Smoothing Methods and Semismooth Methods for Nondifferentiable Operator Equations.
We consider superlinearly convergent analogues of Newton methods for nondifferentiable operator equations in function spaces. The superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally Lipschitzian operator in Rn is based on Rademacher's theorem which does not hold in function spaces. We introduce a concept of slant differentiability and use it to study  superlinear convergence of smoothing methods and semismooth methods in a unified framework. We show that a function is slantly differentiable at a point if and only if it is Lipschitz continuous at that point. An application to the  Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed.
Introduction
. This paper considers the nonlinear operator equation
Y is a continuous mapping, X and Y are Banach spaces, and D
is an open domain in X . In a number of problems, the operator F is nondifferentiable.
For example, a class of such problems arising in optimal control problems for parabolic
partial differential equations with bound constraints on the control [16, 17]:
where K is a completely continuous map from L 1 to C
for some bounded
is the map on C given by
for given l and u in C : A paradigm for such problems is the Urysohn integral
equation of the second kind. Another class of nonsmooth equations related to MHD
(magnetohydrodynamics) equilibria [18, 37] will be discussed in Section 4. A paradigm
for this class is the Dirichlet problem for nonsmooth elliptic partial differential equations
as discussed in Section 4.
The nonsmoothness poses serious difficulties and challenges for devising for non-smooth
problems analogues of existing iterative methods, which use smoothness. For
Department of Mathematics and Computer Science, Shimane University, Matsue 690-8504, Japan
(chen@math.shimane.ac.jp). The work of this author was supported in part by the Japan Society
of the Promotion of Science Grant C11640119.
y Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.
(nashed@math.udel.edu). The research of this author was partially supported by a grant from the
National Science Foundation.
z Department of Applied Mathematics, Hong Kong Polytechnic University, Kowloon, Hong Kong.
(maqilq@polyu.edu.hk). School of Mathematics, University of New South Wales, Sydney 2052.
(L.Qi@unsw.edu.au). The work of this author was partially supported by the Australian Research
Council.
Nashed and L.Qi
example, the Newton method assumes that F is Fr'echet differentiable and is defined
by
where F 0 is the Fr'echet derivative of F . What are suitable analogues of Newton's
method when F is not smooth ?
Iterative methods for nondifferentiable equations have been studied for decades
[7, 10, 15, 16, 31, 34, 38, 40]. Among these methods, smoothing methods and semi-
smooth methods for nondifferentiable equations arising from variational inequalities
and complementarity problems in R n have been studied extensively in the last few
years [2, 4, 5, 9, 11, 14, 34].
Superlinear convergence analysis of semismooth Newton methods for equations
defining a locally Lipschitian operator in R n uses the notions of generalized Jacobian
[12] and semismoothness [32, 35], which are based on the Rademacher theorem. The
Rademacher theorem states that if F : R n locally Lipschitzian function,
then F is differentiable almost everywhere. For a locally Lipschitzian function F :
defined the generalized Jacobian [12] by
where DF is the set of points where F is differentiable. For nonsmooth equations
described by a locally Lipschitzian operator F from R n into R n , the generalized
Newton method is defined by
Qi and Sun [32, 35] established superlinear convergence of (1.3) by using a concept
of semismoothness. The concept of semismoothness was introduced by Mifflin for
real-valued functions [20]. In [35], F is said to be semismooth at x if the limit
lim
exists for any h 2 R n . Local behavior of the generalized Newton method is analyzed
in [13, 31, 32, 35].
The Rademacher theorem does not hold in function spaces. Hence the above
definitions of generalized Jacobian and semismoothness cannot be used in function
spaces.
In this paper, we introduce notions of slanting functions and slant differentiability
of operators in Banach spaces, and use these notions to formulate a concept of semi-
smoothness in infinite dimensional spaces, which coincides with the above notion of
semismoothness in the case of a locally Lipschitzian mapping on R n . These notions
will play a pivotal role in the formulation and convergence analysis of analogues of
Newton's method (smoothing and semismooth methods) for nondifferentiable operator
equations in function spaces.
The main feature of smoothing Newton methods in R n is to approximate F by
a parametric function f(x; ffl) : R n \Theta R++ ! R n which is continuously differentiable
with respect to x, and then to use the partial derivative f x at each step of the
Newton-like iteration. The error of f(\Delta; ffl k ) to F is bounded by
Smoothing Methods and Semismooth Methods 3
1. For complementarity problems, many smoothing
functions have the following properties [9]
exists for every x 2 R n
and
lim
khk
0:
The properties (1.4) and (1.5) suggest a superlinearly convergent Newton method
[6, 11]:
Note that f n\Thetan is a single valued function, and the superlinear convergence
of (1.6) is not based on the Rademacher theorem. This opens a way to study Newton
methods for nonsmooth problems in function spaces.
The organization of the paper is as follows. In Section 2 we introduce the notion
of a slanting function f o and slant differentiability for a general nonsmooth function
F in Banach spaces, and study some of their interesting properties. Using slanting
functions, we extend in Section 3 the semismooth Newton method and the smoothing
Newton method to Banach spaces. An application to a class of nonsmooth Dirichlet
problems is studied in Section 4.
We use ff; to denote scalars. The set of all positive real numbers is
denoted by R++ . Let L(X; Y ) denote the set of all bounded linear operators on X
into Y .
2. Slant differentiability. A function F : D ae X ! Y is said to be (one-
sided) directionally differentiable at x if the limit
exists, in which case is called the (one-sided) directional derivative of F at
x with respect to the direction h. For brevity we will drop "one-sided" in what follows
since this is the only notion of directional derivative that occurs in this paper.
A function F : D ae X ! Y is said to be B-differentiable at a point x if it is
directionally differentiable at x, and
lim
khk
0:
In this case, we call \Delta) the B-derivative of F at x. See [39] for the B-
differentiability (B for Bouligand).
In finite dimensional Euclidean spaces, Shapiro [41] showed that a locally Lipschitzian
function F is B-differentiable at x if and only if it is directionally differentiable
at x. Moreover, Qi and Sun [35] showed that F is semismooth at x if and only if F is
B-differentiable (hence directionally differentiable) at x and for each V 2 @F
4 X.Chen, Z. Nashed and L.Qi
However, these results do not hold in function spaces since the generalized Jacobian
is defined only in finite dimensional spaces. To circumvent this difficulty in infinite
dimensional spaces we introduce the following notion of slant differentiability.
Definition 2.1. A function F : D ae X ! Y is said to be slantly differentiable
at x 2 D if there exists a mapping f such that the family
of bounded linear operators is uniformly bounded in the operator norm
for sufficiently small and
lim
The function f o is called a slanting function for F at x.
Definition 2.2. A function F : D ae X ! Y is said to be slantly differentiable
in an open domain D 0 ae D if there exists a mapping f
f o is a slanting function for F at every point x 2 D 0 . In this case, f o is called a
slanting function for F in D 0 .
Definition 2.3. Suppose that f slanting function for F at
We call the set
xk!x
the slant derivative of F associated with f o at x 2 D. Here, lim xk!x f
the limit of f sequence fx k g ae D such that x k ! x and lim xk!x f
exists, and @S F (x) is the set of all such limits. (Note that f
is always nonempty.)
Slant differentiability captures a property that appears implicitly in some convergence
proofs of Newton-type methods for solving nonsmooth equations as well as
ill-posed smooth equations. For example, consider the parameterized Newton method
for solving ill-posed smooth equations. To overcome ill-posedness and singularity, we
use
is chosen such that F 0 I is nonsingular. Let f
and assume slanting function for F at x   if F 0 (x) is
uniformly bounded in a neighborhood of x   .
We now make a few comments on some unusual properties of slanting functions
which also explain the choice of the terms "slanting function" and "slant derivative".
Remarks
1. Unlike other notions of derivatives, the term "f does not appear in Definition
2.1, so for a slanting function f for F at x, f itself is not characterized
in general by a limit of a quotient or a sequence.
2. A function F may be slantly differentiable at every point of D, but there is
no common slanting function of F at all points of D. For example, if F is
Fr'echet differentiable at x, we take f
slanting function for F at x. But f o in general is not a slanting function of
F at other points of D. If F is continuously differentiable in D and we take
slanting function for F at every
point of D.
Smoothing Methods and Semismooth Methods 5
3. A slanting function f for F at x is a single valued function. A slantly differentiable
function F at x can have infinitely many slanting functions at x.
Even if F is continuously differentiable in D, F still can have infinitely many
slanting functions for all points of D. For example, we may let f o take the
same values of F 0 except at a finite number of points of D, and take arbitrary
values at these finite number of points. Then such f o is still a slanting function
of F for all points of D. One may conjecture that if F is continuously
differentiable in D and f o is a slanting function for F in D, then f
with F 0 except possibly on a set of measure zero.
4. If f are both slanting functions for F at x (in D), then
is also a slanting function for F at x (in D), where - 2 [0; 1]: Moreover,
lim
kf
On the other hand, if f are slanting functions for F and G at x (in
D), respectively, then h slanting function for ffF
at x (in D) where ff and fi are constants. Note that such a result for linear
combination does not hold for the generalized Jacobian [12].
5. f o is not continuous in general. For example, let
be a real number. Then the function
is a slanting function for F in X . The slant derivative of F for x 2 X is
In fact it is easy to see that if f o is continuous at x, then F is differentiable
at x and F 0 (x). The slant derivative of F associated with f o at x
reduces to a singleton @S F
6. For a locally Lipschitzian function F : R n is semismooth at x, then
any single-valued selection of the Clarke Jacobian or the B-subdifferential is
a slanting function of F at x. This may not be true if F is not semismooth
at x. For example, let
ae
The derivative F 0 (x) is discontinuous at 0. The function F is slantly differentiable
at Indeed let f o be any function for which lim h!0 f
Then
lim
0:
Hence F is slantly differentiable at 0 with infinitely many slanting functions
for F at 0. Note that such f o is not a slanting function for F at every point
f0g. If we let f slanting function for F at every
point but not a slanting function for F at 0.
6 X.Chen, Z. Nashed and L.Qi
7. For a slantly differentiable function F at x, the set @S F (x) is dependent on
the choice of a slanting function for F at x. Associated with any slanting
function, the set @S F (x) is bounded, since f uniformly bounded
for h sufficiently small. For example, let
ae
Let
ae
sin 1
slanting function for F at 0 and @S F
that F is neither directionally differentiable at 0 nor Lipschitzian in any
neighbourhood of 0. Note that the function f o in this example is not slantly
differentiable at 0.
8. A continuous function is not necessarily slantly differentiable. For example, let
ae p
jhj, and 1=
there is no
uniformly bounded function f o such that F
Definition 2.4. A function F is said to be Lipschitz continuous at
x if there is a positive constant L such that for all sufficiently small h,
We now present a necessary and sufficient condition for the slant differentiability,
for the proof we need the following lemma, which is a corollary of the Hahn-Banach
theorem.
Lemma 2.5. Let X be a normed space and h be a fixed element of X, h 6= 0.
Then there exists an element g 2 X   , where X   is the (norm) dual of X, such that
(Note by definition of X   , g is a continuous linear functional on X, so it is bounded.)
Theorem 2.6. An operator slantly differentiable at x if and only
if F is Lipschitz continuous at x.
Proof. Suppose that F is slantly differentiable at x. By the definition of slant
differentiability, there are C ? 0 and ffi ? 0 such that for all khk - ffi, kf
and
khk
Hence, for all khk - ffi,
Smoothing Methods and Semismooth Methods 7
Conversely, suppose that F is Lipschitz continuous at x. We shall show that F
is slantly differentiable at x by constructing a slanting function for F at x. For each
fixed h 6= 0, by Lemma 2.5, there exists a continuous linear functional g h 2 X   such
that g h fixed as above, define the following function
on an open domain containing x
khk
for h 6= 0, and define f to be any bounded linear operator on X into Y . Then f
maps D into L(X; Y ) since each g h is in X   . For any z 2 X ,
khk
Thus
kf
khk
khk kzk:
Therefore
sup
z 6=0
kf
kzk
khk
that is
kf
Thus for sufficiently small h,
kf
Now using (2.2), and g h
khk
If X is a Hilbert space, then by the Riesz representation theorem every continuous
linear functional on X can be represented by an inner product. Thus the formula (2.2)
can be written in the form:
khk
Corollary 2.7. (Mean Value Theorem for Slantly Differentiable Functions)
Suppose that F : D ae X ! Y is slantly differentiable at x. Then for any h 6= 0 such
that x + h is in D, there exists a slanting function for F at x such that
8 X.Chen, Z. Nashed and L.Qi
Proof. This follows from the first part of Theorem 2.6 and the proof of the second
part of the same theorem.
Note that the above form of the mean value theorem is in equality form. It is a
selection theorem from the set of slanting functions for F at x. Mean value theorems
for smooth operators whose range is an infinite dimensional space are usually given in
the form of inequalities involving norms or majorants, or an inclusion form involving
the closed convex hull of the set of values of the derivative. For a comprehensive
overview of various types of mean value theorems for smooth operators, see pp. 171-
186 of [23].
Proposition 2.8. Suppose that F is slantly differentiable at x, and let f o be a
slanting function for F at x.
(a) F is directionally differentiable at x if and only if
lim
exists. If F is directionally differentiable at x, then
(b) F has a B-derivative at x if and only if
lim
exists uniformly with respect to h on each bounded set (say on
Proof. (a) Let h 2 X with
lim
is equivalent to
lim
Hence if F is directionally differentiable, then
The converse is also true.
(b) This follows from part (a) and the known (and easy to prove) fact that F has
a B-derivative at x if and only if
lim
exists uniformly with respect to h on each bounded set (see, for example, Nashed [23],
where a hierarchy of notions of differentiability are characterized by convergence of
"remainder" quotients R(th)=t as t approaches zero, uniformly with respect to h in
various classes of subsets).
Theorem 2.9. Suppose F is slantly differentiable at x and let f o be a slanting
function for F at x. Then the following statements are equivalent.
Smoothing Methods and Semismooth Methods 9
(a) For some function which is o(khk), f positively
homogeneous of degree 1 in h.
(b) lim t!0 exists for every h 2 X and
lim
khk
0:
(c) F is B-differentiable at x, and
Proof. (a)) (b): If f positively homogeneous of degree 1 in h,
then for any fixed t ? 0,
so
Note that only if for each fixed h, uniformly in
h on each bounded set. Hence for any h 2 X
lim
uniformly with respect to h on each bounded set. Moreover,
lim
khk
khk
0:
(b) ) (c): By part (b) of Proposition 2.1, statement (b) implies that F is B-
differentiable and lim t!0 statement (c) holds.
(c) positively
homogeneous of degree 1 in h, we have (a).
Proposition 2.10. Suppose that F is slantly differentiable in a neighborhood of
x and let f o be a slanting function for F in the neighborhood of x. Then the following
two statements are equivalent.
(a) There are a neighborhood N x of x and a positive constant C such that for any
(b) There are a neighborhood -
N x and a positive constant -
C such that for any u 2 -
every V 2 @S F (u) is nonsingular and kV
C:
Proof. Part (a) =) part (b): It is straightforward from the definition of @S F (u).
Part (b) =) part (a): It is due to the fact that f
Proposition 2.11. Suppose that F is slantly differentiable at x and let f o be
a slanting function for F at x. If there are a neighborhood N x of x and a positive
constant C such that for any u 2 N x , f o (u) is nonsingular and kf
then there is a positive constant -
C such that every V 2 @S F (x) is nonsingular and
C: Moreover, if Y is a finite dimensional space, the converse holds.
Proof. The first part follows the definition of @S F (x). The second part is due to
the fact that every bounded sequence has convergent subsequence in finite dimensional
spaces. Indeed, if the second part is not true, then there is a sequence fu k g such that
are singular or kf 1. By the definition of
@S F (x), there is a subsequence fu k j g ae fu k g such that f
is singular. This contradicts the assumption.
Nashed and L.Qi
3. Smoothing functions and semismooth functions. We generalize the definition
of smoothing functions for a nonsmooth function and the concept of semism-
mothness of a nonsmooth function in finite dimensional spaces to infinite dimensional
spaces.
Definition 3.1. We say that Y is a smoothing function of F
if f is continuously differentiable with respect to x and for any x 2 D and any ffl ? 0,
where - is a positive constant.
The smoothing function f is said to satisfy the slant derivative consistency
property at - x (in D) if
lim
exists for x in a neighborhood of -
x (in D) and f serves as a slanting function for F
at -
x (in D). Note that the limit in (3.2) is in the topology of the operator norm, so
the pointwise convergence of f x (x; ffl)h to f o (x)h for each fixed h is uniform on the set
Definition 3.2. We say that F is semismooth at x if there is a slanting
function f in a neighborhood N x of x, such that f o and the associated slant
derivative satisfy the following two conditions.
(a) lim t!0 exists for every h 2 X and
lim
(b)
Theorem 3.3. Suppose that F is slantly differentiable in a neighborhood N x of
x, and let f o be a slanting function for F in N x : Then F is semismooth at x if and
only if F is B-differentiable at x and
where @S F is the slant derivative associated with f o in N x .
Proof. Suppose that F is semismooth at x. Then from Theorem 2.9, part (a) of
Definition 3.2 implies that F is B-differentiable and
Thus part (b) of Definition 3.2 implies (3.3).
Now, we suppose that F is B-differentiable at x and (3.3) holds. Then for all
Smoothing Methods and Semismooth Methods 11
Hence part (b) of Definition 3.2 holds. Moreover, we have
By Theorem 2.9, part (a) of Definition 3.2 holds and so F is semismooth at x.
Theorem 3.3 implies that the definition of semismoothness used here coincides
with the definition of Qi-Sun [35] in finite dimensional spaces if we take a single-valued
selection of the Clarke Jacobian or the B-subdifferential as the slanting function.
To illustrate Theorem 3.3, we consider the system of "min" nonsmooth equations
in R n
where p and q are continuously differentiable functions from R n into itself. This
system is equivalent to the complementarity problem
Chen, Qi and Sun showed [9] that every smoothing function f(x; ffl) in the Chen-
Mangasarian smoothing function family [5] for the nonsmooth function F satisfies
(3.2). In particular, for
where ff 2 [0; 1] is dependent on the choice of a smoothing function. Such f
belongs to the set
at every point x 2 R n . (See [33] for @C F (x):) Hence, every smoothing function in the
Chen-Mangasarian smoothing function family satisfies the slant derivative consistency
property in R n . Moreover, the associated slant derivative is, for
which is bounded, nonempty and satisfies
Furthermore, the following fact is known [35]
On the other hand, by Proposition 2.8, we know
Hence, the nonsmooth function F is semismooth in the sense of Definition 3.2.
Now we consider superlinearly convergent Newton-type methods for nonsmooth
equations with slanting differentiable operators.
Nashed and L.Qi
Theorem 3.4. Suppose that F is slantly differentiable at a solution x   of (1.1).
Let f o be a slanting function for F at x   and kf in a neighborhood N
of x   , where M is a positive constant. Then the iterative sequence fx k g generated by
the Newton-type method
superlinearly converges to x   in a neighborhood N 0 of x   . Here A(x) 2 L(X; Y ) and
Proof. By Definition 2.1 and the Banach Lemma [29], there is a neighborhood N 0
of x   , N 0 ae N , and positive constants M
for any x 2 N 0 , A(x) is nonsingular and kA(x)
and
Therefore, starting from any x 0 2 N 0 the Newton method (3.4) is well defined and
the successive iterates satisfy the following inequalities:
Hence the sequence fx k g converges to x   : Moreover, using Definition 2.1 and (3.5),
the inequalities above imply
Using Theorem 3.4 and Proposition 2.10, we can immediately obtain the following
theorem.
Theorem 3.5. Suppose that F is slantly differentiable at a solution x   of (1.1).
Let f o be a slanting function for F at x   and kf in a neighborhood N
of x   , where M is a positive constant. Then the following statements hold.
(a) The Newton-type method (1.6) superlinearly converges to x   in a neighborhood
N 0 of x   .
(b) If f : D \Theta R++ ! Y is a smoothing function of F which satisfies the slant derivative
consistency property (3.2) in N , then the smoothing Newton method
superlinearly converges to x   in a neighborhood N 0 of x   .
(c) If F is semismooth at x   , then the semismooth Newton method
superlinearly converges to x   in a neighborhood N 0 of x   .
Smoothing Methods and Semismooth Methods 13
4. An application to a class of nonsmooth elliptic partial differential
equations.
Let\Omega ae R 2 be a bounded region with piecewise smooth boundary
@\Omega
and let W be the class of functions
from\Omega to R satisfying
Z
R) be the space of functions in W endowed with the norm
Z
\Omega ju(x)jdx:
We consider the following nonsmooth Dirichlet problem
ae
in\Omega
R is a continuous function.
Let
Z
\Omega G(x; y)P (u(y))dy;
where G is the Green function for the boundary value problem (e.g. see [42])
ae
in\Omega
is the Dirac "generalized function" at y in \Omega\Gamma
The nonsmooth integral equation
is equivalent to the nonsmooth Dirichlet problem (4.1).
Theorem 4.1. Suppose that smoothing function of P
satisfying
where - is a positive constant. Then
Z\Omega
is a smoothing function of F and
where
x2\Omega Z\Omega
14 X.Chen, Z. Nashed and L.Qi
Proof. It is easy to see that f is continuously differentiable with respect to u and
Z
Moreover
Z
Z
Z
\Omega kG(x; y)kdy
-ffl:
Theorem 4.2. Suppose that P is slantly differentiable at u, and let p o be a
slanting function for P at u. Then F is slantly differentiable at u, and
Z
\Omega G(x; y)p
is a slanting function for F at u
Proof. Using the definition of f o as given in the statement of the theorem, we
have
'Z
Z\Omega
Z\Omega
'Z\Omega
slanting function for P at u, the above equality implies that f o is a
slanting function for F at u.
By properties of integrals, we have the following proposition.
Proposition 4.3. (1) If smoothing function of P which
satisfies the slant derivative consistency property (3.2) at u, then
is a smoothing function of F which satisfies the slant derivative consistency property
(3.2) at u
(2) If P is semismooth at u, then F is semismooth at u.
The above results demonstrate that superlinearly convergent smooth methods or
semismooth methods can be developed for the nonsmooth Dirichlet problem (4.1).
For example, we consider
that is
ae
where - and ff are constants. Let
Smoothing Methods and Semismooth Methods 15
Then p is a smoothing function of P in X , and satisfies the slant derivative consistency
property in X since
lim
ffl!0
ffl!0
otherwise:
We next show that p o is a slant function for P in X . In fact, for any u; h(6=
we have
Z
-f
Z
Z
u(x)+h(x)=ff
Note that
Letting khk ! 0, we have
khk
f
Z
Z
u(x)+h(x)=ff
dx
0:
Moreover, we can show that P is semismooth in X . Hence, by using
Z\Omega
we can obtain superlinearly convergent smoothing methods and semismooth methods
for the nonsmooth Dirichlet problem
ae
in\Omega
This nonsmooth Dirichlet problem is related to MHD (magnetohydrodynamics) equilibria
[18].
We report numerical results for the following example.
Example 4.1.
ae
where OE(0; problem
has an exact solution
ae
Z. Nashed and L.Qi

Table
Numerical result of Example 4.1: kFn
We use method (1.6) with the five-point finite difference method. The stopping
criterion is kFn Here Fn is the finite difference approximation function
with grids n. We report the value of kFn (x)k 1 at the last five iterations.
Nonsmooth optimization and operator equations involving nonsmooth operators
are becoming crucial in various areas of computational and applied mathematics,
for example in nonsmooth mechanics [21, 22, 30], optimal design of electromagnetic
devices [21], ill-posed problems involving nonsmooth operators and variational inequalities
[19, 24], bounded variation regularization and nondifferentiable optimization
problems in nonreflexive spaces [26, 27], inverse source problems [25], free boundary
problems [28], multi-body system identification [1], and nonlinear complementary
problems (see [8] and references cited therein). Various classes of these problems can
be reformulated as nonsmooth equations with locally Lipschitzian operators. Hence
the smoothing methods and semismooth methods studied in this paper can be applied
to these problems.

Acknowledgement

. This research was initiated while the first author worked in
the School of Applied Mathematics at the University of New South Wales where the
second author visited in 1993 and 1998. We would like to express our appreciation
for the support.



--R

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The global linear convergence of a non-interior path-following algorithm for linear complementarity problem

Smooth approximations to nonlinear complementarity problems
A class of smoothing functions for nonlinear and mixed complementarity problems
Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equa- tions
Global and superlinear convergence of inexact Uzawa methods for saddle point problems with nondifferentiable mappings
Convergence of Newton's method for singular smooth and nonsmooth equations using adaptive outer inverses
Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities
Convergence domains of certain iterative methods for solving nonlinear equations
On homotopy-smoothing methods for variational inequalities
Optimization and Nonsmooth Analysis
Solution of monotone complementarity problems with locally Lipschitzian functions


Identification of the support of nonsmoothness
Multilevel algorithms for constrained compact fixed point prob- lems
Finite element analysis of a nondifferentiable nonlinear problem related to MHD equilibria
Regularization of nonlinear ill-posed variational inequalities and convergence rates
Semismooth and semiconvex functions in constrained optimization
Topics in Nonsmooth Mechanics
Mathematical Theory of Hemivariational Inequalities and Applications
Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear function analysis
On nonlinear ill-posed problems II: Monotone operator equations and monotone variational inequalities
Stable approximation of nondifferentiable optimization problems with variational inequalities
Stable approximation of a minimal surface problem with variational inequalities
Least squares and bounded variation regularization with non-differentiable functionals

Iterative Solution of Nonlinear Equations in Several Variables
Inequality Problems in Mechanics and Applications
motivation and algorithms
Convergence analysis of some algorithms for solving nonsmooth equations

A globally convergent successive approximation method for severely non-smooth equations
A nonsmooth version of Newton's method
Computational Solution of Nonlinear Operator Equations
Approximation of a nondifferentiable nonlinear problem related to MHD equilibria
A unified convergence theory for a class of iterative process
Local structure of feasible sets in nonlinear programming.
Newton's method for a class of nonsmooth functions
On concepts of directional differentiability

--TR

--CTR
B. Mermri , X. Chen, On characterizations and regularity of the solution of bilateral obstacle problems, Journal of Computational and Applied Mathematics, v.152 n.1-2, p.333-345, 1 March
Xiaojun Chen, Applications of smoothing methods in numerical analysis and optimization, Focus on computational neurobiology, Nova Science Publishers, Inc., Commack, NY, 2004
nonsmooth elliptic partial differential equations;superlinear convergence;semismooth methods;nondifferentiable operator equation;smoothing methods