  We consider a family of dense initializations for limited-memory quasi-Newton
methods. The proposed initialization exploits an eigendecomposition-based
separation of the full space into two complementary subspaces, assigning a
different initialization parameter to each subspace. This family of dense
initializations is proposed in the context of a limited-memory
Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) trust-region method that makes use of
a shape-changing norm to define each subproblem. As with L-BFGS methods that
traditionally use diagonal initialization, the dense initialization and the
sequence of generated quasi-Newton matrices are never explicitly formed.
Numerical experiments on the CUTEst test set suggest that this initialization
together with the shape-changing trust-region method outperforms other L-BFGS
methods for solving general nonconvex unconstrained optimization problems.
While this dense initialization is proposed in the context of a special
trust-region method, it has broad applications for more general quasi-Newton
trust-region and line search methods. In fact, this initialization is suitable
for use with any quasi-Newton update that admits a compact representation and,
in particular, any member of the Broyden class of updates.
