Preprint A547/2007
General Projective Splitting Methods for Sums of Maximal

Benar Fux Svaiter | Eckstein, Jonathan

**Keywords: **
Decomposition | splitting | maximal monotone

We describe a general projective framework for finding a zero of the
sum of n maximal monotone operators over a real Hilbert space.
Unlike prior methods for this problem, we neither assume $n=2$ nor
first reduce the problem to the case $n=2$. Our analysis defines a
closed convex extended solution set for which we can construct
a separating hyperplane by individually evaluating the resolvent of
each operator. At the cost of a single, computationally simple
projection step, this framework gives rise to a family of splitting
methods of unprecedented flexibility: numerous parameters, including
the proximal stepsize, may vary by iteration and by operator. The
order of operator evaluation may vary by iteration, and may be either
serial or parallel. The analysis essentially
generalizes our prior results for the case n=2. We also include a
relative error criterion for approximately evaluating resolvents,
which was not present in our earlier work.