A multi-swarm PSO using charged particles in a partitioned search space for continuous optimization

Particle swarm optimization (PSO) is characterized by a fast convergence, which can lead the algorithms of this class to stagnate in local optima. In this paper, a variant of the standard PSO algorithm is presented, called PSO-2S, based on several initializations in different zones of the search space, using charged particles. This algorithm uses two kinds of swarms, a main one that gathers the best particles of auxiliary ones, initialized several times. The auxiliary swarms are initialized in different areas, then an electrostatic repulsion heuristic is applied in each area to increase its diversity. We analyse the performance of the proposed approach on a testbed made of unimodal and multimodal test functions with and without coordinate rotation and shift. The Lennard-Jones potential problem is also used. The proposed algorithm is compared to several other PSO algorithms on this benchmark. The obtained results show the efficiency of the proposed algorithm.     

Differential Evolution algorithm with Separated Groups for multi-dimensional optimization problems

The classical Differential Evolution (DE) algorithm, one of population-based Evolutionary Computation methods, proved to be a successful approach for relatively simple problems, but does not perform well for difficult multi-dimensional non-convex functions. A number of significant modifications of DE have been proposed in recent years, including very few approaches referring to the idea of distributed Evolutionary Algorithms. The present paper presents a new algorithm to improve optimization performance, namely DE with Separated Groups (DE-SG), which distributes population into small groups, defines rules of exchange of information and individuals between the groups and uses two different strategies to keep balance between exploration and exploitation capabilities. The performance of DE-SG is compared to that of eight algorithms belonging to the class of Evolutionary Strategies (Covariance Matrix Adaptation ES), Particle Swarm Optimization (Comprehensive Learning PSO and Efficient Population Utilization Strategy PSO), Differential Evolution (Distributed DE with explorative-exploitative population families, Self-adaptive DE, DE with global and local neighbours and Grouping Differential Evolution) and multi-algorithms (AMALGAM). The comparison is carried out for a set of 10-, 30- and 50-dimensional rotated test problems of varying difficulty, including 10- and 30-dimensional composition functions from CEC2005. Although slow for simple functions, the proposed DE-SG algorithm achieves a great success rate for more difficult 30- and 50-dimensional problems.     

Particle swarm optimization approach to portfolio optimization

The survey of the relevant literature showed that there have been many studies for portfolio optimization problem and that the number of studies which have investigated the optimum portfolio using heuristic techniques is quite high. But almost none of these studies deals with particle swarm optimization (PSO) approach. This study presents a heuristic approach to portfolio optimization problem using PSO technique. The test data set is the weekly prices from March 1992 to September 1997 from the following indices: Hang Seng in Hong Kong, DAX 100 in Germany, FTSE 100 in UK, S&P 100 in USA and Nikkei in Japan. This study uses the cardinality constrained mean-variance model. Thus, the portfolio optimization model is a mixed quadratic and integer programming problem for which efficient algorithms do not exist. The results of this study are compared with those of the genetic algorithms, simulated annealing and tabu search approaches. The purpose of this paper is to apply PSO technique to the portfolio optimization problem. The results show that particle swarm optimization approach is successful in portfolio optimization.     

Hybrid co-evolutionary particle swarm optimization and noising metaheuristics for the delay constrained least cost path problem

This paper presents a co-evolutionary particle swarm optimization (PSO) algorithm, hybridized with noising metaheuristics, for solving the delay constrained least cost (DCLC) path problem, i.e., shortest-path problem with a delay constraint on the total “cost” of the optimal path. The proposed algorithm uses the principle of Lagrange relaxation based aggregated cost. It essentially consists of two concurrent PSOs for solving the resulting minimization-maximization problem. The main PSO is designed as a hybrid PSO-noising metaheuristics algorithm for efficient global search to solve the minimization part of the DCLC-Lagrangian relaxation by finding multiple shortest paths between a source-destination pair. The auxiliary/second PSO is a co-evolutionary PSO to obtain the optimal Lagrangian multiplier for solving the maximization part of the Lagrangian relaxation problem. For the main PSO, a novel heuristics-based path encoding/decoding scheme has been devised for representation of network paths as particles. The simulation results on several networks with random topologies illustrate the efficiency of the proposed hybrid algorithm for the constrained shortest path computation problems.     

New mutation schemes for differential evolution algorithm and their application to the optimization of directional over-current relay settings

Differential evolution is a novel evolutionary approach capable of handling non-differentiable, nonlinear and multimodal objective functions. It has been consistently ranked as one of the best search algorithm for solving global optimization problems in several case studies. In the present study we propose five new mutation schemes for the basic DE algorithm. The corresponding versions are termed as MDE1, MDE2, MDE3, MDE4 and MDE5. These new schemes make use of the absolute weighted difference between the two points and instead of using a fixed scaling factor F, use a scaling factor following the Laplace distribution. The performance of the proposed schemes is validated empirically on a suit of ten benchmark problems having box constraints. Numerical analysis of results shows that the proposed schemes improves the convergence rate of the DE algorithm and also maintains the quality of solution. Efficiency of the proposed schemes is further validated by applying it to a real life electrical engineering problem dealing with the optimization of directional over-current relay settings. It is a highly constrained nonlinear optimization problem. A constraint handling mechanism based on repair methods is used for handling the constraints. Once again the simulation results show the compatibility of the proposed schemes for solving the real life problem.     

Self Balanced Differential Evolution

Differential Evolution (DE) is a well known and simple population based probabilistic approach for global optimization. It has reportedly outperformed a few Evolutionary Algorithms (EAs) and other search heuristics like the Particle Swarm Optimization (PSO) when tested over both benchmark and real world problems. But, DE, like other probabilistic optimization algorithms, sometimes behave prematurely in convergence. Therefore, in order to avoid stagnation while keeping a good convergence speed for DE, two modifications are proposed: one is the introduction of a new control parameter, Cognitive Learning Factor (CLF) and the other is dynamic setting of scale factor. Both modifications are proposed in mutation process of DE. Cognitive learning is a powerful mechanism that adjust the current position of individuals by a means of some specified knowledge. The proposed strategy, named as Self Balanced Differential Evolution (SBDE), balances the exploration and exploitation capability of the DE. To prove efficiency and efficacy of SBDE, it is tested over 30 benchmark optimization problems and compared the results with the basic DE and advanced variants of DE namely, SFLSDE, OBDE and jDE. Further, a real-world optimization problem, namely, Spread Spectrum Radar Polly phase Code Design, is solved to show the wide applicability of the SBDE.     

Particle swarm optimization with time-varying acceleration coefficients for the multidimensional knapsack problem

The multidimensional knapsack problem (MKP) is a difficult combinatorial optimization problem, which has been proven as NP-hard problems. Various population-based search algorithms are applied to solve these problems. The particle swarm optimization (PSO) technique is adapted in our study, which proposes two novel PSO algorithms, namely, the binary PSO with time-varying acceleration coefficients (BPSOTVAC) and the chaotic binary PSO with time-varying acceleration coefficients (CBPSOTVAC). The two proposed methods were tested using 116 benchmark problems from the OR-Library to validate and demonstrate the efficiency of these algorithms in solving multidimensional knapsack problems. The results were then compared with those in the other two existing PSO algorithms. The simulation and evaluation results showed that the proposed algorithms, BPSOTVAC and CBPSOTVAC, are superior over the other methods according to its success rate, mean absolute deviation, mean absolute percentage error, least error, and standard deviation.     

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.     

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.     

A new multi-threshold image segmentation approach using state transition algorithm

Thresholding plays an important role in image segmentation and image analysis. In this paper, the normalized histogram of an image is fitted by a linear combined normal distribution functions and each normal distribution function represents a class of pixels, whereas the parameters like the mean, the variance and the weights in the fitting function are undetermined. By transforming the fitting problem into a nonlinear and non-convex optimization problem, the state transition algorithm (STA) which is a new global optimization method is used to choose the optimal parameters of the fitting function. The effectiveness of proposed approach in multilevel thresholding problems is tested by several experimental results. By comparing with OTSU, particle swarm optimization (PSO), genetic algorithm (GA) and differential evolution (DE) algorithm, it has shown that STA has competitive performance in terms of both optimization results and thresholding segmentation.     

An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters

Real engineering design problems are generally characterized by the presence of many often conflicting and incommensurable objectives. Naturally, these objectives involve many parameters whose possible values may be assigned by the experts. The aim of this paper is to introduce a hybrid approach combining three optimization techniques, dynamic programming (DP), genetic algorithms and particle swarm optimization (PSO). Our approach integrates the merits of both DP and artificial optimization techniques and it has two characteristic features. Firstly, the proposed algorithm converts fuzzy multiobjective optimization problem to a sequence of a crisp nonlinear programming problems. Secondly, the proposed algorithm uses H-SOA for solving nonlinear programming problem. In which, any complex problem under certain structure can be solved and there is no need for the existence of some properties rather than traditional methods that need some features of the problem such as differentiability and continuity. Finally, with different degree of α we get different α-Pareto optimal solution of the problem. A numerical example is given to illustrate the results developed in this paper.     

Advanced particle swarm assisted genetic algorithm for constrained optimization problems

A novel hybrid evolutionary algorithm is developed based on the particle swarm optimization (PSO) and genetic algorithms (GAs). The PSO phase involves the enhancement of worst solutions by using the global-local best inertia weight and acceleration coefficients to increase the efficiency. In the genetic algorithm phase, a new rank-based multi-parent crossover is used by modifying the crossover and mutation operators which favors both the local and global exploration simultaneously. In addition, the Euclidean distance-based niching is implemented in the replacement phase of the GA to maintain the population diversity. To avoid the local optimum solutions, the stagnation check is performed and the solution is randomized when needed. The constraints are handled using an effective feasible population based approach. The parameters are self-adaptive requiring no tuning based on the type of problems. Numerical simulations are performed first to evaluate the current algorithm for a set of 24 benchmark constrained nonlinear optimization problems. The results demonstrate reasonable correlation and high quality optimum solutions with significantly less function evaluations against other state-of-the-art heuristic-based optimization algorithms. The algorithm is also applied to various nonlinear engineering optimization problems and shown to be excellent in searching for the global optimal solutions.     

Multi-objective optimization of water-using systems

Industrial water systems often allow efficient water uses via water reuse and/or recirculation. The design of the network layout connecting water-using processes is a complex problem which involves several criteria to optimize. Most of the time, this design is achieved using Water Pinch technology, optimizing the freshwater flow rate entering the system. This paper describes an approach that considers two criteria: (i) the minimization of freshwater consumption and (ii) the minimization of the infrastructure cost required to build the network. The optimization model considers water reuse between operations and wastewater treatment as the main mechanisms to reduce freshwater consumption. The model is solved using multi-objective distributed Q-learning (MDQL), a heuristic approach based on the exploitation of knowledge acquired during the search process. MDQL has been previously tested on several multi-objective optimization benchmark problems with promising results [C. Mariano, Reinforcement learning in multi-objective optimization, Ph.D. thesis in Computer Science, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Cuernavaca, March, 2002, Cuernavaca, Mor., México, 2001]. In order to compare the quality of the results obtained with MDQL, the reduced gradient method was applied to solve a weighted combination of the two objective functions used in the model. The proposed approach was tested on three cases: (i) a single contaminant four unitary operations problem where freshwater consumption is reduced via water reuse, (ii) a four contaminants real-world case with ten unitary operations, also with water reuse, and (iii) the water distribution network operation of Cuernavaca, Mexico, considering reduction of water leaks, operation of existing treatment plants at their design capacity, and design and construction of new treatment infrastructure to treat 100% of the wastewater produced. It is shown that the proposed approach can solved highly constrained real-world multi-objective optimization problems.     

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.     

Integrating particle swarm optimization with genetic algorithms for solving nonlinear optimization problems

Heuristic optimization provides a robust and efficient approach for solving complex real-world problems. The aim of this paper is to introduce a hybrid approach combining two heuristic optimization techniques, particle swarm optimization (PSO) and genetic algorithms (GA). Our approach integrates the merits of both GA and PSO and it has two characteristic features. Firstly, the algorithm is initialized by a set of random particles which travel through the search space. During this travel an evolution of these particles is performed by integrating PSO and GA. Secondly, to restrict velocity of the particles and control it, we introduce a modified constriction factor. Finally, the results of various experimental studies using a suite of multimodal test functions taken from the literature have demonstrated the superiority of the proposed approach to finding the global optimal solution.     

Robust optimization approximation for joint chance constrained optimization problem

Chance constraint is widely used for modeling solution reliability in optimization problems with uncertainty. Due to the difficulties in checking the feasibility of the probabilistic constraint and the non-convexity of the feasible region, chance constrained problems are generally solved through approximations. Joint chance constrained problem enforces that several constraints are satisfied simultaneously and it is more complicated than individual chance constrained problem. This work investigates the tractable robust optimization approximation framework for solving the joint chance constrained problem. Various robust counterpart optimization formulations are derived based on different types of uncertainty set. To improve the quality of robust optimization approximation, a two-layer algorithm is proposed. The inner layer optimizes over the size of the uncertainty set, and the outer layer optimizes over the parameter t which is used for the indicator function upper bounding. Numerical studies demonstrate that the proposed method can lead to solutions close to the true solution of a joint chance constrained problem.     

A filled function method for constrained global optimization

In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems.     

Particle swarm with radial basis function surrogates for expensive black-box optimization

This paper develops the OPUS (Optimization by Particle swarm Using Surrogates) framework for expensive black-box optimization. In each iteration, OPUS considers multiple trial positions for each particle in the swarm and uses a surrogate model to identify the most promising trial position. Moreover, the current overall best position is refined by finding the global minimum of the surrogate in the neighborhood of that position. OPUS is implemented using an RBF surrogate and the resulting OPUS-RBF algorithm is applied to a 36-D groundwater bioremediation problem, a 14-D watershed calibration problem, and ten mostly 30-D test problems. OPUS-RBF is compared with a standard PSO, CMA-ES, two other surrogate-assisted PSO algorithms, and an RBF-assisted evolution strategy. The numerical results suggest that OPUS-RBF is promising for expensive black-box optimization.     

An innovative flower pollination algorithm for continuous optimization problem

The flower pollination algorithm (FPA) is a relatively new swarm optimization algorithm that inspired by the pollination phenomenon of natural phanerogam. Since its proposed, it has received widespread attention and been applied in various engineering fields. However, the FPA still has certain drawbacks, such as inadequate optimization precision and poor convergence. In this paper, an innovative flower pollination algorithm based on cloud mutation is proposed (CMFPA), which adds information of all dimensions in the global optimization stage and uses the designed cloud mutation method to redistribute the population center. To verify the performance of the CMFPA in solving continuous optimization problems, we test twenty-four well-known functions, composition functions of CEC2013 and all benchmark functions of CEC2017. The results demonstrate that the CMFPA has better performance compared with other state-of-the-art algorithms. In addition, the CMFPA is implemented for five constrained optimization problems in practical engineering, and the performance is compared with state-of-the-art algorithms to further prove the effectiveness and efficiency of the CMFPA.     

Set covering-based surrogate approach for solving sup-{\mathcal{T}} equation constrained optimization problems

This work considers solving the sup-T{\mathcal{T}} equation constrained optimization problems from the integer programming viewpoint. A set covering-based surrogate approach is proposed to solve the sup-T{\mathcal{T}} equation constrained optimization problem with a separable and monotone objective function in each of the variables. This is our first trial of developing integer programming-based techniques to solve sup-T{\mathcal{T}} equation constrained optimization problems. Our computational results confirm the efficiency of the proposed method and show its potential for solving large scale sup-T{\mathcal{T}} equation constrained optimization problems.