Limited-Memory Reduced-Hessian Methods for Large-Scale Unconstrained Optimization

Limited-memory BFGS quasi-Newton methods approximate the Hessian matrix of second derivatives by the sum of a diagonal matrix and a fixed number of rank-one matrices. These methods are particularly effective for large problems in which the approximate Hessian cannot be stored explicitly. It can be shown that the conventional BFGS method accumulates approximate curvature in a sequence of expanding subspaces. This allows an approximate Hessian to be represented using a smaller reduced matrix that increases in dimension at each iteration. When the number of variables is large, this feature may be used to define limited-memory reduced-Hessian methods in which the dimension of the reduced Hessian is limited to save storage. Limited-memory reduced-Hessian methods have the benefit of requiring half the storage of conventional limited-memory methods. In this paper, we propose a particular reduced-Hessian method with substantial computational advantages compared to previous reduced-Hessian methods. Numerical results from a set of unconstrained problems in the CUTE test collection indicate that our implementation is competitive with the limited-memory codes L-BFGS and L-BFGS-B.