Quantifying Double McCormick

When using the standard McCormick inequalities twice to convexify trilinear monomials, as is often the practice in modeling and software, there is a choice of which variables to group first. For the important case in which the domain is a nonnegative box, we calculate the volume of the resulting relaxation, as a function of the bounds defining the box. In this manner, we precisely quantify the strength of the different possible relaxations defined by all three groupings, in addition to the trilinear hull itself. As a by-product, we characterize the best double-McCormick relaxation. We wish to emphasize that, in the context of spatial branch and bound for factorable formulations, our results do not only apply to variables in the input formulation. Our results apply to monomials that involve auxiliary variables as well. So, our results apply to the product of any three (possibly complicated) expressions in a formulation.

[1]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

[2]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[3]  Hermann Schichl,et al.  Interval Analysis on Directed Acyclic Graphs for Global Optimization , 2005, J. Glob. Optim..

[4]  James R. Luedtke,et al.  Some results on the strength of relaxations of multilinear functions , 2012, Math. Program..

[5]  François Margot,et al.  Solving Chance-Constrained Optimization Problems with Stochastic Quadratic Inequalities , 2016, Oper. Res..

[6]  Christodoulos A. Floudas,et al.  Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes , 2004 .

[7]  Leo Liberti,et al.  Branching and bounds tighteningtechniques for non-convex MINLP , 2009, Optim. Methods Softw..

[8]  E. Steingrímsson,et al.  A decomposition of 2-weak vertex-packing polytopes , 1994 .

[9]  Christodoulos A. Floudas,et al.  Convex envelopes for edge-concave functions , 2005, Math. Program..

[10]  Edward M. B. Smith,et al.  A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs , 1999 .

[11]  Hermann Schichl,et al.  Interval propagation and search on directed acyclic graphs for numerical constraint solving , 2009, J. Glob. Optim..

[12]  Robert Weismantel,et al.  The Convex Envelope of (n--1)-Convex Functions , 2008, SIAM J. Optim..

[13]  Nikolaos V. Sahinidis,et al.  Analysis of Bounds for Multilinear Functions , 2001, J. Glob. Optim..

[14]  Han Yu,et al.  Experimental Validation of Volume-Based Comparison for Double-McCormick Relaxations , 2017, CPAIOR.

[15]  Jon Lee,et al.  Mixed-integer nonlinear programming: Some modeling and solution issues , 2007, IBM J. Res. Dev..

[16]  Anatoliy D. Rikun,et al.  A Convex Envelope Formula for Multilinear Functions , 1997, J. Glob. Optim..

[17]  Sonia Cafieri,et al.  Reformulations in Mathematical Programming: A Computational Approach , 2009, Foundations of Computational Intelligence.

[18]  E. Steigrímsson,et al.  A decomposition of 2-weak vertex-packing polytopes , 1994, Discret. Comput. Geom..

[19]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[20]  Jon Lee,et al.  The volume of relaxed Boolean-quadric and cut polytopes , 1997, Discret. Math..

[21]  Jon Lee,et al.  Geometric Comparison of Combinatorial Polytopes , 1994, Discret. Appl. Math..

[22]  Christodoulos A. Floudas,et al.  Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes , 2004, J. Glob. Optim..

[23]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[24]  Sonia Cafieri,et al.  On convex relaxations of quadrilinear terms , 2010, J. Glob. Optim..

[25]  Nikolaos V. Sahinidis,et al.  A branch-and-reduce approach to global optimization , 1996, J. Glob. Optim..

[26]  Sonia Cafieri,et al.  The Reformulation-Optimization Software Engine , 2010, ICMS.

[27]  Nikolaos V. Sahinidis,et al.  Global optimization of nonconvex problems with multilinear intermediates , 2015, Math. Program. Comput..

[28]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[29]  Leo Liberti,et al.  Relaxations of Multilinear Convex Envelopes: Dual Is Better Than Primal , 2012, SEA.