Liquid-Drop-Like Multi-Orbit Topology Versus Ring Topology in PSO for Lennard-Jones Problem

  • Kusum Deep
  • Madhuri
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 202)


The Lennard-Jones (L-J) Potential Problem is a challenging global optimization problem, due to the presence of a large number of local optima that increases exponentially with problem size. The problem is ‘NP-hard’, i.e., it is not possible to design an algorithm which can solve it on a time scale growing linearly with the problem size. For this challenging complexity, a lot of research has been done, to design algorithms to solve it. In this paper, an attempt is made to solve it by incorporating a recently designed multi-orbit (MO) dynamic neighborhood topology in Particle Swarm Optimization (PSO) which is one of the most popular natural computing paradigms. The MO topology is inspired from the cohesive interconnection network of molecules in a drop of liquid. In this topology, the swarm has heterogeneous connectivity with some subsets of the swarm strongly connected while with the others relatively isolated. This heterogeneity of connections balances the exploration–exploitation trade-off in the swarm. Further, it uses dynamic neighborhoods, in order to avoid entrapment in local optima. Simulations are performed with this new PSO on 14 instances of the L-J Problem, and the results are compared with those obtained by commonly used ring topology in conjunction with two adaptive inertia weight variants of PSO, namely Globally adaptive inertia weight and Locally adaptive inertia weight PSO. The results indicate that the L-J problem can be solved more efficiently, by the use of MO topology than the ring topology, with PSO.


Particle swarm optimization Neighborhood topologies Liquid-drop-like topology Multi-orbit topology 


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Authors gratefully acknowledge the help of Mr. Anil Goswami, Junior Research Fellow, Department of Mathmeatics, IIT Roorkee, in writing the codes for the topology. The second author, Madhuri, is highly thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India for providing financial support for this work.


  1. Bansal, J. C., and Deep, K.: A Modified Binary Particle Swarm Optimization for Knapsack Problems. Applied Mathematics and Computation, 218(22), 11042–11061 (2012).Google Scholar
  2. Bharti, Controlled random search technique and their applications, Ph.D. Thesis, Department of Mathematics, University of Roorkee, Roorkee, India, (1994).Google Scholar
  3. Das S., and Suganthan, P. N.: Criteria for CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems, Technical report (2010).Google Scholar
  4. Deep K., and Madhuri: Liquid drop like Multi-orbit Dynamic Neighborhood Topology in Particle Swarm Optimization, Applied Mathematics and Computation (2012) (Communicated).Google Scholar
  5. Deep, K., Madhuri, and Bansal, J. C.: A Non-deterministic Adaptive Inertia Weight in PSO. In proceedings of the 13th annual conference on Genetic and evolutionary computation (GECCO’11), ACM: vol. 131, pp. 1155-1161 (2011).Google Scholar
  6. Deep, K., Shashi, and Katiyar, V. K.: Global Optimization of Lennard-Jones Potential Using Newly Developed Real Coded Genetic Algorithms. In proceedings of IEEE International Conference on Communication Systems and Network Technologies, 614-618 (2011).Google Scholar
  7. Doye, P. K.: The Structure: Thermodynamics and Dynamics of Atomic Clusters. Department of Chemistry, University of Cambridge (1996).Google Scholar
  8. Dugan, N., and Erkoc, S.: Genetic algorithm Monte Carlo hybrid geometry optimization method for atomic clusters. Computational Materials Science 45, 127-132(2009).Google Scholar
  9. Guocheng, Li: An effective Simulated Annealing-based Mathematical Optimization Algorithm for Minimizing the Lennard-Jones Potential. Adv. Materials and Computer Science. Key Engineering Materials Vols. 474-476, pp 2213-2216 (2011).Google Scholar
  10. Hartke, B.: Efficient global geometry optimization of atomic and molecular clusters. Global Optimization 85, 141-168 (2006).Google Scholar
  11. Hoare, M.R.: Structure and dynamics of simple microclusters. Adv. Chem. Phys. 40, 49-135 (1979).Google Scholar
  12. Hodgson, R. J. W.: Particle Swarm Optimization Applied to the Atomic Cluster Optimization Problem. In Proceedings of the Genetic and Evolutionary Computation Conference 2002, 68-73 (2002).Google Scholar
  13. Kennedy, J., and Eberhart, R. C.: Particle Swarm Optimization. In proceedings of 1995 IEEE International Conference on Neural Networks, 4, 1942–1948 (1995).Google Scholar
  14. Kennedy, J., and Eberhart, R. C.: Swarm Intelligence, Morgan Kaufmann, Academic Press (2001).Google Scholar
  15. Kennedy, J.: Small worlds and mega-minds: Effects of Neighborhood Topology on Particle Swarm Performance. In proceedings of Congress on Evolutionary Computation,3, 1931-1938 (1999).Google Scholar
  16. Marques, J. M. C., and Pereira, F. B.: An evolutionary algorithm for global minimum search of binary atomic clusters. Chemical Physics Letters 485, 211-216 (2010).Google Scholar
  17. Moloi N. P., and Ali, M. M.: An Iterative Global Optimization Algorithm for Potential Energy Minimization. J. Comput. Optim. Appl. 30(2), 119-132 (2005).Google Scholar
  18. Northby, J. A.: Structure and binding of Lennard-Jones clusters: 13 ≤ n ≤ 147. J. Chem. Phys. 87, 6166–6178 (1987).Google Scholar
  19. Watts, D. J., and Strogatz, S. H.: Collective Dynamics of ‘Small-World’ Networks. Nat. 393, 440-442 (1998).Google Scholar
  20. Xue, G. L.: Improvements on the Northby Algorithm for molecular conformation: Better solutions. J. Global Optimization 4, 425–440 (1994).Google Scholar

Copyright information

© Springer India 2013

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

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