Positivity 12 (2008), 281–287
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/020281-7, published online January 11, 2008
An Uniparametric Secant–Type Method
for Nonsmooth Generalized Equations
Abstract. We are concerned with the problem of approximating a locally
unique solution of a nonsmooth generalized equation in Banach spaces using
an uniparametric Secant–type method. We provide a local convergence anal-
ysis under ω–conditioned divided difference which generalizes the usual Lips-
chitz continuous and H¨older continuous conditions used in .
Mathematics Subject Classiﬁcation (2000). 47H04, 65K10.
Keywords. Set–valued mapping, generalized equation, q–linear convergence,
Aubin continuity, divided difference.
This paper considers an approximation of the locally unique solution of nondif-
ferentiable generalized equations by an uniparametric Secant–type algorithm. Let
X, Y be two Banach spaces, f is a continuous function from X into Y and G is a
set–valued map from X to the subsets of Y with closed graph and we consider
0 ∈ f(x)+G(x). (1)
Equation (1) is an abstract model including complementarity problems, KKT sys-
tems in mathematical programming problems, variational inequalities, optimal
control and other ﬁelds. Generalized equations (1) was introduced by Robinson
. For more details of this concept and some applications, the reader could be
referred to [18,19].
For approximating the locally unique solution x
of (1), we consider the
are given starting points
; α is ﬁxed in [0, 1[
0 ∈ f(x
; f ](x
; f ] is a ﬁrst order divided difference of f on the points y