  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 only require first-order information to
approximate the true Hessian. Multipoint symmetric secant (MSS) methods can be
thought of as generalizations of quasi-Newton methods in that they attempt to
impose additional requirements on their approximation of the Hessian. Given an
initial Hessian approximation, MSS methods generate a sequence of
possibly-indefinite matrices using rank-2 updates to solve nonconvex
unconstrained optimization problems. 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 for large-scale nonconvex
optimization that allows for dense initializations. Numerical results on the
CUTEst test problems suggest that the MSS method using a dense initialization
outperforms the standard initialization. Numerical results also suggest that
this approach is competitive with both a basic L-SR1 trust-region method and an
L-PSB method.
