🤖 AI Summary
This study investigates whether solutions obtained by reformulating bilevel linear programs into single-level mixed-integer linear programs (MILPs) via the KKT conditions and the Big-M method retain bilevel optimality. We establish, for the first time, that verifying the bilevel optimality of an MILP solution is coNP-complete if even a single Big-M parameter is improperly chosen. Moreover, we show that confirming the global correctness of all Big-M values remains computationally intractable, even when an optimal MILP solution is given. Through complexity-theoretic analysis, we derive two complementary computational lower bounds, demonstrating the inherent intractability of this verification problem. These results apply broadly to uncoupled min-max problems and integer bilevel programs reformulated using strong duality.
📝 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.