A New Multipoint Secant Method with a Dense Initial Matrix

Quasi-Newton methods are able to construct model of the objective function without needing second derivatives of the objective function. In large-scale optimization, when either forming or storing Hessian matrices are prohibitively expensive, quasi-Newton methods are often used in lieu of Newton's method because they make use of first-order information to approximate the true Hessian. Multipoint symmetric secant methods can be thought of as generalizations of quasi-Newton methods in that they have additional requirements on their approximation of the Hessian. Given an initial Hessian approximation, the multipoint symmetric secant (MSS) method generates a sequence of matrices using rank-2 updates. For practical reasons, up to now, the initialization has been a constant multiple of the identity matrix. In this paper, we propose a new limited-memory MSS method that allows for dense initializations. Numerical results on the CUTEst test problems suggest that the \small{MSS} method using a dense initialization outperforms the standard initialization. Numerical results also suggest that this approach is competitive with a basic L-SR1 trust-region method.