Symbolic Preconditioning with Taylor Models: Some Examples

Abstract

Deterministic global optimization with interval analysis involves

• using interval enclosures for ranges of the constraints, objective, and gradient to reject infeasible regions, regions without global optima, and regions without critical points;

• using interval Newton methods to converge on optimum-containing regions and to verify global optima.

There are certain problems for which interval dependency leads to overestimation in the enclosures of the individual components, causing the optimization search to become prohibitively inefficient. As Hansen has observed earlier, in other problems, there is no overestimation in the individual components, but overestimation is introduced in the preconditioning in the interval Newton method.

We examine these issues for a particular nonlinear systems problem that, to date, has defied numerical solution. To reduce overestimation, we use Taylor models. The Taylor models sometimes reduce individual overestimation but, consistent with Hansen's observations, especially reduce the overestimation due to preconditioning. From numerical experiments, we conclude that, in certain instances, Taylor models can greatly reduce both the number of subregions necessary to complete an exhaustive search and the total computational effort.