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Stochastic Optimization and Learning for Two-Stage Supplier Problems

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

TL;DR

The paper studies radius-based coverage in a two-stage stochastic supplier problem (2S-Sup) with recourse, accommodating both homogeneous and inhomogeneous radii $R_j$ and extra first-stage constraints such as matroids and multi-knapsacks. It introduces a black-box generalization framework that combines polynomial-scenarios algorithms with a scenario-discarding Sample Average Approximation (SAA) scheme to optimize the radius under a budget $B$ while ensuring high-probability coverage. Through a robust-outlier reduction and LP-rounding/iterative rounding techniques, it achieves several approximation guarantees: a 3-approximation for homogeneous 2S-Sup-Poly, 5-approximations for homogeneous 2S-MatSup-Poly and 2S-MuSup-Poly, and 9- to 11-approximation results for inhomogeneous 2S-MatSup-Poly. These results support scalable, data-driven design of coverage facilities under uncertainty, with concrete relevance to healthcare accessibility and location planning.

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.

Stochastic Optimization and Learning for Two-Stage Supplier Problems

TL;DR

The paper studies radius-based coverage in a two-stage stochastic supplier problem (2S-Sup) with recourse, accommodating both homogeneous and inhomogeneous radii and extra first-stage constraints such as matroids and multi-knapsacks. It introduces a black-box generalization framework that combines polynomial-scenarios algorithms with a scenario-discarding Sample Average Approximation (SAA) scheme to optimize the radius under a budget while ensuring high-probability coverage. Through a robust-outlier reduction and LP-rounding/iterative rounding techniques, it achieves several approximation guarantees: a 3-approximation for homogeneous 2S-Sup-Poly, 5-approximations for homogeneous 2S-MatSup-Poly and 2S-MuSup-Poly, and 9- to 11-approximation results for inhomogeneous 2S-MatSup-Poly. These results support scalable, data-driven design of coverage facilities under uncertainty, with concrete relevance to healthcare accessibility and location planning.

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 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.

Paper Structure

This paper contains 13 sections, 25 theorems, 26 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our Generalization Scheme and Comparison with Previous Results
  4. Notation and Important Subroutines
  5. Generalizing to the Black-Box Setting
  6. Overview of the Generalization Scheme
  7. Analysis of the Generalization Procedure
  8. Approximation Algorithm for Homogeneous 2S-Sup
  9. A Generic Reduction to Robust Outlier Problems
  10. Approximation Algorithm for Homogeneous 2S-MuSup and 2S-MatSup
  11. Approximation Algorithm for Inhomogeneous 2S-MatSup
  12. Acknowledgements
  13. Standard SAA Methods in the Context of Supplier Problems

Key Result

Theorem 1.1

Suppose we have an efficiently generalizable, $\eta$-approximation for the polynomial-scenarios variant of any of the problems we study. Let $\mathcal{S}$ be the set of all potential black-box solutions its extension process may produce. Then, for any $\gamma, \epsilon, \alpha \in (0,1)$ and with $O

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof : Proof (Sketch)
  • Theorem 2.7: feige
  • ...and 37 more