A Hybrid Simplex Search for Global Optimization with Representation Formula and Genetic Algorithm

Abstract

We consider the problem of minimizing a given function $f:\mathbb {R}^n\longrightarrow \mathbb {R}$ on a regular not empty closed set S. When f attains a global minimum at exactly one point $x^* \in S$, for a convenient random variable X and a convenient function $g:\mathbb {R}^2\longrightarrow \mathbb {R}$. In this paper, we propose to use this Representation Formula (RF) to numerically generate an initial population. In order to obtain a more accurate results, the Representation Formula has been coupled with other algorithms:

• Classical Genetic Algorithm (GA). We obtain a new algorithm called (RFGA),

• Genetic Algorithm using Nelder Mead algorithm at the mutation stage (GANM). We obtain a new algorithm called (RFGANM),

• Nelder Mead Algorithm. We obtain a new algorithm called (RFNM).

All these six algorithms (RF, GA, RFGA, GANM, RFGANM, RFNM) were tested on 21 benchmark functions with a complete analysis of the effect of different parameters of the methods. The experiments show that the RFNM is the most successful algorithm. Its performance was compared with the other algorithms, and observed to be the more effective, robust, and stable than the others.