Pareto最优解的平均数和光滑数的下界
Lower Bounds for the Average and Smoothed Number of Pareto Optima
摘要：
多目标0 - 1线性优化问题的光滑分析近年来引起了人们的广泛关注。帕累托最优解的数量（即，解决方案的性质，没有其他的解决方案是至少在所有的坐标和更好的至少在一个）的多目标优化问题是研究的中心对象。在本文中，我们证明了几个下界的期望数的帕累托最优。我们的基本结果是在相当一般的线性目标分布条件下，对具有d个目标和n个变量的最优化问题，给出了Ω_d（n ^（d-1））的一个下界。我们的证明涉及的问题下界的帕累托最优的数量，结果在几何连接安排的超平面。利用我们的基本结果，我们得到了（1）自然多目标优化问题的第一个下界。我们说明了这一点的最大生成树问题随机选择的边缘权重。我们的技术是足够灵活的，以产生这样的下限在此设置中研究的其他标准目标函数（如，多目标最短路径，TSP旅游，匹配）。(2)对于一类具有φ-半随机分布的0 - 1背包问题，给出了min {Ω_d（n ^（d-1.5）<$^{（d-log d）（1-θ（1/θ））}），2 ^{θ（n）}}$的光滑下界.这改进了Brunsch和Roeglin最近的下界。
英文摘要：
Smoothed analysis of multiobjective 0-1 linear optimization has drawn considerable attention recently. The number of Pareto-optimal solutions (i.e., solutions with the property that no other solution is at least as good in all the coordinates and better in at least one) for multiobjective optimization problems is the central object of study. In this paper, we prove several lower bounds for the expected number of Pareto optima. Our basic result is a lower bound of Ω_d(n^(d-1)) for optimization problems with d objectives and n variables under fairly general conditions on the distributions of the linear objectives. Our proof relates the problem of lower bounding the number of Pareto optima to results in geometry connected to arrangements of hyperplanes. We use our basic result to derive (1) To our knowledge, the first lower bound for natural multiobjective optimization problems. We illustrate this for the maximum spanning tree problem with randomly chosen edge weights. Our technique is sufficiently flexible to yield such lower bounds for other standard objective functions studied in this setting (such as, multiobjective shortest path, TSP tour, matching). (2) Smoothed lower bound of min {Ω_d(n^(d-1.5) ϕ^{(d-log d) (1-Θ(1/ϕ))}), 2^{Θ(n)}}$ for the 0-1 knapsack problem with d profits for phi-semirandom distributions for a version of the knapsack problem. This improves the recent lower bound of Brunsch and Roeglin.
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