Constraint Handling in Particle Swarm Optimization

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Introduction

Recently, there are active studies on using particle swarm optimization (PSO) to solve constrained optimization problems (COPs). Similar to evolutionary algorithms (EAs), the original PSO design lacks a mechanism to handle constraints in an effective manner. Most of the constrained PSO designs adopted the popular constraint handling techniques that are built for EAs (Runarsson & Yao, 2005; Takahama & Sakai, 2006; Cai & Wang, 2006; Wang, Cai, Guo, & Zhou, 2007; Wang, Cai, Zhou, & Zeng, 2008). Evidence shows in recent publications on constraint handling with PSO including penalty methods (Parsopoulus & Vrahatis, 2002), comparison criteria or feasibility tournament (Zielinski & Laur, 2006; He & Wang, 2007; Pulido & Coello Coello, 2004), lagrange-based method (Krohling & Coelho, 2006) lexicographic order (Liu, Wang, & Li, 2008), and multiobjective approach (Lu & Chen, 2006; Li, Li, & Yu, 2008; Liang & Suganthan, 2006; Cushman, 2007), to name a few. The reason of such popularity is credited to its simplicity, easy implementation and rapid convergence capability inherited in PSO design. Most constraint handling techniques such as penalty methods or comparison criteria are treated as an add-on module to be incorporated into any EAs for solving COPs. The PSO’s algorithm is built with mechanisms that can be exploited to handle constraints, without imposing any penalty methods or comparison criteria. Motivated by the advantages and its inherited ability, we propose a constrained PSO with design elements that exploit the key mechanisms to handle constraints as well as optimization of the objective function.

Consider a minimization problem; the general form of the COP is given as follows:Minimize , (1) subject to

(2a) (2b); (2c) where is the decision vector of decision variables. Its upper () and lower () bounds in Equation (2c) define the search space, . represents the jth inequality constraint while represents the jth equality constraint. The inequality constraints that are equal to zero, i.e., , at the global optimum () of a given problem are called active constraints. The feasible region () is defined by satisfying all constraints (Equations (2a)-(2b)). A solution in the feasible region () is called a feasible solution, otherwise it is considered an infeasible solution.