An Interior-Point Path-Following Method for Computing a Perfect Stationary Point of a Polynomial Mapping on a Polytope

Abstract

As a refinement of the concept of stationary point, the notion of perfect stationary point was formulated in the literature. Although simplicial methods could be applied to approximate such a point, these methods do not make use of the possible differentiability of the problem and can be very time-consuming even for small-scale problems. To fully exploit the differentiability of the problem, this paper develops an interior-point path-following method for computing a perfect stationary point of a polynomial mapping on a polytope. By incorporating a logarithmic barrier term into the linear objective function with an appropriate convex combination, the method closely approximates some stationary points of the mapping on a perturbed polytope, especially when the perturbation is sufficiently small. It is proved that there exists a smooth path which starts from a point in the interior of a polytope and ends at a perfect stationary point. A predictor-corrector method is adopted for numerically following the path. Numerical results further confirm the effectiveness of the method.