Nonmonotone curvilinear line search methods for unconstrained optimization

We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary point where the Hessian matrix is positive semidefinite is exhibited for this class of algorithms by means of a nonmonotone stabilization strategy. An implementation using the Bunch-Parlett decomposition is shown to outperform several other techniques on a large class of test problems.The work of this author was based on research supported by the National Science Foundation Grant CCR-9157632, the Air Force Office of Scientific Research Grant F49620-94-1-0036 and the Department of Energy Grant DE-FG03-94ER61915.These authors were partially supported by Agenzia Spaziale Italiana, Roma, Italy.