Geometric and computational hardness of bilevel programming

2024-07-17

arXiv.org

AI Summary

Bilevel optimization suffers from inherent geometric discontinuity and computational intractability, yet its fundamental hardness remains poorly characterized. Method: Integrating semialgebraic geometry, extended-real analysis, polynomial optimization, and polynomial hierarchy complexity theory, the authors conduct a rigorous structural analysis of unconstrained smooth and box-constrained polynomial bilevel problems. Contribution/Results: The work establishes two landmark hardness results: (1) Unconstrained smooth bilevel optimization is equivalent to general extended-real-valued lower-semicontinuous optimization—capable of realizing arbitrarily discontinuous objective functions; (2) Box-constrained polynomial bilevel optimization can encode any semialgebraic function, and its decision version is Σ²ₚ-complete—strictly harder than NP. These findings demonstrate that bilevel programming is intrinsically more complex than standard nonlinear programming along both geometric and computational dimensions. Crucially, the results justify the theoretical necessity of regularity assumptions (e.g., uniqueness of lower-level solutions, continuity of the value function) and delineate fundamental limits for algorithm design.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We first show a simple but striking result in bilevel optimization: unconstrained $C^infty$ smooth bilevel programming is as hard as general extended-real-valued lower semicontinuous minimization. We then proceed to a worst-case analysis of box-constrained bilevel polynomial optimization. We show in particular that any extended-real-valued semi-algebraic function, possibly non-continuous, can be expressed as the value function of a polynomial bilevel program. Secondly, from a computational complexity perspective, the decision version of polynomial bilevel programming is one level above NP in the polynomial hierarchy ($Sigma^p_2$-hard). Both types of difficulties are uncommon in non-linear programs for which objective functions are typically continuous and belong to the class NP. These results highlight the irremediable hardness attached to general bilevel optimization and the necessity of imposing some form of regularity on the lower level.

Problem

• Bilevel Optimization
• Discontinuous Functions
• NP-Hard Problems

Innovation

• Bilevel Optimization
• Complexity Analysis
• Constraint Refinement