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Bi-Parameterized Two-Stage Stochastic Min-Max and Min-Min Mixed Integer Programs

Sumin Kang, Manish Bansal

TL;DR

This work introduces bi-parameterized two-stage stochastic programs (BTSPs) in which the first-stage decisions influence both the second-stage objective and constraints. The authors develop an exact Lagrangian-integrated L-shaped ($L^2$) framework for min-max and min-min BTSPs, with a regularization-augmented variant for mixed-binary first-stage decisions. They prove strong duality under suitable conditions and provide exact reformulations that enable efficient decomposition. Computational experiments on stochastic network interdiction, facility location, and distributionally robust optimization with decision-dependent ambiguity sets demonstrate that the $L^2$ method substantially outperforms benchmarks, solving challenging instances in minutes rather than hours or days. The approach significantly broadens the tractability frontier for BTSPs with decision-dependent uncertainty and offers practical tools for interdiction and robust optimization problems.

Abstract

We introduce two-stage stochastic min-max and min-min integer programs with bi-parameterized recourse (BTSPs), where the first-stage decisions affect both the objective function and the feasible region of the second-stage problem. To solve these programs efficiently, we introduce Lagrangian-integrated $L$-shaped ($L^2$) methods, which guarantee exact solutions when the first-stage decisions are pure binary. For mixed-binary first-stage programs, we present a regularization-augmented variant of this method. Our computational results for a stochastic network interdiction problem show that the $L^2$ method outperforms a benchmark method, solving all instances in 23 seconds on average, while the benchmark method failed to solve any instance within 3600 seconds. The $L^2$ method also achieves optimal solutions, on average, 18.4 times faster for a stochastic facility location problem. Furthermore, we show that the $L^2$ method can effectively address distributionally robust optimization problems with decision-dependent ambiguity sets that may be empty for some first-stage decisions, achieving optimal solutions, on average, 5.3 times faster than existing methods.

Bi-Parameterized Two-Stage Stochastic Min-Max and Min-Min Mixed Integer Programs

TL;DR

This work introduces bi-parameterized two-stage stochastic programs (BTSPs) in which the first-stage decisions influence both the second-stage objective and constraints. The authors develop an exact Lagrangian-integrated L-shaped () framework for min-max and min-min BTSPs, with a regularization-augmented variant for mixed-binary first-stage decisions. They prove strong duality under suitable conditions and provide exact reformulations that enable efficient decomposition. Computational experiments on stochastic network interdiction, facility location, and distributionally robust optimization with decision-dependent ambiguity sets demonstrate that the method substantially outperforms benchmarks, solving challenging instances in minutes rather than hours or days. The approach significantly broadens the tractability frontier for BTSPs with decision-dependent uncertainty and offers practical tools for interdiction and robust optimization problems.

Abstract

We introduce two-stage stochastic min-max and min-min integer programs with bi-parameterized recourse (BTSPs), where the first-stage decisions affect both the objective function and the feasible region of the second-stage problem. To solve these programs efficiently, we introduce Lagrangian-integrated -shaped () methods, which guarantee exact solutions when the first-stage decisions are pure binary. For mixed-binary first-stage programs, we present a regularization-augmented variant of this method. Our computational results for a stochastic network interdiction problem show that the method outperforms a benchmark method, solving all instances in 23 seconds on average, while the benchmark method failed to solve any instance within 3600 seconds. The method also achieves optimal solutions, on average, 18.4 times faster for a stochastic facility location problem. Furthermore, we show that the method can effectively address distributionally robust optimization problems with decision-dependent ambiguity sets that may be empty for some first-stage decisions, achieving optimal solutions, on average, 5.3 times faster than existing methods.
Paper Structure (25 sections, 5 theorems, 49 equations, 1 figure, 4 tables, 3 algorithms)

This paper contains 25 sections, 5 theorems, 49 equations, 1 figure, 4 tables, 3 algorithms.

Table of Contents

  1. Key words:
  2. Introduction
  3. Applications of Bi-Parameterized Stochastic Programs
  4. Contributions and Organization of this Paper
  5. Notation:
  6. An Exact Algorithm for the Min-Max BTSP
  7. A Lagrangian Dual Formulation
  8. The Lagrangian-Integrated L-Shaped Method
  9. An Exact Algorithm for the Min-Min BTSP
  10. A Regularization-Augmented Algorithm for BTSPs
  11. Computational Results
  12. Bi-Parameterized (Min-Min) Facility Location Problem
  13. Bi-Parameterized (Min-Max) Network Interdiction Problem
  14. Distributionally Robust Facility Location with Decision-Dependent Ambiguity Set
  15. Conclusion
  16. ...and 10 more sections

Key Result

Theorem 1

Under assumption:binary_x_component_minmax, $Q(x, \omega) = D(x, \omega)$ for any $\omega\in\Omega$ and $x \in X$, i.e., strong duality holds for the Lagrangian dual eq:dual_Q_om.

Figures (1)

  • Figure 1: An example of $\mathbb{E}[Q(x, \omega)]$.

Theorems & Definitions (13)

  • Example 1
  • Theorem 1
  • Lemma 2
  • Theorem 3
  • Remark 1
  • Proposition 4
  • Remark 2
  • Proposition 5
  • proof
  • proof
  • ...and 3 more