21_hybrid


A Hybrid LBFGS-DE Algorithm

89

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

Ernesto Padernal Adorio
Department of Mathematics,

College of Science, University of the Phillipines,
Diliman, Quezon City 1101

E-mail: adorio@math.upd.edu.ph; eadorio@yahoo.com

Bobby Ondoy Corpus
Department of Computer Science,

Collge of Engineering, University of the Philippines,
Diliman, Quezon City 1101

E-mail: bcorpus jr@yahoo.com

Science Diliman (July–December 2004) 16:2, 89

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.

Various algorithms have been tried for determining
putative global minimum of Lennard-Jones clusters,
for example, simulated annealing and genetic
algorithm. There is a fast multistart two-step algorithm
which can sometimes find the minimum potential
energy clusters in seconds, but it works with a modified
Lennard-Jones potential formula.

In this paper we present a hybrid limited memory BFGS
(L-BFGS) algorithm and a modified differential
evolution (DE) algorithm for determining the global
minimum potential energy configurations of atom

clusters using the unbiased or unmodified potential
Lennard-Jones function. It performed with 100%
reliability for clusters containing 2–50 atoms. The
algorithm has excellent potential for solving other
difficult global nonlinear optimization problems.

The work of the two authors was supported by a
research grant from the Natural Sciences Research
Institute (NSRI).

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