A Survey of Swarm Intelligence Techniques in VLSI Routing Problems <s> 2) MATHEMATICAL REPRESENTATION OF PSO <s> We empirically study the performance of the particle swarm optimizer (PSO). Four different benchmark functions with asymmetric initial range settings are selected as testing functions. The experimental results illustrate the advantages and disadvantages of the PSO. Under all the testing cases, the PSO always converges very quickly towards the optimal positions but may slow its convergence speed when it is near a minimum. Nevertheless, the experimental results show that the PSO is a promising optimization method and a new approach is suggested to improve PSO's performance near the optima, such as using an adaptive inertia weight. <s> BIB001 </s> A Survey of Swarm Intelligence Techniques in VLSI Routing Problems <s> 2) MATHEMATICAL REPRESENTATION OF PSO <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> BIB002 </s> A Survey of Swarm Intelligence Techniques in VLSI Routing Problems <s> 2) MATHEMATICAL REPRESENTATION OF PSO <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> BIB003
In the PSO algorithm, particles dynamically adjust their position information by learning their best individual position and the global best position in the swarm. Consider a minimization problem with D-dimensional search space, assuming that the population size of is M , and the speed and position update formulas of the Standard Particle Swarm Optimization (SPSO) BIB001 are as follows: , ω is the inertia weight to better balance the exploration and exploitation of the algorithm. c 1 and c 2 are acceleration coefficients, which respectively adjust the step size of the particle flying to its personal best position and the global best position. r 1 and r 2 are mutually independent random numbers uniformly distributed in the interval (0,1). P t ij and G t j are the best position of the particle i and the global optimal position of the population respectively, satisfying the following formula: In 1999, Clerc and Kennedy introduced constriction coefficient  in the evolution equation, relaxed the speed limit, and thus improved the convergence speed of the algorithm. The speed update formula is as follows: where ϕ 1 and ϕ 2 are acceleration coefficients, and χ is constriction coefficient satisfying: However, some dimensions that are close to the optimal solution are likely to be far from the optimal solution VOLUME 8, 2020 although the particle position is improved after updating. Therefore, a kind of Cooperative Particle Swarm Optimization (CPSO) BIB002 is proposed to solve this problem. CPSO divides the dimensions of particles into several groups and each group is optimized with a single particle swarm. Then calculate the fitness value after combining the dimensions. Finally, the corresponding update is performed according to the update rules of the SPSO algorithm. In order to solve the problem that PSO is easily trapped into local optimal when dealing with complex multi-peak problems, Liang et al. BIB003 proposed Comprehensive Learning PSO (CLPSO), in which particles have more learning objects and thus have more potential flight space.