On feasibility based bounds tightening

Pietro Belotti (1) , Sonia Cafieri (2) , Jon Lee (3) , Leo Liberti (4)

1 Department of Mathematical Sciences
2 MAIAA - ENAC - Laboratoire de Mathématiques Appliquées, Informatique et Automatique pour l'Aérien
3 University of Michigan [Ann Arbor]
4 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Mathematical programming problems involving nonconvexities are usually solved to optimality using a (spatial) Branch-and-Bound algorithm. Algorithmic e?ciency depends on many factors, among which the widths of the bounding box for the problem variables at each Branch-and-Bound node naturally plays a critical role. The practically fastest box-tightening algorithm is known as FBBT (Feasibility-Based Bounds Tightening): an iterative procedure to tighten the variable ranges. Depending on the instance, FBBT may not converge ?nitely to its limit ranges, even in the case of linear constraints. Tolerance-based termination criteria yield ?nite termination, but not in worstcase polynomial-time. We model FBBT by using ?xed-point equations in terms of the variable bounding box, and we treat these equations as constraints of an auxiliary mathematical program. We demonstrate that the auxiliary mathematical problem is a linear program, which can of course be solved in polynomial time. We demonstrate the usefulness of our approach by improving an existing Branch-and-Bound implementation. global optimization, MINLP, spatial Branch-and-Bound, range reduction.

Mots clés: global optimization; MINLP; spatial Branch-and-Bound; range reduction

Domaines: Optimisation et contrôle [math.OC]

Pietro Belotti, Sonia Cafieri, Jon Lee, Leo Liberti. On feasibility based bounds tightening. 2012. ⟨hal-00935464⟩