We efficiently treat bilinear forms in the context of global optimization, by applying McCormick convexification and by extending an approach of Saxena et al. (Math Prog Ser B 124(1–2):383–411, 2010) for symmetric quadratic forms to bilinear forms. A key application of our work is in treating “structural convexity” in a symmetric quadratic form.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Al-Khayyal, A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)
Boland, N., Dey, S.S., Kalinowski, T., Molinaro, M., Rigterink, F.: Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions. Math. Program. Ser. A 162, 523–535 (2017)
Brimberg, J., Hansen, P., Mladenović, N.: A note on reduction of quadratic and bilinear programs with equality constraints. J. Global Optim. 22(1–4), 39–47 (2002)
Caprara, A., Locatelli, M., Monaci, M.: Bidimensional packing by bilinear programming. In: Integer Programming and Combinatorial Optimization. Volume 3509 of Lecture Notes in Computer Science, pp. 377–391. Springer, Berlin (2005)
Castro, P.M., Grossmann, I.E.: Optimality-based bound contraction with multiparametric disaggregation for the global optimization of mixed-integer bilinear problems. J. Global Optim. 59(2–3), 277–306 (2014)
Dey, Santanu S., Santana, Asteroide, Wang, Yang: New SOCP relaxation and branching rule for bipartite bilinear programs. Optim. Eng. 20, 307–336 (2019)
Fampa, M., Lee, J., Melo, W.: On global optimization with indefinite quadratics. EURO J. Comput. Optim. 5(3), 309–337 (2017)
Fuentes, V.K., Fampa, M., Lee, J.: Sparse pseudoinverses via LP and SDP relaxations of Moore–Penrose. In: Maturana, S. (ed.) Proceedings of the XVIII Latin-Iberoamerican Conference on Operations Research (CLAIO 2016), pp. 342–350. Instituto Chileno de Investigación Operativa (ICHIO) (2016)
Günlük, O., Lee, J., Leung, J.: A polytope for a product of real linear functions in 0/1 variables. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming Volume 154 of IMA Journal of Applied Mathematics, pp. 513–529. Springer, New York (2012)
Gupte, A., Ahmed, S., Seok Cheon, M., Dey, S.: Solving mixed integer bilinear problems using MILP formulations. SIAM J. Optim. 23(2), 721–744 (2013)
Kolodziej, S., Castro, P.M., Grossmann, I.E.: Global optimization of bilinear programs with a multiparametric disaggregation technique. J. Glob. Optim. 57(4), 1039–1063 (2013)
Locatelli, M.: Polyhedral subdivisions and functional forms for the convex envelopes of bilinear, fractional and other bivariate functions over general polytopes. J. Glob. Optim. 66(4), 629–668 (2016)
Nahapetyan, A., Pardalos, P.M.: A bilinear relaxation based algorithm for concave piecewise linear network flow problems. J. Ind. Manag. Optim. 3(1), 71–85 (2007)
Rebennack, S., Nahapetyan, A., Pardalos, P.M.: Bilinear modeling solution approach for fixed charge network flow problems. Optim. Lett. 3(3), 347–355 (2009)
Ruiz, J.P., Grossmann, I.E.: Exploiting vector space properties to strengthen the relaxation of bilinear programs arising in the global optimization of process networks. Optim. Lett. 5(1), 1–11 (2011)
Saxena, A., Bonami, P., Lee, J.: Disjunctive cuts for non-convex mixed integer quadratically constrained programs. In: Integer Programming and Combinatorial Optimization, 13th International Conference, IPCO 2008, Bertinoro, Italy, May 26–28, 2008, Proceedings, pp. 17–33 (2008)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. Ser. B 124, 383–411 (2010)
Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. Ser. B 130, 359–413 (2010)
Xia, Wei, Vera, Juan C., Zuluaga, Luis F.: Globally solving nonconvex quadratic programs via linear integer programming techniques. INFORMS J. Comput. 32(1), 40–56 (2020)
Zorn, K., Sahinidis, N.V.: Global optimization of general non-convex problems with intermediate bilinear substructures. Optim. Methods Softw. 29(3), 442–462 (2014)
The authors are grateful to the anonymous referees for making several points which significantly improved the presentation.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
M. Fampa was supported in part by CNPq Grant 303898/2016-0. J. Lee was supported in part by ONR Grant N00014-17-1-2296. Additionally, part of this work was done while J. Lee was visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant #CCF-1740425.
About this article
Cite this article
Fampa, M., Lee, J. Convexification of bilinear forms through non-symmetric lifting. J Glob Optim (2021). https://doi.org/10.1007/s10898-020-00975-z
- Global optimization