Generalizing single-level relaxations for bilevel linear programs

Network Relaxations for Combinatorial Bilevel Optimization Under Linear Interactions

A Network View of Bilevel Optimization Many decision problems are inherently hierarchical; a leader acts first while anticipating that a follower will respond optimally. Determining the leader’s best decision is notoriously difficult in areas such as infrastructure design, transportation, and security, especially when decisions are discrete. In “Network Relaxations for Combinatorial Bilevel Optimization under Linear Interactions,” Leonardo Lozano, David Bergman, and Andre Cire introduce a new representation for problems in which leader-follower interactions are captured by linear inequalities involving binary leader decisions. Their approach uses a layered decision-diagram network whose paths encode the leader’s choices and whose terminal values represent the follower’s optimal objective value. The network reveals symmetries, supports strong flow-based formulations, and can be compressed through node aggregation to create tractable approximations. Computational tests on previously unresolved benchmark problems show substantial improvements over leading bilevel solvers, proving optimality for many open instances and pointing to promising directions for future research in the field.

A true single–level reformulation for pessimistic bilevel optimization

Multi-parametric approach for nonlinear bilevel optimization problems and beyond

One of the peculiar features of multi-parametric approach for bilevel programs is that most methods using this approach can be extended to tri-level (and generally to $k$-level) programs, which is not always the case with other non-heuristic solution methods. However, most of existing multi-parametric methods work well when the constraint of the lower-level problem is polyhedral. In this article we propose a multi-parametric programming based solution algorithm for bilevel optimization problems whose lower-level problem involves convex smooth nonlinear constraints. The method is also extended to solve some classes of $k$-level convex optimization problems with nonlinear constraints. The algorithm recasts the lower-level problem as a multi-parametric problem and employs an equivalent barrier problem reformulation. The solution obtained through multi-parametric programming is incorporated in the upper-level problem to create a set of single-level optimization problems which are solved using standard global optimization techniques. The proposed algorithm can give an exact global solution to some class of nonlinear Multi-level problems with convex nonlinear constraints.

Adaptive Scenario Partitioning for Stochastic Bilevel Linear Programs

Exact and Heuristic Methods for $\Gamma$-Robust Min-Max Problems

Bilevel optimization is a powerful tool for modeling hierarchical decision-making processes, which arise in various real-world applications. Due to their nested structure, however, bilevel problems are intrinsically hard to solve, even if all variables are continuous and all parameters of the problem are exactly known. Further challenges arise if mixed-integer aspects and problems under uncertainty are considered. In this article, we summarize selected results from the author's dissertation. We study mixed-integer linear min-max problems with a $\Gamma$-robust treatment of uncertain data, for which we present exact and heuristic solution approaches. The performance of the methods is assessed in a computational study on 560 instances of the knapsack interdiction problem. Our results show that the heuristic closes the optimality gap for a significant portion of the considered instances and often practically outperforms both heuristic and exact benchmark approaches.

Computing diverse solutions to optimization problems