%A Spyridon Kontogiorgis
%A Robert R. Meyer
%T A Variable-Penalty Alternating Directions Method for Convex Optimization
%D November 1995
%R 95-18
%I COMPUTER SCIENCES DEPARTMENT, UNIVERSITY OF WISCONSIN
%C MADISON, WI
%X We study a generalized version of the
method of alternating directions
as applied to the minimization of the sum of two
convex functions subject to linear constraints.
The method consists of solving consecutively
in each iteration two optimization problems
which contain in the objective function
both Lagrangian and proximal terms.
The minimizers determine the new proximal terms
and a simple update of the Lagrangian terms follows.
We prove a convergence theorem which extends existing
results by relaxing the assumption of uniqueness
of minimizers. Another novelty is that we allow
penalty matrices, and these may vary per iteration.
This can be beneficial in applications, since it allows
additional tuning of the method to the problem and can
lead to faster convergence relative to fixed penalties.
As an application, we derive a decomposition scheme
for block angular optimization
and present computational results
on a class of dual block angular problems.