Validated Probing with Linear Relaxations

During branch and bound search in deterministic global optimization, adaptive subdivision is used to produce subregions x, which are then eliminated, shown to contain an optimal point, reduced in size, or further subdivided. The various techniques used to reduce or eliminate a subregion x determine the efficiency and practicality of the algorithm. Ryoo and Sahinidis have proposed a “probing” technique, involving the dual variables of a linear relaxation, to reduce the size of subregions x. This technique, combined with others, has been successful in the BARON global optimization software. Here, we show how the Ryoo and Sahinidis technique can be used in a validated context, without significant performance penalty. Our validation technique involves a mathematically rigorous regularization process and use of an interval Newton method on the Kuhn-Tucker conditions (complementarity conditions). This allows us to obtain rigorous bounds on dual variables. We compare the process, implemented within our GlobSol environment, to an algorithm using LP relaxations but no “probing,” on a standard test set. The results indicate that use of the “probing” technique does not significantly benefit the overall branch and bound process, although there is evidence that GlobSol’s performance can depend crucially on the problem formulation and on values of heuristically set algorithm parameters. In any case, the regularization process we propose here, a relatively simple technique that results in a rigorous relaxation, is potentially of wider use in validated computations, where validated bounds on selected dual variables are desired.