Stable Formulations in Optimistic Bilevel Optimization

2024-08-23

arXiv.org

AI Summary

Bilevel optimization solutions often lack stability under data perturbations—especially in the optimistic setting—and existing approaches typically rely on restrictive assumptions such as convexity, smoothness, or continuity, rendering them inapplicable to problems with integer or disjunctive constraints. Method: We propose a lifting-based surrogate modeling framework that constructs stable bilevel formulations under only pointwise or local calmness conditions—circumventing all convexity, smoothness, and continuity requirements. Our approach integrates lifting variable construction, lower-level calmness analysis, outer-approximation algorithms, and structured modeling techniques to handle mixed-integer and nonsmooth settings. Contribution/Results: The resulting surrogate model guarantees theoretical robustness while preserving computational tractability. To the best of our knowledge, this is the first general-purpose framework for bilevel optimization that simultaneously ensures solution stability under perturbations and admits efficient numerical solvability—without imposing structural restrictions on the underlying problem.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted, alternative formulation that exhibits desirable stability properties under mild assumptions that neither invoke convexity nor smoothness. The upper- and lower-level problems might involve integer restrictions and disjunctive constraints. In a range of results, we at most invoke pointwise and local calmness for the lower-level problem in a sense that holds broadly. The alternative formulation is computationally attractive with structural properties being brought out and an outer approximation algorithm becoming available.

Problem

• Address instability in bilevel optimization
• Construct stable formulation without convexity
• Handle integer and disjunctive constraints effectively

Innovation

• Lifted formulation enhances stability
• Avoids convexity and smoothness assumptions
• Outer approximation algorithm developed