具有混沌局部搜索策略的粒子群优化算法研究
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摘要
如何改进粒子群优化(Particle swarm optimization, PSO)算法,并将其用于解决现实世界的各类优化问题已成为当前的研究热点。在众多的改进方法中,一种重要的形式就是混沌PSO算法。本文对混沌PSO算法进行了定义和分类,分析了各类混沌PSO算法的特点,指出了混沌局部搜索的PSO算法是各类混沌PSO算法中最为有效的一种方法。目前,研究者们虽然提出了很多形式的混沌局部搜索算法,但是这些算法都有一个共同的特点,即每次混沌搜索时,被指定的搜索解向量的每一维都会改变。当指定的搜索解向量维数较多时,这种每维改变的方式很难实现精细搜索,为此提出了一种单维的混沌局部搜索算法。基于单维混沌局部搜索算法、PSO算法以及其它一些策略,提出了4种混合算法以解决不同类型的优化问题,具体如下：
1、提出了一种单维混沌局部搜索和多样性维持策略的混沌PSO算法用于解决无约束的单目标连续函数优化问题,其中采用单维的混沌局部搜索算法以增强PSO算法的局部搜索能力,采用混沌序列替换PSO算法的某些参数以增强PSO算法的全局搜索能力。同时,当混合算法陷入早熟收敛时,采用多样度维持策略改进PSO算法的多样性。提出的算法与其它三种混沌PSO算法分别对10维和30维的8个经典测试函数进行了50次求解,仿真实验结果表明了提出的算法在两种情况下所获最优解的平均误差精度和标准差均好于其它三种算法。
2、提出了一种混沌全局搜索和局部搜索相结合的PSO算法用于解决整数规划问题(属于约束的单目标离散函数优化问题)和混合整数规划问题(属于约束的同时包含离散变量和连续变量的单目标函数优化问题),其中混沌全局搜索算法用于增强PSO算法的全局搜索能力,混沌局部搜索算法用于增强PSO算法的局部搜索能力,这种混沌局部搜索算法与其它学者提出混沌局部搜索算法最本质的区别在于：前者能同时执行多维和单维的混沌局部搜索,后者只能执行多维的混沌局部搜索。14个整数和混合整数问题求解结果表明：提出的算法对整数规划问题能获得100%的成功率,而对混合整数规划问题不能获得100%的成功率。
3、考虑到提出的混沌全局搜索和局部搜索相结合的PSO算法解决混合整数规划问题时不能获得100%的成功率,提出一种融合差分进化、混沌局部搜索和PSO的混合算法用于解决一类混合整数规划问题,即可靠性冗余分配问题,其中引入差分进化算法以间接增强PSO算法的全局搜索能力,混沌局部搜索算法也是采用多维和单维结合的形式。实验比较了新算法与其它6种改进的元启发式算法对4个典型系统的可靠性冗余分配问题的求解情况,结果表明了新算法比其它6种算法中最好的算法能获得更好的或同样的系统可靠性。此外,针对已有的算法性能评价指标MPI的不足,提出了一个新的性能指标SR, SR对新算法与其它6种算法的评价结果表明了新算法是所有算法中最好的一种算法。
4、提出了一种基于混沌局部搜索的多目标PSO算法用于解决无约束的多目标连续函数优化问题,其中采用个体档案文件和全局档案文件分别保存个体非支配解和全局非支配解,且当个体或全局档案文件超过最大容量时为使非支配解均匀分布在解空间,采用基于相邻个体间距离之和最小删除法处理多余的非支配解。同时,为找到更多或更为接近的Pareto最优解,每代采用混沌局部搜索算法对所有粒子产生的支配解(相对于全局档案文件中非支配解)和全局档案文件中的所有非支配解进行搜索。实验对提出的算法与其它2种多目标PSO算法在解决9个基本的多目标函数优化问题时进行了比较,结果表明：在最终代距和间距性能指标的评价下,提出的算法在每个问题上都好于其它两种算法。
The researches on improved types of particle swarm optimization (PSO) to solve all kinds of optimization problems in real life have become hot topics. Chaos PSO is an important one of all improved types of PSO. In this paper, the definition of chaos PSO is given, and the chaos PSO algorithms are classified, and the characteristics of chaos PSO algorithms are analyzed. In all of chaos PSO algorithms, PSO with chaotic local search is the most effective method. Nowadays, although many chaotic local search algorithms are proposed, these proposed algorithms have the same characteristic, namely, each dimension of the specified solution vector is changed during one search. However, this way can't perform refine search when the dimensions of the specified solution vector are many, a single dimensional chaotic local search is proposed to solve this problem. Based on the single dimensional chaotic local search, PSO and some other strategies, the following four algorithms are proposed to solve different types of optimization problems.
1) A kind of chaos PSO with single dimensional chaotic local search and diversity maintenance strategy is proposed to solve single objective continuous function optimization problems without constraints, where single dimensional chaotic local search is employed to increase the local search ability of PSO, and chaotic sequences are used to substitute some parameters of PSO to improve the global search ability of PSO. In addition, the diversity maintenance strategy is introduced into the proposed algorithm to prevent from premature convergence. Ten dimensions and thirty dimensions of eight benchmark functions are solved by the proposed algorithm and the other three chaos PSO algorithms in50times, respectively. The simulation results show that for each function, the average error accuracy and standard deviation of the optimal solutions obtained by proposed algorithm in both cases are better than by the other three algorithms.
2) A new algorithm which hybrids PSO with chaotic global and local searches is proposed to solve integer programming problems (which belong to single objective function optimization problems with constraints, where all the decision variables are discrete) and mixed integer programming problems (which belong to single objective function optimization problems with constraints, where some decision variables are discrete, and others are continuous). In the new algorithm, the chaotic global search is applied to increase the global search ability of PSO, and the chaotic local search is used to improve the local search ability of PSO. The most important difference between the chaotic local search in the proposed algorithm and other chaotic local search algorithms is that the former can simultaneously perform multi-dimensional and single dimensional chaotic local search, but the latter can only perform multi-dimensional chaotic local search. The simulation results indicate that among fourteen problems, the proposed algorithm can obtain100%success in solving integer problems, but can't obtain100%success in solving mixed integer problems.
3) Because the proposed algorithm, PSO combined with chaotic global and local searches, can't obtain100%success in solving mixed integer problems, a novel algorithm is proposed to solve a class of mixed integer programming problems, namely, reliability-redundancy allocation problem. The novel algorithm is a combination of differential evolution, chaotic local search and PSO, where differential evolution is used to indirectly increase global search ability of PSO, and the chaotic local search is a combination of a multi-dimensional one and a single dimensional one. The experiments compared the novel algorithm with the other six improved meta-heuristic algorithms in solving four typical system reliability-redundancy allocation problems, and the results show that the novel algorithm can obtain the best system reliabilities or the same system reliabilities as the best one of six algorithms. In addition, considering the deficiency of the performance evaluation index MPI, a new performance evaluation index SR is proposed. The new algorithm and the other six algorithms are evaluated by SR, and the results show that the new algorithm is the best one of all compared algorithms.
4) A multi-objective PSO algorithm based on chaotic local search is proposed to solve the multi-objective continuous function optimization problems without constraints, where individual archive and global archive save individual non-dominated solutions and global non-dominated solutions, respectively. In order to make non-dominated solutions uniformly distribute in the solutions space, when the individual or global archive exceeds the maximum capacity, excessive non-dominated solutions will be deleted by the smallest value of the sum of the adjacent individuals distances. Also, in order to find more or closer to the Pareto optimal solutions, dominated solutions (with respect to non-dominated solutions in global archive) generated by all particles and all non-dominated solutions in global archive are performed by chaotic local search in each generation. The proposed algorithm was compared with the other two multi-objective PSO algorithms in solving nine basic multi-objective function optimization problems, and the results indicate that for each problem, the proposed algorithm is better than the other two algorithms using two performance evaluation indexes (final generational distance and spacing).
1、提出了一种单维混沌局部搜索和多样性维持策略的混沌PSO算法用于解决无约束的单目标连续函数优化问题,其中采用单维的混沌局部搜索算法以增强PSO算法的局部搜索能力,采用混沌序列替换PSO算法的某些参数以增强PSO算法的全局搜索能力。同时,当混合算法陷入早熟收敛时,采用多样度维持策略改进PSO算法的多样性。提出的算法与其它三种混沌PSO算法分别对10维和30维的8个经典测试函数进行了50次求解,仿真实验结果表明了提出的算法在两种情况下所获最优解的平均误差精度和标准差均好于其它三种算法。
2、提出了一种混沌全局搜索和局部搜索相结合的PSO算法用于解决整数规划问题(属于约束的单目标离散函数优化问题)和混合整数规划问题(属于约束的同时包含离散变量和连续变量的单目标函数优化问题),其中混沌全局搜索算法用于增强PSO算法的全局搜索能力,混沌局部搜索算法用于增强PSO算法的局部搜索能力,这种混沌局部搜索算法与其它学者提出混沌局部搜索算法最本质的区别在于：前者能同时执行多维和单维的混沌局部搜索,后者只能执行多维的混沌局部搜索。14个整数和混合整数问题求解结果表明：提出的算法对整数规划问题能获得100%的成功率,而对混合整数规划问题不能获得100%的成功率。
3、考虑到提出的混沌全局搜索和局部搜索相结合的PSO算法解决混合整数规划问题时不能获得100%的成功率,提出一种融合差分进化、混沌局部搜索和PSO的混合算法用于解决一类混合整数规划问题,即可靠性冗余分配问题,其中引入差分进化算法以间接增强PSO算法的全局搜索能力,混沌局部搜索算法也是采用多维和单维结合的形式。实验比较了新算法与其它6种改进的元启发式算法对4个典型系统的可靠性冗余分配问题的求解情况,结果表明了新算法比其它6种算法中最好的算法能获得更好的或同样的系统可靠性。此外,针对已有的算法性能评价指标MPI的不足,提出了一个新的性能指标SR, SR对新算法与其它6种算法的评价结果表明了新算法是所有算法中最好的一种算法。
4、提出了一种基于混沌局部搜索的多目标PSO算法用于解决无约束的多目标连续函数优化问题,其中采用个体档案文件和全局档案文件分别保存个体非支配解和全局非支配解,且当个体或全局档案文件超过最大容量时为使非支配解均匀分布在解空间,采用基于相邻个体间距离之和最小删除法处理多余的非支配解。同时,为找到更多或更为接近的Pareto最优解,每代采用混沌局部搜索算法对所有粒子产生的支配解(相对于全局档案文件中非支配解)和全局档案文件中的所有非支配解进行搜索。实验对提出的算法与其它2种多目标PSO算法在解决9个基本的多目标函数优化问题时进行了比较,结果表明：在最终代距和间距性能指标的评价下,提出的算法在每个问题上都好于其它两种算法。
The researches on improved types of particle swarm optimization (PSO) to solve all kinds of optimization problems in real life have become hot topics. Chaos PSO is an important one of all improved types of PSO. In this paper, the definition of chaos PSO is given, and the chaos PSO algorithms are classified, and the characteristics of chaos PSO algorithms are analyzed. In all of chaos PSO algorithms, PSO with chaotic local search is the most effective method. Nowadays, although many chaotic local search algorithms are proposed, these proposed algorithms have the same characteristic, namely, each dimension of the specified solution vector is changed during one search. However, this way can't perform refine search when the dimensions of the specified solution vector are many, a single dimensional chaotic local search is proposed to solve this problem. Based on the single dimensional chaotic local search, PSO and some other strategies, the following four algorithms are proposed to solve different types of optimization problems.
1) A kind of chaos PSO with single dimensional chaotic local search and diversity maintenance strategy is proposed to solve single objective continuous function optimization problems without constraints, where single dimensional chaotic local search is employed to increase the local search ability of PSO, and chaotic sequences are used to substitute some parameters of PSO to improve the global search ability of PSO. In addition, the diversity maintenance strategy is introduced into the proposed algorithm to prevent from premature convergence. Ten dimensions and thirty dimensions of eight benchmark functions are solved by the proposed algorithm and the other three chaos PSO algorithms in50times, respectively. The simulation results show that for each function, the average error accuracy and standard deviation of the optimal solutions obtained by proposed algorithm in both cases are better than by the other three algorithms.
2) A new algorithm which hybrids PSO with chaotic global and local searches is proposed to solve integer programming problems (which belong to single objective function optimization problems with constraints, where all the decision variables are discrete) and mixed integer programming problems (which belong to single objective function optimization problems with constraints, where some decision variables are discrete, and others are continuous). In the new algorithm, the chaotic global search is applied to increase the global search ability of PSO, and the chaotic local search is used to improve the local search ability of PSO. The most important difference between the chaotic local search in the proposed algorithm and other chaotic local search algorithms is that the former can simultaneously perform multi-dimensional and single dimensional chaotic local search, but the latter can only perform multi-dimensional chaotic local search. The simulation results indicate that among fourteen problems, the proposed algorithm can obtain100%success in solving integer problems, but can't obtain100%success in solving mixed integer problems.
3) Because the proposed algorithm, PSO combined with chaotic global and local searches, can't obtain100%success in solving mixed integer problems, a novel algorithm is proposed to solve a class of mixed integer programming problems, namely, reliability-redundancy allocation problem. The novel algorithm is a combination of differential evolution, chaotic local search and PSO, where differential evolution is used to indirectly increase global search ability of PSO, and the chaotic local search is a combination of a multi-dimensional one and a single dimensional one. The experiments compared the novel algorithm with the other six improved meta-heuristic algorithms in solving four typical system reliability-redundancy allocation problems, and the results show that the novel algorithm can obtain the best system reliabilities or the same system reliabilities as the best one of six algorithms. In addition, considering the deficiency of the performance evaluation index MPI, a new performance evaluation index SR is proposed. The new algorithm and the other six algorithms are evaluated by SR, and the results show that the new algorithm is the best one of all compared algorithms.
4) A multi-objective PSO algorithm based on chaotic local search is proposed to solve the multi-objective continuous function optimization problems without constraints, where individual archive and global archive save individual non-dominated solutions and global non-dominated solutions, respectively. In order to make non-dominated solutions uniformly distribute in the solutions space, when the individual or global archive exceeds the maximum capacity, excessive non-dominated solutions will be deleted by the smallest value of the sum of the adjacent individuals distances. Also, in order to find more or closer to the Pareto optimal solutions, dominated solutions (with respect to non-dominated solutions in global archive) generated by all particles and all non-dominated solutions in global archive are performed by chaotic local search in each generation. The proposed algorithm was compared with the other two multi-objective PSO algorithms in solving nine basic multi-objective function optimization problems, and the results indicate that for each problem, the proposed algorithm is better than the other two algorithms using two performance evaluation indexes (final generational distance and spacing).
引文
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