一个求解多目标问题的算法：Pareto-MEC
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摘要
在现实世界中，很多问题都要涉及对多个目标进行优化。因此多目标优化也成为近30年来迅速发展起来的一门新兴学科。
本文提出了一种新的方法来进行多目标优化。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)。本文将Pareto理论引入到基本MEC中形成了该方法。
Pareto-MEC的基本思想是：(1)首先在整个解空间散布一些个体，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Pareto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。
本文分别将Pareto-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是这几种比较算法中性能最好的。实验结果表明Pareto-MEC在所有的测试
太原理工大学硕士研究生学位论文
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;在最后
一个测试问题上，该算法的结果好于SPEA的结果。此外其它
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
果从数量的角度上说明了Pareto一MEC这种算法与当前最好的算
法之一SPEA相比，具有相当甚至略好的性能。
本算法是使用MEC进行多目标优化的第一次尝试，也是首
次把Parcto的概念引入到MEC之中。从获得的实验结果来看，
本算法与SPEA各有所长。而本算法的计算效率却比较高。而
且与那些比较算法不同，该算法有自己客观的停止准则。可以
看出Pareto一MEC十分适合处理多目标优化问题。
In the real world, there are many multi-objective optimization problems. The multi-objective optimization is a rising subject in the recent 30 years.
In this paper a new approach is proposed. It introduces the conception of Pareto into the Mind Evolutionary Computation (MEC) which was first proposed by Chengyi Sun in 1998. MEC imitates two phenomena of human society - similartaxis and dissimilation and overcomes the problems of GA successfully.
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.
Pareto-MEC is 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, Pareto-MEC outperforms Rand, VEGA and NSGA; Pareto-MEC is as good as SPEA on the first three test problems; yet it beats SPEA on the last test problem. Different from the reference algorithms that use the pre-specified generation number as their terminations, Pareto-MEC has 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 that the performance of Pareto-MEC is as good as SPEA which is one of the excellence algorithms in the world.
This is the first time that using MEC resolving multi-objective optimization problems. And the MEC had never used the pareto conception before. The experiment results of Pareto-MEC are as good as that of SPEA. Yet Pareto-MEC has less number of generations. Moreover, this approach has an impersonal stopping condition not as other methods mentions here which use a given iteration times as their teminatind conditions. It can be seen that Pareto-MEC is very suit multi-objective optimization problems.
本文提出了一种新的方法来进行多目标优化。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)。本文将Pareto理论引入到基本MEC中形成了该方法。
Pareto-MEC的基本思想是：(1)首先在整个解空间散布一些个体，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Pareto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。
本文分别将Pareto-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是这几种比较算法中性能最好的。实验结果表明Pareto-MEC在所有的测试
太原理工大学硕士研究生学位论文
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;在最后
一个测试问题上，该算法的结果好于SPEA的结果。此外其它
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
果从数量的角度上说明了Pareto一MEC这种算法与当前最好的算
法之一SPEA相比，具有相当甚至略好的性能。
本算法是使用MEC进行多目标优化的第一次尝试，也是首
次把Parcto的概念引入到MEC之中。从获得的实验结果来看，
本算法与SPEA各有所长。而本算法的计算效率却比较高。而
且与那些比较算法不同，该算法有自己客观的停止准则。可以
看出Pareto一MEC十分适合处理多目标优化问题。
In the real world, there are many multi-objective optimization problems. The multi-objective optimization is a rising subject in the recent 30 years.
In this paper a new approach is proposed. It introduces the conception of Pareto into the Mind Evolutionary Computation (MEC) which was first proposed by Chengyi Sun in 1998. MEC imitates two phenomena of human society - similartaxis and dissimilation and overcomes the problems of GA successfully.
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.
Pareto-MEC is 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, Pareto-MEC outperforms Rand, VEGA and NSGA; Pareto-MEC is as good as SPEA on the first three test problems; yet it beats SPEA on the last test problem. Different from the reference algorithms that use the pre-specified generation number as their terminations, Pareto-MEC has 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 that the performance of Pareto-MEC is as good as SPEA which is one of the excellence algorithms in the world.
This is the first time that using MEC resolving multi-objective optimization problems. And the MEC had never used the pareto conception before. The experiment results of Pareto-MEC are as good as that of SPEA. Yet Pareto-MEC has less number of generations. Moreover, this approach has an impersonal stopping condition not as other methods mentions here which use a given iteration times as their teminatind conditions. It can be seen that Pareto-MEC is very suit multi-objective optimization problems.
引文
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