Constrained Group Counseling Optimization

Group Counseling Optimization (GCO) has recently been proposed in an attempt to emulate the human social behavior in solving life problems through counseling within a group. After its promising results in solving unconstrained singleobjective and multi-objective optimization problems, in this paper, GCO is extended to solve the constrained optimization problems for the first time. Also, a hybrid parameter-less constraint handling technique is proposed, which uses two wellknown constraint handling techniques: feasible rules and penalty function. The Constrained Group Counseling Optimization (CGCO) uses gradient-based mutation only in the case of all-equality-constraints COPs to reach the extremely small feasible region easily. Moreover, CGCO performance is tested by solving the constrained benchmarks function of the CEC 2010 competition. The results demonstrate that CGCO is competitive to other state-of-the-art algorithms and consistently reaches feasible solutions. Introduction Many real-world applications necessitate the solution of Constrained Optimization Problems (COPs). The solution of such problems means optimizing a given objective function while satisfying a set of imposed constraints. A COP can be defined as follows (Mezura-Montes and Coello C., 2011): minimize f (x) sub ject to g j(x)≤ 0, j = 1, ...,q (1) h j(x) = 0, j = q+1, ...,m Ld ≤ xd ≤Ud , d = 1, ...,D where x = (x1,x2, ...,xD) ∈ R D is a real-valued Ddimensional vector, f is the real valued objective function, g j and h j are q inequality constraints and (m− q) equality constraints, respectively, Ld and Ud are the lower and upper bounds of xd , respectively. To solve COPs, researchers used well-known natureinspired algorithms such as Particle Swarm optimization (PSO) (Kennedy et al., 1995), and Differential Evolution (DE) (Storn and Price, 1997), etc. Originally, all of these algorithms are mainly proposed to deal with unconstrained optimization problems so that it should add a Constraint Handling Technique (CHT) to enable them to deal with COPs. Currently, seven categories of CHTs are known in the literature (Mezura-Montes and Coello C., 2011): Feasibility Rules (FR) (Deb, 2000), Stochastic Ranking (SR) (Runarsson and Yao, 2000), ε-constrained method (Takahama et al., 2005), Novel Penalty Functions, Novel special operators, Multi-objective concepts, Ensemble of constraint-handling techniques. In FR, a solution with less constraint violation is preferred. If two solutions have the same value of constraint violation, the fitter solution is preferred. In SR, a user-defined parameter called p f determines which criterion to use when comparing infeasible solutions: (1) based on their sum of constraints violation or (2) based only on their objective function values. In ε-constrained method, the value of ε> 0 relaxes the limit of considering a solution as feasible. In Novel Penalty Functions, researchers recently proposed two penalty-based approaches namely adaptive penalty function and dynamic penalty function. Adaptive penalty function (Tessema and Yen, 2009) calculates the penalty factor based on the status of the candidate solutions in the search space. Dynamic penalty functions (Tasgetiren and Suganthan, 2006) adopts the current generation number to decrease the penalty factor. In Novel special operators, operator such as boundary operator (Leguizamón and Coello C., 2009) is suggested. In Multi-objective concepts, a COP is turned into a bi-objective optimization problem (objective function and sum of constraints violation) (Wang et al., 2007). In ensemble of constraint-handling techniques (Mallipeddi and Suganthan, 2010a), more than one of the aforementioned categories are hybridized to get the advantages of each category. For simplicity, the Constrained Group Counseling Optimization (CGCO) proposes the use of two parameter-less CHTs: FR and penalty function without any penalty factor. Gradient-based mutation operator (Takahama and Sakai, 2006) is also used to deal with COPs that have only equality constraints. CGCO is applied to a set of standard benchmark ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems COPs of the IEEE CEC 2010 competition (Mallipeddi and Suganthan, 2010b) and the results are compared with εDEag (Takahama and Sakai, 2010), the winner of this competition, and another recently proposed algorithm Co-CLPSO (Liang et al., 2010). This paper is organized as follows: section 2 outlines the related work. Section 3 provides an overview of GCO. Section 4 presents an overview of the CHTs and gradient-based mutation used by CGCO. Section 5 introduces the proposed CGCO. In section 6, CGCO is tested on COPs to evaluate its performance. Finally, the conclusions and future work are put forward in section 7. Related Works Here, some well-known nature-inspired algorithms, namely DE (Storn and Price, 1997), PSO (Kennedy et al., 1995) are briefly addressed. In (Takahama and Sakai, 2006), an approach, called εDE, has been suggested to solve COPs using ε-constrained method as CHT and gradient-based mutation as a repair operator. Gradient-based mutation helps εDE handle COPs whose constraints are equality. This kind of COPs is difficult because the feasible region is very small. Afterwards, Takahama and Sakai added the concept of archive in their new approach εDEag (Takahama and Sakai, 2010). The archive increases the diversity of candidate solutions so that it gives better stability. In εDEag, a new controlling method of ε level is adopted. εDEag yielded promising results and was the winner of the CEC2010 competition (Mallipeddi and Suganthan, 2010b). Liang et al. proposed an approach using PSO to solve COPs which is called cooperation comprehensive learning PSO (Co-CLPSO) (Liang et al., 2010). In Co-CLPSO, a novel CHT in which the population is divided into two subswarms is used. These two sub-swarms cooperate with each other in an attempt to solve the COPs. Particles of each swarm are responsible to deal with different constraints. Two swarms exchange their experiences to benefit each other. Sequential quadratic programming (SQP) is used as a local optimizer to improve the obtained solutions. CoCLPSO ranked the fifth position in the CEC2010 competition. Group Counseling Optimization “Instead of mimicking the behavior of biological organisms such as birds, fish, ants, and bees, GCO is inspired by the human social behavior in solving life problems through counseling within a group” (Eita and Fahmy, 2010, 2014). Counseling (Burnard, 2002) is a well-established branch in sociology and psychology. This was the first time that a connection is found between population-based optimization and group counseling (Berg et al., 2006). Based on the group counseling concept, GCO is developed (Eita and Fahmy, 2010, 2014). Four parameters affect the behavior of GCO: • Number of group members acting as counselors, c, (c≤ m–1). • Counseling probability, cp. • Search range reduction coefficient, red, set into the range [0,1]. • Transition rate from the stage of exploration to that of exploitation, tr. The GCO algorithm is illustrated in the following steps: Step 1 At the very beginning, the algorithm initializes randomly a population with m D-dimensional candidate solutions X i in the search space according to a beta distribution (Gentle, 2003); (Owen, 2008), β (x) = xa−1(1− x)b−1 B(a,b) 0 < x < 1