Patricia Tossings

The Perturbed Proximal Point Algorithm and Some of its Applications

Following the works of R. T. Rockafellar, to search for a zero of a maximal monotone operator, and of B. Lemaire, to solve convex optimization problems, we present a perturbed version of the proximal point algorithm. We apply this new algorithm to convex optimization and to variational inclusions or, more particularly, to variational inequalities.

Variational metric and exponential penalization

Attouch and Wets have introduced recently a variational metric between closed proper convex functions. The aim of this note is to give an estimation of this metric in the case of the exponential penalties. We can therefore recover some convergence results for the exponential penalty method.

Mixing proximal regularization, penalization and parallel decomposition in convex programming

Let A and B be two mutually complementary subspaces of a real Hubert space H, T being a maximal monotone operator on H. The method of partial inverses has been introduced by J. Spingarn in 1983 for solving problems in which the object is to find x ∈ A and y ∈ B such that y ∈ Tx. We present a perturbed version of this method and its most important application: a perturbed method of parallel decomposition in convex programming.1

Relaxed Assumptions for Iteration Methods

We consider a general inexact iterative method for finding a fixed point of a nonexpansive mapping on a real Hilbert space. Proofs of convergence of sequences generated by such a method generally require at least that the error term goes to zero. The aim of the present paper is to weaken this nonrealistic theoretical assumption. The obtained result is applied to the proximal point algorithm.