Multiterm polyhedral relaxations for nonconvex, quadratically constrained quadratic programs

GloMIQO: Global mixed-integer quadratic optimizer

“…Although αBB addresses the broader class of MINLP, it has specialized routines to handle MIQCQP via the convex envelopes of bilinear terms [11,91]. -BARON [26,133,134,135] Like αBB, the BARON code base addresses general MINLP to ε-global optimality but specializes its approach for MIQCQP. In addition to relaxing bilinear terms using the convex hull, the BARON preprocessing routines detect connected multivariable terms within quadratic equations [26].…”

“…-BARON [26,133,134,135] Like αBB, the BARON code base addresses general MINLP to ε-global optimality but specializes its approach for MIQCQP. In addition to relaxing bilinear terms using the convex hull, the BARON preprocessing routines detect connected multivariable terms within quadratic equations [26]. -Branch-and-cut for QCQP [18,19,20,21] Audet et al [18] discuss their implementation of a branch-and-cut global optimization algorithm for QCQP which made contributions to generating cutting planes and boundupdating strategies.…”

“…The GloMIQO reformulation uses the observation that disaggregating bilinear terms tightens the relaxation of MIQCQP and actively takes advantage of any redundant linear constraints added to the model. It is standard to use termwise convex/concave envelopes [11,91] to relax MIQCQP, but many tighter relaxations have been developed based on: polyhedral facets of edge-concave multivariable term aggregations [17,26,34,94,95,96,99,111,130,131,132], eigenvector projections [38,106,113,122], piecewise-linear underestimators [29,65,66,73,93,98,99,100,101,107,119,139], outer approximation of convex terms [32,48,47], and semidefinite programming (SDP) relaxations [16,25,35,122,121]. GloMIQO incorporates several of these advanced relaxations.…”

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

“…Cutting planes developed for MINLP include those based on: pseudo-convex MINLP problems , outer approximation of convex terms and linearization of other convex underestimators (Tawarmalani and Sahinidis, 2005;, multi-term quadratic expressions (Bao et al, 2009;Luedtke et al, 2012;, multilinear functions (Rikun, 1997;Belotti et al, 2010b;Qualizza et al, 2012), optimizing convex quadratic functions over nonconvex sets (Bienstock and Michalka, 2014), and other cutting plane classes based on nonlinear functional forms (D'Ambrosio et al, 2010;Richard and Tawarmalani, 2010). A review on cutting plane methods for MINLP can be found in Nowak (2005); multivariable and multiterm relaxations are typically favoured for MINLP because the tightest convex relaxation of each individual is not typically equivalent to the tightest possible relaxation of the entire MINLP (Westerlund et al, 2011).…”

“…products adhering to the rules in the manuscripts can be more tightly underestimated Meyer and Floudas (2005b) Piecewise αBB convexification Gounaris and Floudas (2008b,c) Gentilini et al (2013) Undercover branching Berthold and Gleixner (2013b) (2000); Sherali and Tuncbilek (1995) Reduced-cost bounds tightening Sahinidis (1995, 1996) Quadratic equation constraint satisfaction Domes andNeumaier (2010, 2011) Low-dimensional edge-concave aggregations Misener and Floudas (2012b) Nonlinearities removal Caprara and Locatelli (2010) Propagating Lagrangian bounds Gleixner and Weltge (2013) Sahinidis (1996); Sahinidis (2002a,b, 2004) Global MINLP framework Sahinidis (1995, 1996) Range reduction (Duality-based) Ryoo and Sahinidis (2001) Range reduction (FBBT) Tawarmalani and Sahinidis (2001) Relaxations for fractional terms Tawarmalani and Sahinidis (2005) Polyhedral Branch-&-Cut Bao et al (2009) ;Sahinidis (2013, 2014a) Bilinear cutting planes (including RLT) Zorn and Sahinidis (2014b) Polynomial cutting planes …”