离散重心问题的一种列生成方法

IF 0.9 4区 数学 Q3 MATHEMATICS, APPLIED
Steffen Borgwardt , Stephan Patterson
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引用次数: 5

摘要

离散Wasserstein重心问题是一组离散概率测度的最小代价质量传递问题。虽然通过线性规划可以计算出精确的重心,但潜在的线性规划可能非常大。对于最坏情况输入,最著名的线性规划公式是变量数量呈指数增长,但约束数量很少,这使其成为列生成的有趣候选。在本文中,我们设计并研究了两种列生成策略:一种是基于简化成本计算的自然列生成策略,另一种是基于dantzigg - wolfe分解的列生成策略。对于后者,我们产生了有效可解的子问题,即经典运输问题形式的定价问题。这两种策略从初始可行解的有效计算开始。虽然约束的结构导致计算所有剩余变量的成本降低,但这两种方法在速度上都可能优于使用完整程序的计算,并且在内存需求方面明显优于使用完整程序的计算。在我们的计算实验中,我们证明,根据输入,任何一种策略都可以成为最佳选择。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A column generation approach to the discrete barycenter problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation.

In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig–Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.

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来源期刊
Discrete Optimization
Discrete Optimization 管理科学-应用数学
CiteScore
2.10
自引率
9.10%
发文量
30
审稿时长
>12 weeks
期刊介绍: Discrete Optimization publishes research papers on the mathematical, computational and applied aspects of all areas of integer programming and combinatorial optimization. In addition to reports on mathematical results pertinent to discrete optimization, the journal welcomes submissions on algorithmic developments, computational experiments, and novel applications (in particular, large-scale and real-time applications). The journal also publishes clearly labelled surveys, reviews, short notes, and open problems. Manuscripts submitted for possible publication to Discrete Optimization should report on original research, should not have been previously published, and should not be under consideration for publication by any other journal.
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