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

Sergey S. Ketkov, Oleg A. Prokopyev

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.

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- parameters. While sufficiently large and bilevel-correct big-s can be computed in polynomial time, verifying a priori that given big-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- parameter, it is -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- correctness remains computationally difficult a posteriori, even when an optimal solution of the MILP model is available.
Paper Structure (8 sections, 13 theorems, 38 equations)

This paper contains 8 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. KKT-based reformulation
  3. Strong duality-based reformulation
  4. Analysis of min-max case
  5. Global bilevel correctness of big-$M$
  6. Valid bounds for dual feasible solutions
  7. Valid bounds for bilevel feasible solutions
  8. Conclusion

Key Result

Theorem 1

The optimality verification problem [BLP-D] is $coNP$-complete even when there is only one global big-$M$ parameter, i.e., $M^p = M^d$.

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2
  • Theorem 3
  • ...and 15 more