### A Hybrid LBFGS-DE Algorithm for Global Optimization of the Lennard-Jones Cluster Problem

#### Abstract

Excerpt

The Lennard-Jones cluster conformation problem is to determine a configuration of n atoms in three-dimensional space where the sum of the nonlinear pairwise potential function is at a minimum. In this formula, ri,j is the distance between atoms i and j. This optimization problem is a severe test for global optimization algorithms due to its computational complexity: the number of local minima grows exponentially large as the number of atoms in the cluster is increased. As a specific test case, a better cluster configuration than the previously published putative minimum for the 38-atom case was found in the mid-1990s.

The Lennard-Jones cluster conformation problem is to determine a configuration of n atoms in three-dimensional space where the sum of the nonlinear pairwise potential function is at a minimum. In this formula, ri,j is the distance between atoms i and j. This optimization problem is a severe test for global optimization algorithms due to its computational complexity: the number of local minima grows exponentially large as the number of atoms in the cluster is increased. As a specific test case, a better cluster configuration than the previously published putative minimum for the 38-atom case was found in the mid-1990s.