Abstract
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.
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References
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)
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The authors are grateful to the anonymous referees for making several points which significantly improved the presentation.
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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.
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Fampa, M., Lee, J. Convexification of bilinear forms through non-symmetric lifting. J Glob Optim 80, 287–305 (2021). https://doi.org/10.1007/s10898-020-00975-z
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DOI: https://doi.org/10.1007/s10898-020-00975-z