Abstract
Nonconvex mixed-binary nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2instances, where our methods are able to produce feasible solutions for \({60}{\%}\) of all instances in at most \({10}\,{\hbox {s}}\).
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Notes
For CONOPT4, the failed instances are crudeoil_lee1_08, crudeoil_lee2_05, crudeoil_lee2_06, crudeoil_lee3_05, crudeoil_lee3_06, crudeoil_lee4_10, nuclear25a, telecomsp_njlata, telecomsp_pacbell for the reformulation scheme of Scholtes (9); crudeoil_lee1_08, crudeoil_lee3_06, crudeoil_lee3_10, gasprod_sarawak81, nuclear25a, sepasequ_convent, telecomsp_njlata, telecomsp_pacbell for the Fischer–Burmeister reformulation scheme (8) and crudeoil_lee1_06, crudeoil_lee1_09, crudeoil_lee2_06, crudeoil_lee2_07, crudeoil_lee2_08, crudeoil_lee2_09, crudeoil_lee2_10, crudeoil_lee3_06, crudeoil_lee3_08, crudeoil_lee4_05, crudeoil_lee4_06, crudeoil_lee4_07, crudeoil_lee4_08, nuclear49a, nuclear49b, squfl025-040persp, telecomsp_njlata, telecomsp_pacbell for the penalty-based reformulation (18). For Ipopt, the failed instances are faclay60, faclay70, faclay80 for the reformulation scheme of Scholtes (3), whereas it fails on faclay75 for the Fischer–Burmeister reformulation and on faclay33 and for the penalty-based reformulation.
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Acknowledgements
This research has been performed as part of the Energie Campus Nürnberg and supported by funding through the “Aufbruch Bayern (Bavaria on the move)” initiative of the state of Bavaria. Both authors acknowledge funding through the DFG Transregio TRR 154, subprojects A05, B07, and B08. Last but not least, we want to express our sincere gratefulness to Stefan Vigerske from GAMS. Without his patient help, the implementations underlying this paper would not have been possible. Thanks a lot, Stefan.
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Schewe, L., Schmidt, M. Computing feasible points for binary MINLPs with MPECs. Math. Prog. Comp. 11, 95–118 (2019). https://doi.org/10.1007/s12532-018-0141-x
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DOI: https://doi.org/10.1007/s12532-018-0141-x