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