On the composition of convex envelopes for quadrilinear terms

Within the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting.

[1]  N. Maculan,et al.  Global optimization : from theory to implementation , 2006 .

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

[3]  Nikolaos V. Sahinidis,et al.  Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs , 2009, Optim. Methods Softw..

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

[5]  Leo Liberti,et al.  Convex Envelopes of Monomials of Odd Degree , 2003, J. Glob. Optim..

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

[7]  Fabio Tardella,et al.  Existence and sum decomposition of vertex polyhedral convex envelopes , 2008, Optim. Lett..

[8]  Linus Schrage,et al.  The global solver in the LINDO API , 2009, Optim. Methods Softw..

[9]  Kurt M. Anstreicher,et al.  Institute for Mathematical Physics Semidefinite Programming versus the Reformulation–linearization Technique for Nonconvex Quadratically Constrained Quadratic Programming Semidefinite Programming versus the Reformulation-linearization Technique for Nonconvex Quadratically Constrained , 2022 .

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

[11]  F. Tardella.  On the existence of polyhedral convex envelopes , 2004 .

[12]  M. Kojima,et al.  Second order cone programming relaxation of nonconvex quadratic optimization problems , 2001 .

[13]  Leo Liberti,et al.  Molecular distance geometry methods: from continuous to discrete , 2010, Int. Trans. Oper. Res..

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

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

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

[17]  Hoang Tuy,et al.  Optimization under Composite Monotonic Constraints and Constrained Optimization over the Efficient Set , 2006 .

[18]  Jon Lee,et al.  Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations , 2011, Math. Program..

[19]  Jon Lee,et al.  Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations , 2010, Math. Program..

[20]  Leo Liberti,et al.  Writing Global Optimization Software , 2006 .

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

[22]  Hanif D. Sherali,et al.  CONVEX ENVELOPES OF MULTILINEAR FUNCTIONS OVER A UNIT HYPERCUBE AND OVER SPECIAL DISCRETE SETS , 1997 .

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

[24]  Nikolaos V. Sahinidis,et al.  Convex extensions and envelopes of lower semi-continuous functions , 2002, Math. Program..

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

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

[27]  Richard M. Karp,et al.  On linear characterizations of combinatorial optimization problems , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[28]  Panos M. Pardalos,et al.  Frontiers in Global Optimization , 2012 .

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

[30]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[31]  Nikolaos V. Sahinidis,et al.  Exact Algorithms for Global Optimization of Mixed-Integer Nonlinear Programs , 2002 .

[32]  Nikolaos V. Sahinidis,et al.  Global optimization of mixed-integer nonlinear programs: A theoretical and computational study , 2004, Math. Program..