Hypervolume Subset Selection for Triangular and Inverted Triangular Pareto Fronts of Three-Objective Problems
Hypervolume Subset Selection for Triangular and Inverted Triangular Pareto Fronts of Three-Objective Problems
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DOI：
10.1145/3040718.3040730
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发表时间：
2017-01
期刊：
影响因子：
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通讯作者：
H. Ishibuchi;Ryo Imada;Yu Setoguchi;Y. Nojima
中科院分区：
文献类型：
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作者：
H. Ishibuchi;Ryo Imada;Yu Setoguchi;Y. Nojima
超体积子集选择是为了找到一个预先指定的数量的解决方案的超体积最大化。文献中对双目标问题的Pareto前沿的最优解分布进行了理论研究。本文讨论了具有三角和倒三角Pareto前沿的三目标问题的超体积子集选择问题。我们的贡献是表明，超体积计算的参考点的位置上的最佳分布的解决方案是完全不同的三角和倒三角帕累托前沿之间。当参考点远离Pareto前沿时，大多数解位于倒三角Pareto前沿的两侧，而它们均匀地分布在整个三角Pareto前沿上。这些属性似乎在具有四个或更多目标的多目标问题中成立。我们还表明，一个参考点的位置上的最优分布的效果是完全不同的最大化和最小化问题具有相同的三角帕累托前沿。这个属性是支持的事实，即最大化问题与三角帕累托阵线是等价的最小化问题与倒三角帕累托阵线。还讨论了当参考点接近Pareto前沿（即，当它的位置在最低点和帕累托前沿之间时）。
Hypervolume subset selection is to find a pre-specified number of solutions for hypervolume maximization. The optimal distribution of solutions on the Pareto front has been theoretically studied for two-objective problems in the literature. In this paper, we discuss hypervolume subset selection for three-objective problems with triangular and inverted triangular Pareto fronts. Our contribution is to show that the effect of the location of a reference point for hypervolume calculation on the optimal distribution of solutions is totally different between triangular and inverted triangular Pareto fronts. When the reference point is far from the Pareto front, most solutions are on the sides of the inverted triangular Pareto front while they are evenly distributed over the entire triangular Pareto front. These properties seem to hold in multiobjective problems with four or more objectives. We also show that the effect of the location of a reference point on the optimal distribution is totally different between maximization and minimization problems with the same triangular Pareto fronts. This property is supported by the fact that maximization problems with triangular Pareto fronts are equivalent to minimization problems with inverted triangular Pareto fronts. The optimal distribution of solutions is also discussed when the reference point is close to the Pareto front (i.e., when its location is between the nadir point and the Pareto front).