Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

Thomas Kleinert,Martine Labbé,Fränk Plein,Martin Schmidt

doi:10.1007/s11590-020-01660-6

Abstract

Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch-and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.

Highlights

Speaking, branch-and-bound algorithms solve mathematical optimization problems by successively finding lower and upper bounds on the optimal objective function value
Every primal feasible solution provides a valid upper bound on the objective function value
In mixed-integer programming, the discussed obstacle has been tackled by subsequently adding valid inequalities that cut off integer-infeasible points

Summary

A new valid primal-dual inequality

All cuts reviewed in the last section have in common that they exploit the explicit disjunctive structure of the complementarity conditions. Using dual feasibility (3c), we can substitute λ D with f in (3d) to obtain λ b − λ C x − f y ≤ 0 This is exactly the strong-duality condition of the lower-level problem (2), as shown in the following. Even because of its simplicity, this inequality can be very useful It explicitly couples the primal lowerlevel variable y to the dual lower-level variable λ—a coupling that is missing in the root node problem of branch-and-bound approaches. Whenever a (maybe locally valid) bound for λi is available by chance, e.g., due to a combination of branching and node presolve, the overestimator (9c) can be used to potentially tighten the valid inequality (6) In this light, the derivation via McCormick envelopes (8) may provide tighter versions of Inequality (6). In order to streamline the presentation, we will stick to the discussion of the optimistic case

Computational study

Findings

Conclusion