Bound tightening in lifted formulations: (sub)solver-dependent impact on performance in RLT-based algorithms

Abstract

In this paper we explore a relevant aspect of the interplay between two core elements of global optimization algorithms for nonconvex nonlinear programming problems, which we believe has been overlooked by past literature. The first one is the reformulation of the original problem, which requires the introduction of auxiliary variables with the goal of defining convex relaxations that can be solved both reliably and efficiently on a node-by-node basis. The second one, bound tightening or, more generally, domain reduction, allows to reduce the search space to be explored by the branch-and-bound algorithm. We are interested in the performance implications of propagating the bounds of the original variables to the auxiliary ones in the lifted space: does this propagation reduce the overall size of the tree? does it improve the efficiency at solving the node relaxations? To better understand the above interplay, we focus on the reformulation-linearization technique for polynomial optimization. In this setting we are able to obtain a theoretical result on the implicit bounds of the auxiliary variables in the RLT relaxations, which sets the stage for the ensuing computational study, whose goal is to assess to what extent the performance of an RLT-based algorithm may be affected by the decision to explicitly propagate the bounds on the original variables to the auxiliary ones.