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Top1. Introduction
In real world applications, most optimization problems are subject to different types of constraints. These problems are known as the constrained optimization problems (COPs) or constrained multiobjective optimization problems (CMOPs) if more than one objective function is involved. Comprehensive survey (Michalewicz & Schoenauer, 1996; Mezura-Montes & Coell Coello, 2006) shows a variety of constraint handling techniques have been developed to counter the deficiency of evolutionary algorithms (EAs), in which, their original design are unable to deal with constraints in an effective manner. These techniques are mainly targeted at EAs, particularly genetic algorithms (GAs), to solve COPs (Runarsson & Yao, 2005; Takahama & Sakai, 2006; Cai & Wang, 2006; Wang et al., 2007, 2008; Oyama et al., 2007; Tessema & Yen, 2009) and CMOPs (Fonseca & Fleming, 1998; Coello Coello & Christiansen, 1999; Binh & Korn, 1997; Deb et al., 2002; Kurpati et al., 2002; Hingston et al., 2006; Jimenéz et al., 2002; Ray & Won, 2005; Harada et al., 2007; Geng et al., 2006; Zhang et al., 2006; Chafekar, Xuan & Rasheed, 2003; Woldesenbet, Tessema, & Yen, 2009). During the past few years, due to the success of particle swarm optimization (PSO) in solving many unconstrained optimization problems, research on incorporating existing constraint handling techniques in PSO for solving COPs is steadily gaining attention (Parsopoulus & Vrahatis, 2002; Zielinski & Laur, 2006; He & Wang, 2007; Pulido & Coello Coello, 2004; Liu, Wang, & Li, 2008; Lu & Chen, 2006; Li, Li, & Yu, 2008; Liang & Suganthan, 2006; Cushman, 2007; Wei & Wang, 2006). Nevertheless, many real world problems are often multiobjective in nature. The ultimate goal is to develop multiobjective particle swarm optimization algorithms (MOPSOs) that effectively solve CMOPs. In addition to this perspective, the recent successes of MOPSOs in solving unconstrained MOPs have further motivated us to design a constrained MOPSO to solve CMOPs.
Considering a minimization problem, the general form of the CMOP with 
objective functions 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.