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On the ReLU Lagrangian Cuts for Stochastic Mixed Integer Programming

Haoyun Deng, Weijun Xie

Abstract

We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. A new family of Lagrangian cuts, termed ``ReLU Lagrangian cuts," is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition methods. We show that including ReLU Lagrangian cuts is sufficient to achieve optimality in the original stochastic mixed integer programs. Without solving the Lagrangian dual problems, we derive closed-form expressions for these cuts. Furthermore, to speed up the cut-generating procedures, we introduce linear programming-based methods to enhance the cut coefficients. Numerical studies demonstrate the effectiveness of the proposed cuts compared to existing cut families.

On the ReLU Lagrangian Cuts for Stochastic Mixed Integer Programming

Abstract

We study stochastic mixed integer programs with both first-stage and recourse decisions involving mixed integer variables. A new family of Lagrangian cuts, termed ``ReLU Lagrangian cuts," is introduced by reformulating the nonanticipativity constraints using ReLU functions. These cuts can be integrated into scenario decomposition methods. We show that including ReLU Lagrangian cuts is sufficient to achieve optimality in the original stochastic mixed integer programs. Without solving the Lagrangian dual problems, we derive closed-form expressions for these cuts. Furthermore, to speed up the cut-generating procedures, we introduce linear programming-based methods to enhance the cut coefficients. Numerical studies demonstrate the effectiveness of the proposed cuts compared to existing cut families.

Paper Structure

This paper contains 36 sections, 35 theorems, 97 equations, 5 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related literature
  3. Summary of contributions
  4. Organization.
  5. Notation.
  6. ReLU Lagrangian Cuts
  7. Preliminary: ordinary Lagrangian cuts
  8. ReLU Lagrangian cuts
  9. Related cut families
  10. Purely Binary First-stage Decisions
  11. Dual optimal solution set ${\Pi_{\widehat{\bm x}}^s}$
  12. Cut strengthening: theoretical foundation
  13. Cut strengthening -- practical implementations based on LP relaxation
  14. Implementation details
  15. General Mixed Integer First-stage Decisions
  16. ...and 21 more sections

Key Result

Proposition 1

[Theorem 3.9, fullner2024lipschitz] The primal characterization where the convex envelope of $Q_s$, denoted by $\mathrm{co}(Q_s): \mathop{\mathrm{conv}}\nolimits(\widebar{X})\rightarrow{\mathbb R}$, is defined as $\mathrm{co}(Q_s)(\bm x) = \sup\{g(\bm x): g\text{ is convex and } g(\bm z)\leq Q_s(\bm z), \forall \bm z\in\widebar{X}\}$.

Figures (5)

  • Figure 1: The illustration of Example \ref{['eg: conv_env']}
  • Figure 2: The illustration of Example \ref{['alc_vs_ReLU']}.
  • Figure 3: The illustration of Example \ref{['example_conv_hull_ReLU']}.
  • Figure 4: The illustration of Examples \ref{['eg:unbounded_stren_prob']} and \ref{['eg:strategy1']}
  • Figure 5: Comparisons of cut combinations for the MARM instances

Theorems & Definitions (85)

  • Definition 1: Lagrangian Cuts
  • Definition 2: Tight Cuts
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • proof
  • Corollary 3
  • Example 1
  • Theorem 1
  • ...and 75 more