The Tunneling Algorithm for the Global Minimization of Functions

This paper considers the problem of finding the global minima of a function $f(x):\Omega \subset \mathcal{R}^n \to \mathcal{R}$. For this purpose we present an algorithm composed of a sequence of cycles, each cycle consisting of two phases: (a) a minimization phase having the purpose of lowering the current function value until a local minimizer is found and, (b) a tunneling phase that has the purpose of finding a point $x \in \Omega $, other than the last minimizer found, such that when employed as starting point for the next minimization phase, the new stationary point will have a function value no greater than the previous minimum found.In order to test the algorithm, several numerical examples are presented. The functions considered are such that the number of relative minima varies between a few and several thousand; in all cases, the algorithm presented here was able to find the global minimizer(s). When compared with alternate procedures, the results show that the new algorithm converges more often...