A FAMILY OF VARIABLE-METRIC PROXIMAL METHODS

We consider conceptual optimization methods combining two ideas: the M oreau-Yosida regularization in convex analysis, and quasi-Newton appro ximations of smooth functions. We outline several approaches based on this combination, and establish their global convergence. Then we stud y theoretically the local convergence properties of one of these appro aches, which uses quasi-Newton updates of the objective function itsel f. Also, we obtain a globally and superlinearly convergent BFGS proxim al method. At each step of our study, we single out the assumptions th at are useful to derive the result concerned.