Feasibility-Guided Constraint-Handling Techniques for Engineering Optimization Problems

The particle swarm optimization (PSO) algorithm is an established nature-inspired population-based meta-heuristic that replicates the synchronizing movements of birds and fish. PSO is essentially an unconstrained algorithm and requires constraint handling techniques (CHTs) to solve constrained optimization problems (COPs). For this purpose, we integrate two CHTs, the superiority of feasibility (SF) and the violation constraint-handling (VCH), with a PSO. These CHTs distinguish feasible solutions from infeasible ones. Moreover, in SF, the selection of infeasible solutions is based on their degree of constraint violations, whereas in VCH, the number of constraint violations by an infeasible solution is of more importance. Therefore, a PSO is adapted for constrained optimization, yielding two constrained variants, denoted SF-PSO and VCH-PSO. Both SF-PSO and VCH-PSO are evaluated with respect to five engineering problems: the Himmelblau’s nonlinear optimization, the welded beam design, the spring design, the pressure vessel design, and the three-bar truss design. The simulation results show that both algorithms are consistent in terms of their solutions to these problems, including their different available versions. Comparison of the SF-PSO and the VCH-PSO with other existing algorithms on the tested problems shows that the proposed algorithms have lower computational cost in terms of the number of function evaluations used. We also report our disagreement with some unjust comparisons made by other researchers regarding the tested problems and their different variants.     

基于智能混合策略的测试集优化

利用基于粒子群和蚁群算法的智能混合优化策略,删除冗余测试向量以解决测试集的优化问题. 利用蚁群算法的并行搜索能力构造初始解集,通过粒子群优化算法将解集维数降低,确定每次迭代的个体最优解和全局最优解,并利用新粒子信息更新信息素,最终通过多次迭代找到一个或多个最优测试集. 通过多组数据实例分析可知: 该智能混合优化策略与蚁群算法等其他测试集优化算法相比,可得到多个可行性最优测试集;与蚁群算法相比可提高收敛速度,并降低蚁群算法参数选取对收敛结果的影响,从而避免次优解的出现.     

Evolutionary computation of unconstrained and constrained problems using a novel momentum-type particle swarm optimization

This study proposes a novel momentum-type particle swarm optimization (PSO) method, which will find good solutions of unconstrained and constrained problems using a delta momentum rule to update the particle velocity. The algorithm modifies Shi and Eberhart's PSO to enhance the computational efficiency and solution accuracy. This study also presents a continuous non-stationary penalty function, to force design variables to satisfy all constrained functions. Several well-known and widely used benchmark problems were employed to compare the performance of the proposed PSO with Kennedy and Eberhart's PSO and Shi and Eberhart's modified PSO. Additionally, an engineering optimization task for designing a pressure vessel was applied to test the three PSO algorithms. The optimal solutions are presented and compared with the data from other works using different evolutionary algorithms. To show that the proposed momentum-type PSO algorithm is robust, its convergence rate, solution accuracy, mean absolute error, standard deviation, and CPU time were compared with those of both the other PSO algorithms. The experimental results reveal that the proposed momentum-type PSO algorithm can efficiently solve unconstrained and constrained engineering optimization problems.     

Particle swarm optimization: an alternative in marine propeller optimization?

This article deals with improving and evaluating the performance of two evolutionary algorithm approaches for automated engineering design optimization. Here a marine propeller design with constraints on cavitation nuisance is the intended application. For this purpose, the particle swarm optimization (PSO) algorithm is adapted for multi-objective optimization and constraint handling for use in propeller design. Three PSO algorithms are developed and tested for the optimization of four commercial propeller designs for different ship types. The results are evaluated by interrogating the generation medians and the Pareto front development. The same propellers are also optimized utilizing the well established NSGA-II genetic algorithm to provide benchmark results. The authors' PSO algorithms deliver comparable results to NSGA-II, but converge earlier and enhance the solution in terms of constraints violation.     

Guided particle swarm optimization method to solve general nonlinear optimization problems

The development of hybrid algorithms is becoming an important topic in the global optimization research area. This article proposes a new technique in hybridizing the particle swarm optimization (PSO) algorithm and the Nelder–Mead (NM) simplex search algorithm to solve general nonlinear unconstrained optimization problems. Unlike traditional hybrid methods, the proposed method hybridizes the NM algorithm inside the PSO to improve the velocities and positions of the particles iteratively. The new hybridization considers the PSO algorithm and NM algorithm as one heuristic, not in a sequential or hierarchical manner. The NM algorithm is applied to improve the initial random solution of the PSO algorithm and iteratively in every step to improve the overall performance of the method. The performance of the proposed method was tested over 20 optimization test functions with varying dimensions. Comprehensive comparisons with other methods in the literature indicate that the proposed solution method is promising and competitive.     

Hybrid simplex search and particle swarm optimization for the global optimization of multimodal functions

This article proposes the hybrid Nelder–Mead (NM)–Particle Swarm Optimization (PSO) algorithm based on the NM simplex search method and PSO for the optimization of multimodal functions. The hybrid NM–PSO algorithm is very easy to implement, in practice, since it does not require gradient computation. This hybrid procedure performed the exploration with PSO and the exploitation with the NM simplex search method. In a suite of 17 multi-optima test functions taken from the literature, the computational results via various experimental studies showed that the hybrid NM–PSO approach is superior to the two original search techniques (i.e. NM and PSO) in terms of solution quality and convergence rate. In addition, the presented algorithm is also compared with eight other published methods, such as hybrid genetic algorithm (GA), continuous GA, simulated annealing (SA), and tabu search (TS) by means of a smaller set of test functions. On the whole, the new algorithm is demonstrated to be extremely effective and efficient at locating best-practice optimal solutions for multimodal functions.     

A new optimization algorithm based on a combination of particle swarm optimization,convergence and divergence operators for single-objective and multi-objective problems

Particle swarm optimization (PSO) is a randomized and population-based optimization method that was inspired by the flocking behaviour of birds and human social interactions. In this work, multi-objective PSO is modified in two stages. In the first stage, PSO is combined with convergence and divergence operators. Here, this method is named CDPSO. In the second stage, to produce a set of Pareto optimal solutions which has good convergence, diversity and distribution, two mechanisms are used. In the first mechanism, a new leader selection method is defined, which uses the periodic iteration and the concept of the particle's neighbour number. This method is named periodic multi-objective algorithm. In the second mechanism, an adaptive elimination method is employed to limit the number of non-dominated solutions in the archive, which has influences on computational time, convergence and diversity of solution. Single-objective results show that CDPSO performs very well on the complex test functions in terms of solution accuracy and convergence speed. Furthermore, some benchmark functions are used to evaluate the performance of periodic multi-objective CDPSO. This analysis demonstrates that the proposed algorithm operates better in three metrics through comparison with three well-known elitist multi-objective evolutionary algorithms. Finally, the algorithm is used for Pareto optimal design of a two-degree of freedom vehicle vibration model. The conflicting objective functions are sprung mass acceleration and relative displacement between sprung mass and tyre. The feasibility and efficiency of periodic multi-objective CDPSO are assessed in comparison with multi-objective modified NSGAII.     

基于改进PSO算法的结构损伤检测

结构的损伤检测常转化为求解约束优化问题,针对粒子群算法容易出现早熟问题,增大算法后期的粒子位置的改变量,从而增加粒子位置的差异,因而能够增强其在求解约束优化问题时抵抗局部极小的能力。两层刚架单损伤和多损伤识别的数值结果和收敛曲线表明了改进后的粒子群算法优于传统的带惯性因子的粒子群算法。三层框架结构的4种损伤工况的试验研究进一步说明了该算法应用于结构损伤检测领域的有效性。     

Probabilistic optimal design of laminates using improved particle swarm optimization

A new approach to the particle swarm optimization (PSO) is proposed for the solution of non-linear optimization problems with constraints, and is applied to the reliability-based optimum design of laminated composites. Special mutation-interference operators are introduced to increase swarm variety and improve the convergence performance of the algorithm. The reliability-based optimum design of laminated composites is modelled and solved using the improved PSO. The maximization of structural reliability and the minimization of total weight of laminates are analysed. The stacking sequence optimization is implemented in the improved PSO by using a special coding technique. Examples show that the improved PSO has high convergence and good stability and is efficient in dealing with the probabilistic optimal design of composite structures.     

Comparison of linear and classical velocity update rules in particle swarm optimization: notes on diversity

In this study, we investigate the significance of diversity in the particle swarm optimization (PSO) algorithm. To do so, we study two different implementations of the PSO, being the so‐called linear and classical PSO formulations. While the behaviour of these two implementations is markedly different, they only differ in the formulation of the velocity update rule. In fact, the differences are merely due to subtle differences in the introduction of randomness into the algorithm. In this paper, we show that in algorithms employing the linear PSO velocity update rule, particle trajectories collapse to line searches in n‐dimensional space. The classical formulation does not suffer this drawback. Instead, directional diverse stochastic search trajectories are retained, which in turn helps to alleviate premature convergence. The performance of the classical implementation is therefore superior for all test problems considered, due to the presence of adequate diversity. Copyright © 2006 John Wiley & Sons, Ltd.     

RDS-NSGA-II: a memetic algorithm for reference point based multi-objective optimization

Reference point based optimization offers tools for the effective treatment of preference based multi-objective optimization problems, e.g. when the decision-maker has a rough idea about the target objective values. For the numerical solution of such problems, specialized evolutionary strategies have become popular, despite their possible slow convergence rates. Hybridizing such evolutionary algorithms with local search techniques have been shown to produce faster and more reliable algorithms. In this article, the directed search (DS) method is adapted to the context of reference point optimization problems, making this variant, called RDS, a well-suited option for integration into evolutionary algorithms. Numerical results on academic test problems with up to five objectives demonstrate the benefit of the novel hybrid (i.e. the same approximation quality can be obtained more efficiently by the new algorithm), using the state-of-the-art algorithm R-NSGA-II for this coupling. This represents an advantage when treating costly-to-evaluate real-world engineering design problems.     

Swarm algorithms with chaotic jumps for optimization of multimodal functions

In this article, the use of some well-known versions of particle swarm optimization (PSO) namely the canonical PSO, the bare bones PSO (BBPSO) and the fully informed particle swarm (FIPS) is investigated on multimodal optimization problems. A hybrid approach which consists of swarm algorithms combined with a jump strategy in order to escape from local optima is developed and tested. The jump strategy is based on the chaotic logistic map. The hybrid algorithm was tested for all three versions of PSO and simulation results show that the addition of the jump strategy improves the performance of swarm algorithms for most of the investigated optimization problems. Comparison with the off-the-shelf PSO with local topology (l _best model) has also been performed and indicates the superior performance of the standard PSO with chaotic jump over the standard both using local topology (l _best model).     

一种改进的雁群扩展粒子群算法研究

针对粒子群算法易陷入局部最优的问题,结合雁群启示粒子群算法和扩展粒子群算法提出了基于雁群启示的扩展粒子群(GeEPSO)算法。该算法在利用雁群飞行方向的多样性同时融合了所有粒子的个体极值信息,提高了种群多样性。为进一步提高改进算法的收敛速度,引入简化粒子群提出了 GeESPSO算法。基准函数的仿真表明:改进算法GeESPSO较好地平衡了收敛速度和局部最优两个矛盾,总体较优。为进一步验证算法在实际应用中的有效性,又分别用两种改进算法优化BP神经网络,并用相关气象数据对PM2.5的值进行预测。     

A particle swarm optimizer for constrained multi-objective engineering design problems

This article presents a particle swarm optimizer (PSO) capable of handling constrained multi-objective optimization problems. The latter occur frequently in engineering design, especially when cost and performance are simultaneously optimized. The proposed algorithm combines the swarm intelligence fundamentals with elements from bio-inspired algorithms. A distinctive feature of the algorithm is the utilization of an arithmetic recombination operator, which allows interaction between non-dominated particles. Furthermore, there is no utilization of an external archive to store optimal solutions. The PSO algorithm is applied to multi-objective optimization benchmark problems and also to constrained multi-objective engineering design problems. The algorithmic effectiveness is demonstrated through comparisons of the PSO results with those obtained from other evolutionary optimization algorithms. The proposed particle swarm optimizer was able to perform in a very satisfactory manner in problems with multiple constraints and/or high dimensionality. Promising results were also obtained for a multi-objective engineering design problem with mixed variables.     

Drilling path optimization by the particle swarm optimization algorithm with global convergence characteristics

Drilling path optimization is one of the key problems in holes-machining. This paper presents a new approach to solve the drilling path optimization problem belonging to discrete space, based on the particle swarm optimization (PSO) algorithm. Since the standard PSO algorithm is not guaranteed to be global convergent or local convergent, based on the mathematical model, the algorithm is improved by adopting the method to generate the stop evolution particle once again to obtain the ability of convergence on the global optimization solution. Also, the operators are proposed by establishing the Order Exchange Unit (OEU) and the Order Exchange List (OEL) to satisfy the need of integer coding in drilling path optimization. The experimentations indicate that the improved algorithm has the characteristics of easy realization, fast convergence speed, and better global convergence capability. Hence the new PSO can play a role in solving the problem of drilling path optimization.     

Applying multiple objective tabu search to continuous optimization problems with a simple neighbourhood strategy

One of the first multiple objective versions of the tabu search (TS) algorithm is proposed by the author. The idea of applying TS to multiple objective optimization is inspired from its solution structure. TS works with more than one solution (neighbourhood solutions) at a time and this situation gives the opportunity to evaluate multiple objectives simultaneously in one run. The selection and updating stages are modified to enable the original TS algorithm to work with more than one objective. In this paper, the multiple objective tabu search (MOTS) algorithm is applied to multiple objective non‐linear optimization problems with continuous variables using a simple neighbourhood strategy. The algorithm is applied to four mechanical components design problems. The results are compared with several other solution techniques including multiple objective genetic algorithms. It is observed that MOTS is able to find better and much wider spread of solutions than the reported ones. Copyright © 2005 John Wiley & Sons, Ltd.     

基于混合PSO算法的桁架动力响应优化

摘 要：本文针对以结构动力响应为约束,最小重量为目标的桁架拓扑优化问题,提出了一种将微粒群算法和优化准则法结合的混合PSO算法。利用优化准则法的迭代关系找出群体中适应度最好的微粒,将其作为特殊微粒,其他微粒的寻优采用PSO的基本进化规则,位移响应约束利用特殊微粒的灵敏度信息近似计算。算例的计算结果表明,混合PSO算法适用于受简谐荷载以及脉冲荷载作用桁架结构的拓扑优化。混合PSO的计算效率比PSO算法高,其优化效果比优化准则法好。     

A new hybrid imperialist competitive algorithm on data clustering

Clustering is a process for partitioning datasets. This technique is very useful for optimum solution. k-means is one of the simplest and the most famous methods that is based on square error criterion. This algorithm depends on initial states and converges to local optima. Some recent researches show that k-means algorithm has been successfully applied to combinatorial optimization problems for clustering. In this paper, we purpose a novel algorithm that is based on combining two algorithms of clustering; k-means and Modify Imperialist Competitive Algorithm. It is named hybrid K-MICA. In addition, we use a method called modified expectation maximization (EM) to determine number of clusters. The experimented results show that the new method carries out better results than the ACO, PSO, Simulated Annealing (SA), Genetic Algorithm (GA), Tabu Search (TS), Honey Bee Mating Optimization (HBMO) and k-means.     

Swarm algorithms for single- and multi-objective optimization problems incorporating sensitivity analysis

Swarm algorithms such as particle swarm optimization (PSO) are non-gradient probabilistic optimization algorithms that have been successfully applied for global searches in complex problems such as multi-peak problems. However, application of these algorithms to structural and mechanical optimization problems still remains a complex matter since local optimization capability is still inferior to general numerical optimization methods. This article discusses new swarm metaphors that incorporate design sensitivities concerning objective and constraint functions and are applicable to structural and mechanical design optimization problems. Single- and multi-objective optimization techniques using swarm algorithms are combined with a gradient-based method. In the proposed techniques, swarm optimization algorithms and a sequential linear programming (SLP) method are conducted simultaneously. Finally, truss structure design optimization problems are solved by the proposed hybrid method to verify the optimization efficiency.     

混合粒子群算法在混流装配线优化调度中的应用

应用粒子群算法求解混流装配线的优化调度问题,给出粒子的构造方法,并针对算法中存在过早收敛的问题,提出了一种与局部优化和粒子微变异方法相结合的混合粒子群算法.给出了一个实例,实例应用粒子群算法和混合粒子群算法分别进行求解,与其他一些方法比较表明,混合粒子群算法可以有效、快速地求得混流装配线优化调度问题的解.