Preprint A222/2003
Proximal methods in vector optimization

B. F. Svaiter | Bonnel, Henri | Iusem, Alfredo

**Keywords: **
vector optimization | multi-objective optimization | proximal point

We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y, with respect to the partial order induced by a closed,
convex and pointed cone C in Y, with nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact version and an inexact one, in which the subproblems are solved only approximately, within a constant relative tolerance.In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generated sequence is a minimizing one,in a suitable sense.