A Globally Convergent Method for Computing B-stationary Points of Mathematical Programs with Equilibrium Constraints

This paper introduces a computationally efficient method that converges globally to B-stationary points of mathematical programs with equilibrium constraints (MPECs). B-stationarity is necessary for optimality and means that no feasible first-order direction can improve the objective. It can be certified by solving a linear program with equilibrium constraints (LPEC) constructed at a given feasible point. The proposed method solves a finite sequence of LPECs, which either certify B-stationarity or provide an active set estimate for the complementarity constraints, along with branch nonlinear programs (BNLPs) obtained by fixing the complementarity active set in the MPEC. In particular, the method proceeds in two phases: the first identifies a feasible BNLP or a stationary point of a constraint infeasibility minimization problem, and the second solves a sequence of BNLPs until a B-stationary point of the MPEC is found. We prove that under the MPEC-MFCQ, the method requires solving only a finite number of BNLPs and LPECs for convergence. Moreover, we show that, unless the current iterate is B-stationary, the combinatorial LPECs need not be solved to optimality. For convergence, it suffices to compute a nonzero feasible point, which in practice often requires solving a single linear program, yielding significant computational savings. Numerical experiments show that the proposed method is more robust and faster than relaxation-based methods and mixed-integer NLP reformulations (which, in contrast to the proposed approach, do not provide a certificate of B-stationarity), even on medium- to large-scale instances.

💡 Research Summary

This paper presents a novel algorithm, called MPECopt, that globally converges to B‑stationary points of mathematical programs with equilibrium constraints (MPECs). B‑stationarity is a strong first‑order optimality condition requiring that no feasible descent direction exists; it can be verified by solving a linear program with equilibrium constraints (LPEC) at a given feasible point. Traditional NLP reformulations of MPECs suffer from degeneracy because the Mangasarian‑Fromovitz constraint qualification fails everywhere, making standard KKT‑based methods ineffective. Existing approaches fall into two categories: regularization/penalty methods that smooth or relax the complementarity constraints, and active‑set/combinatorial methods that fix an active set of complementarity constraints and solve the resulting branch NLPs (BNLPs). The latter often require the restrictive MPEC‑LICQ and lack a certificate of B‑stationarity.

MPECopt combines the strengths of both families in a two‑phase hybrid scheme. Phase I first obtains a feasible BNLP by solving relaxed MPECs (e.g., Scholtes’ global relaxation) and then constructs an LPEC at the current point. Rather than solving the LPEC to optimality, it suffices to find any non‑zero feasible direction; this direction both reduces the objective and suggests a new active‑set partition for the complementarity constraints. Fixing this partition yields a BNLP that is a standard NLP and can be solved with any off‑the‑shelf NLP solver. Phase I therefore guarantees a feasible BNLP (or a stationary point of an infeasibility‑minimization problem) after a finite number of such steps.

Phase II starts from the feasible BNLP obtained in Phase I and repeatedly solves LPECs at the BNLP’s stationary points. If the LPEC returns the zero vector as its minimizer, the current point is certified B‑stationary and the algorithm terminates. If not, the LPEC provides a non‑zero feasible descent direction and a revised active‑set guess, leading to a new BNLP with a strictly lower objective value. Crucially, the LPECs need not be solved to global optimality except when certifying B‑stationarity; a feasible point (often obtainable by solving a single linear program) is enough to guarantee progress.

The convergence analysis assumes the MPEC‑MFCQ (a mild constraint qualification). Under this condition, the algorithm requires solving only a finite number of LPECs and BNLPs. The authors prove that each time a non‑zero feasible LPEC solution is found, the objective strictly decreases, and because there are only finitely many possible active‑set partitions (2^{|I_{00}|}), the process must terminate at a B‑stationary point. Moreover, the method does not rely on MPEC‑LICQ, allowing it to converge to B‑stationary points that are not S‑stationary.

From a computational standpoint, the authors discuss two practical ways to solve LPECs. One is to reformulate the LPEC as a mixed‑integer linear program (MILP) and use commercial MILP solvers; the other exploits the structure of LPECs that involve only bound and complementarity constraints, which can be solved in linear time. In their implementation, they typically solve a single LP (or a small MILP) per LPEC, dramatically reducing overhead.

Extensive numerical experiments are conducted on two benchmark suites: the MacMPEC collection and a synthetic large‑scale nonlinear MPEC set. MPECopt is compared against several state‑of‑the‑art relaxation‑based methods and a mixed‑integer nonlinear programming (MINLP) reformulation. Results show that MPECopt achieves higher success rates, lower average solution times, and, importantly, provides a certificate of B‑stationarity—something the competing methods lack. Even for medium‑ to large‑scale instances, solving the LPECs never becomes a bottleneck; most instances require only one LP solve per phase.

The paper’s contributions are fourfold: (1) a globally convergent algorithm that directly targets B‑stationarity, (2) a theoretical justification that LPECs need not be solved to optimality except for the final certification step, (3) an efficient practical scheme for active‑set estimation via inexpensive LPEC solves, and (4) an open‑source implementation (available at github.com/nosnoc/mpecopt) that facilitates reproducibility. By delivering both robustness and computational efficiency, MPECopt advances the state of the art in solving MPECs, especially in applications where a reliable certificate of optimality is essential.