One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables $x$ and $y$ over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of $xy$ can be much tighter when computed over a strict, nonrectangular subset of the box. In order to exploit this in practice, we propose computing valid linear inequalities for the projection of the feasible region onto the $x$-$y$-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and-bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for $x$, $y$, and $xy$ over the available projections. Our computational evaluations using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.


  1. mixed-integer quadratically constrained programs
  2. nonconvex
  3. global optimization
  4. separation
  5. propagation
  6. projection
  7. bilinear terms

MSC codes

  1. 90-08
  2. 90C27
  3. 90C26

Get full access to this article

View all available purchase options and get full access to this article.


T. Achterberg, Constraint Integer Programming, Ph.D. thesis, Technische Universität Berlin, 2007, https://doi.org/10.14279/depositonce-1634, URNnbn:de:kobv:83-opus-16117.
T. Achterberg and R. Wunderling, Mixed integer programming: Analyzing 12 years of progress, in Facets of Combinatorial Optimization, Springer, Berlin, Heidelberg, 2013, pp. 449--481, https://doi.org/10.1007/978-3-642-38189-8_18.
F. A. Al-Khayyal and J. E. Falk, Jointly constrained biconvex programming, Math. Oper. Res., 8 (1983), pp. 273--286, https://doi.org/10.1287/moor.8.2.273.
P. R. Amestoy, I. S. Duff, J.-Y. L'Excellent, and J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23 (2001), pp. 15--41, https://doi.org/10.1137/s0895479899358194.
K. M. Anstreicher, Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming, J. Global Optim., 43 (2008), pp. 471--484, https://doi.org/10.1007/s10898-008-9372-0.
K. M. Anstreicher, On convex relaxations for quadratically constrained quadratic programming, Math. Program., 136 (2012), pp. 233--251, https://doi.org/10.1007/s10107-012-0602-3.
ANTIGONE -- Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations, http://ares.tamu.edu/ANTIGONE.
E. Balas, Projection, lifting and extended formulation in integer and combinatorial optimization, Ann. Oper. Res., 140 (2005), pp. 125--161, https://doi.org/10.1007/s10479-005-3969-1.
E. Balas, S. Ceria, and G. Cornuéjols, A lift-and-project cutting plane algorithm for mixed 0--1 programs, Math. Programming, 58 (1993), pp. 295--324, https://doi.org/10.1007/bf01581273.
E. Balas and M. Perregaard, Lift-and-project for mixed 0--1 programming: Recent progress, Discrete Appl. Math., 123 (2002), pp. 129--154, https://doi.org/10.1016/S0166-218X(01)00340-7.
X. Bao, N. V. Sahinidis, and M. Tawarmalani, Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons, Math. Program., 129 (2011), pp. 129--157, https://doi.org/10.1007/s10107-011-0462-2.
P. Belotti, S. Cafieri, J. Lee, and L. Liberti, On Feasibility Based Bounds Tightening, Tech. Report 3325, Optimization Online, 2012, http://www.optimization-online.org/DB_HTML/2012/01/3325.html.
P. Belotti, J. Lee, L. Liberti, F. Margot, and A. Wächter, Branching and bounds tightening techniques for non-convex MINLP, Optim. Methods Softw., 24 (2009), pp. 597--634, https://doi.org/10.1080/10556780903087124.
A. Ben-Tal and A. Nemirovski, On polyhedral approximations of the second-order cone, Math. Oper. Res., 26 (2001), pp. 193--205, https://doi.org/10.1287/moor.
S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
CMU-IBM Cyber-Infrastructure for MINLP, http://www.minlp.org/.
COIN-OR, Couenne, an exact solver for nonconvex MINLPs, http://www.coin-or.org/Couenne.
COIN-OR, Ipopt, Interior Point Optimizer, http://www.coin-or.org/Ipopt, 2018.
COIN-OR, CppAD, a Package for Differentiation of p̧p algorithms, http://www.coin-or.org/CppAD (accessed December 2019).
T. Fujie and M. Kojima, Semidefinite programming relaxation for nonconvex quadratic programs, J. Global Optim., 10 (1997), pp. 367--380, https://doi.org/10.1023/A:1008282830093.
A. Gleixner, T. Berthold, B. Müller, and S. Weltge, Three enhancements for optimization-based bound tightening, J. Global Optim., 67 (2016), pp. 731--757, https://doi.org/10.1007/s10898-016-0450-4.
A. Gleixner and S. Weltge, Learning and propagating Lagrangian variable bounds for mixed-integer nonlinear programming, in Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Lecture Notes in Comput. Sci. 7874, Springer, Berlin, Heidelberg, 2013, pp. 355--361, https://doi.org/10.1007/978-3-642-38171-3_26.
I. E. Grossmann and N. V. Sahinidis, Special issue on mixed integer programming and its application to engineering, part I, Optim. Eng., 3 (2002), pp. i--ii and 233--346.
C. Helmberg, F. Rendl, and R. Weismantel, A semidefinite programming approach to the quadratic knapsack problem, J. Comb. Optim., 4 (2000), pp. 197--215, https://doi.org/10.1023/2Fa/3A1009898604624.
G. Hendel, Empirical Analysis of Solving Phases in Mixed Integer Programming, Master's thesis, Technische Universität Berlin, 2014, URNnbn:de:0297-zib-54270.
H. Hijazi, Perspective envelopes for bilinear functions, in AIP Conference Proceedings, 2019, 020017, https://doi.org/10.1063/1.5089984.
B. Hunsaker, E. L. Johnson, and C. A. Tovey, Polarity and the complexity of the shooting experiment, Discrete Optim., 5 (2008), pp. 541--549, https://doi.org/10.1016/j.disopt.2006.12.001.
ILOG Inc., CPLEX: High-Performance Software for Mathematical Programming and Optimization, http://www.ilog.com/products/cplex/ (accessed December 2019).
W. Karush, Minima of functions of several variables with inequalities as side conditions, in Traces and Emergence of Nonlinear Programming, Springer, Basel, 2013, pp. 217--245, https://doi.org/10.1007/978-3-0348-0439-4_10.
T. Koch, T. Achterberg, E. Andersen, O. Bastert, T. Berthold, R. E. Bixby, E. Danna, G. Gamrath, A. Gleixner, S. Heinz, A. Lodi, H. Mittelmann, T. Ralphs, D. Salvagnin, D. E. Steffy, and K. Wolter, MIPLIB 2010, Math. Program. Comput., 3 (2011), pp. 103--163, https://doi.org/10.1007/s12532-011-0025-9.
H. W. Kuhn and A. W. Tucker, Nonlinear programming, in Traces and Emergence of Nonlinear Programming, Springer Basel, 2014, pp. 247--258, https://doi.org/10.1007/978-3-0348-0439-4_11.
J. Lee and W. D. Morris, Geometric comparison of combinatorial polytopes, Discrete Appl. Math., 55 (1994), pp. 163--182, https://doi.org/10.1016/0166-218X(94)90006-X.
C. Lemaréchal and F. Oustry, SDP relaxations in combinatorial optimization from a Lagrangian viewpoint, in Advances in Convex Analysis and Global Optimization, Nonconvex Optim. Appl. 54, Springer, Boston, 2001, pp. 119--134, https://doi.org/10.1007/978-1-4613-0279-7_6.
L. Liberti, Symmetry in mathematical programming, in Mixed Integer Nonlinear Programming, Springer, New York, 2011, pp. 263--283, https://doi.org/10.1007/978-1-4614-1927-3_9.
J. Linderoth, A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs, Math. Program., 103 (2005), pp. 251--282, https://doi.org/10.1007/s10107-005-0582-7.
M. Locatelli, Convex envelopes of bivariate functions through the solution of KKT systems, J. Global Optim., 72 (2018), pp. 277--303, https://doi.org/10.1007/s10898-018-0626-1.
A. Lodi and A. Tramontani, Performance variability in mixed-integer programming, in Theory Driven by Influential Applications, INFORMS, 2013, pp. 1--12, https://doi.org/10.1287/educ.2013.0112.
Z. Luo, W. Ma, A. M. So, Y. Ye, and S. Zhang, Semidefinite relaxation of quadratic optimization problems, IEEE Signal Processing Mag., 27 (2010), pp. 20--34, https://doi.org/10.1109/msp.2010.936019.
G. P. McCormick, Computability of global solutions to factorable nonconvex programs: Part I \textemdash Convex underestimating problems, Math. Programming, 10 (1976), pp. 147--175, https://doi.org/10.1007/bf01580665.
C. A. Meyer and C. A. Floudas, Global optimization of a combinatorially complex generalized pooling problem, AIChE J., 52 (2006), pp. 1027--1037, https://doi.org/10.1002/aic.10717.
A. J. Miller, P. Belotti, and M. Namazifar, Linear inequalities for bounded products of variables, SIAG/OPT Views and News, 22 (2011), pp. 1--8, http://wiki.siam.org/siag-op/index.php/View_and_News.
B. Müller, F. Serrano, and A. Gleixner, Using Two-Dimensional Projections for Stronger Separation and Propagation of Bilinear Terms, ZIB-Report 19-15, Zuse Institute Berlin, Berlin, 2019.
J. L. Nazareth, The homotopy principle and algorithms for linear programming, SIAM J. Optim., 1 (1991), pp. 316--332, https://doi.org/10.1137/0801021.
S. Poljak, F. Rendl, and H. Wolkowicz, A recipe for semidefinite relaxation for $(0,1)$-quadratic programming, J. Global Optim., 7 (1995), pp. 51--73, https://doi.org/10.1007/bf01100205.
Y. Puranik and N. V. Sahinidis, Domain reduction techniques for global NLP and MINLP optimization, Constraints, 22 (2017), pp. 338--376, https://doi.org/10.1007/s10601-016-9267-5.
I. Quesada and I. E. Grossmann, A global optimization algorithm for linear fractional and bilinear programs, J. Global Optim., 6 (1995), pp. 39--76, https://doi.org/10.1007/bf01106605.
H. Ryoo and N. Sahinidis, Global optimization of nonconvex NLPs and MINLPs with applications in process design, Comput. Chem. Eng., 19 (1995), pp. 551--566, https://doi.org/10.1016/0098-1354(94)00097-2.
N. V. Sahinidis, BARON 17.8.9: Global Optimization of Mixed-Integer Nonlinear Programs, User's Manual, 2017; available at ŭlhttp://www.minlp.com/baron.
SCIP -- Solving Constraint Integer Programs, http://scip.zib.de.
T. Serra, Essays on Postoptimality, Lift-and-Project, and Scheduling, Ph.D. thesis, Tepper School of Business, Carnegie Mellon, Pittsburgh, PA, 2018, https://doi.org/10.1184/R1/6716444.v1.
H. D. Sherali and W. P. Adams, A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Nonconvex Optim. Appl. 31, Kluwer, Dordrecht, 1999, https://doi.org/10.1007/978-1-4757-4388-3.
H. D. Sherali and B. M. P. Fraticelli, Enhancing RLT relaxations via a new class of semidefinite cuts, J. Global Optim., 22 (2002), pp. 233--261, https://doi.org/10.1023/A:1013819515732.
E. M. Smith and C. C. Pantelides, Global optimisation of nonconvex MINLPs, Comput. Chem. Eng., 21 (1997), pp. S791--S796, https://doi.org/10.1016/S0098-1354(97)87599-0.
E. Speakman and J. Lee, Quantifying double McCormick, Math. Oper. Res., 42 (2017), pp. 1230--1253, https://doi.org/10.1287/moor.2017.0846.
L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49--95, https://doi.org/10.1137/1038003.
S. Vigerske, Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to Mixed-Integer Nonlinear Programming, Ph.D. thesis, Mathematisch-Naturwissenschaftliche Fakultät II, Humboldt-Universität zu Berlin, 2013, URNnbn:de:kobv:11-100208240.
S. Vigerske and A. Gleixner, SCIP: Global optimization of mixed-integer nonlinear programs in a branch-and-cut framework, Optim. Methods Softw., 33 (2017), pp. 563--593, https://doi.org/10.1080/10556788.2017.1335312.
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math. Program., 106 (2005), pp. 25--57, https://doi.org/10.1007/s10107-004-0559-y.
J. M. Zamora and I. E. Grossmann, A branch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms, J. Global Optim., 14 (1999), pp. 217--249, https://doi.org/10.1023/A:1008312714792.

Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1339 - 1365
ISSN (online): 1095-7189


Submitted: 13 March 2019
Accepted: 10 January 2020
Published online: 11 May 2020


  1. mixed-integer quadratically constrained programs
  2. nonconvex
  3. global optimization
  4. separation
  5. propagation
  6. projection
  7. bilinear terms

MSC codes

  1. 90-08
  2. 90C27
  3. 90C26



Funding Information

Bundesministerium für Wirtschaft und Energie : 03ET1549D
Bundesministerium für Bildung und Forschung https://doi.org/10.13039/501100002347 : 05M14ZAM

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media