Self-adaptive particle swarm optimization: a review and analysis of convergence

Particle swarm optimization (PSO) is a population-based, stochastic search algorithm inspired by the flocking behaviour of birds. The PSO algorithm has been shown to be rather sensitive to its control parameters, and thus, performance may be greatly improved by employing appropriately tuned parameters. However, parameter tuning is typically a time-intensive empirical process. Furthermore, a priori parameter tuning makes the implicit assumption that the optimal parameters of the PSO algorithm are not time-dependent. To address these issues, self-adaptive particle swarm optimization (SAPSO) algorithms adapt their control parameters throughout execution. While there is a wide variety of such SAPSO algorithms in the literature, their behaviours are not well understood. Specifically, it is unknown whether these SAPSO algorithms will even exhibit convergent behaviour. This paper addresses this lack of understanding by investigating the convergence behaviours of 18 SAPSO algorithms both analytically and empirically. This paper also empirically examines whether the adapted parameters reach a stable point and whether the final parameter values adhere to a well-known convergence criterion. The results depict a grim state for SAPSO algorithms; over half of the SAPSO algorithms exhibit divergent behaviour while many others prematurely converge.     

A competitive clustering particle swarm optimizer for dynamic optimization problems

Optimization in dynamic optimization problems (DOPs) requires the optimization algorithms not only to locate, but also to continuously track the moving optima. Particle swarm optimization (PSO) is a population-based optimization algorithm, originally developed for static problems. Recently, several researchers have proposed variants of PSO for optimization in DOPs. This paper presents a novel multi-swarm PSO algorithm, namely competitive clustering PSO (CCPSO), designed specially for DOPs. Employing a multi-stage clustering procedure, CCPSO splits the particles of the main swarm over a number of sub-swarms based on the particles positions and on their objective function values. The algorithm automatically adjusts the number of sub-swarms and the corresponding region of each sub-swarm. In addition to the sub-swarms, there is also a group of free particles that explore the environment to locate new emerging optima or exploit the current optima which are not followed by any sub-swarm. The adaptive search strategy adopted by the sub-swarms improves both the exploitation and tracking characteristics of CCPSO. A set of experiments is conducted to study the behavior of the proposed algorithm in different DOPs and to provide guidelines for setting the algorithm’s parameters in different problems. The results of CCPSO on a variety of moving peaks benchmark (MPB) functions are compared with those of several state-of-the-art PSO algorithms, indicating the efficiency of the proposed model.     

Optimized Particle Swarm Optimization (OPSO) and its application to artificial neural network training

Background  

Particle Swarm Optimization (PSO) is an established method for parameter optimization. It represents a population-based adaptive optimization technique that is influenced by several "strategy parameters". Choosing reasonable parameter values for the PSO is crucial for its convergence behavior, and depends on the optimization task. We present a method for parameter meta-optimization based on PSO and its application to neural network training. The concept of the Optimized Particle Swarm Optimization (OPSO) is to optimize the free parameters of the PSO by having swarms within a swarm. We assessed the performance of the OPSO method on a set of five artificial fitness functions and compared it to the performance of two popular PSO implementations.     

Hierarchical winner-take-all particle swarm optimization social network for neural model fitting

Particle swarm optimization (PSO) has gained widespread use as a general mathematical programming paradigm and seen use in a wide variety of optimization and machine learning problems. In this work, we introduce a new variant on the PSO social network and apply this method to the inverse problem of input parameter selection from recorded auditory neuron tuning curves. The topology of a PSO social network is a major contributor to optimization success. Here we propose a new social network which draws influence from winner-take-all coding found in visual cortical neurons. We show that the winner-take-all network performs exceptionally well on optimization problems with greater than 5 dimensions and runs at a lower iteration count as compared to other PSO topologies. Finally we show that this variant of PSO is able to recreate auditory frequency tuning curves and modulation transfer functions, making it a potentially useful tool for computational neuroscience models.     

Stochastic stability of particle swarm optimisation

Particle swarm optimisation (PSO) is a metaheuristic algorithm used to find good solutions in a wide range of optimisation problems. The success of metaheuristic approaches is often dependent on the tuning of the control parameters. As the algorithm includes stochastic elements that effect the behaviour of the system, it may be studied using the framework of random dynamical systems (RDS). In PSO, the swarm dynamics are quasi-linear, which enables an analytical treatment of their stability. Our analysis shows that the region of stability extends beyond those predicted by earlier approximate approaches. Simulations provide empirical backing for our analysis and show that the best performance is achieved in the asymptotic case where the parameters are selected near the margin of instability predicted by the RDS approach.     

Continuous optimization algorithms for tuning real and integer parameters of swarm intelligence algorithms

The performance of optimization algorithms, including those based on swarm intelligence, depends on the values assigned to their parameters. To obtain high performance, these parameters must be fine-tuned. Since many parameters can take real values or integer values from a large domain, it is often possible to treat the tuning problem as a continuous optimization problem. In this article, we study the performance of a number of prominent continuous optimization algorithms for parameter tuning using various case studies from the swarm intelligence literature. The continuous optimization algorithms that we study are enhanced to handle the stochastic nature of the tuning problem. In particular, we introduce a new post-selection mechanism that uses F-Race in the final phase of the tuning process to select the best among elite parameter configurations. We also examine the parameter space of the swarm intelligence algorithms that we consider in our study, and we show that by fine-tuning their parameters one can obtain substantial improvements over default configurations.     

Particle Swarm Optimization with Scale-Free Interactions

The particle swarm optimization (PSO) algorithm, in which individuals collaborate with their interacted neighbors like bird flocking to search for the optima, has been successfully applied in a wide range of fields pertaining to searching and convergence. Here we employ the scale-free network to represent the inter-individual interactions in the population, named SF-PSO. In contrast to the traditional PSO with fully-connected topology or regular topology, the scale-free topology used in SF-PSO incorporates the diversity of individuals in searching and information dissemination ability, leading to a quite different optimization process. Systematic results with respect to several standard test functions demonstrate that SF-PSO gives rise to a better balance between the convergence speed and the optimum quality, accounting for its much better performance than that of the traditional PSO algorithms. We further explore the dynamical searching process microscopically, finding that the cooperation of hub nodes and non-hub nodes play a crucial role in optimizing the convergence process. Our work may have implications in computational intelligence and complex networks.     

A new particle swarm optimization algorithm for noisy optimization problems

We propose a new particle swarm optimization algorithm for problems where objective functions are subject to zero-mean, independent, and identically distributed stochastic noise. While particle swarm optimization has been successfully applied to solve many complex deterministic nonlinear optimization problems, straightforward applications of particle swarm optimization to noisy optimization problems are subject to failure because the noise in objective function values can lead the algorithm to incorrectly identify positions as the global/personal best positions. Instead of having the entire swarm follow a global best position based on the sample average of objective function values, the proposed new algorithm works with a set of statistically global best positions that include one or more positions with objective function values that are statistically equivalent, which is achieved using a combination of statistical subset selection and clustering analysis. The new PSO algorithm can be seamlessly integrated with adaptive resampling procedures to enhance the capability of PSO to cope with noisy objective functions. Numerical experiments demonstrate that the new algorithm is able to consistently find better solutions than the canonical particle swarm optimization algorithm in the presence of stochastic noise in objective function values with different resampling procedures.     

Artificial bee colonies for continuous optimization: Experimental analysis and improvements

The artificial bee colony (ABC) algorithm is a recent class of swarm intelligence algorithms that is loosely inspired by the foraging behavior of honeybee swarms. It was introduced in 2005 using continuous optimization problems as an example application. Similar to what has happened with other swarm intelligence techniques, after the initial proposal, several researchers have studied variants of the original algorithm. Unfortunately, often these variants have been tested under different experimental conditions and different fine-tuning efforts for the algorithm parameters. In this article, we review various variants of the original ABC algorithm and experimentally study nine ABC algorithms under two settings: either using the original parameter settings as proposed by the authors, or using an automatic algorithm configuration tool using a same tuning effort for each algorithm. We also study the effect of adding local search to the ABC algorithms. Our experimental results show that local search can improve considerably the performance of several ABC variants and that it reduces strongly the performance differences between the studied ABC variants. We also show that the best ABC variants are competitive with recent state-of-the-art algorithms on the benchmark set we used, which establishes ABC algorithms as serious competitors in continuous optimization.     

Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions

This paper presents glowworm swarm optimization (GSO), a novel algorithm for the simultaneous computation of multiple optima of multimodal functions. The algorithm shares a few features with some better known swarm intelligence based optimization algorithms, such as ant colony optimization and particle swarm optimization, but with several significant differences. The agents in GSO are thought of as glowworms that carry a luminescence quantity called luciferin along with them. The glowworms encode the fitness of their current locations, evaluated using the objective function, into a luciferin value that they broadcast to their neighbors. The glowworm identifies its neighbors and computes its movements by exploiting an adaptive neighborhood, which is bounded above by its sensor range. Each glowworm selects, using a probabilistic mechanism, a neighbor that has a luciferin value higher than its own and moves toward it. These movements—based only on local information and selective neighbor interactions—enable the swarm of glowworms to partition into disjoint subgroups that converge on multiple optima of a given multimodal function. We provide some theoretical results related to the luciferin update mechanism in order to prove the bounded nature and convergence of luciferin levels of the glowworms. Experimental results demonstrate the efficacy of the proposed glowworm based algorithm in capturing multiple optima of a series of standard multimodal test functions and more complex ones, such as stair-case and multiple-plateau functions. We also report the results of tests in higher dimensional spaces with a large number of peaks. We address the parameter selection problem by conducting experiments to show that only two parameters need to be selected by the user. Finally, we provide some comparisons of GSO with PSO and an experimental comparison with Niche-PSO, a PSO variant that is designed for the simultaneous computation of multiple optima. This work is partially supported by a project grant from the Ministry of Human Resources Development, India and by DRDO-IISc Mathematical Engineering Program.     

Optimal Allocation of Node Capacity in Cascade-Robustness Networks

The robustness of large scale critical infrastructures, which can be modeled as complex networks, is of great significance. One of the most important means to enhance robustness is to optimize the allocation of resources. Traditional allocation of resources is mainly based on the topology information, which is neither realistic nor systematic. In this paper, we try to build a framework for searching for the most favorable pattern of node capacity allocation to reduce the vulnerability to cascading failures at a low cost. A nonlinear and multi-objective optimization model is proposed and tackled using a particle swarm optimization algorithm (PSO). It is found that the network becomes more robust and economical when less capacity is left on the heavily loaded nodes and the optimized network performs better resisting noise. Our work is helpful in designing a robust economical network.     

Inertia weight control strategies for particle swarm optimization

Particle swarm optimization (PSO) is a population-based, stochastic optimization technique inspired by the social dynamics of birds. The PSO algorithm is rather sensitive to the control parameters, and thus, there has been a significant amount of research effort devoted to the dynamic adaptation of these parameters. The focus of the adaptive approaches has largely revolved around adapting the inertia weight as it exhibits the clearest relationship with the exploration/exploitation balance of the PSO algorithm. However, despite the significant amount of research efforts, many inertia weight control strategies have not been thoroughly examined analytically nor empirically. Thus, there are a plethora of choices when selecting an inertia weight control strategy, but no study has been comprehensive enough to definitively guide the selection. This paper addresses these issues by first providing an overview of 18 inertia weight control strategies. Secondly, conditions required for the strategies to exhibit convergent behaviour are derived. Finally, the inertia weight control strategies are empirically examined on a suite of 60 benchmark problems. Results of the empirical investigation show that none of the examined strategies, with the exception of a randomly selected inertia weight, even perform on par with a constant inertia weight.     

Parameter Identification of Robot Manipulators: A Heuristic Particle Swarm Search Approach

Parameter identification of robot manipulators is an indispensable pivotal process of achieving accurate dynamic robot models. Since these kinetic models are highly nonlinear, it is not easy to tackle the matter of identifying their parameters. To solve the difficulty effectively, we herewith present an intelligent approach, namely, a heuristic particle swarm optimization (PSO) algorithm, which we call the elitist learning strategy (ELS) and proportional integral derivative (PID) controller hybridized PSO approach (ELPIDSO). A specified PID controller is designed to improve particles’ local and global positions information together with ELS. Parameter identification of robot manipulators is conducted for performance evaluation of our proposed approach. Experimental results clearly indicate the following findings: Compared with standard PSO (SPSO) algorithm, ELPIDSO has improved a lot. It not only enhances the diversity of the swarm, but also features better search effectiveness and efficiency in solving practical optimization problems. Accordingly, ELPIDSO is superior to least squares (LS) method, genetic algorithm (GA), and SPSO algorithm in estimating the parameters of the kinetic models of robot manipulators.     

Experimental multivariable adaptive optimization of the steady-state cellular productivity of a continuous baker's yeast culture

A multivariable adaptive optimization algorithm that uses transient data to improve the optimization speed was successfully implemented on-line to maximize the steady-state cellular productivity of a continuous culture of baker's yeast. The algorithm was shown to be stable even during periods of oscillatory growth and was able to reoptimize the culture when planned disturbances were introduced. Although adaptive tuning of the forgetting factor improved the performance, further refinements in the adaptive forgetting factor algorithm are necessary for completely satisfactory results.     

Set-based particle swarm optimization applied to the multidimensional knapsack problem

Particle swarm optimization algorithms have been successfully applied to discrete/valued optimization problems. However, in many cases the algorithms have been tailored specifically for the problem at hand. This paper proposes a generic set-based particle swarm optimization algorithm for use in discrete-valued optimization problems that can be formulated as set-based problems. A detailed sensitivity analysis of the parameters of the algorithm is conducted. The performance of the proposed algorithm is then compared against three other discrete particle swarm optimization algorithms from literature using the multidimensional knapsack problem and is shown to statistically outperform the existing algorithms.     

Biomimicry of symbiotic multi-species coevolution for discrete and continuous optimization in RFID networks

In recent years, symbiosis as a rich source of potential engineering applications and computational model has attracted more and more attentions in the adaptive complex systems and evolution computing domains. Inspired by different symbiotic coevolution forms in nature, this paper proposed a series of multi-swarm particle swarm optimizers called PS²Os, which extend the single population particle swarm optimization (PSO) algorithm to interacting multi-swarms model by constructing hierarchical interaction topologies and enhanced dynamical update equations. According to different symbiotic interrelationships, four versions of PS²O are initiated to mimic mutualism, commensalism, predation, and competition mechanism, respectively. In the experiments, with five benchmark problems, the proposed algorithms are proved to have considerable potential for solving complex optimization problems. The coevolutionary dynamics of symbiotic species in each PS²O version are also studied respectively to demonstrate the heterogeneity of different symbiotic interrelationships that effect on the algorithm’s performance. Then PS²O is used for solving the radio frequency identification (RFID) network planning (RNP) problem with a mixture of discrete and continuous variables. Simulation results show that the proposed algorithm outperforms the reference algorithms for planning RFID networks, in terms of optimization accuracy and computation robustness.     

Multi-Hop Routing Optimization Method Based on Improved Ant Algorithm for Vehicle to Roadside Network

This paper proposes a route optimization method to improve the performance of route selection in Vehicle Ad-hoc Network （VANET）. A novel bionic swarm intelligence algorithm, which is called ant colony algorithm, was introduced into a traditional ad-hoc route algorithm named AODV. Based on the analysis of movement characteristics of vehicles and according to the spatial relationship between the vehicles and the roadside units, the parameters in ant colony system were modified to enhance the performance of the route selection probability rules. When the vehicle moves into the range of several different roadsides, it could build the route by sending some route testing packets as ants, so that the route table can be built by the reply information of test ants, and then the node can establish the optimization path to send the application packets. The simulation results indicate that the proposed algorithm has better performance than the traditional AODV algorithm, especially when the vehicle is in higher speed or the number of nodes increases.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.     

Particle swarm optimizer for weighting factor selection in intensity-modulated radiation therapy optimization algorithms

In inverse treatment planning of intensity-modulated radiation therapy (IMRT), the objective function is typically the sum of the weighted sub-scores, where the weights indicate the importance of the sub-scores. To obtain a high-quality treatment plan, the planner manually adjusts the objective weights using a trial-and-error procedure until an acceptable plan is reached. In this work, a new particle swarm optimization (PSO) method which can adjust the weighting factors automatically was investigated to overcome the requirement of manual adjustment, thereby reducing the workload of the human planner and contributing to the development of a fully automated planning process. The proposed optimization method consists of three steps. (i) First, a swarm of weighting factors (i.e., particles) is initialized randomly in the search space, where each particle corresponds to a global objective function. (ii) Then, a plan optimization solver is employed to obtain the optimal solution for each particle, and the values of the evaluation functions used to determine the particle’s location and the population global location for the PSO are calculated based on these results. (iii) Next, the weighting factors are updated based on the particle’s location and the population global location. Step (ii) is performed alternately with step (iii) until the termination condition is reached. In this method, the evaluation function is a combination of several key points on the dose volume histograms. Furthermore, a perturbation strategy – the crossover and mutation operator hybrid approach – is employed to enhance the population diversity, and two arguments are applied to the evaluation function to improve the flexibility of the algorithm. In this study, the proposed method was used to develop IMRT treatment plans involving five unequally spaced 6 MV photon beams for 10 prostate cancer cases. The proposed optimization algorithm yielded high-quality plans for all of the cases, without human planner intervention. A comparison of the results with the optimized solution obtained using a similar optimization model but with human planner intervention revealed that the proposed algorithm produced optimized plans superior to that developed using the manual plan. The proposed algorithm can generate admissible solutions within reasonable computational times and can be used to develop fully automated IMRT treatment planning methods, thus reducing human planners’ workloads during iterative processes.