6.7.1
Background
The Lennard-Jones potential is a mathematical model that closely approximates the interaction between
neutral atoms[133]. Determining the structural arrangement of multiple atoms that minimizes this po-
tential energy function presents an ideal problem for global optimization, in that it is simultaneously
easy to describe and formulate, but difficult to solve. This problem has existed in the literature for quite
some time and is well understood [135][136]. Its exponentially increasing number of local minima with
the number of atoms makes it a straightforward matter to attempt to solve formulations of increasing
difficulty.
On a practical note, chemistry literature commonly cites the knowledge of the global minima of
Lennard-Jones clusters as “a fundamental step towards a better understanding of some molecular con-
formation problems”[137]. Because of this a great deal of effort has gone into determining these minima,
and to date they are believed to be known for all cluster sizes of up to 250 atoms[138].
Lennard-Jones problems are fairly well known in the global optimization literature, and are used to
demonstrate the applicability of algorithms to “real world” problems. Purpose-built genetic algorithms
have been demonstrated to be able to reliably solve the problem for up to at least N = 110, when the
algorithm was specifically designed for solving only this specific problem[139][140]. PSO algorithms
have also been applied in varying capacities, from GA-comparative works[141] to use as examples of
multi-funnel landscapes[142]. These examinations were limited to at most N = 15, and showed mostly
poor performance – [142] was in fact an attempted demonstration of the unsuitability of the standard
PSO algorithm for this type of landscape.
Better results were obtained for the N = 26 Lennard-Jones problem using an enhanced and exten-
sively specialized form of the inertia-weight formulation of PSO[143], indicating that the algorithm is
not inherently unsuited to Lennard-Jones problems, but can be adapted from the default configuration
cations allowed the algorithm to reliably find the global optimum of the 26-atom Lennard-Jones problem
(it was not tested on any other configuration), but restricted its applicability entirely to the optimization
of this and similar atomic structuring problems[143].
6.7.2
Definition
The potential energy in a Lennard-Jones atomic cluster can be represented by the equation:
E = 4 N −1 X i=1 N X j=i+1 ( σ rij )12− ( σ rij )6) (6.2)
where N is the number of atoms in the cluster and rij is the distance between two atoms i and
j. and σ represent specific features of the atomic structure, and here are set to 1, as per previous studies[140][143].
The large number of very deep minima requires an optimization algorithm to be able to both ef-
ficiently explore the landscape, and then rapidly exploit the discovered basin of attraction prior to pre-
mature convergence. 21 different formulations of the problem were used here, of increasing difficulty:
starting with the simplest configuration, 2 atoms, all configurations up to 20 atoms were tested, along
with the 26-atom configuration, and the extremely complex 38-atom configuration. Minimum potential
energies for all 21 configurations are shown in table 6.1. Each atom was represented in 3-dimensional
space, giving problems with dimensionalities ranging from 6 for the 2-atom configuration, to 114 for the
38-atom configuration.
Problem Energy Problem Energy Problem Energy
LJ2 -1.000000 LJ9 -24.113360 LJ16 -56.815742 LJ3 -3.000000 LJ10 -28.422532 LJ17 -61.317995 LJ4 -6.000000 LJ11 -32.765970 LJ18 -66.530949 LJ5 -9.103852 LJ12 -37.967600 LJ19 -72.659782 LJ6 -12.712062 LJ13 -44.326801 LJ20 -77.177043 LJ7 -16.505384 LJ14 -47.845157 LJ26 -108.315616 LJ8 -19.821489 LJ15 -52.322627 LJ38 -173.928427
Table 6.5: Minimum potential energy for tested Lennard-Jones configurations.
The 2-atom through 20-atom problems provide both a sizable sample for obtaining a full view of
the performance of each algorithm, and a good point of comparison for other algorithms in the literature,
most of which test on this range or a subset. The 26-atom configuration was used purely for comparison
to the aforementioned PSO variation that reliably solved it due to having been explicitly built and tuned
for this specific problem and configuration[143]. The 38-atom configuration of the problem was chosen
This quality both simplifies the problem by providing two available solutions, and increases the difficulty
by making convergence to and exploitation of only one of those peaks by the entire swarm more difficult.
There is no commonly-defined initialization or bounding strategy for an optimization population
on the Lennard-Jones class of problems. Global optimization literature favors initializing the popula-
tion with a uniform spread throughout the search space[139], or a region-scaled approach[142], while
publications concerned with the real-world chemical aspect of the problems are more forgiving, disal-
lowing unfeasible solutions and enforcing minimum distances between atoms[143][140]. No bounds are
explicitly defined either; none of the referenced literature makes mention of them.
As both of these properties can influence the performance of the algorithm, a conservative approach
was taken here. The population was uniformly initialized throughout the space, but the initialization
space was defined to extend beyond the optimal points in each dimension by 10 times the maximum
space between them. For example, the optimal solution for the 3-atom configuration is shown in table
6.2 (precision has been abridged for the sake of this example).
Dim 1 Dim 2 Dim 3
Atom 1 0.44 0.11 -0.46
Atom 2 -0.52 0.39 0.05
Atom 3 0.08 -0.50 0.40
Table 6.6: Optimal structure for the 3-atom Lennard-Jones problem
In the first dimension, the optimal points for each atom are 0.44, -0.52, and 0.08. The maximum
distance is therefore 0.44 + 0.52 = 0.96, and hence the initialization region in that dimension will cover
a space with a size of 9.6, i.e. [−4.8..4.8]. Because this value is tied to the configuration of the specific
problem under optimization, initializing the swarm in this manner will not predispose any configuration
to a performance result that is better or worse than that of any other configuration, as could be the case
with bounds of initialization that were fixed and common to all configurations.
Feasible bounds during optimization also vary in the literature. Again, the strategy used here can
greatly influence overall behaviour and performance – a feasible search space that only barely encom-
passed the optimal solution would render the problem much simpler than a space in which the range
containing all components of the optimal solution represented only a small area of the whole. To avoid
any semblance of problem structure influencing algorithm performance, no boundaries were defined for
the atomic configurations tested here. All particle positions were considered feasible and evaluated.
These conservative approaches to the problem search space will almost certainly negatively affect
the performance of the algorithms tested here, at least when compared to that of the algorithms proposed
in other studies. As the focus of this work is mostly confined to the algorithms defined and tested within,
however, this only needs to be taken into account in comparisons to the other studies, none of which
Number of atoms − Mean error Minim um potential energy 2 5 10 15 20 26 38 0.00 0.25 0.50 0.75
Figure 6.11: Fixed-parameter PSO-DRS v SPSO on full range of Lennard-Jones