Preprint C4/2001
Inexact versions of proximal point and cone-constrained augmented Lagrangians in Banach spaces

Rolando Garciga Otero

**Keywords: **
proximal point method | augmented Lagrangian | inexact solutions | maximal monotone operator | cone-constrained optimization

We extend to Banach spaces the hybrid proximal-extragradient
and proximal-projection methods
for finding zeroes of maximal monotone operators,
recently proposed
by Solodov and Svaiter
in finite dimension and
Hilbert spaces respectively. The generalization of the
hybrid-projection method makes it possible the use of regularizations
other than the quadratic by using an appropriate error criterion,
which allows for bounded relative error,
and a Bregman projection
instead of the metric projection. Boundedness of the sequence generated
by both methods and optimality of the weak accumulation points are
established under suitable assumptions on the regularizing function, which hold for any power
greater than one
of the norm of any uniformly smooth and uniformly convex Banach space,
without any assumption
on the operator other
than existence of zeroes. A variant of the error for the
hybrid-projection method let us establish superlinear convergence
even with inexact solutions of the proximal subproblem
in Hilbert spaces. We show that the hybrid steps of
the Proximal Point methods, allowing for
constant relative errors, are necessary in order to ensure boundedness of
the generated sequence, even in the optimization case. Moreover,
we show that such conditions
do not imply that the sequence of errors results summable ``a posteriori".
We then transpose such methods to generate augmented Lagrangian methods
for the following cone-constrained
convex optimization problem in Banach spaces: $\min g(x)$
subject to $-G(x)\in K$,
with $g:B_1\to\re$, $G:B_1 \to\B_2$,
where
$B_1$ and $B_2$ are real reflexive Banach
spaces and
$K$ is a nonempty closed convex cone in $B_2$. Two alternative procedures
are developed which allow for
inexact solutions of the primal subproblems. Boundedness of both
the primal and the dual sequences, and optimality of
primal and dual weak accumulation points, are then established,
assuming only existence of Karush-Kuhn-Tucker pairs. Finally we add to the
hybrid-extragradient method a penalization effect
for solving variational inequality problems in Banach spaces,
by introducing a boundary coercive condition
on the regularizing function. We give examples of regularizing functions
for the cases of the feasible set being closed balls and polyhedra. We get
convergence results similar to those of the methods without penalization
under the assumptions of pseudo- and paramonotonicity of the involved operator
and existence of solutions.