Solving Unconstrained Global Optimization Problems via Hybrid Swarm Intelligence Approaches

Solving Unconstrained Global Optimization Problems via Hybrid Swarm Intelligence Approaches Jui-Yu Wu Department of Business Administration, Lunghwa University of Science and Technology, No. 300, Section 1, Wanshou Road, Guishan, Taoyuan County 33306, Taiwan Received 7 September 2012; Revised 3 December 2012; Accepted 4 December 2012 Academic Editor: Baozhen Yao Copyright © 2013 Jui-Yu Wu. 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. Abstract Stochastic global optimization (SGO) algorithms such as the particle swarm optimization (PSO) approach have become popular for solving unconstrained global optimization (UGO) problems. The PSO approach, which belongs to the swarm intelligence domain, does not require gradient information, enabling it to overcome this limitation of traditional nonlinear programming methods. Unfortunately, PSO algorithm implementation and performance depend on several parameters, such as cognitive parameter, social parameter, and constriction coefficient. These parameters are tuned by using trial and error. To reduce the parametrization of a PSO method, this work presents two efficient hybrid SGO approaches, namely, a real-coded genetic algorithm-based PSO (RGA-PSO) method and an artificial immune algorithm-based PSO (AIA-PSO) method. The specific parameters of the internal PSO algorithm are optimized using the external RGA and AIA approaches, and then the internal PSO algorithm is applied to solve UGO problems. The performances of the proposed RGA-PSO and AIA-PSO algorithms are then evaluated using a set of benchmark UGO problems. Numerical results indicate that, besides their ability to converge to a global minimum for each test UGO problem, the proposed RGA-PSO and AIA-PSO algorithms outperform many hybrid SGO algorithms. Thus, the RGA-PSO and AIA-PSO approaches can be considered alternative SGO approaches for solving standard-dimensional UGO problems. 1. Introduction An unconstrained global optimization (UGO) problem can generally be formulated as follows:  ? M i n i m i z e ? ( ? ) , ? = 1 , ? 2 , … , ? ?  ? ∈ ℜ ? , ( 1 ) where ? ( ? )   is an objective function and ?   represents a decision variable vector. Additionally, ? ∈ ? , ? ⊆ ℜ ?    denotes search space ( ? ), which is ?   dimensional and bounded by parametric constraints as follows: ? ? ? ≤ ? ? ≤ ? ? ? , ? = 1 , 2 , … , ? , ( 2 ) where ? ? ?   and ? ? ?   are the lower and upper boundaries of the decision variables ? ? , respectively. Many conventional nonlinear programming (NLP) techniques, such as the golden search, quadratic approximation, Nelder-Mead, steepest descent, Newton, and conjugate gradient methods, have been used to solve UGO problems [1]. Unfortunately, such NLP methods have difficulty in solving UGO problems when an objective function of an UGO problem is nondifferential. Many stochastic global optimization (SGO) approaches developed to overcome this limitation of the traditional NLP methods include genetic algorithms (GAs), particle swarm optimization (PSO), ant colony optimization (ACO), and artificial immune algorithms (AIAs). For instance, Hamzaçebi [2] developed an enhanced GA incorporating a local random search algorithm for eight continuous functions. Furthermore, Chen [3] presented a two-layer PSO method to solve nine UGO problems. Zhao [4] presented a perturbed PSO approach for 12 UGO problems. Meanwhile, Toksari [5] developed an ACO algorithm for solving UGO problems. Finally, Kelsey and Timmis [6] presented an AIA method based on the clonal selection principle for solving 12 UGO problems. This work focuses on a PSO algorithm, based on it is being effective, robust and easy to use in the SGO methods. Research on the PSO method has considered many critical issues such as parameter selection, integration of the PSO algorithm with the approaches of self-adaptation, and integration with other intelligent optimizing methods [7]. This work surveys two issues: first is a PSO approach that integrates with other intelligent optimizing methods and second is parameter selection for use in a PSO approach. Regarding the first issue, the conventional PSO algorithm lacks evolution operators of GAs, such as crossover and mutation operations. Therefore, PSO has premature convergence, that is, a rapid loss of diversity during optimization [4]. To overcome this limitation, many hybrid SGO methods have been developed to create diverse candidate solutions to enhance the performance of a PSO approach. Hybrid algorithms have some advantages; for instance, hybrid algorithms outperform individual algorithms in solving certain problems and thus can solve general problems more efficiently [8]. Kao and Zahara [9] presented a hybrid GA and PSO algorithm to solve 17 multimodal test functions. Their study used the operations of GA and P (...truncated)