Constrained structural design optimization via a parallel augmented Lagrangian particle swarm optimization approach

This paper presents an extension to the basic particle swarm optimization approach for the solution of constrained engineering design optimization problems. The approach takes advantage of the PSO ability to find global optimum in problems with complex design spaces while directly enforcing feasibility of constraints using an augmented Lagrange multiplier method. Details in the algorithm implementation and properties are presented and the effectiveness of the approach is illustrated in different benchmark structural optimization test cases. Results show the ability of the proposed methodology to find better solutions for structural optimization tasks as compared to other optimization algorithms.     

A parallel boundary search particle swarm optimization algorithm for constrained optimization problems

During the past decade, considerable research has been conducted on constrained optimization problems (COPs) which are frequently encountered in practical engineering applications. By introducing resource limitations as constraints, the optimal solutions in COPs are generally located on boundaries of feasible design space, which leads to search difficulties when applying conventional optimization algorithms, especially for complex constraint problems. Even though penalty function method has been frequently used for handling the constraints, the adjustment of control parameters is often complicated and involves a trial-and-error approach. To overcome these difficulties, a modified particle swarm optimization (PSO) algorithm named parallel boundary search particle swarm optimization (PBSPSO) algorithm is proposed in this paper. Modified constrained PSO algorithm is adopted to conduct global search in one branch while Subset Constrained Boundary Narrower (SCBN) function and sequential quadratic programming (SQP) are applied to perform local boundary search in another branch. A cooperative mechanism of the two branches has been built in which locations of the particles near boundaries of constraints are selected as initial positions of local boundary search and the solutions of local boundary search will lead the global search direction to boundaries of active constraints. The cooperation behavior of the two branches effectively reinforces the optimization capability of the PSO algorithm. The optimization performance of PBSPSO algorithm is illustrated through 13 CEC06 test functions and 5 common engineering problems. The results are compared with other state-of-the-art algorithms and it is shown that the proposed algorithm possesses a competitive global search capability and is effective for constrained optimization problems in engineering applications.     

An effective co-evolutionary particle swarm optimization for constrained engineering design problems

Many engineering design problems can be formulated as constrained optimization problems. So far, penalty function methods have been the most popular methods for constrained optimization due to their simplicity and easy implementation. However, it is often not easy to set suitable penalty factors or to design adaptive mechanism. By employing the notion of co-evolution to adapt penalty factors, this paper proposes a co-evolutionary particle swarm optimization approach (CPSO) for constrained optimization problems, where PSO is applied with two kinds of swarms for evolutionary exploration and exploitation in spaces of both solutions and penalty factors. The proposed CPSO is population based and easy to implement in parallel. Especially, penalty factors also evolve using PSO in a self-tuning way. Simulation results based on well-known constrained engineering design problems demonstrate the effectiveness, efficiency and robustness on initial populations of the proposed method. Moreover, the CPSO obtains some solutions better than those previously reported in the literature.     

A hybrid differential evolution augmented Lagrangian method for constrained numerical and engineering optimization

We present a new hybrid method for solving constrained numerical and engineering optimization problems in this paper. The proposed hybrid method takes advantage of the differential evolution (DE) ability to find global optimum in problems with complex design spaces while directly enforcing feasibility of constraints using a modified augmented Lagrangian multiplier method. The basic steps of the proposed method are comprised of an outer iteration, in which the Lagrangian multipliers and various penalty parameters are updated using a first-order update scheme, and an inner iteration, in which a nonlinear optimization of the modified augmented Lagrangian function with simple bound constraints is implemented by a modified differential evolution algorithm. Experimental results based on several well-known constrained numerical and engineering optimization problems demonstrate that the proposed method shows better performance in comparison to the state-of-the-art algorithms.     

An interval space reducing method for constrained problems with particle swarm optimization

In this paper, we propose a method for solving constrained optimization problems using interval analysis combined with particle swarm optimization. A set inverter via interval analysis algorithm is used to handle constraints in order to reduce constrained optimization to quasi unconstrained one. The algorithm is useful in the detection of empty search spaces, preventing useless executions of the optimization process. To improve computational efficiency, a space cleaning algorithm is used to remove solutions that are certainly not optimal. As a result, the search space becomes smaller at each step of the optimization procedure. After completing pre-processing, a modified particle swarm optimization algorithm is applied to the reduced search space to find the global optimum. The efficiency of the proposed approach is demonstrated through comprehensive experimentation involving 100 000 runs on a set of well-known benchmark constrained engineering design problems. The computational efficiency of the new method is quantified by comparing its results with other PSO variants found in the literature.     

Penalty function approach for the mixed discrete nonlinear problems by particle swarm optimization

In this paper, the basic characteristics of particle swarm optimization (PSO) for the global search are discussed at first, and then the PSO for the mixed discrete nonlinear problems (MDNLP) is suggested. The penalty function approach to handle the discrete design variables is employed, in which the discrete design variables are handled as the continuous ones by penalizing at the intervals. As a result, a useful method to determine the penalty parameter of penalty term for the discrete design variables is proposed. Through typical mathematical and structural optimization problems, the validity of the proposed approach for the MDNLP is examined.     

求解约束优化问题的动态邻域粒子群算法*

粒子群算法(PSO)求解约束优化问题存在较严重的早熟收敛现象,为了有效抑制早熟收敛,提出了基于改进的约束自适应方法的动态邻域粒子群算法(IPSO)。算法采用动态邻域策略提高算法的全局搜索能力,设计了一种改进的自适应约束处理方法,根据迭代代数线性增加搜索偏向系数,在早期偏向于搜索可行解,在后期偏向于搜索最优解,并引入序列二次规划增强算法的局部搜索能力。通过基准测试函数实验对比分析,表明该算法对于约束优化问题具有较好的全局收敛性。     

Coevolutionary particle swarm optimization using Gaussian distribution for solving constrained optimization problems.

In this correspondence, an approach based on coevolutionary particle swarm optimization to solve constrained optimization problems formulated as min-max problems is presented. In standard or canonical particle swarm optimization (PSO), a uniform probability distribution is used to generate random numbers for the accelerating coefficients of the local and global terms. We propose a Gaussian probability distribution to generate the accelerating coefficients of PSO. Two populations of PSO using Gaussian distribution are used on the optimization algorithm that is tested on a suite of well-known benchmark constrained optimization problems. Results have been compared with the canonical PSO (constriction factor) and with a coevolutionary genetic algorithm. Simulation results show the suitability of the proposed algorithm in terms of effectiveness and robustness.     

Constrained particle swarm optimization using a bi-objective formulation

This paper introduces an approach for dealing with constraints when using particle swarm optimization. The constrained, single objective optimization problem is converted into an unconstrained, bi-objective optimization problem that is solved using a multi-objective implementation of the particle swarm optimization algorithm. A specialized bi-objective particle swarm optimization algorithm is presented and an engineering example problem is used to illustrate the performance of the algorithm. An additional set of 13 test problems from the literature is used to further validate the performance of the newly proposed algorithm. For the example problems considered here, the proposed algorithm produced promising results, indicating that it is an approach that deserves further consideration. The newly proposed algorithm provides performance similar to that of a tuned penalty function approach, without having to tune any penalty parameters.     

A hybrid multi-swarm particle swarm optimization to solve constrained optimization problems

In the real-world applications, most optimization problems are subject to different types of constraints. These problems are known as constrained optimization problems (COPs). Solving COPs is a very important area in the optimization field. In this paper, a hybrid multi-swarm particle swarm optimization (HMPSO) is proposed to deal with COPs. This method adopts a parallel search operator in which the current swarm is partitioned into several subswarms and particle swarm optimization (PSO) is severed as the search engine for each sub-swarm. Moreover, in order to explore more promising regions of the search space, differential evolution (DE) is incorporated to improve the personal best of each particle. First, the method is tested on 13 benchmark test functions and compared with three stateof-the-art approaches. The simulation results indicate that the proposed HMPSO is highly competitive in solving the 13 benchmark test functions. Afterward, the effectiveness of some mechanisms proposed in this paper and the effect of the parameter setting were validated by various experiments. Finally, HMPSO is further applied to solve 24 benchmark test functions collected in the 2006 IEEE Congress on Evolutionary Computation (CEC2006) and the experimental results indicate that HMPSO is able to deal with 22 test functions.     

一种求解约束优化问题的混合算法

提出一种基于修改增广Lagrange函数和PSO的混合算法用于求解约束优化问题。将约束优化问题转化为界约束优化问题,混合算法由两层迭代结构组成,在内层迭代中,利用改进PSO算法求解界约束优化问题得到下一个迭代点。外层迭代主要修正Lagrange乘子和罚参数,检查收敛准则是否满足,重构下次迭代的界约束优化子问题,检查收敛准则是否满足。数值实验结果表明该混合算法的有效性。     

A ranking selection-based particle swarm optimizer for engineering design optimization problems

Particle swarm optimization (PSO) algorithms have been proposed to solve optimization problems in engineering design, which are usually constrained (possibly highly constrained) and may require the use of mixed variables such as continuous, integer, and discrete variables. In this paper, a new algorithm called the ranking selection-based PSO (RSPSO) is developed. In RSPSO, the objective function and constraints are handled separately. For discrete variables, they are partitioned into ordinary discrete and categorical ones, and the latter is managed and searched directly without the concept of velocity in the standard PSO. In addition, a new ranking selection scheme is incorporated into PSO to elaborately control the search behavior of a swarm in different search phases and on categorical variables. RSPSO is relatively simple and easy to implement. Experiments on five engineering problems and a benchmark function with equality constraints were conducted. The results indicate that RSPSO is an effective and widely applicable optimizer for optimization problems in engineering design in comparison with the state-of-the-art algorithms in the area.     

用并行化的QPSO解决有约束的优化问题

采用粒子群系统的并行化的量子化模型提高全局搜寻能力,在解决约束问题时采用不固定的多阶段任务补偿函数以提高收敛性,并获得更准确的结果,提出了并行化的QPSO（PQPSO）算法。此算法在几个可信赖的基准函数中被测试,并且实验结果显示PQPSO的最优值和运行时间比QPSO和传统的PSO有很大的提高,而且运行所用的时间资源接近线性减少。     

求解工程约束优化问题的PSO-ABC混合算法*

针对包含约束条件的工程优化问题,提出了基于人工蜂群的粒子群优化PSO-ABC算法。将PSO中较优的粒子作为ABC算法的蜜源,并使用禁忌表存储其局部极值,克服粒子群优化算法易陷入局部最优的缺陷。采用可行性规则进行约束处理,将粒子种群分为可行子群和不可行子群,并在ABC算法产生蜜源的过程中保留部分较优的可行解和不可行解的信息,弥补了可行性规则处理最优点位于约束边界附近的问题时存在的不足。四个典型工程优化设计的实验结果表明,该算法能够寻得更优的约束最优化解,且稳健性更强。     

Dynamic particle swarm optimizer with escaping prey for solving constrained non-convex and piecewise optimization problems

This paper presents a novel meta-heuristic algorithm, dynamic particle swarm optimizer with escaping prey (DPSOEP), for solving constrained non-convex and piecewise optimization problems. In DPSOEP, the particles developed from two different species are classified into three different types, consisting of preys, strong particles and weak particles, to simulate the behavior of hunting and escaping characteristics observed in nature. Compared to other variants of particle swarm optimizer (PSO), the proposed algorithm takes account of an escaping mechanism for the preys to circumvent the problem of local optimum and also develops a classification mechanism to cope with different situations in the search space so as to achieve a good balance between its global exploration and local exploitation abilities. Simulation results obtained based on thirteen benchmark functions and two practical economic dispatch problems prove the effectiveness and applicability of the DPSOEP to deal with non-convex and piecewise optimization problem, considering the integration of linear equality and inequality constraints.     

Co-evolutionary particle swarm optimization to solve constrained optimization problems

This paper presents a co-evolutionary particle swarm optimization (CPSO) algorithm to solve global nonlinear optimization problems. A new co-evolutionary PSO (CPSO) is constructed. In the algorithm, a deterministic selection strategy is proposed to ensure the diversity of population. Meanwhile, based on the theory of extrapolation, the induction of evolving direction is enhanced by adding a co-evolutionary strategy, in which the particles make full use of the information each other by using gene-adjusting and adaptive focus-varied tuning operator. Infeasible degree selection mechanism is used to handle the constraints. A new selection criterion is adopted as tournament rules to select individuals. Also, the infeasible solution is properly accepted as the feasible solution based on a defined threshold of the infeasible degree. This diversity mechanism is helpful to guide the search direction towards the feasible region. Our approach was tested on six problems commonly used in the literature. The results obtained are repeatedly closer to the true optimum solution than the other techniques.     

改进的微粒群优化算法在过程综合中的应用

针对过程综合中的混合整数非线性规划(Mixed Integer Non-Linear Programming，MINLP)问题，利用改进的微粒群优化(Particle Swarm Optimization，PSO)算法对其进行求解。在基本的PSO算法的基础上，通过利用罚函数和引入sigmoid函数把PSO算法应用到MINLP问题的求解中，利用两个测试函数和一个过程综合的实例对其进行了测试并与其它算法所得的结果进行了比较，结果表明，PSO算法在使用的普遍性、求解的准确性方面都优于一般的算法，是一种有效的求解MINLP问题的方法。     

Non-parametric particle swarm optimization for global optimization

In recent years, particle swarm optimization (PSO) has extensively applied in various optimization problems because of its simple structure. Although the PSO may find local optima or exhibit slow convergence speed when solving complex multimodal problems. Also, the algorithm requires setting several parameters, and tuning the parameters is a challenging for some optimization problems. To address these issues, an improved PSO scheme is proposed in this study. The algorithm, called non-parametric particle swarm optimization (NP-PSO) enhances the global exploration and the local exploitation in PSO without tuning any algorithmic parameter. NP-PSO combines local and global topologies with two quadratic interpolation operations to increase the search ability. Nineteen (19) unimodal and multimodal nonlinear benchmark functions are selected to compare the performance of NP-PSO with several well-known PSO algorithms. The experimental results showed that the proposed method considerably enhances the efficiency of PSO algorithm in terms of solution accuracy, convergence speed, global optimality, and algorithm reliability.     

A modified particle swarm optimization for economic dispatch with non-smooth cost functions

This paper presents a new approach to economic dispatch (ED) problems with non-smooth cost functions using a particle swarm optimization (PSO) technique. The practical ED problems have non-smooth cost functions with equality and inequality constraints, which makes the problem of finding the global optimum difficult when using any mathematical approaches. Since, standard PSO may converge at the early stage, in this paper, a modified PSO (MPSO) mechanism is suggested to deal with the equality and inequality constraints in the ED problems. To validate the results obtained by MPSO, standard particle swarm optimization (PSO) and guaranteed convergence particle swarm optimization (GCPSO) are applied for comparison. Also, the results obtained by MPSO, PSO and GCPSO are compared with the previous approaches reported in the literature. The results show that the MPSO produces optimal or nearly optimal solutions for the study systems.     

基于PSO的确定性问题优化研究

粒子群算法原理简单，易于实现，是进化算法中优化效率很高的算法。针对确定环境下的问题优化，提出采用粒子群算法对其进行优化求解。通过对确定性环境下的Benchmark函数的算法仿真研究，表明粒子群算法在确定性问题优化中具有快速收敛性和精确性的特点。