Abstract
It is well known that the convex hull of \(\{{(x,y,xy)}\}\), where (x, y) is constrained to lie in a box, is given by the reformulation-linearization technique (RLT) constraints. Belotti et al. (Electron Notes Discrete Math 36:805–812, 2010) and Miller et al. (SIAG/OPT Views News 22(1):1–8, 2011) showed that if there are additional upper and/or lower bounds on the product \(z=xy\), then the convex hull can be represented by adding an infinite family of inequalities, requiring a separation algorithm to implement. Nguyen et al. (Math Progr 169(2):377–415, 2018) derived convex hulls for \(\{(x,y,z)\}\) with bounds on \(z=xy^b\), \(b\ge 1\). We focus on the case where \(b=1\) and show that the convex hull with either an upper bound or lower bound on the product is given by RLT constraints, the bound on z and a single second-order cone (SOC) constraint. With both upper and lower bounds on the product, the convex hull can be represented using no more than three SOC constraints, each applicable on a subset of (x, y) values. In addition to the convex hull characterizations, volumes of the convex hulls with either an upper or lower bound on z are calculated and compared to the relaxation that imposes only the RLT constraints. As an application of these volume results, we show how spatial branching can be applied to the product variable so as to minimize the sum of the volumes for the two resulting subproblems.
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References
Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273–286 (1983)
Alfaki, M., Haugland, D.: Strong formulations for the pooling problem. J. Glob. Optim. 56, 897–916 (2013)
Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43(2–3), 471–484 (2009)
Anstreicher, K.M., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Progr. 124(1–2), 33–43 (2010)
Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex minlp. Optim. Methods Softw. 24(4–5), 597–634 (2009)
Belotti, P., Miller, A.J., Namazifar, M.: Valid inequalities and convex hulls for multilinear functions. Electron. Notes Discrete Math. 36, 805–812 (2010)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)
Dey, S.S., Santana, A., Wang, Y.: New SOCP relaxation and branching rule for bipartite bilinear programs. Optim. Eng. 20(2), 307–336 (2019)
Jach, M., Michaels, D., Weismantel, R.: The convex envelope of (\(n-1\))-convex functions. SIAM. J. Optim. 19, 1451–1466 (2008)
Lee, J., Skipper, D., Speakman, E.: Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations. Math. Progr. B 170, 121–140 (2018)
Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Progr. 103(2), 251–282 (2005)
McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I–convex underestimating problems. Math. Progr. 10(1), 147–175 (1976)
Miller, A.J., Belotti, P., Namazifar, M.: Linear inequalities for bounded products of variables. SIAG/OPT Views News 22(1), 1–8 (2011)
Nguyen, T.T., Richard, J.P.P., Tawarmalani, M.: Deriving convex hulls through lifting and projection. Math. Progr. 169(2), 377–415 (2018)
Sahinidis, N.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)
Santana, A., Dey, S.S.: The convex hull of a quadratic constraint over a polytope. SIAM J. Optim. 30(4), 2983–2997 (2020)
Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Springer (2013)
Sherali, H.D., Alameddine, A.: A new reformulation-linearization technique for bilinear programming problems. J. Glob. Opt. 2(4), 379–410 (1992)
Speakman, E., Lee, J.: On branching-point selection for trilinear monomials in spatial branch-andbound: the hull relaxation. J. Glob. Optim. 72(2), 129–153 (2018)
Acknowledgements
The authors are grateful to three anonymous referees for their careful readings of the paper and suggestions to improve it. Kurt Anstreicher would like to thank Pietro Belotti, Jeff Linderoth and Mohit Tawarmalani for very helpful conversations on the topic of this paper.
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Anstreicher, K.M., Burer, S. & Park, K. Convex hull representations for bounded products of variables. J Glob Optim 80, 757–778 (2021). https://doi.org/10.1007/s10898-021-01046-7
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DOI: https://doi.org/10.1007/s10898-021-01046-7