用于区间参数多目标优化问题的遗传算法
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摘要
区间参数多目标优化问题是普遍存在，且非常重要的不确定优化问题。由于该问题的参数取值为区间，且含有多个目标函数，因此，有效的解决方法非常少见。论文根据不同的实际需求，提出有效解决该问题的3类遗传算法。
首先，面向多目标优化问题的一般需求，给出一种求取近似Pareto最优解集的遗传算法。该算法通过定义区间占优可信度下界，给出基于该下界的区间多目标优化问题的占优关系，及其相应的Pareto最优解集的性质；利用提出的占优关系，修改NSGA-Ⅱ的快速非被占优解排序方法，开发一种新的解决区间参数多目标优化问题的遗传算法，并从理论上分析该算法的性能；将所提方法应用于6个区间参数多目标优化问题，并与2个典型的优化方法比较，实验结果表明所提方法的优越性。
然后，根据在实际应用中，决策者往往仅需要一个最满意解(集)的要求，研究2种偏好表示方式下，采用边优化边决策的方法，解决区间多目标优化问题的进化优化方法。通过建立用于区间参数优化问题的偏好多面体理论，提出一种基于偏好多面体的区间多目标交互式遗传算法，该算法定期将部分非被占优解提交给决策者，以最差解为顶点，在目标空间中构建偏好多面体；利用该多面体，进一步区分具有相同序值的进化个体。进一步地，从偏好多面体中提取决策者的偏好方向；基于该偏好方向，设计反映进化个体逼近性能的测度，将具有相同序值、相同偏好的个体排序，开发一种基于偏好方向的区间多目标交互式遗传算法。将上述2种方法应用于4个区间参数2目标优化问题，并与后验方法比较，实验结果表明，2种方法皆优于后验方法，可以得到符合决策者偏好的优化解。此外，利用目标的相对重要性，提出一种交互式遗传算法，以得到一个符合决策者偏好的最满意解集。在该算法中，决策者根据需要，交互式输入代表其偏好的目标间的相对重要性关系；由该关系得到其在目标空间的偏好区域；基于该偏好区域，进一步比较具有相同序值进化个体的性能，指导算法向决策者真正的偏好区域搜索。将所提方法应用于2个区间参数2目标优化问题和2个区间参数3目标优化问题，并与先验方法和后验方法比较，实验结果证实所提方法是有效的，能够找到更多符合决策者偏好的优化解。
最后，基于对不确定优化问题的特殊要求，提出一种有效解决区间参数很多目标优化问题的集合进化遗传算法。该方法以超体积和不确定度为目标，将原优化问题转化为精确参数2目标优化问题；定义基于集合的Pareto占优关系，并修改NSGA-II的快速非被占优解排序方法；此外，还提出集合进化策略。将所提方法应用于4个区间参数很多目标优化问题，并与已有的方法比较，结果表明，所提方法能够得到收敛性和不确定性均衡的Pareto最优解集。
所提3类遗传算法不仅为区间参数多目标优化问题的求解提供了切实可行的途径，而且丰富了区间数学的研究内容。
Multi-objective optimization problems (MOPs) with interval parameters are verypopular and important in real-world applications. However, there has been very feweffective method of solving these problems up to date because of their intervalparameters and multiple objectives. According to different actual demands, threekinds of genetic algorithms (GAs) were proposed to solve the above problems in thisdissertation.
First, a GA for obtaining a Pareto set approximation was given to face thecommon demand of a MOP. In this method, through defining a lower limit ofpossibility degree of interval dominance, the dominance relation of an intervalmulti-objective optimization problem (IMOP) based on the above lower limit and theproperty of the corresponding Pareto set are given; the proposed dominance relation isemployed to modify the fast non-dominated sorting approach in NSGA-II, and a novelGA for an IMOP is then developed. Further, after theoretically analyzing theperformance of the proposed method, it was applied to six IMOPs and compared withtwo typical optimization methods. The experimental results confirm its advantages.
Then, to meet a decision maker (DM)’s demand for a most preferred solution(set), three GAs incorporating a search-cum-decision-making procedure for solvingIMOPs were studied in the context of two kinds of preference presentations. Byconstructing the theory of a preference polyhedron for an optimization problem withinterval parameters, an interactive genetic algorithm (IGA) for an IMOP based onpreference polyhedron was proposed. The algorithm periodically provides a part ofnon-dominated solutions to a DM, and a preference polyhedron, based on whichdifferent evolutionary individuals with the same ranks are distinguished, is createdtaking the worst solution as vertex in the objective space. Further, a preferencedirection is elicited from the above preference polyhedron, and a metric based on thepreference direction, reflecting the approximation performance of an evolutionaryindividual, is designed to rank different individuals with the same ranks andpreferences. Consequently, an IGA for an IMOP based on the preference direction wasdeveloped. Both of the above methods were applied to four interval bi-objectiveoptimization problems, and compared with an a posterior method. The experimentalresults show that both of the methods are superior than the a posterior method, andthe most preferred solution that fits the DM’s preferences can be obtained. Additionally, an IGA was presented to obtain a most preferred solution set meetingthe DM’s preferences by utilizing the relative importance of objectives. In thisalgorithm, the preference informations are interactively input, and the preferenceregion in the objective space is then obtained; based on this region, differentevolutionary individuals with the same ranks are further discriminated, to guide thepopulation to evolve towards the DM’s really preference regions. The proposedmethod was applied to four IMOPs, and compared with an a priori method and an aposterior method. The experimental results confirm its advantages of the proposedalgorithm, and more optimal solutions which fit the DM’s preferences can be derived.
Finally, to satisfy the special request of uncertain optimization problems, aset-based GA was proposed to effectively solve many-objective optimizationproblems with interval parameters. In this method, the original optimization problemis transformed into a bi-objective one with deterministic parameters by takinghyper-volume and imprecision as two new objectives; a set-based Pareto dominancerelation is defined to modify the fast non-dominated sorting approach in NSGA-II;moreover, set-based evolutionary schemes are suggested. The proposed method wasapplied to four benchmark many-objective optimization problems with intervalparameters and compared with a state-of-the-art method. The experimental resultsindicate that a trade-off Pareto set approximation between convergence anduncertainty can be found by the proposed method.
Three kinds of GAs proposed in this dissertation not only offer several practicalways to solve IMOPs, but also enrich the researches of interval mathematics.
首先，面向多目标优化问题的一般需求，给出一种求取近似Pareto最优解集的遗传算法。该算法通过定义区间占优可信度下界，给出基于该下界的区间多目标优化问题的占优关系，及其相应的Pareto最优解集的性质；利用提出的占优关系，修改NSGA-Ⅱ的快速非被占优解排序方法，开发一种新的解决区间参数多目标优化问题的遗传算法，并从理论上分析该算法的性能；将所提方法应用于6个区间参数多目标优化问题，并与2个典型的优化方法比较，实验结果表明所提方法的优越性。
然后，根据在实际应用中，决策者往往仅需要一个最满意解(集)的要求，研究2种偏好表示方式下，采用边优化边决策的方法，解决区间多目标优化问题的进化优化方法。通过建立用于区间参数优化问题的偏好多面体理论，提出一种基于偏好多面体的区间多目标交互式遗传算法，该算法定期将部分非被占优解提交给决策者，以最差解为顶点，在目标空间中构建偏好多面体；利用该多面体，进一步区分具有相同序值的进化个体。进一步地，从偏好多面体中提取决策者的偏好方向；基于该偏好方向，设计反映进化个体逼近性能的测度，将具有相同序值、相同偏好的个体排序，开发一种基于偏好方向的区间多目标交互式遗传算法。将上述2种方法应用于4个区间参数2目标优化问题，并与后验方法比较，实验结果表明，2种方法皆优于后验方法，可以得到符合决策者偏好的优化解。此外，利用目标的相对重要性，提出一种交互式遗传算法，以得到一个符合决策者偏好的最满意解集。在该算法中，决策者根据需要，交互式输入代表其偏好的目标间的相对重要性关系；由该关系得到其在目标空间的偏好区域；基于该偏好区域，进一步比较具有相同序值进化个体的性能，指导算法向决策者真正的偏好区域搜索。将所提方法应用于2个区间参数2目标优化问题和2个区间参数3目标优化问题，并与先验方法和后验方法比较，实验结果证实所提方法是有效的，能够找到更多符合决策者偏好的优化解。
最后，基于对不确定优化问题的特殊要求，提出一种有效解决区间参数很多目标优化问题的集合进化遗传算法。该方法以超体积和不确定度为目标，将原优化问题转化为精确参数2目标优化问题；定义基于集合的Pareto占优关系，并修改NSGA-II的快速非被占优解排序方法；此外，还提出集合进化策略。将所提方法应用于4个区间参数很多目标优化问题，并与已有的方法比较，结果表明，所提方法能够得到收敛性和不确定性均衡的Pareto最优解集。
所提3类遗传算法不仅为区间参数多目标优化问题的求解提供了切实可行的途径，而且丰富了区间数学的研究内容。
Multi-objective optimization problems (MOPs) with interval parameters are verypopular and important in real-world applications. However, there has been very feweffective method of solving these problems up to date because of their intervalparameters and multiple objectives. According to different actual demands, threekinds of genetic algorithms (GAs) were proposed to solve the above problems in thisdissertation.
First, a GA for obtaining a Pareto set approximation was given to face thecommon demand of a MOP. In this method, through defining a lower limit ofpossibility degree of interval dominance, the dominance relation of an intervalmulti-objective optimization problem (IMOP) based on the above lower limit and theproperty of the corresponding Pareto set are given; the proposed dominance relation isemployed to modify the fast non-dominated sorting approach in NSGA-II, and a novelGA for an IMOP is then developed. Further, after theoretically analyzing theperformance of the proposed method, it was applied to six IMOPs and compared withtwo typical optimization methods. The experimental results confirm its advantages.
Then, to meet a decision maker (DM)’s demand for a most preferred solution(set), three GAs incorporating a search-cum-decision-making procedure for solvingIMOPs were studied in the context of two kinds of preference presentations. Byconstructing the theory of a preference polyhedron for an optimization problem withinterval parameters, an interactive genetic algorithm (IGA) for an IMOP based onpreference polyhedron was proposed. The algorithm periodically provides a part ofnon-dominated solutions to a DM, and a preference polyhedron, based on whichdifferent evolutionary individuals with the same ranks are distinguished, is createdtaking the worst solution as vertex in the objective space. Further, a preferencedirection is elicited from the above preference polyhedron, and a metric based on thepreference direction, reflecting the approximation performance of an evolutionaryindividual, is designed to rank different individuals with the same ranks andpreferences. Consequently, an IGA for an IMOP based on the preference direction wasdeveloped. Both of the above methods were applied to four interval bi-objectiveoptimization problems, and compared with an a posterior method. The experimentalresults show that both of the methods are superior than the a posterior method, andthe most preferred solution that fits the DM’s preferences can be obtained. Additionally, an IGA was presented to obtain a most preferred solution set meetingthe DM’s preferences by utilizing the relative importance of objectives. In thisalgorithm, the preference informations are interactively input, and the preferenceregion in the objective space is then obtained; based on this region, differentevolutionary individuals with the same ranks are further discriminated, to guide thepopulation to evolve towards the DM’s really preference regions. The proposedmethod was applied to four IMOPs, and compared with an a priori method and an aposterior method. The experimental results confirm its advantages of the proposedalgorithm, and more optimal solutions which fit the DM’s preferences can be derived.
Finally, to satisfy the special request of uncertain optimization problems, aset-based GA was proposed to effectively solve many-objective optimizationproblems with interval parameters. In this method, the original optimization problemis transformed into a bi-objective one with deterministic parameters by takinghyper-volume and imprecision as two new objectives; a set-based Pareto dominancerelation is defined to modify the fast non-dominated sorting approach in NSGA-II;moreover, set-based evolutionary schemes are suggested. The proposed method wasapplied to four benchmark many-objective optimization problems with intervalparameters and compared with a state-of-the-art method. The experimental resultsindicate that a trade-off Pareto set approximation between convergence anduncertainty can be found by the proposed method.
Three kinds of GAs proposed in this dissertation not only offer several practicalways to solve IMOPs, but also enrich the researches of interval mathematics.
引文
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