solving continuous non-linear optimization problem is probably the most cited and the most used method in the optimization literature and in practical applications, too. It belongs to the direct search methods, those which do not use the first and the second order derivatives. The popularity of NM is based on its simplicity. In this paper we propose even more simple algorithm for larger instances that follows NM idea. We call it Simplified NM (SNM): instead of generating all $n + 1$ simplex points in $\mathcal{R}^n$, we perform search using just $q + 1$ vertices, where $q$ is usually much smaller than $n$. Though the results cannot be better than after performing calculations in $n+1$ points as in NM, significant speed-up allows to run many times SNM from different starting solutions, usually getting better results than those obtained by NM within the same cpu time. Computational analysis is performed on 10 classical con vex and non-convex instances, where the number of variables n can be arbitrarily large. The obtained results show that SNM is more effective than the original NM, confirming that LIMA yields good results when solving a continuous optimization problem.