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On Dual Convergence of the Generalized Proximal
Point Method with Bregman Distances

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A. Iusem and R.D.C. Monteiro

The use of generalized distances (e.g.\ Bregman distances), instead
of the Euclidean one, in the proximal point method for convex
optimization, allows for elimination of the inequality constraints
from the subproblems. In this paper we consider the proximal point method
with Bregman distances
applied to linearly constrained convex optimization problems, and study the
behavior of the dual sequence obtained from the optimal
multipliers of the linear constraints of each subproblem. Under
rather general assumptions, which cover most Bregman distances
of interest, we obtain an ergodic convergence result, namely that a
sequence of weighted averages of the dual sequence converges to
the centroid of the dual optimal set. As an intermediate result, we prove
under the same assumptions that the dual central path generated by
a large class of barriers, including the generalized Bregman distances,
converges to the same point.
manuscript, School of ISyE, Georgia Tech, Atlanta,
GA 30332, October 1997.

Contact: monteiro@isye.gatech.edu