Armijo-type condition for the determination of a generalized Cauchy point in trust region algorithms using exact or inexact projections on convex constraints

This paper considers some aspects of two classes of trust region methods for solving constrained optimization problems. The first class proposed by Toint uses techniques based on the explicitly calculated projected gradient while the second class proposed by Conn, Gould, Sartenaer and Toint allows for inexact projections on the constraints. We propose and analyze, for each class, a step-size rule in the spirit of the Armijo rule for the determination of a Generalized Cauchy Point. We then prove under mild assumptions that, in both cases, the classes preserve their theoretical properties of global convergence and identification of the correct active set in a finite number of iterations. Numerical issues are also discussed for both classes.