Generalized Proximal Point Algorithms and Bundle Implementations

In this paper, we present a study of the proximal point algorithm using very general regularizations for minimizing possibly nondifferentiable and nonconvex locally Lipschitz functions. We deduce from the proximal point scheme simple and implementable bundle methods for the convex and nonconvex cases. The originality of our bundle method is that the bundle information incorporates the subgradients of both the objective and the regularization function. The resulting method opens up a broad class of regularizations which are not restricted to quadratic, convex or even differentiable functions