We study a class of bilevel spanning tree (BST) problems that involve two independent decision‐makers (DMs), the leader and the follower with different objectives, who jointly construct a spanning tree in a graph. The leader, who acts first, selects an initial subset of edges that do not contain a cycle, from the set under her control. The follower then selects the remaining edges to complete the construction of a spanning tree, but optimizes his own objective function. If there exist multiple optimal solutions for the follower that result in different objective function values for the leader, then the follower may choose either the one that is the most (optimistic version) or least (pessimistic version) favorable to the leader. We study BST problems with the sum‐ and bottleneck‐type objective functions for the DMs under both the optimistic and pessimistic settings. The polynomial‐time algorithms are then proposed in both optimistic and pessimistic settings for BST problems in which at least one of the DMs has the bottleneck‐type objective function. For BST problem with the sum‐type objective functions for both the leader and the follower, we provide an equivalent single‐level linear mixed‐integer programming formulation. A computational study is then presented to explore the efficacy of our reformulation.
On Bilevel Optimization with Inexact Follower
Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.
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- Award ID(s):
- 1634835
- NSF-PAR ID:
- 10206044
- Date Published:
- Journal Name:
- Decision Analysis
- Volume:
- 17
- Issue:
- 1
- ISSN:
- 1545-8490
- Page Range / eLocation ID:
- 74 to 95
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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This work develops analysis and algorithms for solving a class of bilevel optimization problems where the lower-level (LL) problems have linear constraints. Most of the existing approaches for constrained bilevel problems rely on value function-based approximate reformulations, which suffer from issues such as non-convex and non-differentiable constraints. In contrast, in this work, we develop an implicit gradient-based approach, which is easy to implement, and is suitable for machine learning applications. We first provide an in-depth understanding of the problem, by showing that the implicit objective for such problems is in general non-differentiable. However, if we add some small (linear) perturbation to the LL objective, the resulting implicit objective becomes differentiable almost surely. This key observation opens the door for developing (deterministic and stochastic) gradient-based algorithms similar to the state-of-the-art ones for unconstrained bi-level problems. We show that when the implicit function is assumed to be stronglyconvex, convex, and weakly-convex, the resulting algorithms converge with guaranteed rate. Finally, we experimentally corroborate the theoretical findings and evaluate the performance of the proposed framework on numerical and adversarial learning problems.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1287/deca.2019.0392
