Positivity 13 (2009), 385–398
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020385-14, published online October 28, 2008
Tikhonov regularization of metrically regular
Micha¨el Gaydu and Michel H. Geoﬀroy
Abstract. We present a Tikhonov regularization method for inclusions of the
form T (x) 0whereT is a set-valued mapping deﬁned on a Banach space
that enjoys metric regularity properties. We investigate, subsequently, the
case when the mapping T is metrically regular, strongly metrically regular,
strongly subregular and Lipschitz continuous and show the strong convergence
of the solutions of regularized problems to a solution to the original inclusion
T (x) 0. We also prove that the method has ﬁnite termination under some
special conditioning assumptions on T and we study its stability with respect
to some variational perturbations.
Mathematics Subject Classiﬁcation (2000). Primary 49J53; Secondary 49J40.
Keywords. Set-valued mapping, metric regularity, strong subregularity, exact
regularization, variational convergences.
The concept of regularization we are concerned with goes back to the works of
Tikhonov and Arsenine  and is known as Tikhonov regularization.Itwasﬁrst
considered for convex minimization and consisted in replacing an ill-posed problem
by a family (more often a sequence) of well-posed ones. More precisely, consider-
ing a proper lower semicontinuous convex function f on a Hilbert space H,the
regularized problem of the minimization of f is the minimization of the function
= f +
∈ H and t is a positive number. The function f
being proper, lower semicontinuous and strongly convex it has a unique minimizer
(usually denoted by prox
). It is well known that if argmin f = ∅ and t goes
to 0 then prox
strongly converges to the unique minimizer of the function
x →x − ¯x
over the set argminf (see [17, 28]).
In  Tossings extended in a very natural way this regularization method
to inclusion problems. When T is a maximal monotone operator on H, Tossings
proposed the following regularized method of the inclusion T (x) 0:
These authors are supported by Contract EA3591 (France).