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Applied Computational Intelligence and Soft Computing
Volume 2011, Article ID 980216, 19 pages
Research Article

A Probability Collectives Approach with a Feasibility-Based Rule for Constrained Optimization

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798

Received 6 May 2011; Accepted 6 September 2011

Academic Editor: R. Saravanan

Copyright © 2011 Anand J. Kulkarni and K. Tai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper demonstrates an attempt to incorporate a simple and generic constraint handling technique to the Probability Collectives (PC) approach for solving constrained optimization problems. The approach of PC optimizes any complex system by decomposing it into smaller subsystems and further treats them in a distributed and decentralized way. These subsystems can be viewed as a Multi-Agent System with rational and self-interested agents optimizing their local goals. However, as there is no inherent constraint handling capability in the PC approach, a real challenge is to take into account constraints and at the same time make the agents work collectively avoiding the tragedy of commons to optimize the global/system objective. At the core of the PC optimization methodology are the concepts of Deterministic Annealing in Statistical Physics, Game Theory and Nash Equilibrium. Moreover, a rule-based procedure is incorporated to handle solutions based on the number of constraints violated and drive the convergence towards feasibility. Two specially developed cases of the Circle Packing Problem with known solutions are solved and the true optimum results are obtained at reasonable computational costs. The proposed algorithm is shown to be sufficiently robust, and strengths and weaknesses of the methodology are also discussed.