Tensor Methods for Large, Sparse Unconstrained Optimization

Abstract

Tensor methods for unconstrained optimization were first introduced by Schnabel and Chow [SIAM J. Optim., 1 (1991), pp. 293--315], who described these methods for small- to moderate-sized problems. The major contribution of this paper is the extension of these methods to large, sparse unconstrained optimization problems. This extension requires an entirely new way of solving the tensor model that makes the methods suitable for solving large, sparse optimization problems efficiently. We present test results for sets of problems where the Hessian at the minimizer is nonsingular and where it is singular. These results show that tensor methods are significantly more efficient and more reliable than standard methods based on Newton's method.