Robust Bilevel Optimization for Near-Optimal Lower-Level Solutions

Abstract

Bilevel optimization studies problems where the optimal response to a second mathematical optimization problem is integrated in the constraints. Such structure arises in a variety of decision-making problems in areas such as market equilibria, policy design or product pricing. We introduce near-optimal robustness for bilevel problems, protecting the upper-level decision-maker from bounded rationality at the lower level and show it is a restriction of the corresponding pessimistic bilevel problem. Essential properties are derived in generic and specific settings. This model finds a corresponding and intuitive interpretation in various situations cast as bilevel optimization problems. We develop a duality-based solution method for cases where the lower level is convex, leveraging the methodology from robust and bilevel literature. The models obtained are tested numerically using different solvers and formulations, showing the successful implementation of the near-optimal robust bilevel problem.

Complexity of near-optimal robust versions of multilevel optimization problems

“…the set of near-optimal lower-level solutions, depending on both the upper-level decision x and δ. (NORBiP) is a generalization of the pessimistic bilevel problem since the latter is both a special case and a relaxation of (NORBiP) [2]. We refer to (BiP) as the canonical problem for (NORBiP) (or equivalently Problem (4)) and (NORBiP) as the near-optimal robust version of (BiP).…”

“…In the formulation of (NORBiP), the upper-level objective depends on decision variables of both levels, but is not protected against near-optimal deviations. A more conservative formulation also protecting the objective by moving it to the constraints in an epigraph formulation [2] is given by: (NORBiP-Alt): min…”

“…A variant is also defined, where the lower level may pick a suboptimal response only impacting the upper-level objective. This variant is comparable to the Objective-Robust Near-Optimal Bilevel Problem defined in [2]. In [4], the independent case of the pessimistic bilevel problem is studied, corresponding to a special case of (NORBiP) with δ = 0 and all lower-level constraints independent of the upper-level variables.…”

“…The adversarial problem can be split into m u adversarial problems as done in [2], each finding the worst-case with respect to one of the upper-level constraints. The canonical problem refers to the optimistic bilevel problem without near-optimal robustness constraints.…”

“…Near-optimal robustness, defined in [2], is an extension of bilevel optimization. In this setting, the upper level anticipates limited deviations of the lower level from an optimal solution and aims at a solution that remains feasible for any feasible and near-optimal solution of the lower level.…”