A Family of Trust-Region-Based Algorithms for Unconstrained Minimization with Strong Global Convergence Properties

This paper has two aims: to exhibit very general conditions under which members of a broad class of unconstrained minimization algorithms are globally convergent in a strong sense, and to propose several new algorithms that use second derivative information and achieve such convergence. In the first part of the paper we present a general trust-region-based algorithm schema that includes an undefined step selection strategy. We give general conditions on this step selection strategy under which limit points of the algorithm will satisfy first and second order necessary conditions for unconstrained minimization. Our algorithm schema is sufficiently broad to include line search algorithms as well. Next, we show that a wide range of step selection strategies satisfy the requirements of our convergence theory. This leads us to propose several new algorithms that use second derivative information and achieve strong global convergence, including an indefinite line search algorithm, several indefinite dogleg algorithms, and a modified "optimal-step" algorithm. Finally, we propose an implementation of one such indefinite dogleg algorithm.