{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-globally-convergent-method-for-computing-b","title":"A Globally Convergent Method for Computing B-stationary Points of Mathematical Programs with Equilibrium Constraints","arxiv_id":"2501.13835","date":"2025-01-23","proceeding":null,"authors":["Armin Nurkanović","Sven Leyffer"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2501.13835v1","url_pdf":"https://arxiv.org/pdf/2501.13835v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-globally-convergent-method-for-computing-b","repo_url":"https://github.com/nosnoc/mpecopt","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}