Quasi-Newton methods are reliable and efficient on a wide range of problems, but they can require many iterations if no good estimate of the Hessian is available or the problem is ill-conditioned. Methods that are less susceptible to ill-conditioning can be formulated by exploiting the fact that quasi-Newton methods accumulate second-derivative information in a sequence of expanding manifolds. These modified methods represent the approximate second derivatives by a smaller reduced approximate Hessian. The availability of a reduced Hessian allows conventional quasi-Newton methods to be improved in two ways. First, it is possible to reduce the objective while the iterates are forced to linger on a manifold whose dimension can be significantly smaller than the manifold on which curvature is being accumulated. Second, approximate curvature in directions off the manifold can be reinitialized as the iterations proceed, thereby reducing the influence of a poor initial estimate of the Hessian.

We conclude by presenting extensive numerical results from problems in the CUTE test set. Our experiments provide strong evidence that reduced Hessian quasi-Newton are more robust and more efficient than conventional quasi-Newton methods on small- to medium-sized problems.

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