求解多目标问题的思维进化算法
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摘要
在现实世界中，大多数优化问题都要涉及多个目标。多目标优化是近30年来迅速发展起来的一门新兴学科。多年来许多研究者的工作就是寻找一些重要技术来处理多目标优化问题。
进化算法(Evolutionary Algorithm：EA)作为优化算法来解决复杂的多目标优化问题具有一定的优势。Schaffer在二十世纪80年代中期提出了VEGA，它是使用进化算法来解决多目标优化问题的第一次实现。现在的进化多目标算法大致可以被分为三类：(1)聚合法：这种方法是将多个目标聚合成一个目标函数来进行优化。它虽然简单且容易执行，但是当我们对问题了解较少时，就很难为各个目标产生合适的权值。(2)基于群体的非Pareto法，如VEGA；(3)基于Pareto的方法，如SPEA(Strength Pareto Evolutionary Algorithms)，它是最好的进化多目标优化算法之一。
根据对GA存在问题的思考以及对人类思维进步的分析，模仿人类社会中存在的趋同和异化现象，孙承意教授于1998年8月提出了思维进化计算(MEC)。经过几年来的理论和实验研究，目前思维进化计算在理论上已经有了很大的发展，同时也广泛应用于一些实际问题，所有这些工作已经为MEC建立了一个初步完整的体系。
本文提出两种多目标优化算法：Pareto-MEC和SP-MEC。它们都是将Pareto理论引入到基本MEC中来解决多目标优化问题的。
Pareto-MEC的基本思想是：(1)首先在整个解空间散布一
太原理工大学硕士研究生学位论文
些个体，选择最好的一些个体作为子群体的初始中心。(2)每个
子群体从这些初始中心出发，每个子群体仅搜索一个局部区域
并逐渐向Paret。前沿漂移。(3)在漂移的过程中，算法调整各子
群体的搜索范围和方向。上述的(1)和(3)称为异化操作，(2)称为
趋同操作。
SP一MEC的基本思想是:(l)首先在整个解空间散布一些个
体，根据它们的得分选择一些最好个体作为子群体的初始中心，
该得分反映个体间的统治关系和密度信息。(2)每个子群体从这
些初始中心出发，仅搜索一个局部区域来逐渐地向Paret。前沿
漂移。(3)在漂移的过程中，算法会调整各子群体的搜索范围和
前进方向。上述(1)和(3)称为异化操作，(2)称为趋同操作。
分别将Pareto一MEC和SP一MEC与Rand，VEGA，NSGA和
SPEA这四种算法进行了比较实验。并且在凸的，非凸的，离散
的及分布不均匀的测试问题上进行了测试，其中SPEA是这些
比较算法中性能较好的。实验结果表明Pareto一MEC和SP一MEC
在所有的测试函数上都超越了Rand，VEGA和NSGA这三种算
法:在第三个测试问题上，Pareto一MEC与SPEA具有同样优越的
性能，SP一MEC略好于SPEA;在最后一个测试问题上，两种算
法的结果都好于SPEA的结果。其它算法预先给定了迭代次数，
而这里提出的两种算法可以有客观的停止准则，这样即保证了
解的质量又提高了计算效率。
在凸的及分布不均匀的这两个测试函数上，我们还使用了
两个度量标准Cover和Spacing对算法进行定量的评价。实验结
果从数学的角度上说明了Pareto一MEC和SP一MEC这两种算法与
当前最好的算法SPEA相比，具有更好的性能。
There are many multi-objective optimization problems in the real world. The multi-objective optimization is a rising subject in the recent 30 years. Many researchers have been finding some important techniques to deal with multi-objective optimization problems.
Evolutionary Algorithms (EA) have some advantages in the complex multi-objective optimization problems. Schaffer firstly used VEGA to optimize multi-objective problems in the mid -1980s. Most of the approaches can be put into three categories: (1) Aggregating approaches: this approach combines objectives into a single function to optimize. Although it is very simple and easy to implement, this approach may be difficult to generate a set of weights that properly scales the objectives when little is known about the problem. (2) Population-based non-Pareto approaches, e.g. Vector Evaluated Genetic Algorithms (VEGA): this approach performs the proportional selection according to each objective function. Although it modifies the selection mechanism of the GA, this approach behaves as an aggregating approach. (3) Pareto-based approaches, e.g. Strength Pareto Evolutionary Algorithm (SPEA): this approach is outstanding in the approaches to evolutionary
multi-objective optimization.
To overcome the problems of GA, Mind Evolutionary Computation (MEC) was proposed by Chengyi Sun in 1998, which imitates two phenomena of human society - similartaxis and dissimilation. After several years' studies on its theory and experiment, MEC has been made great progress. So far, a preliminary system has already been established for MEC.
This paper proposes two kinds of multi-objective optimization algorithms, respectively called Pareto Mind Evolutionary Computation (Pareto-MEC) and Scored Pareto Mind Evolutionary Computation (SP-MEC). They use MEC to solve the multi-objective optimization problems.
The principles of Pareto-MEC are: (1) A number of individuals are scattered in the whole solution space, and then some better individuals of them are selected as the initial centers for every group. (2) Each group only searches a local area and gradually shifts from its initial center to the Pareto front. (3) During the process of shift to this front, this algorithm would bound the searching region of the group and control the shifting direction of the group. Both of above function (1) and function (3) are called as dissimilation, and function (2) is called as similartaxis.
The principles of SP-MEC are: (1) A number of individuals are scattered in the whole solution space, and then some better individuals of them are selected as the initial centers for every group according to their scores. (2) Each group only searches a local area and gradually shifts from its initial center to the Pareto front. (3) During the process of shift to this front, this algorithm would bound the searching region of the group and control the
shifting direction of the group. Both of above function (1) and function (3) are called as dissimilation, and function (2) is called as similartaxis.
Pareto-MEC and SP-MEC are respectively compared with the reference algorithms of Rand, VEGA, NSGA and SPEA. The test functions used in the experiment are a suit of four different test problems: convexity, non-convexity, discreteness and non-uniformity. On all test problems, both Pareto-MEC and SP-MEC outperform Rand, VEGA and NSGA; Pareto-MEC is as good as SPEA on the first three test problems, but SP-MEC is better than SPEA; They beat SPEA on the last test problem. Different from the reference algorithms that use the pre-specified generation number as their terminations, Pareto-MEC and SP-MEC have an objective termination criterion that can ensure the quality of solutions and the computational efficiency.
Two evaluative methods: Cover and Spacing are used as the quantificational criterion for our algorithms on the test functions: convexity, non-convexity and non-uniformity. The experiment shows from the angle of arith that the performances of Pareto-MEC and SP-MEC are better than SPEA which is one of the excellenc
进化算法(Evolutionary Algorithm：EA)作为优化算法来解决复杂的多目标优化问题具有一定的优势。Schaffer在二十世纪80年代中期提出了VEGA，它是使用进化算法来解决多目标优化问题的第一次实现。现在的进化多目标算法大致可以被分为三类：(1)聚合法：这种方法是将多个目标聚合成一个目标函数来进行优化。它虽然简单且容易执行，但是当我们对问题了解较少时，就很难为各个目标产生合适的权值。(2)基于群体的非Pareto法，如VEGA；(3)基于Pareto的方法，如SPEA(Strength Pareto Evolutionary Algorithms)，它是最好的进化多目标优化算法之一。
根据对GA存在问题的思考以及对人类思维进步的分析，模仿人类社会中存在的趋同和异化现象，孙承意教授于1998年8月提出了思维进化计算(MEC)。经过几年来的理论和实验研究，目前思维进化计算在理论上已经有了很大的发展，同时也广泛应用于一些实际问题，所有这些工作已经为MEC建立了一个初步完整的体系。
本文提出两种多目标优化算法：Pareto-MEC和SP-MEC。它们都是将Pareto理论引入到基本MEC中来解决多目标优化问题的。
Pareto-MEC的基本思想是：(1)首先在整个解空间散布一
太原理工大学硕士研究生学位论文
些个体，选择最好的一些个体作为子群体的初始中心。(2)每个
子群体从这些初始中心出发，每个子群体仅搜索一个局部区域
并逐渐向Paret。前沿漂移。(3)在漂移的过程中，算法调整各子
群体的搜索范围和方向。上述的(1)和(3)称为异化操作，(2)称为
趋同操作。
SP一MEC的基本思想是:(l)首先在整个解空间散布一些个
体，根据它们的得分选择一些最好个体作为子群体的初始中心，
该得分反映个体间的统治关系和密度信息。(2)每个子群体从这
些初始中心出发，仅搜索一个局部区域来逐渐地向Paret。前沿
漂移。(3)在漂移的过程中，算法会调整各子群体的搜索范围和
前进方向。上述(1)和(3)称为异化操作，(2)称为趋同操作。
分别将Pareto一MEC和SP一MEC与Rand，VEGA，NSGA和
SPEA这四种算法进行了比较实验。并且在凸的，非凸的，离散
的及分布不均匀的测试问题上进行了测试，其中SPEA是这些
比较算法中性能较好的。实验结果表明Pareto一MEC和SP一MEC
在所有的测试函数上都超越了Rand，VEGA和NSGA这三种算
法:在第三个测试问题上，Pareto一MEC与SPEA具有同样优越的
性能，SP一MEC略好于SPEA;在最后一个测试问题上，两种算
法的结果都好于SPEA的结果。其它算法预先给定了迭代次数，
而这里提出的两种算法可以有客观的停止准则，这样即保证了
解的质量又提高了计算效率。
在凸的及分布不均匀的这两个测试函数上，我们还使用了
两个度量标准Cover和Spacing对算法进行定量的评价。实验结
果从数学的角度上说明了Pareto一MEC和SP一MEC这两种算法与
当前最好的算法SPEA相比，具有更好的性能。
There are many multi-objective optimization problems in the real world. The multi-objective optimization is a rising subject in the recent 30 years. Many researchers have been finding some important techniques to deal with multi-objective optimization problems.
Evolutionary Algorithms (EA) have some advantages in the complex multi-objective optimization problems. Schaffer firstly used VEGA to optimize multi-objective problems in the mid -1980s. Most of the approaches can be put into three categories: (1) Aggregating approaches: this approach combines objectives into a single function to optimize. Although it is very simple and easy to implement, this approach may be difficult to generate a set of weights that properly scales the objectives when little is known about the problem. (2) Population-based non-Pareto approaches, e.g. Vector Evaluated Genetic Algorithms (VEGA): this approach performs the proportional selection according to each objective function. Although it modifies the selection mechanism of the GA, this approach behaves as an aggregating approach. (3) Pareto-based approaches, e.g. Strength Pareto Evolutionary Algorithm (SPEA): this approach is outstanding in the approaches to evolutionary
multi-objective optimization.
To overcome the problems of GA, Mind Evolutionary Computation (MEC) was proposed by Chengyi Sun in 1998, which imitates two phenomena of human society - similartaxis and dissimilation. After several years' studies on its theory and experiment, MEC has been made great progress. So far, a preliminary system has already been established for MEC.
This paper proposes two kinds of multi-objective optimization algorithms, respectively called Pareto Mind Evolutionary Computation (Pareto-MEC) and Scored Pareto Mind Evolutionary Computation (SP-MEC). They use MEC to solve the multi-objective optimization problems.
The principles of Pareto-MEC are: (1) A number of individuals are scattered in the whole solution space, and then some better individuals of them are selected as the initial centers for every group. (2) Each group only searches a local area and gradually shifts from its initial center to the Pareto front. (3) During the process of shift to this front, this algorithm would bound the searching region of the group and control the shifting direction of the group. Both of above function (1) and function (3) are called as dissimilation, and function (2) is called as similartaxis.
The principles of SP-MEC are: (1) A number of individuals are scattered in the whole solution space, and then some better individuals of them are selected as the initial centers for every group according to their scores. (2) Each group only searches a local area and gradually shifts from its initial center to the Pareto front. (3) During the process of shift to this front, this algorithm would bound the searching region of the group and control the
shifting direction of the group. Both of above function (1) and function (3) are called as dissimilation, and function (2) is called as similartaxis.
Pareto-MEC and SP-MEC are respectively compared with the reference algorithms of Rand, VEGA, NSGA and SPEA. The test functions used in the experiment are a suit of four different test problems: convexity, non-convexity, discreteness and non-uniformity. On all test problems, both Pareto-MEC and SP-MEC outperform Rand, VEGA and NSGA; Pareto-MEC is as good as SPEA on the first three test problems, but SP-MEC is better than SPEA; They beat SPEA on the last test problem. Different from the reference algorithms that use the pre-specified generation number as their terminations, Pareto-MEC and SP-MEC have an objective termination criterion that can ensure the quality of solutions and the computational efficiency.
Two evaluative methods: Cover and Spacing are used as the quantificational criterion for our algorithms on the test functions: convexity, non-convexity and non-uniformity. The experiment shows from the angle of arith that the performances of Pareto-MEC and SP-MEC are better than SPEA which is one of the excellenc
引文
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