Computing Points that Satisfy Second Order Necessary Optimality Conditions for Unconstrained Minimization

In this paper we present a new class of algorithms for computing points that satisfy second order necessary optimality conditions for unconstrained minimization problems. We introduce a framework based on a curvilinear line search using a combination of a direction of negative curvature and a Newton-type descent direction which covers the scheme proposed by More and Sorensen [Math. Program., 16 (1979), pp. 1-20]. We then propose two kinds of descent direction from which two new algorithms emerge. One is a Levenberg-Marquardt direction, while the other is a cubic regularized Newton direction. Global convergence results and asymptotic quadratic rate are proven under mild assumptions.