Validated Linear Relaxations and Preprocessing: Some Experiments

Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation process. This process decomposes the objective and constraints (if any) into convex and nonconvex unary and binary operations. The convex operations can be approximated arbitrarily well by appending additional constraints, while the domain must somehow be subdivided (in an overall branch-and-bound process or in some other local process) to handle nonconvex constraints. In general, a problem can be hard if even a single nonconvex term appears. However, certain nonconvex terms lead to easier-to-solve problems than others. Recently, Neumaier, Lebbah, Michel, ourselves, and others have paved the way to utilizing such techniques in a validated context. In this paper, we present a symbolic preprocessing step that provides a measure of the intrinsic difficulty of a problem. Based on this step, one of two methods can be chosen to relax nonconvex terms. This preprocessing step is similar to a method previously proposed by Epperly and Pistikopoulos [J. Global Optim., 11 (1997), pp. 287–311] for determining subspaces in which to branch, but we present it from a different point of view that is amenable to simplification of the problem presented to the linear programming solver, and within a validated context. Besides an illustrative example, we have implemented general relaxations in a validated context, as well as the preprocessing technique, and we present experiments on a standard test set. Finally, we present conclusions.