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L2O-CCG: Adversarial Learning with Set Generalization for Adaptive Robust Optimization

Zhiyi Zhou, Ján Drgoňa, Yury Dvorkin

Abstract

The adversarial subproblem in two-stage adaptive robust optimization (ARO), which identifies the worst-case uncertainty realization, is a major computational bottleneck. This difficulty is exacerbated when the recourse value function is non-concave and the uncertainty set shifts across applications. Existing approaches typically exploit specific structural assumptions on the value function or the uncertainty set geometry to reformulate this subproblem, but degrade when these assumptions are violated or the geometry changes at deployment. To address this challenge, we propose L2O-CCG, a bi-level framework that enables the integration of structure-aware adversarial solvers within the constraint-and-column generation (CCG) algorithm. As one instantiation, we develop a generalizable adversarial learning method, which replaces solver-based adversarial search with a learned proximal gradient optimizer that can generalize across uncertainty set geometries without retraining. Here, an inner-level neural network approximates the recourse value function from offline data, while an outer-level pre-trained mapping generates iteration-dependent step sizes for a proximal gradient scheme. We also establish out-of-distribution convergence bounds under uncertainty set parameter shifts, showing how the trajectory deviation of the learned optimizer is bounded by the uncertainty set shift. We illustrate performance of the L2O-CCG method on a building HVAC management task.

L2O-CCG: Adversarial Learning with Set Generalization for Adaptive Robust Optimization

Abstract

The adversarial subproblem in two-stage adaptive robust optimization (ARO), which identifies the worst-case uncertainty realization, is a major computational bottleneck. This difficulty is exacerbated when the recourse value function is non-concave and the uncertainty set shifts across applications. Existing approaches typically exploit specific structural assumptions on the value function or the uncertainty set geometry to reformulate this subproblem, but degrade when these assumptions are violated or the geometry changes at deployment. To address this challenge, we propose L2O-CCG, a bi-level framework that enables the integration of structure-aware adversarial solvers within the constraint-and-column generation (CCG) algorithm. As one instantiation, we develop a generalizable adversarial learning method, which replaces solver-based adversarial search with a learned proximal gradient optimizer that can generalize across uncertainty set geometries without retraining. Here, an inner-level neural network approximates the recourse value function from offline data, while an outer-level pre-trained mapping generates iteration-dependent step sizes for a proximal gradient scheme. We also establish out-of-distribution convergence bounds under uncertainty set parameter shifts, showing how the trajectory deviation of the learned optimizer is bounded by the uncertainty set shift. We illustrate performance of the L2O-CCG method on a building HVAC management task.
Paper Structure (13 sections, 4 theorems, 28 equations, 3 figures, 5 tables, 1 algorithm)

This paper contains 13 sections, 4 theorems, 28 equations, 3 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed L2O-CCG Framework
  4. Bi-Level Learning Structure
  5. Adversarial Learning
  6. Inner-Level Surrogate
  7. Outer-Level Optimizer
  8. Formal OOD Guarantees for Uncertainty Sets
  9. Experiments
  10. Problem Settings
  11. In-Distribution Results
  12. Out-of-Distribution Results
  13. Conclusions

Key Result

Proposition 1

For the composite $F(\xi) = f(\xi) + r(\xi)$ with piecewise $L$-smooth $f(\cdot)$, the optimal L2O configuration is: Under these optimal settings, when crossing from piece $\mathcal{P}_i$ to $\mathcal{P}_j$ at boundary $\xi_b$, define gradient jump as $\Delta \xi_b \triangleq ||\nabla f_j(\xi_b) - \nabla f_i(\xi_b)||$. The per-iteration bound is given as: which can be accumulated across multiple

Figures (3)

  • Figure 1: Diagram for the proposed L2O-CCG
  • Figure 2: Relative error (compared with benchmark CCG) across out-of-distribution scenarios for different uncertainty sets. We evaluate four different uncertainty sets: box, polyhedral, elliposid, and GMM. L2O-CCG consistently achieves comparable or lower error than Neur2RO across all settings, while maintaining tighter distributions and fewer extreme deviations. The results indicate strong generalization performance of the proposed method under distributional shifts.
  • Figure 3: Computation time across out-of-distribution scenarios for different uncertainty sets. We evaluate four different uncertainty sets: box, polyhedral, elliposid, and GMM. L2O-CCG achieves orders-of-magnitude speedup compared to CCG and outperforms Neur2RO in most cases.

Theorems & Definitions (6)

  • Definition 1: InD Domain and OOD Domain
  • Proposition 1: InD Convergence with Boundary Transition song2024towards
  • Definition 2: Virtual Feature
  • Theorem 1: Per-Iteration OOD Descent Bound
  • Corollary 1: Subgradient Difference
  • Theorem 2: Multi-Iteration OOD Rate