GLOPTLAB: a configurable framework for the rigorous global solution of quadratic constraint satisfaction problems

Abstract

GloptLab is an easy-to-use testing and development platform for solving quadratic constraint satisfaction problems, written in Matlab. The algorithms implemented in GloptLab are used to reduce the search space: scaling, constraint propagation, linear relaxations, strictly convex enclosures, conic methods, and branch and bound. All these methods are rigorous; hence, it is guaranteed that no feasible point is lost. Finding and verifying feasible points complement the reduction methods. From the method repertoire custom-made strategies can be built, with a user-friendly graphical interface. GloptLab was tested on a large test set of constraint satisfaction problems, and the results show the importance of composing a clever strategy.

GloMIQO: Global mixed-integer quadratic optimizer

“…There are several novel branching strategies within Couenne [27]. -GloptLAB [42,43,44,45] GloptLAB is a Matlab-based framework for solving quadratic constraint satisfaction problems [42]. The GloptLAB bounding and scaling strategies are particularly interesting [43,44,45].…”

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

“…While some work has been done in rigorous global optimization that formally verifies nonlinear functions including semialgebraic and transcendental functions (Domes, 2009;Domes and Neumaier, 2014), the most commonly-used solver software generates relaxations and cutting planes via floating point arithmetic and then uses floating point-based LP and NLP solvers for finding underestimators and heuristic solutions. Numerical instability can be at least partially mitigated using validated interval arithmetic (Brönnimann et al, 2003(Brönnimann et al, , 2006de Moura and Passmore, 2013) for FBBT but, especially for badlyscaled optimization problems, combining diverent solving strategies may induce numerical trouble because of the variance in tolerances between solvers (including clash between different sub-solvers of a single meta-solver).…”

“…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 …”

Interval unions

“…[8]). Global optimization solvers like Gloptlab [5] and COCONUT [28,29] rely heavily on interval analysis to guarantee rigorous solutions, even non-rigorous solvers like BARON [27] and α-Branch and bound [1] use rigorous computations in some steps of the search.…”