A New Multipoint Symmetric Secant Method with a Dense Initial Matrix

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.

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Report 2021-1, Wake Forest University, July 2021.

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