求解约束优化问题的几种智能算法
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摘要
智能计算也称为“软计算”,是人们受自然(生物界)规律的启迪,根据其原理,模仿其某些规律而设计的求解实际问题的一类算法。它将复杂任务交给大量的群体合作完成,具有概念简单、实现方便的特点。由于群体智能优化算法具有的分布性、简单性、灵活性和健壮性,已在计算机科学、知识发现、通信网络、机器人等研究领域显示出潜力和魅力,成为智能算法领域一个研究热点。
实际遇到的数值优化问题绝大多数是有约束的,我们求解约束优化问题时首先必须处理好约束。罚函数法是处理约束最常用的方法之一,罚函数法简单易行,但困难在于实际操作时要仔细调整罚因子,用以确定对不可行个体的合适的惩罚力度,才能使进化算法获得好的效果。
为了有效解决约束优化问题,本文分别研究用进化算法与粒子群算法两种智能算法来处理约束优化问题。从约束优化智能算法=约束处理技术+智能算法的研究框架出发,对约束处理技术和智能算法分别进行改进,从而设计出几种新的智能算法。本文的主要工作如下:
1罚函数法是进化算法中解决约束优化问题最常用的方法之一,它通过对不可行解进行惩罚使得搜索逐步进入可行域。罚函数常定义为目标函数与惩罚项之和,其缺陷一方面在于罚因子难以控制,另一方面当目标函数值与惩罚项的函数值的差值很大时,此模型不能有效地区分可行解与不可行解,从而不能有效处理约束。为了克服这些缺点,首先引入了目标满意度函数与约束满意度函数,前者是根据目标函数对解的满意度给出的一个度量,而后者是根据约束违反度对解的满意度给出的一个度量。然后定义了一种新的罚函数建立新罚函数模型。并且设置了自适应动态罚因子,其随当前种群的质量及进化代数的改变而改变。进一步,设计了新的杂交和变异算子。在此基础上,提出了解决约束优化问题的一种新的进化算法。
2首先,为了利用可行域附近的不可行点的信息,构造了自适应动态的扩展可行域,不仅包括所有可行点,还包括可行域附近的不可行点。其次,为了使得不同约束优化问题采取统一比较标准,提出了以个体序值构造适应度函数,即以个体序值代替个体适应度值评价个体。最后,提出了改进的算术杂交算子,但比杂交算子能产生更多的好点。
3首先提出了不带参数的罚函数,它能有效处理约束,由目标函数和罚函数构造一个双适应度函数。此双适应度函数能有效区分可行解与不可行解。而且还能合理评价可行解和不可行解。同时提出了单纯形杂交算子和PSO变异算子,两类新算子能更有效地开发搜索空间,从而有利的搜索方向,因此更易产生好解。
4.粒子群算法是解决优化问题的有效工具,但是用其解决约束优化问题时,容易产生早熟问题,从而得到局部最优解而非全局最优解。根据约束优化问题的特点,本文提出的双粒子群算法可以克服这一缺陷。首先为了挖掘较好非可行解(违反度较小且目标值很好)的信息,提出扩展的动态优化域(在可行域中添加高质量的不可行点的同时抛弃低质量的可行点形成的区域),使得搜索从可行域内外两个方向进行,从而增强算子的搜索最优解的能力。其次为了增强种群多样性,产生更多好解以避免早熟,本算法设计了两个进化方向,依据个体可行与否而采取不同的进化方向。
Intelligent computing originated from natural (biological) rules, also known as "soft computing", is a kind of computational algorithms simulating these rules for solving practical problems efficiently. Its related concepts are easy to understand and it is usually convenient to execute. Moreover, it is usually efficient because a population of individuals cooperate to complete the task. Due to the advantages of its simplicity, flexibility and robustness, intelligent computing has become a hot research area and has been widely used in many fields such as computer science, telecommunication network, knowledge discovery, robots and so on.
Most real-world optimization problems involve constraints. It is necessary for us to deal with constraints firstly when we are faced with a constrained optimization problem. The methods based on penalty are one of the most common approaches to deal with constrained optimization problems. In spite of their simplicity they have several drawbacks in which the main one is the difficulty to tune the penalty parameters. A careful tuning scheme is required to the penalty parameters that can accurately estimate the degree of penalization so that the approach can get a proper balance between the feasible solutions and infeasible solutions and become more efficient.
In order to solve constrained optimization problems effectively, evolutionary algorithms and particle swarm optimization algorithms are applied to deal with them. Starting from the framework that the constrained optimization algorithm = constraint processing technology + intelligent algorithms, some improved approaches on both basic aspects: constraint handling technology and intelligent algorithms, have been proposed. The main contributions of this thesis can be summarized as follows:
1. Penalty function method is one of the most widely used methods for constrained optimization problems in evolutionary algorithms. It makes the search approach to the feasible region gradually by the way to punish the infeasible solutions. The penalty functions are usually defined as the sum of the objective function and the penalty terms. This kind of the methods are of two main drawbacks. Firstly, it is difficult to control penalty parameters. Secondly, when the difference between the objective function value and the constrained function value is great, the algorithm can not effectively distinguish feasible solutions from infeasible ones, and thus can not handle the constraints effectively. To overcome the shortcomings, two satisfaction degree functions defined by the objective function and the constraints function are designed, respectively. A new penalty function is constructed by these two satisfaction degree functions. Moreover, an adaptive penalty parameter is designed, which is varying with the quality of the population and the number of the generations. As a result, the penalty parameter can be easily controlled. Based on these, a new penalty function optimization model is proposed. Furthermore, a new crossover operator and a new mutation operator are designed. Finally, a new evolutionary algorithm for constrained optimization problems is proposed.
2 Firstly, in order to make use of the information of the infeasible solutions near the feasible region, a self-adaptively extended-feasible region is constructed. This region includes not only all feasible solutions, but also some infeasible solutions near the boundary of the feasible region. Furthermore, in order to design a universal and fair solution quality measure for different constrained optimization problem, a new fitness function based on stochastic ranking is constructed. Finally, a new crossover operator, which is the improvement of the arithmetic crossover operator, is proposed. It can produce individuals with better diversity than the arithmetic crossover operator can.
3 Firstly, a new penalty function which has no parameter and can effectively handle constraints is proposed, then a hybrid fitness function defined by the penalty function and objective function is designed. The new fitness function not only can distinguish feasible individuals from infeasible ones, but also can evaluate both feasible and infeasible individuals reasonably. Moreover, a new crossover operator based on simplex scheme and a PSO mutation operator are also proposed. These new operators can exploit the search space more efficiently, and can provide potential search direction. As a result, the better individuals can often be generated.
4 Particle swarm algorithm is efficient to solve optimization problems. However, for constrained optimization problems, it usually cause premature and can trap into a local optimum easily. To overcome this shortcoming, a bi-particle swarm algorithm for constrained optimization problems is proposed. Firstly, In order to use the information of the good infeasible solutions (i.e. with low constrained violation and good objective function value), a dynamic extended optimization region (deleting poor feasible solutions and adding good infeasible solutions in feasible region) is designed which can search from both the feasible region and the infeasible region. Secondly, two search directions are designed for feasible and infeasible solutions, respectively. They are efficient to exploit the search space and increase diversity of the population.
实际遇到的数值优化问题绝大多数是有约束的,我们求解约束优化问题时首先必须处理好约束。罚函数法是处理约束最常用的方法之一,罚函数法简单易行,但困难在于实际操作时要仔细调整罚因子,用以确定对不可行个体的合适的惩罚力度,才能使进化算法获得好的效果。
为了有效解决约束优化问题,本文分别研究用进化算法与粒子群算法两种智能算法来处理约束优化问题。从约束优化智能算法=约束处理技术+智能算法的研究框架出发,对约束处理技术和智能算法分别进行改进,从而设计出几种新的智能算法。本文的主要工作如下:
1罚函数法是进化算法中解决约束优化问题最常用的方法之一,它通过对不可行解进行惩罚使得搜索逐步进入可行域。罚函数常定义为目标函数与惩罚项之和,其缺陷一方面在于罚因子难以控制,另一方面当目标函数值与惩罚项的函数值的差值很大时,此模型不能有效地区分可行解与不可行解,从而不能有效处理约束。为了克服这些缺点,首先引入了目标满意度函数与约束满意度函数,前者是根据目标函数对解的满意度给出的一个度量,而后者是根据约束违反度对解的满意度给出的一个度量。然后定义了一种新的罚函数建立新罚函数模型。并且设置了自适应动态罚因子,其随当前种群的质量及进化代数的改变而改变。进一步,设计了新的杂交和变异算子。在此基础上,提出了解决约束优化问题的一种新的进化算法。
2首先,为了利用可行域附近的不可行点的信息,构造了自适应动态的扩展可行域,不仅包括所有可行点,还包括可行域附近的不可行点。其次,为了使得不同约束优化问题采取统一比较标准,提出了以个体序值构造适应度函数,即以个体序值代替个体适应度值评价个体。最后,提出了改进的算术杂交算子,但比杂交算子能产生更多的好点。
3首先提出了不带参数的罚函数,它能有效处理约束,由目标函数和罚函数构造一个双适应度函数。此双适应度函数能有效区分可行解与不可行解。而且还能合理评价可行解和不可行解。同时提出了单纯形杂交算子和PSO变异算子,两类新算子能更有效地开发搜索空间,从而有利的搜索方向,因此更易产生好解。
4.粒子群算法是解决优化问题的有效工具,但是用其解决约束优化问题时,容易产生早熟问题,从而得到局部最优解而非全局最优解。根据约束优化问题的特点,本文提出的双粒子群算法可以克服这一缺陷。首先为了挖掘较好非可行解(违反度较小且目标值很好)的信息,提出扩展的动态优化域(在可行域中添加高质量的不可行点的同时抛弃低质量的可行点形成的区域),使得搜索从可行域内外两个方向进行,从而增强算子的搜索最优解的能力。其次为了增强种群多样性,产生更多好解以避免早熟,本算法设计了两个进化方向,依据个体可行与否而采取不同的进化方向。
Intelligent computing originated from natural (biological) rules, also known as "soft computing", is a kind of computational algorithms simulating these rules for solving practical problems efficiently. Its related concepts are easy to understand and it is usually convenient to execute. Moreover, it is usually efficient because a population of individuals cooperate to complete the task. Due to the advantages of its simplicity, flexibility and robustness, intelligent computing has become a hot research area and has been widely used in many fields such as computer science, telecommunication network, knowledge discovery, robots and so on.
Most real-world optimization problems involve constraints. It is necessary for us to deal with constraints firstly when we are faced with a constrained optimization problem. The methods based on penalty are one of the most common approaches to deal with constrained optimization problems. In spite of their simplicity they have several drawbacks in which the main one is the difficulty to tune the penalty parameters. A careful tuning scheme is required to the penalty parameters that can accurately estimate the degree of penalization so that the approach can get a proper balance between the feasible solutions and infeasible solutions and become more efficient.
In order to solve constrained optimization problems effectively, evolutionary algorithms and particle swarm optimization algorithms are applied to deal with them. Starting from the framework that the constrained optimization algorithm = constraint processing technology + intelligent algorithms, some improved approaches on both basic aspects: constraint handling technology and intelligent algorithms, have been proposed. The main contributions of this thesis can be summarized as follows:
1. Penalty function method is one of the most widely used methods for constrained optimization problems in evolutionary algorithms. It makes the search approach to the feasible region gradually by the way to punish the infeasible solutions. The penalty functions are usually defined as the sum of the objective function and the penalty terms. This kind of the methods are of two main drawbacks. Firstly, it is difficult to control penalty parameters. Secondly, when the difference between the objective function value and the constrained function value is great, the algorithm can not effectively distinguish feasible solutions from infeasible ones, and thus can not handle the constraints effectively. To overcome the shortcomings, two satisfaction degree functions defined by the objective function and the constraints function are designed, respectively. A new penalty function is constructed by these two satisfaction degree functions. Moreover, an adaptive penalty parameter is designed, which is varying with the quality of the population and the number of the generations. As a result, the penalty parameter can be easily controlled. Based on these, a new penalty function optimization model is proposed. Furthermore, a new crossover operator and a new mutation operator are designed. Finally, a new evolutionary algorithm for constrained optimization problems is proposed.
2 Firstly, in order to make use of the information of the infeasible solutions near the feasible region, a self-adaptively extended-feasible region is constructed. This region includes not only all feasible solutions, but also some infeasible solutions near the boundary of the feasible region. Furthermore, in order to design a universal and fair solution quality measure for different constrained optimization problem, a new fitness function based on stochastic ranking is constructed. Finally, a new crossover operator, which is the improvement of the arithmetic crossover operator, is proposed. It can produce individuals with better diversity than the arithmetic crossover operator can.
3 Firstly, a new penalty function which has no parameter and can effectively handle constraints is proposed, then a hybrid fitness function defined by the penalty function and objective function is designed. The new fitness function not only can distinguish feasible individuals from infeasible ones, but also can evaluate both feasible and infeasible individuals reasonably. Moreover, a new crossover operator based on simplex scheme and a PSO mutation operator are also proposed. These new operators can exploit the search space more efficiently, and can provide potential search direction. As a result, the better individuals can often be generated.
4 Particle swarm algorithm is efficient to solve optimization problems. However, for constrained optimization problems, it usually cause premature and can trap into a local optimum easily. To overcome this shortcoming, a bi-particle swarm algorithm for constrained optimization problems is proposed. Firstly, In order to use the information of the good infeasible solutions (i.e. with low constrained violation and good objective function value), a dynamic extended optimization region (deleting poor feasible solutions and adding good infeasible solutions in feasible region) is designed which can search from both the feasible region and the infeasible region. Secondly, two search directions are designed for feasible and infeasible solutions, respectively. They are efficient to exploit the search space and increase diversity of the population.
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