Runtime analysis of a binary particle swarm optimizer

We investigate the runtime of a binary Particle Swarm Optimizer (PSO) for optimizing pseudo-Boolean functions f:{0,1}ⁿ→R. The binary PSO maintains a swarm of particles searching for good solutions. Each particle consists of a current position from {0,1}ⁿ, its own best position and a velocity vector used in a probabilistic process to update its current position. The velocities for a particle are then updated in the direction of its own best position and the position of the best particle in the swarm.We present a lower bound for the time needed to optimize any pseudo-Boolean function with a unique optimum. To prove upper bounds we transfer a fitness-level argument that is well-established for evolutionary algorithms (EAs) to PSO. This method is applied to estimate the expected runtime for the class of unimodal functions. A simple variant of the binary PSO is considered in more detail for the test function OneMax, showing that there the binary PSO is competitive to EAs. An additional experimental comparison reveals further insights.     

粒子群优化算法分析及研究进展

粒子群优化算法是一类基于群体智能的启发式全局优化技术,群体中的每一个微粒代表待解决问题的一个候选解,算法通过粒子间信息素的交互作用发现复杂搜索空间中的最优区域。本文介绍了粒子群优化算法的基本原理,并通过建立记忆表,详尽描述了粒子群优化算法中个体极优和全局极优的搜寻求解过程。同时,文章给出了多种改进形式以及研究现状,并提出了未来可能的研究方向。     

A study of particle swarm optimization particle trajectories

Particle swarm optimization (PSO) has shown to be an efficient, robust and simple optimization algorithm. Most of the PSO studies are empirical, with only a few theoretical analyses that concentrate on understanding particle trajectories. These theoretical studies concentrate mainly on simplified PSO systems. This paper overviews current theoretical studies, and extend these studies to investigate particle trajectories for general swarms to include the influence of the inertia term. The paper also provides a formal proof that each particle converges to a stable point. An empirical analysis of multi-dimensional stochastic particles is also presented. Experimental results are provided to support the conclusions drawn from the theoretical findings.     

Quantum-behaved particle swarm optimization: analysis of individual particle behavior and parameter selection

Quantum-behaved particle swarm optimization (QPSO), motivated by concepts from quantum mechanics and particle swarm optimization (PSO), is a probabilistic optimization algorithm belonging to the bare-bones PSO family. Although it has been shown to perform well in finding the optimal solutions for many optimization problems, there has so far been little analysis on how it works in detail. This paper presents a comprehensive analysis of the QPSO algorithm. In the theoretical analysis, we analyze the behavior of a single particle in QPSO in terms of probability measure. Since the particle's behavior is influenced by the contraction-expansion (CE) coefficient, which is the most important parameter of the algorithm, the goal of the theoretical analysis is to find out the upper bound of the CE coefficient, within which the value of the CE coefficient selected can guarantee the convergence or boundedness of the particle's position. In the experimental analysis, the theoretical results are first validated by stochastic simulations for the particle's behavior. Then, based on the derived upper bound of the CE coefficient, we perform empirical studies on a suite of well-known benchmark functions to show how to control and select the value of the CE coefficient, in order to obtain generally good algorithmic performance in real world applications. Finally, a further performance comparison between QPSO and other variants of PSO on the benchmarks is made to show the efficiency of the QPSO algorithm with the proposed parameter control and selection methods.     

一种基于栅格的动态粒子数微粒群算法

微粒群算法的全局搜索性能容易受到局部极值点的影响,对此,提出一种基于栅格的动态粒子数微粒群算法(GB-DPPPSO).通过设计栅格信息更新策略、粒子产生策略和粒子消灭策略,可以根据种群搜索情况动态控制粒子数变化,以保持种群多样性,提高全局搜索性能,通过对4个典型数学验证函数的仿真实验,表明了该算法相对于DPPPSO)在全局搜索成功率和搜索效率两方面均有明显改进.     

A discrete version of particle swarm optimization for flowshop scheduling problems

Particle swarm optimization (PSO) is a novel metaheuristic inspired by the flocking behavior of birds. The applications of PSO to scheduling problems are extremely few. In this paper, we present a PSO algorithm, extended from discrete PSO, for flowshop scheduling. In the proposed algorithm, the particle and the velocity are redefined, and an efficient approach is developed to move a particle to the new sequence. To verify the proposed PSO algorithm, comparisons with a continuous PSO algorithm and two genetic algorithms are made. Computational results show that the proposed PSO algorithm is very competitive. Furthermore, we incorporate a local search scheme into the proposed algorithm, called PSO-LS. Computational results show that the local search can be really guided by PSO in our approach. Also, PSO-LS performs well in flowshop scheduling with total flow time criterion, but it requires more computation times.     

Cellular particle swarm optimization

This paper proposes a cellular particle swarm optimization (CPSO), hybridizing cellular automata (CA) and particle swarm optimization (PSO) for function optimization. In the proposed CPSO, a mechanism of CA is integrated in the velocity update to modify the trajectories of particles to avoid being trapped in the local optimum. With two different ways of integration of CA and PSO, two versions of CPSO, i.e. CPSO-inner and CPSO-outer, have been discussed. For the former, we devised three typical lattice structures of CA used as neighborhood, enabling particles to interact inside the swarm; and for the latter, a novel CA strategy based on “smart-cell” is designed, and particles employ the information from outside the swarm. Theoretical studies are made to analyze the convergence of CPSO, and numerical experiments are conducted to compare the proposed algorithm with different variants of PSO. According to the experimental results, the proposed method performs better than other variants of PSO on benchmark test functions.     

一种改进的微粒群优化算法

基于惯性权重对微粒群优化算法（Particle Swarm Optimization,PSO）优化性能的显著影响,提出了一种改变惯性权重的方法以改进PSO算法的优化性能。算法中惯性权重的动态改变是通过对其进行PSO寻优来控制的。经过对标准函数的测试计算,无论是二维还是多维的问题,这种改变惯性权重的PSO算法的寻优结果的准确度和精度均得以提高,收到了良好的效果,尤其在高维情况下,显示出算法性能得到了明显改善。     

Accurate sub-swarms particle swarm optimization algorithm for service composition

Service composition (SC) generates various composite applications quickly by using a novel service interaction model. Before composing services together, the most important thing is to find optimal candidate service instances compliant with non-functional requirements. Particle swarm optimization (PSO) is known as an effective and efficient algorithm, which is widely used in this process. However, the premature convergence and diversity loss of PSO always results in suboptimal solutions. In this paper, we propose an accurate sub-swarms particle swarm optimization (ASPSO) algorithm by adopting parallel and serial niching techniques. The ASPSO algorithm locates optimal solutions by using sub-swarms searching grid cells in which the density of feasible solutions is high. Simulation results demonstrate that the proposed algorithm improves the accuracy of the standard PSO algorithm in searching the optimal solution of service selection problem.     

改进的多种群协同进化微粒群优化算法

提出一种改进的基于多种群协同进化的微粒群优化算法（ＰＳＯ）．该算法首先利用免疫算法实现解空间的均匀划分,增加了算法稳定性和全局搜索能力．在运行过程中,通过种群进化信息生成解优胜区域,指导变异生成的微粒群向最优解子空间逼近,提高算法逃出局部最优的能力．将此算法与ＰＳＯ 算法和多种群协同进化微粒群算法进行比较,数据实验证明,该算法不仅能有效地克服其他算法易陷入局部极小值的缺点,而且全局收敛能力和稳定性均有显著提高．     

粒子带质量的粒子群算法

针对粒子群算法在陷入局部最优时难于跳出的缺陷提出了一种带有质量的粒子群算法。该算法受运动学原理启发,粒子位置的更新不仅受自身最优和种群最优的影响,还受到由粒子质量引起的梯度场的影响。当粒子群出现早熟现象时,用电磁学原理与动量守恒定理更新种群的最优位置,使群体能及时摆脱局部最优区域。仿真结果表明,该算法优化4种具有代表性的基准函数,无论是在优化精度方面还是在优化效率方面,均较以往提出的改进粒子群算法在性能上有所改进。     

结合梯度法的混合微粒群优化算法

在微粒群优化算法PSO中引入梯度算法,提出了一种新型的混合微粒群优化算法——GPSO。该混合优化算法是对PSO每一次进化后的所有微粒进一步执行梯度法寻优操作,并以寻找到的更优个体替代当前个体参与群体的下一代进化。GPSO既利用了PSO出色的全局搜索能力,又借助梯度法的快速局部寻优能力,很好地将两者的优势结合在一起。数值实验表明：无论是对于低维的多峰函数,还是高维的多峰和单峰病态函数,GPSO都表现出很强的优化效率、适用性和鲁棒性。     

A perturbed particle swarm algorithm for numerical optimization

The canonical particle swarm optimization (PSO) has its own disadvantages, such as the high speed of convergence which often implies a rapid loss of diversity during the optimization process, which inevitably leads to undesirable premature convergence. In order to overcome the disadvantage of PSO, a perturbed particle swarm algorithm (pPSA) is presented based on the new particle updating strategy which is based upon the concept of perturbed global best to deal with the problem of premature convergence and diversity maintenance within the swarm. A linear model and a random model together with the initial max–min model are provided to understand and analyze the uncertainty of perturbed particle updating strategy. pPSA is validated using 12 standard test functions. The preliminary results indicate that pPSO performs much better than PSO both in quality of solutions and robustness and comparable with GCPSO. The experiments confirm us that the perturbed particle updating strategy is an encouraging strategy for stochastic heuristic algorithms and the max–min model is a promising model on the concept of possibility measure.     

A new particle swarm optimization for the open shop scheduling problem

This paper presents a new particle swarm optimization (PSO) for the open shop scheduling problem. Compared with the original PSO, we modified the particle position representation using priorities, and the particle movement using an insert operator. We also implemented a modified parameterized active schedule generation algorithm (mP-ASG) to decode a particle position into a schedule. In mP-ASG, we can reduce or increase the search area between non-delay schedules and active schedules by controlling the maximum delay time allowed. Furthermore, we hybridized our PSO with beam search. The computational results show that our PSO found many new best solutions of the unsolved problems.     

Unified particle swarm delivers high efficiency to particle swarm optimization

This paper suggests integrating a unification factor into particle swarm optimization (PSO) to balance the effects of cognitive and social terms. The resultant unified particle swarm (UPS) moves particles toward the center of its personal best and the global best. This improves on PSO, which moves particles far beyond the center. Widely used benchmark functions and four types of experiments demonstrate that the proposed UPS uses slightly more computational time than PSO to attain significantly higher efficiency and, usually, better solution effectiveness and consistency than PSO. Robust performance was further demonstrated by the significantly higher efficiency and better solution effectiveness and stability achieved by the UPS, as compared to the PSO and its variants. Outstandingly, convergence speeds for the proposed UPS were very good on the 13 benchmark functions examined in experiment 1, demonstrating the correct movement of UPS particles toward convergence.     

粒子群优化粒子滤波方法

针对粒子滤波方法存在粒子贫乏以及初始状态未知时需要大量粒子才能进行鲁棒状态预估等问题,将粒子群优化思想引入粒子滤波中．该方法将最新观测值融合到采样过程中,并对采样过程利用粒子群优化算法进行优化．通过优化,可使粒子集朝后验概率密度分布取值较大的区域运动,从而克服了粒子贫乏问题,并极大地降低了精确预估所需的粒子数．实验结果表明,该算法具有较高的预估精度和较好的鲁棒性．     

嵌入层叠混沌策略的随机粒子群算法

鉴于求解复杂问题时粒子群优化算法易出现早熟收敛的问题,通过引入轨迹扰动因子,提出随机粒子群进化迭代方程.该方程在统计行为中保证粒子向特定的收敛中心逼近,但对“旧址”的依赖性呈现出随机特性,从而使粒子群的快速跳转和迁移成为可能,避免过早落入局部陷阱.同时该进化方程还利用层叠混沌策略和对称极值扰动策略进一步增强算法的局部收敛性和全局搜索性.实验表明,由上述进化方程和改进策略构成的随机混沌粒子群算法具有鲁棒性较强、收敛速度较快和精度较高等优势,性能优于其他同源粒子群算法.     

A new controller for DC-DC converters based on particle swarm optimization

We propose a new controller for DC-DC converters based on particle swarm optimization (PSO). This new converter controller uses the PSO optimization method to directly control, by itself, the output voltage of a boost DC-DC converter. In order to validate and qualify the proposed converter controller, we analyzed and implemented some variants of the PSO algorithm, namely the standard PSO and the global local best PSO. The proposed converter controller was then compared with a variant of the classic PI controller with anti-windup, for different operational conditions. The three controllers compared in this work were implemented in the microcontroller TMS320F28027 by using the code composer studio from Texas Instruments. The results show that the proposed controller exhibits better behavior in terms of settling time and overshoot. Unlike most popular DC-DC converter controllers, the proposed controller does not require any sort of optimal parameter determination.     

A hybrid particle swarm optimization for job shop scheduling problem

A hybrid particle swarm optimization (PSO) for the job shop problem (JSP) is proposed in this paper. In previous research, PSO particles search solutions in a continuous solution space. Since the solution space of the JSP is discrete, we modified the particle position representation, particle movement, and particle velocity to better suit PSO for the JSP. We modified the particle position based on preference list-based representation, particle movement based on swap operator, and particle velocity based on the tabu list concept in our algorithm. Giffler and Thompson’s heuristic is used to decode a particle position into a schedule. Furthermore, we applied tabu search to improve the solution quality. The computational results show that the modified PSO performs better than the original design, and that the hybrid PSO is better than other traditional metaheuristics.     

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.