Curvilinear Linesearch for Tensor Methods

Abstract

This paper presents a curvilinear linesearch for use when solving nonlinear equations with tensor methods. Standard tensor methods use a combination of the tensor step and Newton step in a linesearch, but they are handled separately and in an ad hoc manner. Our curvilinear linesearch combines the two directions in a single parametric step, which guarantees a monotonic decrease on the tensor model and also asymptotically approaches the Newton direction as the step length shrinks to zero, thus guaranteeing descent on the nonlinear equations. Numerical experiments on a set of 35 small-scale problems, drawn primarily from the constrained and unconstrained testing environment, show an 18--23% improvement (in terms of function evaluations) over previous tensor linesearches when using quadratic backtracking and a 41--83% improvement when halving $\lambda$ before each trial. Our results also suggest that the curvilinear linesearch is more robust than linesearch-based Newton's method or the standard tensor linesearch, producing fewer failures than either method. Experiments on two large-scale fluid flow problems complement the small-scale results and give preliminary indication of the applicability of the tensor method and the efficiency of the curvilinear linesearch. The theoretical properties, coupled with the better performance, make this a desirable improvement over previous tensor method linesearches.