Self-adaptive velocity particle swarm optimization for solving constrained optimization problems

Particle swarm optimization (PSO) is originally developed as an unconstrained optimization technique, therefore lacks an explicit mechanism for handling constraints. When solving constrained optimization problems (COPs) with PSO, the existing research mainly focuses on how to handle constraints, and the impact of constraints on the inherent search mechanism of PSO has been scarcely explored. Motivated by this fact, in this paper we mainly investigate how to utilize the impact of constraints (or the knowledge about the feasible region) to improve the optimization ability of the particles. Based on these investigations, we present a modified PSO, called self-adaptive velocity particle swarm optimization (SAVPSO), for solving COPs. To handle constraints, in SAVPSO we adopt our recently proposed dynamic-objective constraint-handling method (DOCHM), which is essentially a constituent part of the inherent search mechanism of the integrated SAVPSO, i.e., DOCHM + SAVPSO. The performance of the integrated SAVPSO is tested on a well-known benchmark suite and the experimental results show that appropriately utilizing the knowledge about the feasible region can substantially improve the performance of the underlying algorithm in solving COPs.     

Chaotic catfish particle swarm optimization for solving global numerical optimization problems

Chaotic catfish particle swarm optimization (C-CatfishPSO) is a novel optimization algorithm proposed in this paper. C-CatfishPSO introduces chaotic maps into catfish particle swarm optimization (CatfishPSO), which increase the search capability of CatfishPSO via the chaos approach. Simple CatfishPSO relies on the incorporation of catfish particles into particle swarm optimization (PSO). The introduced catfish particles improve the performance of PSO considerably. Unlike other ordinary particles, the catfish particles initialize a new search from extreme points of the search space when the gbest fitness value (global optimum at each iteration) has not changed for a certain number of consecutive iterations. This results in further opportunities of finding better solutions for the swarm by guiding the entire swarm to promising new regions of the search space and accelerating the search. The introduced chaotic maps strengthen the solution quality of PSO and CatfishPSO significantly. The resulting improved PSO and CatfishPSO are called chaotic PSO (C-PSO) and chaotic CatfishPSO (C-CatfishPSO), respectively. PSO, C-PSO, CatfishPSO, C-CatfishPSO, as well as other advanced PSO procedures from the literature were extensively compared on several benchmark test functions. Statistical analysis of the experimental results indicate that the performance of C-CatfishPSO is better than the performance of PSO, C-PSO, CatfishPSO and that C-CatfishPSO is also superior to advanced PSO methods from the literature.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

Enhanced speciation in particle swarm optimization for multi-modal problems

In this paper, we present a novel multi-modal optimization algorithm for finding multiple local optima in objective function surfaces. We build from Species-based particle swarm optimization (SPSO) by using deterministic sampling to generate new particles during the optimization process, by implementing proximity-based speciation coupled with speciation of isolated particles, and by including “turbulence regions” around already found solutions to prevent unnecessary function evaluations. Instead of using error threshold values, the new algorithm uses the particle’s experience, geometric mean, and “exclusion factor” to detect local optima and stop the algorithm. The performance of each extension is assessed with leave-it-out tests, and the results are discussed. We use the new algorithm called Isolated-Speciation-based particle swarm optimization (ISPSO) and a benchmark algorithm called Niche particle swarm optimization (NichePSO) to solve a six-dimensional rainfall characterization problem for 192 rain gages across the United States. We show why it is important to find multiple local optima for solving this real-world complex problem by discussing its high multi-modality. Solutions found by both algorithms are compared, and we conclude that ISPSO is more reliable than NichePSO at finding optima with a significantly lower objective function value.     

A novel hybrid differential evolution and particle swarm optimization algorithm for unconstrained optimization

An algorithm called DE-PSO is proposed which incorporates concepts from DE and PSO, updating particles not only by DE operators but also by mechanisms of PSO. The proposed algorithm is tested on several benchmark functions. Numerical comparisons with different hybrid meta-heuristics demonstrate its effectiveness and efficiency.     

Memetic particle swarm optimization

We propose a new Memetic Particle Swarm Optimization scheme that incorporates local search techniques in the standard Particle Swarm Optimization algorithm, resulting in an efficient and effective optimization method, which is analyzed theoretically. The proposed algorithm is applied to different unconstrained, constrained, minimax and integer programming problems and the obtained results are compared to that of the global and local variants of Particle Swarm Optimization, justifying the superiority of the memetic approach.     

A rank based particle swarm optimization algorithm with dynamic adaptation

The particle swarm optimization (PSO) technique is a powerful stochastic evolutionary algorithm that can be used to find the global optimum solution in a complex search space. This paper presents a variation on the standard PSO algorithm called the rank based particle swarm optimizer, or PSO_rank, employing cooperative behavior of the particles to significantly improve the performance of the original algorithm. In this method, in order to efficiently control the local search and convergence to global optimum solution, the γ best particles are taken to contribute to the updating of the position of a candidate particle. The contribution of each particle is proportional to its strength. The strength is a function of three parameters: strivness, immediacy and number of contributed particles. All particles are sorted according to their fitness values, and only the γ best particles will be selected. The value of γ decreases linearly as the iteration increases. A time-varying inertia weight decreasing non-linearly is introduced to improve the performance. PSO_rank is tested on a commonly used set of optimization problems and is compared to other variants of the PSO algorithm presented in the literature. As a real application, PSO_rank is used for neural network training. The PSO_rank strategy outperformed all the methods considered in this investigation for most of the functions. Experimental results show the suitability of the proposed algorithm in terms of effectiveness and robustness.     

Balanced fuzzy particle swarm optimization

In the present study an extension of particle swarm optimization (PSO) algorithm which is in conformity with actual nature is introduced for solving combinatorial optimization problems. Development of this algorithm is essentially based on balanced fuzzy sets theory. The classical fuzzy sets theory cannot distinguish differences between positive and negative information of membership functions, while in the new method both kinds of information “positive and negative” about membership function are equally important. The balanced fuzzy particle swarm optimization algorithm is used for fundamental optimization problem entitled traveling salesman problem (TSP). For convergence inspecting of new algorithm, method was used for TSP problems. Convergence curves were represented fast convergence in restricted and low iterations for balanced fuzzy particle swarm optimization algorithm (BF-PSO) comparison with fuzzy particle swarm optimization algorithm (F-PSO).     

A hybrid simplex search and particle swarm optimization for unconstrained optimization

This paper proposes the hybrid NM-PSO algorithm based on the Nelder–Mead (NM) simplex search method and particle swarm optimization (PSO) for unconstrained optimization. NM-PSO is very easy to implement in practice since it does not require gradient computation. The modification of both the Nelder–Mead simplex search method and particle swarm optimization intends to produce faster and more accurate convergence. The main purpose of the paper is to demonstrate how the standard particle swarm optimizers can be improved by incorporating a hybridization strategy. In a suite of 20 test function problems taken from the literature, computational results via a comprehensive experimental study, preceded by the investigation of parameter selection, show that the hybrid NM-PSO approach outperforms other three relevant search techniques (i.e., the original NM simplex search method, the original PSO and the guaranteed convergence particle swarm optimization (GCPSO)) in terms of solution quality and convergence rate. In a later part of the comparative experiment, the NM-PSO algorithm is compared to various most up-to-date cooperative PSO (CPSO) procedures appearing in the literature. The comparison report still largely favors the NM-PSO algorithm in the performance of accuracy, robustness and function evaluation. As evidenced by the overall assessment based on two kinds of computational experience, the new algorithm has demonstrated to be extremely effective and efficient at locating best-practice optimal solutions for unconstrained optimization.     

Improved particle swarm algorithm for hydrological parameter optimization

In this paper, a new method named MSSE-PSO (master-slave swarms shuffling evolution algorithm based on particle swarm optimization) is proposed. Firstly, a population of points is sampled randomly from the feasible space, and then partitioned into several sub-swarms (one master swarm and other slave swarms). Each slave swarm independently executes PSO or its variants, including the update of particles’ position and velocity. For the master swarm, the particles enhance themselves based on the social knowledge of master swarm and that of slave swarms. At periodic stage in the evolution, the master swarm and the whole slave swarms are forced to mix, and points are then reassigned to several sub-swarms to ensure the share of information. The process is repeated until a user-defined stopping criterion is reached. The tests of numerical simulation and the case study on hydrological model show that MSSE-PSO remarkably improves the accuracy of calibration, reduces the time of computation and enhances the performance of stability. Therefore, it is an effective and efficient global optimization method.     

Tribal particle swarm optimization for neurofuzzy inference systems and its prediction applications

This study presents tribal particle swarm optimization (TPSO) to optimize the parameters of the functional-link-based neurofuzzy inference system (FLNIS) for prediction applications. The proposed TPSO uses particle swarm optimization (PSO) as evolution strategies of the tribes optimization algorithm (TOA) to balance local and global exploration of the search space. The proposed TPSO uses a self-clustering algorithm to divide the particle swarm into multiple tribes, and selects suitable evolution strategies to update each particle. The TPSO also uses a tribal adaptation mechanism to remove and generate particles and reconstruct tribal links. The tribal adaptation mechanism can improve the qualities of the tribe and the tribe adaptation. Finally, the FLNIS model with the proposed TPSO (FLNIS-TPSO) was used in several predictive applications. Experimental results demonstrated that the proposed TPSO method converges quickly and yields a lower RMS error than other current methods.     

An efficient co-swarm particle swarm optimization for non-linear constrained optimization

This paper proposes a new co-swarm PSO (CSHPSO) for constrained optimization problems, which is obtained by hybridizing the recently proposed shrinking hypersphere PSO (SHPSO) with the differential evolution (DE) approach. The total swarm is subdivided into two sub swarms in such a way that the first sub swarms uses SHPSO and second sub swarms uses DE. Experiments are performed on a state-of-the-art problems proposed in IEEE CEC 2006. The results of the CSHPSO is compared with SHPSO and DE in a variety of fashions. A statistical approach is applied to provide the significance of the numerical experiments. In order to further test the efficacy of the proposed CSHPSO, an economic dispatch (ED) problem with valve points effects for 40 generating units is solved. The results of the problem using CSHPSO is compared with SHPSO, DE and the existing solutions in the literature. It is concluded that CSHPSO is able to give the minimal cost for the ED problem in comparison with the other algorithms considered. Hence, CSHPSO is a promising new co-swarm PSO which can be used to solve any real constrained optimization problem.     

A discrete particle swarm optimization method for feature selection in binary classification problems

This paper investigates the feature subset selection problem for the binary classification problem using logistic regression model. We developed a modified discrete particle swarm optimization (PSO) algorithm for the feature subset selection problem. This approach embodies an adaptive feature selection procedure which dynamically accounts for the relevance and dependence of the features included the feature subset. We compare the proposed methodology with the tabu search and scatter search algorithms using publicly available datasets. The results show that the proposed discrete PSO algorithm is competitive in terms of both classification accuracy and computational performance.     

A nonlinear interval number programming method for uncertain optimization problems

In this paper, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method.     

Hybrid procedures for solving some unconstrained nonlinear optimization problems

The vector -algorithm is a generalization of Aitken Δ²-process to the vector case. It has been used for solving a system of linear and nonlinear equations. The acceleration properties of the hybrid procedures have been also studied for solving a system of linear equations. In this paper, we consider the vector -algorithm and hybrid procedures for solving some unconstrained nonlinear optimization problems. A convergence acceleration result is established and numerical examples are given.     

An improved particle swarm optimization algorithm combined with piecewise linear chaotic map

Particle swarm optimization (PSO) has gained increasing attention in tackling complex optimization problems. Its further superiority when hybridized with other search techniques is also shown. Chaos, with the properties of ergodicity and stochasticity, is definitely a good candidate, but currently only the well-known logistic map is prevalently used. In this paper, the performance and deficiencies of schemes coupling chaotic search into PSO are analyzed. Then, the piecewise linear chaotic map (PWLCM) is introduced to perform the chaotic search. An improved PSO algorithm combined with PWLCM (PWLCPSO) is proposed subsequently, and experimental results verify its great superiority.     

A novel particle swarm optimization algorithm based on particle migration

Inspired by the migratory behavior in the nature, a novel particle swarm optimization algorithm based on particle migration (MPSO) is proposed in this work. In this new algorithm, the population is randomly partitioned into several sub-swarms, each of which is made to evolve based on particle swarm optimization with time varying inertia weight and acceleration coefficients (LPSO-TVAC). At periodic stage in the evolution, some particles migrate from one complex to another to enhance the diversity of the population and avoid premature convergence. It further improves the ability of exploration and exploitation. Simulations for benchmark test functions illustrate that the proposed algorithm possesses better ability to find the global optima than other variants and is an effective global optimization tool.     

The fitness evaluation strategy in particle swarm optimization

The particle swarm optimization (PSO) computational method has recently become popular. However, it has limitations. It may trap into local optima and cause the premature convergence phenomenon, especially for multimodal and high-dimensional problems. In this paper, we focus on investigating the fitness evaluation in terms of a particle’s position. Particularly, we find that the fitness evaluation strategy in the standard PSO has two drawbacks, i.e., “two steps forward and one step back” and “two steps back and one step forward”. In addition, we propose a general fitness evaluation strategy (GFES), by which a particle is evaluated in multiple subspaces and different contexts in order to take diverse paces towards the destination position. As demonstrations of GFES, a series of PSOs with GFES are presented. Experiments are conducted on several benchmark optimization problems. The results show that GFES is effective at handling multimodal and high-dimensional problems.     

Quantum-behaved particle swarm optimization with Gaussian distributed local attractor point

This paper proposes a novel variant of quantum-behaved particle swarm optimization (QPSO) algorithm with the local attractor point subject to a Gaussian probability distribution (GAQPSO). The local attractor point in QPSO plays an important in that determining the convergence behavior of an individual particle. As such, the mean value and standard deviation of the proposed Gaussian probability distribution in GAQPSO are carefully selected. The distributions and diversities of the local attractor points in GAQPSO and QPSO are evaluated and compared. For the purpose of comparison, two variants of the GAQPSO algorithm are proposed by using a mutation probability and other types of probability distribution. The GAQPSO has been comprehensively evaluated on the suite of CEC2005 benchmark functions, and the experimental results are compared with those of the PSO and QPSO algorithms based on different probability distributions. It is shown by the results that the GAQPSO algorithm is an effective approach that can improve the QPSO performance considerably, that is, the GAQPSO algorithm is less likely to be stuck in local optima and hence it can achieve better solutions in most cases.     

A new discrete particle swarm optimization approach for the single-machine total weighted tardiness scheduling problem with sequence-dependent setup times

In this paper we present a new Discrete Particle Swarm Optimization (DPSO) approach to face the NP-hard single machine total weighted tardiness scheduling problem in presence of sequence-dependent setup times. Differently from previous approaches the proposed DPSO uses a discrete model both for particle position and velocity and a coherent sequence metric. We tested the proposed DPSO mainly over a benchmark originally proposed by Cicirello in 2003 and available online. The results obtained show the competitiveness of our DPSO, which is able to outperform the best known results for the benchmark. In addition, we also tested the DPSO on a set of benchmark instances from ORLIB for the single machine total weighted tardiness problem, and we analysed the role of the DPSO swarm intelligence mechanisms as well as the local search intensification phase included in the algorithm.