Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> A concept for the optimization of nonlinear functions using particle swarm methodology is introduced. The evolution of several paradigms is outlined, and an implementation of one of the paradigms is discussed. Benchmark testing of the paradigm is described, and applications, including nonlinear function optimization and neural network training, are proposed. The relationships between particle swarm optimization and both artificial life and genetic algorithms are described. <s> BIB001 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> The study manipulated the neighborhood topologies of particle swarms optimizing four test functions. Several social network structures were tested, with "small-world" randomization of a specified number of links. Sociometric structure and the small-world manipulation interacted with function to produce a significant effect on performance. <s> BIB002 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> The performance of particle swarm optimization using an inertia weight is compared with performance using a constriction factor. Five benchmark functions are used for the comparison. It is concluded that the best approach is to use the constriction factor while limiting the maximum velocity Vmax to the dynamic range of the variable Xmax on each dimension. This approach provides performance on the benchmark functions superior to any other published results known by the authors. <s> BIB003 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> The effects of various population topologies on the particle swarm algorithm were systematically investigated. Random graphs were generated to specifications, and their performance on several criteria was compared. What makes a good population structure? We discovered that previous assumptions may not have been correct. <s> BIB004 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> The particle swarm optimizer (PSO) is a stochastic, population-based optimization technique that can be applied to a wide range of problems, including neural network training. This paper presents a variation on the traditional PSO algorithm, called the cooperative particle swarm optimizer, or CPSO, employing cooperative behavior to significantly improve the performance of the original algorithm. This is achieved by using multiple swarms to optimize different components of the solution vector cooperatively. Application of the new PSO algorithm on several benchmark optimization problems shows a marked improvement in performance over the traditional PSO. <s> BIB005 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> This paper presents a variant of particle swarm optimizers (PSOs) that we call the comprehensive learning particle swarm optimizer (CLPSO), which uses a novel learning strategy whereby all other particles' historical best information is used to update a particle's velocity. This strategy enables the diversity of the swarm to be preserved to discourage premature convergence. Experiments were conducted (using codes available from http://www.ntu.edu.sg/home/epnsugan) on multimodal test functions such as Rosenbrock, Griewank, Rastrigin, Ackley, and Schwefel and composition functions both with and without coordinate rotation. The results demonstrate good performance of the CLPSO in solving multimodal problems when compared with eight other recent variants of the PSO. <s> BIB006 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> We vary the way an individual in the particle swarm interacts with its neighbors. Performance depends on population topology as well as algorithm version. <s> BIB007 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> This study presents an evolutionary neural fuzzy network, designed using the functional-link-based neural fuzzy network (FLNFN) and a new evolutionary learning algorithm. This new evolutionary learning algorithm is based on a hybrid of cooperative particle swarm optimization and cultural algorithm. It is thus called cultural cooperative particle swarm optimization (CCPSO). The proposed CCPSO method, which uses cooperative behavior among multiple swarms, can increase the global search capacity using the belief space. Cooperative behavior involves a collection of multiple swarms that interact by exchanging information to solve a problem. The belief space is the information repository in which the individuals can store their experiences such that other individuals can learn from them indirectly. The proposed FLNFN model uses functional link neural networks as the consequent part of the fuzzy rules. This study uses orthogonal polynomials and linearly independent functions in a functional expansion of the functional link neural networks. The FLNFN model can generate the consequent part of a nonlinear combination of input variables. Finally, the proposed FLNFN with CCPSO (FLNFN-CCPSO) is adopted in several predictive applications. Experimental results have demonstrated that the proposed CCPSO method performs well in predicting the time series problems. <s> BIB008 </s> Hybridizing Differential Evolution and Particle Swarm Optimization to Design Powerful Optimizers: A Review and Taxonomy <s> B. Classical Particle Swarm Optimization <s> This paper develops bidding strategy for operating multiunit pumped-storage power plant in a day-ahead electricity market. Based on forecasted hourly market clearing price, the objective is to self-schedule and maximize the expected profit of the pumped-storage plant, considering both spinning and nonspinning reserve bids and meeting the technical operating constraints. Evolutionary tristate particle swarm optimization (ETPSO) based approach is proposed to solve the problem, combining basic particle swarm optimization (PSO) with tristate coding technique and genetics-based mutation operation. The discrete characteristic of a pumped-storage plant is modeled using tristate coding technique and mutation operation is used for faster convergence. The proposed model is adaptive for nonlinear 3-D relationship between the power produced, the energy stored, and the head of the associated reservoir. The proposed approach is applied for a practical utility consisting of four units. Simulation results for different operating cycles of the storage plant indicate the attractive properties of ETPSO approach with highly optimal solution and robust convergence behavior. <s> BIB009
The particle swarm optimizer that is proposed by Kennedy and Eberhart is also a powerful tool in search and optimization BIB001 . During the past decade, PSO has been successfully and widely applied in the practice of science and engineering (e.g.,   BIB009 BIB008 ), which demonstrates the superiority of this algorithm. Originally, PSO was inspired by the cooperative behavior of animal groups, such as bird flocks. In PSO, many particles form a swarm, which flies through D-dimensional problem space. Each particle has its own velocity and fitness value. Particles accumulate their own experiences about the problem space and, meanwhile, learn from each other according to their fitness values. The iteration equations for the velocity and position in PSO with inertia weight (PSO-w) BIB003 are given as follows: where T represent the position and velocity of the ith particle at the kth generation (i = 1, 2, . . . , PS; PS is the population size), respectively; pbest T is the best position that is found so far by the ith particle; nbest T is the best position that is found by particles in the neighborhood of the ith particle; and rand1 and rand2 are the random numbers that are uniformly distributed in [0,1]. The PSO parameter w is the so-called inertia weight, and a common setting of this parameter is w * = 0.7298, which corresponds to the constriction factor that is proposed by Clerc and Kennedy  . Concerning another setting that is frequently adopted in the literature, the inertia weight is dynamically adjusted, decreasing from a number around 1 to a smaller number around zero BIB005 , BIB006 . c1 and c2 are acceleration coefficients, which are also termed as the cognitive factor and the social factor, respectively. Both of them are set to 0.7298 × 2.05 ≈ 1.496 in the PSO with the constriction factor (PSO-cf)  . One kernel of PSO is population topology, which is also termed as neighborhood or information topology, specifying the social structure of particle swarm BIB002 BIB004 BIB007 . Information such as the best position that is found by some particle can only be transferred to its neighbors that are determined by population structure. The population topology has obvious effects on the spreading speed of information and the convergence of particle swarm BIB002 . Different population topologies correspond to different PSO versions BIB004 , BIB007 . The most well-known topology models are Gbest and Lbest. In the Gbest model, each particle's neighborhood covers the whole swarm and all particles are connected with each other. Usually, "gbest"is used to represent the best position found by the whole swarm. In the Lbest model, particles are not fully connected and each particle has fewer neighbors than its Gbest counterpart. The "Ring" topology in which each particle is connected to two neighbors to form a "ring," and the von Neumann topology in which each particle has four neighbors on a two-dimensional (2-D) lattice (left, right, above, and below) are typical Lbest topologies BIB007 . Obviously, the Gbest model can be regarded as a special case of the Lbest model.