多目标优化的改进光滑分析
Improved Smoothed Analysis of Multiobjective Optimization
摘要：
本文给出了多目标优化问题光滑分析的几个新结果。由于最坏情况分析与实际经验之间的差异，这一研究领域在过去十年中受到了广泛关注。我们考虑的问题，其中d线性和一个任意的目标函数是优化的子集S的{0，1}^n的可行的解决方案。我们改进了以前最好的已知边界的帕累托最优解的平滑的数量为O（n^{2d} phi^d），其中phi表示扰动参数。此外，我们还证明了对于任何常数c，Pareto最优解的平滑数的c阶矩的界为O（（n^{2d} phi^d）^c）。这显著地改进了先前已知的边界。此外，我们解决的批评，在平滑分析的扰动破坏零结构的问题表明，帕累托最优解的平滑数量保持多项式有界，即使为保零扰动。这扩大了平滑分析所捕获的问题的类别，并且对非线性目标函数具有影响。我们的结果的一个推论是，帕累托最优解的平滑数量是多项式有界的多项式目标函数。
英文摘要：
We present several new results about smoothed analysis of multiobjective optimization problems. Motivated by the discrepancy between worst-case analysis and practical experience, this line of research has gained a lot of attention in the last decade. We consider problems in which d linear and one arbitrary objective function are to be optimized over a subset S of {0,1}^n of feasible solutions. We improve the previously best known bound for the smoothed number of Pareto-optimal solutions to O(n^{2d} phi^d), where phi denotes the perturbation parameter. Additionally, we show that for any constant c the c-th moment of the smoothed number of Pareto-optimal solutions is bounded by O((n^{2d} phi^d)^c). This improves the previously best known bounds significantly. Furthermore, we address the criticism that the perturbations in smoothed analysis destroy the zero-structure of problems by showing that the smoothed number of Pareto-optimal solutions remains polynomially bounded even for zero-preserving perturbations. This broadens the class of problems captured by smoothed analysis and it has consequences for non-linear objective functions. One corollary of our result is that the smoothed number of Pareto-optimal solutions is polynomially bounded for polynomial objective functions.
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