A Projection-Proximal Bundle Method for Convex Nondifferentiable Minimization

We consider the problem inf fx ≔ f + i x, where f is a closed proper convex function and i x is the indicator function of a closed convex set X in a real Hilbert space. We study an inexact proximal point algorithm under mild conditions on accuracy tolerances that ensure weak convergence of its iterates to a solution, or strong convergence when fx is boundedly well-set. We also give a bundle method for minimizing fx. At each iteration, it finds a trial point by minimizing a simple “unconstrained” polyhedral model of fx augmented with a proximal term, and projecting a certain point onto X. We establish convergence of descent and nondescent versions. Finally, we discuss an application in Tikhonov’s regularization of ill-posed minimization problems.

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