Stochastic Optimization and Learning for Two-Stage Supplier Problems

Brian Brubach; Nathaniel Grammel; David G. Harris; Aravind Srinivasan; Leonidas Tsepenekas; Anil Vullikanti

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arXiv · 2008.03325

Stochastic Optimization and Learning for Two-Stage Supplier Problems

Brian Brubach·Nathaniel Grammel·David G. Harris·Aravind Srinivasan·Leonidas Tsepenekas·Anil Vullikanti

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Abstract

The main focus of this paper is radius-based (supplier) clustering in the two-stage stochastic setting with recourse, where the inherent stochasticity of the model comes in the form of a budget constraint. In addition to the standard (homogeneous) setting where all clients must be within a distance $R$ of the nearest facility, we provide results for the more general problem where the radius demands may be inhomogeneous (i.e., different for each client). We also explore a number of variants where additional constraints are imposed on the first-stage decisions, specifically matroid and multi-knapsack constraints, and provide results for these settings. We derive results for the most general distributional setting, where there is only black-box access to the underlying distribution. To accomplish this, we first develop algorithms for the polynomial scenarios setting; we then employ a novel scenario-discarding variant of the standard Sample Average Approximation (SAA) method, which crucially exploits properties of the restricted-case algorithms. We note that the scenario-discarding modification to the SAA method is necessary in order to optimize over the radius.

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Brian Brubach, Nathaniel Grammel, David G. Harris, Aravind Srinivasan, Leonidas Tsepenekas, Anil Vullikanti. 2020-08-07. Stochastic Optimization and Learning for Two-Stage Supplier Problems. https://arxiv.org/abs/2008.03325

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