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A family of variable metric proximal methods. (English) Zbl 0832.90102
Summary: We consider conceptual optimization methods combining two ideas: the Moreau-Yosida regularization in convex analysis, and quasi-Newton approximations of smooth functions. We outline several approaches based on this combination, and establish their global convergence. Then we study theoretically the local convergence properties of one of these approaches, which uses quasi-Newton updates of the objective function itself. Also, we obtain a globally and superlinearly convergent BFGS proximal method. At each step of our study, we single out the assumptions that are useful to derive the result concerned.

MSC:
90C30 Nonlinear programming
65K05 Numerical mathematical programming methods
52A41 Convex functions and convex programs in convex geometry
90C25 Convex programming
49J52 Nonsmooth analysis
Software:
Modulopt
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References:
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