Network Relaxations for Discrete Bilevel Optimization under Linear Interactions
We investigate relaxations for a class of discrete bilevel programs where the interaction constraints linking the leader and the follower are linear. Our approach reformulates the upper-level optimality constraints by projecting the leader's decisions onto vectors that map to distinct follower...
Saved in:
Main Authors | , , |
---|---|
Format | Journal Article |
Language | English |
Published |
25.07.2024
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | We investigate relaxations for a class of discrete bilevel programs where the
interaction constraints linking the leader and the follower are linear. Our
approach reformulates the upper-level optimality constraints by projecting the
leader's decisions onto vectors that map to distinct follower solution values,
each referred to as a state. Based on such a state representation, we develop a
network-flow linear program via a decision diagram that captures the convex
hull of the follower's value function graph, leading to a new single-level
reformulation of the bilevel problem. We also present a reduction procedure
that exploits symmetry to identify the reformulation of minimal size. For large
networks, we introduce parameterized relaxations that aggregate states by
considering tractable hyperrectangles based on lower and upper bounds
associated with the interaction constraints, and can be integrated into
existing mixed-integer bilevel linear programming (MIBLP) solvers. Numerical
experiments suggest that the new relaxations, whether used within a simple
cutting-plane procedure or integrated into state-of-the-art MIBLP solvers,
significantly reduce runtimes or solve additional benchmark instances. Our
findings also highlight the correlation between the quality of relaxations and
the properties of the interaction matrix, underscoring the potential of our
approach in enhancing solution methods for structured bilevel optimization
instances. |
---|---|
DOI: | 10.48550/arxiv.2407.18193 |