Preprint A144/2002
An inexact method of partial inverses. Application to Splitting methods.

Susana Scheinberg de Makler | Burachik, Regina S. | Sagastizábal, Claudia

**Keywords: **
maximal monotone operator | Splitting methods | proximal point algorithm | enlargement of a maximal monotoneoperator | hybrid proximal methods.

For a maximal monotone operator $T$ on a Hilbert space $H$ and a
closed subspace $A$ of $H$, we consider the problem of finding
$(x,y\in T(x))$ satisfying $x\in A$ and $y\in A^\perp$. An equivalent
formulation of this problem makes use of the partial inverse operator
of Spingarn. The resulting generalized equation can be solved by using
the proximal point algorithm. We consider instead the use of hybrid
proximal methods. Hybrid methods use enlargements of operators, close
in spirit to the concept of $\ve$-subdifferentials. We characterize
the enlargement of the partial inverse operator in terms of the
enlargement of $T$ itself. We present a new algorithm of resolution
that combines Spingarn and hybrid methods, we prove for this method
global convergence only assuming existence of solutions
and maximal monotonicity of $T$. We also
show that, under standard assumptions, the method has a linear rate of
convergence.
For the important problem of finding a zero of a sum of maximal
monotone operators $T_1,\ldots,T_m$, we present a highly paralellizable
scheme. Unlike the standard requirements,
we only assume maximal monotonicity
of each term $T_i$, as in Spingarn's pioneer's work.
We also illustrate the applicability of this approach to some
interesting problems.