Tensor Methods for Large, Sparse Unconstrained Optimization

Tensor methods for unconstrained optimization were rst introduced by Schn-abel and Chow SIAM J. Optimization, 1 (1991), pp. 293-315], who describe these methods for small to moderate-size 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 eeciently. 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 signiicantly more eecient and more reliable than standard methods based on Newton's method.