A Decomposition Algorithm for Convex Nondifferentiable Minimization with Errors

A decomposition algorithm based on proximal bundle-type method with inexact data is presented for minimizing an unconstrained nonsmooth convex functionf. At each iteration, only the approximate evaluation offand its approximate subgradients are required which make the algorithm easier to implement. It is shown that every cluster of the sequence of iterates generated by the proposed algorithm is an exact solution of the unconstrained minimization problem. Numerical tests emphasize the theoretical findings.

Stable Approximations of a Minimal Surface Problem with Variational Inequalities

In this paper we develop a new approach for the stable approximation of a minimal surface problem associated with a relaxed Dirichlet problem in the spaceBV(Ω)of functions of bounded variation. The problem can be reformulated as an unconstrained minimization problem of a functional𝒥onBV(Ω)defined by𝒥(u)=𝒜(u)+∫∂Ω|Tu−Φ|, where𝒜(u)is the “area integral” ofuwith respect toΩ,Tis the “trace operator” fromBV(Ω)intoL i(∂Ω), andϕis the prescribed data on the boundary ofΩ. We establish convergence and stability of approximate regularized solutions which are solutions of a family of variational inequalities. We also prove convergence of an iterative method based on Uzawa's algorithm for implementation of our regularization procedure.