PSOHS: an efficient two-stage approach for data clustering

Cluster analysis is an important task in data mining and refers to group a set of objects such that the similarities among objects within the same group are maximal while similarities among objects from different groups are minimal. The particle swarm optimization algorithm (PSO) is one of the famous metaheuristic optimization algorithms, which has been successfully applied to solve the clustering problem. However, it has two major shortcomings. The PSO algorithm converges rapidly during the initial stages of the search process, but near global optimum, the convergence speed will become very slow. Moreover, it may get trapped in local optimum if the global best and local best values are equal to the particle’s position over a certain number of iterations. In this paper we hybridized the PSO with a heuristic search algorithm to overcome the shortcomings of the PSO algorithm. In the proposed algorithm, called PSOHS, the particle swarm optimization is used to produce an initial solution to the clustering problem and then a heuristic search algorithm is applied to improve the quality of this solution by searching around it. The superiority of the proposed PSOHS clustering method, as compared to other popular methods for clustering problem is established for seven benchmark and real datasets including Iris, Wine, Crude Oil, Cancer, CMC, Glass and Vowel.     

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.     

Global optimization of general constrained grey-box models: new method and its application to constrained PDEs for pressure swing adsorption

This paper introduces a novel methodology for the global optimization of general constrained grey-box problems. A grey-box problem may contain a combination of black-box constraints and constraints with a known functional form. The novel features of this work include (i) the selection of initial samples through a subset selection optimization problem from a large number of faster low-fidelity model samples (when a low-fidelity model is available), (ii) the exploration of a diverse set of interpolating and non-interpolating functional forms for representing the objective function and each of the constraints, (iii) the global optimization of the parameter estimation of surrogate functions and the global optimization of the constrained grey-box formulation, and (iv) the updating of variable bounds based on a clustering technique. The performance of the algorithm is presented for a set of case studies representing an expensive non-linear algebraic partial differential equation simulation of a pressure swing adsorption system for \(\hbox {CO}_{2}\). We address three significant sources of variability and their effects on the consistency and reliability of the algorithm: (i) the initial sampling variability, (ii) the type of surrogate function, and (iii) global versus local optimization of the surrogate function parameter estimation and overall surrogate constrained grey-box problem. It is shown that globally optimizing the parameters in the parameter estimation model, and globally optimizing the constrained grey-box formulation has a significant impact on the performance. The effect of sampling variability is mitigated by a two-stage sampling approach which exploits information from reduced-order models. Finally, the proposed global optimization approach is compared to existing constrained derivative-free optimization algorithms.     

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 global optimization method for the molecular replacement problem in X-ray crystallography

The primary technique for determining the three-dimensional structure of a protein molecule is X-ray crystallography, from which the molecular replacement (MR) problem often arises as a critical step. The MR problem is a global optimization problem to locate an optimal position of a model protein so that at this position the model will produce calculated intensities closest to those observed from an X-ray crystallography experiment involving a protein with unknown but similar atomic structure. Improving the applicability and robustness of MR methods is an important research topic because commonly used traditional MR methods, though often successful, have their limitations in solving difficult problems.We introduce a new global optimization strategy that combines a coarse-grid search, using a surrogate function, with extensive multi-start local optimization. A new MR code, called SOMoRe, based on this strategy is developed and tested on four realistic problems, including two difficult problems that traditional MR codes failed to solve directly. SOMoRe was able to solve each test problem without any complication, and SOMoRe solved an MR problem using a less complete model than the models required by three other programs. These results indicate that the new method is promising and should enhance the applicability and robustness of the MR methodology.     

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.     

A Kriging-based constrained global optimization algorithm for expensive black-box functions with infeasible initial points

In many engineering optimization problems, the objective and the constraints which come from complex analytical models are often black-box functions with extensive computational effort. In this case, it is necessary for optimization process to use sampling data to fit surrogate models so as to reduce the number of objective and constraint evaluations as soon as possible. In addition, it is sometimes difficult for the constrained optimization problems based on surrogate models to find a feasible point, which is the premise of further searching for a global optimal feasible solution. For this purpose, a new Kriging-based Constrained Global Optimization (KCGO) algorithm is proposed. Unlike previous Kriging-based methods, this algorithm can dispose black-box constrained optimization problem even if all initial sampling points are infeasible. There are two pivotal phases in KCGO algorithm. The main task of the first phase is to find a feasible point when there is no feasible data in the initial sample. And the aim of the second phase is to obtain a better feasible point under the circumstances of fewer expensive function evaluations. Several numerical problems and three design problems are tested to illustrate the feasibility, stability and effectiveness of the proposed method.     

An efficient multi-objective PSO algorithm assisted by Kriging metamodel for expensive black-box problems

The huge computational overhead is the main challenge in the application of community based optimization methods, such as multi-objective particle swarm optimization and multi-objective genetic algorithm, to deal with the multi-objective optimization involving costly simulations. This paper proposes a Kriging metamodel assisted multi-objective particle swarm optimization method to solve this kind of expensively black-box multi-objective optimization problems. On the basis of crowding distance based multi-objective particle swarm optimization algorithm, the new proposed method constructs Kriging metamodel for each expensive objective function adaptively, and then the non-dominated solutions of the metamodels are utilized to guide the update of particle population. To reduce the computational cost, the generalized expected improvements of each particle predicted by metamodels are presented to determine which particles need to perform actual function evaluations. The suggested method is tested on 12 benchmark functions and compared with the original crowding distance based multi-objective particle swarm optimization algorithm and non-dominated sorting genetic algorithm-II algorithm. The test results show that the application of Kriging metamodel improves the search ability and reduces the number of evaluations. Additionally, the new proposed method is applied to the optimal design of a cycloid gear pump and achieves desirable results.     

基于粒子群算法的非线性二层规划问题的求解算法

粒子群算法（Particle Swarm Optimization，PSO）是一种新兴的优化技术，其思想来源于人工生命和演化计算理论。PSO通过粒子追随自己找到的最好解和整个群的最好解来完成优化。该算法简单易实现，可调参数少，已得到了广泛研究和应用。本文根据该算法能够有效的求出非凸数学规划全局最优解的特点，对非线性二层规划的上下层问题求解，并根据二层规划的特点，给出了求解非线性二层规划问题全局最优解的有效算法。数值计算结果表明该算法有效。     

基于PSO-AFSA混合算法的模糊投资组合问题的研究

针对模糊不确定的证券市场,用可能性均值、下可能性方差和协方差分别替换了投资组合模型中概率均值、方差和协方差,构建了双目标均值-方差投资组合模型。然后采用线性加权法将双目标模型转化为单目标模型,进而提出了一个PSO-AFSA混合算法对其求解。该混合算法中,将粒子群算法搜索的结果作为人工鱼群算法初始鱼群,进一步搜索,这样能有效的避免粒子群算法陷入局部最优。同时,将人工鱼群中的最好位置反馈到粒子群算法的速度更新公式中,指引粒子运动,加快算法收敛。最后,进行实例分析,结果表明:PSO-AFSA混合算法是有效的,混合算法搜索到的全局最优值好于基本粒子群算法搜索到的全局最优值。     

Influence of ensemble surrogate models and sampling strategy on the solution quality of algorithms for computationally expensive black-box global optimization problems

This paper examines the influence of two major aspects on the solution quality of surrogate model algorithms for computationally expensive black-box global optimization problems, namely the surrogate model choice and the method of iteratively selecting sample points. A random sampling strategy (algorithm SO-M-c) and a strategy where the minimum point of the response surface is used as new sample point (algorithm SO-M-s) are compared in numerical experiments. Various surrogate models and their combinations have been used within the SO-M-c and SO-M-s sampling frameworks. The Dempster–Shafer Theory approach used in the algorithm by Müller and Piché (J Glob Optim 51:79–104, 2011) has been used for combining the surrogate models. The algorithms are numerically compared on 13 deterministic literature test problems with 2–30 dimensions, an application problem that deals with groundwater bioremediation, and an application that arises in energy generation using tethered kites. NOMAD and the particle swarm pattern search algorithm (PSWARM), which are derivative-free optimization methods, have been included in the comparison. The algorithms have also been compared to a kriging method that uses the expected improvement as sampling strategy (FEI), which is similar to the Efficient Global Optimization (EGO) algorithm. Data and performance profiles show that surrogate model combinations containing the cubic radial basis function (RBF) model work best regardless of the sampling strategy, whereas using only a polynomial regression model should be avoided. Kriging and combinations including kriging perform in general worse than when RBF models are used. NOMAD, PSWARM, and FEI perform for most problems worse than SO-M-s and SO-M-c. Within the scope of this study a Matlab toolbox has been developed that allows the user to choose, among others, between various sampling strategies and surrogate models and their combinations. The open source toolbox is available from the authors upon request.     

SO-I: a surrogate model algorithm for expensive nonlinear integer programming problems including global optimization applications

This paper presents the surrogate model based algorithm SO-I for solving purely integer optimization problems that have computationally expensive black-box objective functions and that may have computationally expensive constraints. The algorithm was developed for solving global optimization problems, meaning that the relaxed optimization problems have many local optima. However, the method is also shown to perform well on many local optimization problems, and problems with linear objective functions. The performance of SO-I, a genetic algorithm, Nonsmooth Optimization by Mesh Adaptive Direct Search (NOMAD), SO-MI (Müller et al. in Comput Oper Res 40(5):1383–1400, 2013), variable neighborhood search, and a version of SO-I that only uses a local search has been compared on 17 test problems from the literature, and on eight realizations of two application problems. One application problem relates to hydropower generation, and the other one to throughput maximization. The numerical results show that SO-I finds good solutions most efficiently. Moreover, as opposed to SO-MI, SO-I is able to find feasible points by employing a first optimization phase that aims at minimizing a constraint violation function. A feasible user-supplied point is not necessary.     

Comparative analysis of the machine repair Problem with imperfect coverage and service pressure condition

In this paper we analyze the warm-standby M/M/R machine repair problem with multiple imperfect coverage which involving the service pressure condition. When an operating machine (or warm standby) fails, it may be immediately detected, located, and replaced with a coverage probability c by a standby if one is available. We use a recursive method to develop the steady-state analytic solutions which are used to calculate various system performance measures. The total expected profit function per unit time is derived to determine the joint optimal values at the maximum profit. We first utilize the direct search method to measure the various characteristics of the profit function followed by Quasi-Newton method to search the optimal solutions. Furthermore, the particle swarm optimization (PSO) algorithm is implemented to find the optimal combinations of parameters in the pursuit of maximum profit. Finally, a comparative analysis of the Quasi-Newton method with the PSO algorithm has demonstrated that the PSO algorithm provides a powerful tool to perform the optimization problem.     

Particle swarm optimization with age-group topology for multimodal functions and data clustering

This paper proposes particle swarm optimization with age-group topology (PSOAG), a novel age-based particle swarm optimization (PSO). In this work, we present a new concept of age to measure the search ability of each particle in local area. To keep population diversity during searching, we separate particles to different age-groups by their age and particles in each age-group can only select the ones in younger groups or their own groups as their neighbourhoods. To allow search escape from local optima, the aging particles are regularly replaced by new and randomly generated ones. In addition, we design an age-group based parameter setting method, where particles in different age-groups have different parameters, to accelerate convergence. This algorithm is applied to nonlinear function optimization and data clustering problems for performance evaluation. In comparison against several PSO variants and other EAs, we find that the proposed algorithm provides significantly better performances on both the function optimization problems and the data clustering tasks.     

PSO with surrogate models for feature selection: static and dynamic clustering-based methods

Feature selection is an important but often expensive process, especially with a large number of instances. This problem can be addressed by using a small training set, i.e. a surrogate set. In this work, we propose to use a hierarchical clustering method to build various surrogate sets, which allows to analyze the effect of surrogate sets with different qualities and quantities on the feature subsets. Further, a dynamic surrogate model is proposed to automatically adjust surrogate sets for different datasets. Based on this idea, a feature selection system is developed using particle swarm optimization as the search mechanism. The experiments show that the hierarchical clustering method can build better surrogate sets to reduce the computational time, improve the feature selection performance, and alleviate overfitting. The dynamic method can automatically choose suitable surrogate sets to further improve the classification accuracy.     

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.     

GOSAC: global optimization with surrogate approximation of constraints

We introduce GOSAC, a global optimization algorithm for problems with computationally expensive black-box constraints and computationally cheap objective functions. The variables may be continuous, integer, or mixed-integer. GOSAC uses a two-phase optimization approach. The first phase aims at finding a feasible point by solving a multi-objective optimization problem in which the constraints are minimized simultaneously. The second phase aims at improving the feasible solution. In both phases, we use cubic radial basis function surrogate models to approximate the computationally expensive constraints. We iteratively select sample points by minimizing the computationally cheap objective function subject to the constraint function approximations. We assess GOSAC’s efficiency on computationally cheap test problems with integer, mixed-integer, and continuous variables and two environmental applications. We compare GOSAC to NOMAD and a genetic algorithm (GA). The results of the numerical experiments show that for a given budget of allowed expensive constraint evaluations, GOSAC finds better feasible solutions more efficiently than NOMAD and GA for most benchmark problems and both applications. GOSAC finds feasible solutions with a higher probability than NOMAD and GOSAC.     

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.     

A novel chaotic particle swarm optimization approach using Hénon map and implicit filtering local search for economic load dispatch

Particle swarm optimization (PSO) is a population-based swarm intelligence algorithm driven by the simulation of a social psychological metaphor instead of the survival of the fittest individual. Based on the chaotic systems theory, this paper proposed a novel chaotic PSO combined with an implicit filtering (IF) local search method to solve economic dispatch problems. Since chaotic mapping enjoys certainty, ergodicity and the stochastic property, the proposed PSO introduces chaos mapping using Hénon map sequences which increases its convergence rate and resulting precision. The chaotic PSO approach is used to produce good potential solutions, and the IF is used to fine-tune of final solution of PSO. The hybrid methodology is validated for a test system consisting of 13 thermal units whose incremental fuel cost function takes into account the valve-point loading effects. Simulation results are promising and show the effectiveness of the proposed approach.