On Big-M Reformulations of Bilevel Linear Programs: Hardness of A Posteriori Verification

Abstract

A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary binary variables. This yields a single-level mixed-integer linear programming (MILP) model involving big-$M$ parameters. While sufficiently large and bilevel-correct big-$M$s can be computed in polynomial time, verifying a priori that given big-$M$s do not cut off any feasible or optimal lower-level solutions is known to be computationally difficult. In this paper, we establish two complementary hardness results. First, we show that, even with a single potentially incorrect big-$M$ parameter, it is $coNP$-complete to verify a posteriori whether the optimal solution of the resulting MILP model is bilevel optimal. In particular, this negative result persists for min-max problems without coupling constraints and applies to strong-duality-based reformulations of mixed-integer BLPs. Second, we show that verifying global big-$M$ correctness remains computationally difficult a posteriori, even when an optimal solution of the MILP model is available.