A family of variable metric proximal methods

The authors consider conceptual optimization methods combining two ideas: the Moreau-Yosida regularization in convex analysis, and quasi-Newton approximations of smooth functions. They outline several approaches based on this combination, and establish their global convergence. Then the authors study theoretically the local convergence properties of one of these approaches, whch uses quasi-Newton updates of the object function itself. Also, they obtain a globally and superlinearly convergent BFGS proximal method. At each step of the present study, the authors single out the assumptions that are useful to derive the result concerned.