Global optimization methods for high-dimensional problems

In this paper, we show how stochastic techniques coupled with deterministic local search methods can be successfully applied to solve moderately sized multimodal optimization problems. A crucial point in the definition of these kinds of algorithms is the correct calibration of the global phase, consisting of random sampling, and the local phase. Theoretical results are developed which guarantee a good behavior even for high-dimensional problems. In the proposed approach, the global phase consists of performing a random sample in the feasible region, while the local phase is accomplished through a deterministic local optimization algorithm applied to carefully selected points in the sample. A set of computational experiments have been performed using as a test the minimization of the Lennard–Jones potential energy of a cluster of atoms, a well-known and extremely hard global optimization problem. It is shown that the algorithm performs significantly better than other well-known methods in the literature on most of the tests.