In this paper we have addressed the problem of unboundedness in the search direction when the Hessian is indefinite or near singular. A new algorithm has been proposed which naturally handles singular Hessian matrices, and is theoretically equivalent to the trust-region approach. This is accomplished by performing explicit matrix modifications adaptively that mimic the implicit modifications used by trust-region methods. Further, we provide a new variant of modified conjugate gradient algorithms which implements this strategy in a robust and efficient way. Numerical results are provided demonstrating the effectiveness of this approach in the context of a line-search method for large-scale unconstrained nonconvex optimization.
Technical Report 2009-01, SAS Institute Inc., 100 SAS Campus Dr, Cary, NC 27513 USA.