On Two Applications of H-Differentiability to Optimization and Complementarity Problems.
In a recent paper, Gowda and Ravindran (Algebraic univalence theorems for nonsmooth functions, Research Report, Department of Mathematics and Statistics, University of Maryland, Baltimore, MD 21250, March 15, 1998) introduced the concepts of H-differentiability and H-differential for a function f : Rn  Rn and showed that the Frchet derivative of a Frchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function, the Bouligand subdifferential of a semismooth function, and the C-differential of a C-differentiable function are particular instances of H-differentials.In this paper, we consider two applications of H-differentiability. In the first application, we derive a necessary optimality condition for a local minimum of an H-differentiable function. In the second application, we consider a nonlinear complementarity problem corresponding to an H-differentiable function f and show how, under appropriate conditions on an H-differential of f, minimizing a merit function corresponding to f leads to a solution of the nonlinear complementarity problem. These two applications were motivated by numerous studies carried out for C1, convex, locally Lipschitzian, and semismooth function by various researchers.
Introduction
In a recent paper [10], Gowda and Ravindran introduced the concepts of the H-
differentiability and H-differential for a function f They showed that
Fr'echet differentiable (locally Lipschitzian, semismooth, C-differentiable) functions
are H-differentiable (at given  x) with an H-differential given by frf(x)g (respec-
tively, @f(x), @B f(x), C-differential). In their paper, Gowda and Ravindran investigated
the injectivity of an H-differentiable function based on conditions on
H-differentials. Also, in [25], H-differentials were used to characterize P(P 0 )-
functions.
In this paper, we consider two applications of H-differentiability. In the first
application, we derive a necessary optimality condition for a local minimum of an
H-differentiable real valued function. Specifically, we show in Theorem 3 that if x
is a local minimum of such a function f , then
an H-differential of f at x   .
In the second application, we consider a nonlinear complementarity problem
NCP(f) corresponding to an H-differentiable function f
such that
By considering an NCP function \Phi associated with NCP(f) so that
x solves NCP(f);
and the corresponding merit function
in this paper (see Sections 6,7, and 8), we show how, under appropriate P 0 (P,
regularity)-conditions on an H-differential of f , finding local/global minimum of
\Psi (or a 'stationary point' of \Psi) leads to a solution of the given nonlinear complementarity
problem. Our results unify/extend various similar results proved in the
literature for C 1 , locally Lipschitzian, and semismooth functions [1], [5], [6], [7], [8],
[9], [11], [12], [13], [14].
2. Preliminaries
We regard vectors in R n as column vectors. We denote the inner-product between
two vectors x and y in R n by either x T y or hx; yi. Vector inequalities are interpreted
componentwise. For a set E ' R n , co E denotes the convex hull of E and co E
denotes the closure of co E. For a differentiable function f : R n
denotes the Jacobian matrix of f at  x. For a matrix A, A i denotes the ith row of
A.
A function OE is called an NCP function if OE(a;
0: For the problem NCP(f ), we define
APPLICATIONS OF H-DIFFERENTIABILITY 3
(2)
and, by abuse of language, call \Phi(x) an NCP function for NCP(f ).
We now recall the following definition and examples from Gowda and Ravindran
[10].
Definition 1. Given a function f
where\Omega is an open set in
R n and x   2 \Omega\Gamma we say that a nonempty subset T (x   ) (also denoted by T f (x   )) of
R m\Thetan is an H-differential of f at x   if for every sequence fx k g
'\Omega converging to
x   ; there exist a subsequence fx k j g and a matrix A 2 T (x   ) such that
We say that f is H-differentiable at x   if f has an H-differential at x   .
A useful equivalent definition of an H-differential T (x   ) is: For any sequence x k :=
all k, there exist convergent subsequences
lim
Remarks As noted by a referee, it is easily seen that if a function f
R m is H-differentiable at a point  x, then there exist a constant L ? 0 and a
neighbourhood B(x; ffi) of
x with
Conversely, if condition (4) holds, then T (x) := R m\Thetan can be taken as an H-
differential of f at  x. We thus have, in (4), an alternate description of H-differentiability.
But, as we see in the sequel, it is the identification of an appropriate H-differential
that becomes important and relevant.
any function locally Lipschitzian at
x will satisfy (4). For real valued func-
tions, condition (4) is known as the 'calmness' of f at
x. This concept has been
well studied in the literature of nonsmooth analysis (see [24], Chapter 8).
As noted in [10], (i) any superset of an H-differential is an H-differential, (ii)
H-differentiability implies continuity, and (iii) H-differentials enjoy simple sum,
product and chain rules.
We include the following examples from [10].
differentiable at x   2 R n , then f is H-
differentiable with frf(x   )g as an H-differential.
Example 2 Let f
locally Lipschitzian at each point of an open
Let\Omega f be the set of all points
in\Omega where f is Fr'echet differentiable. For
denote the Bouligand subdifferential of f at x   . Then, the (Clarke) generalized
Jacobian [2]
is an H-differential of f at x   .
Example 3 Consider a locally Lipschitzian function f
that is
semismooth at x
2\Omega [17], [20], [22]. This means for any sequence x k ! x   , and
for any V k 2 @f(x k );
Then the Bouligand subdifferential
is an H-differential of f at x   . In particular, this holds if f is piecewise smooth,
i.e., there exist continuously differentiable functions
Example 4 C-differentiable in a neighborhood D of x   .
This means that there is a compact upper semicontinuous multivalued mapping
n\Thetan satisfying the following condition at any
a 2 D: For
Then, f is H-differentiable at x   with T (x   ) as an H-differential. See [21] for
further details on C-differentiability.
We recall the definitions of P 0 and P-functions (matrices).
Definition 2. For a function f : R n ! R n , we say that f is a P 0
(P)-function if,
for any x 6= y in R n ,
A matrix M 2 R n\Thetan is said to be a P 0 (P)-matrix if the function
is a P 0
(P)-function or equivalently, every principle minor of M is nonnegative
(respectively, positive [3]).
APPLICATIONS OF H-DIFFERENTIABILITY 5
We note that every monotone (strictly monotone) function is a P 0 (P)-function.
The following result is from [18] and [25].
Theorem 1 Under each the following
function.
(a) f is Fr'echet differentiable on R n and for every x 2 R n , the Jacobian matrix
rf(x) is a P 0 (P)-matrix.
(b) f is locally Lipschitzian on R n and for every x 2 R n , the generalized Jacobian
@f(x) consists of P 0 (P)-matrices.
(c) f is semismooth on R n (in particular, piecewise affine or piecewise smooth)
and for every x 2 R n , the Bouligand subdifferential @B f(x) consists of P 0 (P)-
matrices.
(d) f is H-differentiable on R n and for every x 2 R n , an H-differential T f (x)
consists of P 0 (P)-matrices.
3. Necessary optimality conditions in H-differentiable optimization
In this section, we derive necessary optimality conditions for optimization problems
involving H-differentiable functions. We first consider the H-differentiability of
minimum/maximum of several H-differentiable functions.
Theorem 2 For be H-differentiable at x   with an
H-differential R be defined by
where I(x
is H-differentiable at x   with T f
as an H-differential. Also, a similar statement holds if 'min' in (6) is replaced by
'max'.
Proof. We prove the result for the min-function; the proof of the max-function is
similar. Consider a sequence fx k g converging to x   in R n . Then there exist l 2
ng and a subsequence fx k j g such that f(x k j
We have f(x   (by the continuity of f l and f ). Now because of the H-
differentiability of f l at x   , there is a subsequence of fx k j g, which we continue to
write as fx k j g for simplicity, and a matrix A l 2 T f l (x   ) such that
which leads to
that f is H-differentiable at x
with (defined in (7)) as an H-differential. This completes the proof.
Remark In the above theorem, we considered real valued functions. With obvious
modifications, one can consider vector valued functions. See Example 8 for an
Theorem 3 Suppose f : R n ! R and x   is a local optimal solution of the problem
min
If f is H-differentiable at x   and T (x   ) is any H-differential, then
Proof. Suppose, if possible, that 0 62 co T (x   closed and convex,
by the strict separation theorem (see p.50, [15]), there exists a nonzero vector d in
R n such that Ad ! 0 for all A 2 co T (x   From the H-differentiability of f , for the
sequence fx
k dg, there exist a subsequence fx
dg and
Ad:
Since f(x)  f(x   ) for all x near x   , we see that
Ad  0 reaching a contradiction.
Hence
Remarks When f is differentiable at x   with T (x   )g, the above optimality
condition reduces to the familiar condition rf(x   locally
Lipschitzian at
x, the above result reduces to Proposition 2.3.2 in [2] that
see also, Theorem 7 in [17].
The above theorem motivates us to define a stationary point of the problem min f(x)
as a point x   such that is an H-differential of f at x   .
By weakening this condition, we may call a point x   a quasi-stationary point (semi-
stationary point) of the problem min f(x) if
While local/global minimizers of min f(x) are stationary points, it is not clear how
to get or describe semi- and quasi- stationary points. However, as we shall see in
Sections 6, 7, and 8, they are used in formulating conditions for a point x   to be a
solution of a nonlinear complementarity problem.
We now describe a necessary optimality condition for inequality constrained optimization
problems.
APPLICATIONS OF H-DIFFERENTIABILITY 7
Theorem 4 Suppose that f and g i are real valued functions defined
on R n and x   is a local optimal solution of the problem
minimize f(x)
subject to g i (x)  0 for
Suppose that f and g i are H-differentiable at x   with H-differentials
respectively, by T f
and I(x
Proof. We see that x   is a local optimal solution of the problem
minimize f(x)
subject to g(x)  0: (10)
From Theorem 2, we see that g is H-differentiable with T g (x
as an H-differential. We have to show that
statement is false. Then by the strict separation theorem (see p.50, [15]), there
exists a nonzero vector d in R n such that Ad ! 0 for all A 2 T f
From the H-differentiability of f and g, for the sequence fx
k dg, there exist a
subsequence fx
dg, matrices
Ad
and
Bd:
From
we see that f(x
We reach a contradiction since x   is assumed
to be locally optimal to the given problem. Thus we have the stated conclusion.
4. H-differentials of some NCP functions associated with H-differentiable
functions
In this section, we describe the H-differentials of some well known NCP functions.
Example has an H-differential T (x) at  x 2 R n . Consider
the associated Fischer-Burmeister function [7]
where all the operations are performed componentwise. Let
Consider the set \Gamma of all quadruples (A; V; W; d) with A 2 T (x),
are diagonal matrices satisfying the conditions
and
when i 62 J(x)
when
arbitrary when i 2 J(x) and d 2
when i 62 J(x)
when
arbitrary when i 2 J(x) and d 2
We now claim that \Phi F (or \Phi for simplicity) has an H-differential at
x given by
To see this claim, let  x 1: By the H-
differentiability of f , there exist a subsequence ft k j
g of ft k g, d k j ! d, and A 2 T (x)
such that f(x
Let, for ease of notation,
d k j . With A and d, define V and W satisfying (11) and (12); let
We claim that \Phi(y j
To see this, we
fix an index i and show that \Phi i (y

loss
of generality, let 1. We consider two cases:
Case
In this case we have
T is the first row of the identity matrix and
APPLICATIONS OF H-DIFFERENTIABILITY 9
Case
Subcase
In this case,
and an easy calculation shows
In this case d
These arguments prove that \Phi i (y

holds for all i.
Thus we have the H-differentiability of \Phi with S(x) as an H-differential.
Remarks We observe that in the above example, if T (x) consists of P-matrices
then S(x) consists of P-matrices. To see this, suppose that every A 2 T (x) is a P-matrix
and consider any A is a P-matrix, there exists an
index j with x j 6= 0 such that x j in (12) are nonnegative
and their sum is positive, x j [Bx]
It follows that B is a P-matrix.
This observation together with Theorem 1 says that if T (x) consists of P-matrices
then the function \Phi F is a P-function. (In fact, \Phi F is a P-function
whenever f is a continuous P-function, see [23].)
We note that S(x) may not consist of P-matrices if f is merely a P-function on
R n . This can be seen by the following example. Let
P-function and \Phi F
. By a simple calculation, we see that
the f2; 0g is an H-differential of \Phi F at zero and that it contains a singular object.
Example 6 In the previous example, we described the H-differential of Fischer-
Burmeister function. A similar analysis can be carried out for the NCP function
[13]
where  is a fixed parameter in (0; 4). We note that when reduces to
the Fischer-Burmeister function, while as  ! 0; \Phi(x) becomes
Let
An H-differential of \Phi in (15) is given by
is the set of all quadruples (A; V; W; d) with A 2 T (x),
are diagonal matrices satisfying the conditions
and
when i 62 J(x)
when
arbitrary when i 2 J(x) and (d
when i 62 J(x)
when
arbitrary when i 2 J(x) and (d
Example 7 The following NCP function is called the penalized Fischer-Burmeister
function [1]
is a fixed parameter. Let
For \Phi  in (18), a straightforward calculation shows that an H-differential is given
by
is the set of all quadruples (A; V; W; d) with A 2 T (x),
are diagonal matrices with
when
when
arbitrary when i 2 J(x) and d 2
APPLICATIONS OF H-DIFFERENTIABILITY 11
when
when
arbitrary when i 2 J(x) and d 2
The above calculation relies on the observation that the following is an H-
differential of the one variable function t 7! t + at any  t:
\Delta(
Example 8 For an H-differentiable function f consider the NCP
function
We claim that the H-differential of \Phi is given by
To see this claim, let x k !  x: By the H-differentiability of f , there exist a sub-sequence
of fx k g, which we continue to write as fx k g for simplicity, and a matrix
By considering a
suitable subsequence, if necessary, we may write ng as a disjoint union of
sets ff and fi where
Put
We show that \Phi(x k
To see this, we fix an
index j and show that \Phi j
simplicity). We have two cases:
Case
\Theta \Phi(x k
\Theta f(x k
Case It is easy to verify that \Phi 1
This proves the above claim.
5. The H-differentiability of the merit function
In this section, we consider an NCP function \Phi corresponding to NCP(f) and let
Theorem 5 Suppose \Phi is H-differentiable at  x with S(x) as an H-differential.
Then \Psi := 1jj\Phijj 2 is H-differentiable at  x with an H-differential given by
Proof. Consider a sequence fx
there exist d
We have
ff \Gamma2 h\Phi(x); \Phi(x)i
ff
\Gamma2
This gives us
lim
This completes the proof.
6. Minimizing the merit function under P 0
-conditions
For a given function f consider the associated NCP function \Phi and
the corresponding merit function It should be recalled that
x solves NCP(f):
One very popular method of finding zeros of \Phi is to find the local/global minimum
points or 'stationary' points of \Psi. Various researchers have shown, under certain
that when f is continuously differentiable or more generally locally
Lipschitzian, 'stationary' points of \Psi are the zeros of \Psi. In what follows, starting
with an H-differentiable function f , we show that under appropriate conditions, a
vector  x is a solution of the NCP(f) if and only if zero belongs to one of the sets
Theorem 6 Suppose f : R n ! R n is H-differentiable at
x with an H-differential
is an NCP function of f: Assume that \Psi := 1jj\Phijj 2 is H-differentiable
at
x with an H-differential given by
APPLICATIONS OF H-DIFFERENTIABILITY 13
Further suppose that T (x) consists of P 0 -matrices. Then
Proof. Clearly, implies that T \Psi by the description of T \Psi (x):
Conversely, suppose that 0 2 T \Psi (x), so that for some \Phi(x) T [V A +W
yielding A T y Note that for any index
which
case y i
0, contradicting the P 0 -property of A. We conclude that
In the next two successive theorems, we replace the condition
conditions relaxations come at the
expense of imposing either stronger or different conditions on the H-differential of
f .
First we recall a definition from [26].
Definition 3. Consider a nonempty set C in R n\Thetan . We say that a matrix A is
a row representative of C if for each index row of A is the
ith row of some matrix C 2 C. We say that C has the row-P 0 -property (row-P-
property) if every row representative of C is a P 0 -matrix (P-matrix). We say that
C has the column-P 0 -property (column-P-property) if C Cg has the
We have the following result from [26].
Proposition 1 A set C has the row-P 0 -property (row-P-property) if and only if
for each nonzero x in R n there is an index i such that x i
for all C 2 C.
A simple consequence of this proposition is the following.
Corollary 1 The following statements hold:
(i) Suppose the set of matrices fA has the row-P 0 -property. Then
for any collection fV of nonnegative diagonal matrices, the sum
A
is a P 0 -matrix. In particular, any convex combination of the A i s is a P 0 -matrix.
(ii) Suppose the set of matrices fA has the row-P-property. Then for
any collection fY of nonnegative diagonal matrices with
A
is a P-matrix.
Proof. (i) Let x 6= 0 in R n . By the above proposition, there exists an index
i such that x i 6= 0 and x i
This proves the P 0 -property of A   . By specializing
we get the additional statement.
(ii) Let x 6= 0: By Proposition 1, there exists an index i such that x i 6= 0 and
Now we have x i
the terms of the above sum are nonnegative. If (Z
which means that
we see that x i
is a P-matrix.
Remark We note that the implications in the above corollary can be reversed: if
every A   in (i) ((ii)) is a P 0 -matrix (respectively, P-matrix), then fA
has the row-P 0 -property (respectively, row-P-property). Peng [19] proves results
similar to Corollary 1 under additional/different hypotheses.
Theorem 7 Suppose is H-differentiable at
x with an H-differential
T (x). Suppose that \Psi is H-differentiable at
x with an H-differential given by
Further suppose that T (x) has the row-P 0 -property. Then
Proof. Suppose and we have co T \Psi
versely, suppose 0 2 co T \Psi (x). Then by Carath'eodory's theorem [15], there exist
that
where
Lg. We rewrite (22) as
APPLICATIONS OF H-DIFFERENTIABILITY 15
reduces to
where
diagonal matrix
and jZ i the equality unchanged if we
replace Y i by jY i j and Z i by jZ i j, we may assume that Y i and Z i are nonnegative
for all i. Now suppose, if possible, that By the above corollary, the
matrices M and M T are P 0 -matrices. Therefore, there exists an index i   such that
0. From \Phi(x) i
6= 0, we see that (W j
and so (Z
0: But
(\GammaZ   u) i
is clearly a contradiction since u i
This proves that
Remarks We note that Theorems 6 and 7 are applicable to the Fischer-Burmeister
function
This is because, the set T \Psi (x)
described in Theorems 6 and 7 is a superset of the H-differential T \Psi
described in Example 5. (Note that [\Phi F (x)] i
J(x) and hence from (12), v Similarly, we see that Theorems 6 and 7 are
applicable to the following NCP functions:
(Clarification Example
(Clarification Example 7)
We state the next result for the Fischer-Burmeister function \Phi. However, as in
Theorems 6 and 7, it is possible to state a very general result for any NCP function
\Phi. For simplicity, we avoid dealing in such a generality.
Theorem is H-differentiable at
x with an H-differential
T (x) which is compact and having the row-P 0 -property. Let \Phi be the Fischer-
Burmeister function as in Example 5 and \Psi := 1jj\Phijj 2 . Let S(x) and T \Psi (x) be as
in Example 5 and Theorem 5. Then the following are equivalent:
(a)
x is a local minimizer of \Psi.
(c)
x solves NCP(f).
Proof. The implication (a) ) (b) follows from Theorem 3. The implication
(c) ) (a) is obvious. We now prove that (b) ) (c). Suppose 0 2 co T \Psi (x) and
assume that there exists a sequence fC k g of matrices in
co S(x) such that Now each C k is a convex combination of at most
matrices of the form V A+W 2 S(x) where A 2 T (x), V and W satisfy (11)
and (12). Since T (x) is compact and the entries of V and W vary over bounded
sets in R, we may assume that C k ! C where C is a convex combination of at most
matrices of the form V A +W where A 2 T (x), V and W are nonnegative
diagonal matrices satisfying a condition like (11) with
and
when
g. From
an equation similar to (22) but now with V i , A i , and W i in place of V i , A i , and
respectively. By repeating the argument given in the proof of the previous
theorem, we arrive at a contradiction. Hence proving (b) ) (c) .
We now state two consequences of the above theorems for the Fischer-Burmeister
function (for the sake of simplicity).
differentiable and \Phi(x) be the Fischer-Burmeister
function and
x is a local minimizer to \Psi
if and only if  x solves NCP(f).
This corollary is seen from the above theorem by taking T frf(x)g. If we
assume the continuous differentiability of f in the above corollary, we get a result
of Facchinei and Soares [5]: For a continuously differentiable P 0 -function f , every
stationary point of \Psi solves NCP(f ). (This is because, when f is C 1 , \Psi becomes
continuously differentiable, see Prop. 3.4 in [5].) See [9] for the monotone case.
locally Lipschitzian. Let \Phi be the Fischer-
Burmeister function and the equivalence
holds under each of the following conditions.
(a) @f(x) consists of P 0 -matrices;
(b) @B f(x) has the row-P 0 -property.
Proof. The stated equivalence under (a) has already been established by Fischer
[8]. In fact, by applying Theorem 6 with T f using his result
that @\Psi(x) ' T \Psi (x) for all x, we get the equivalence in (a). Now to see the
equivalence under (b), assume (b) holds. Then by Corollary 1, every matrix in
Now we have condition (a) and hence the
stated equivalence.
Remark The condition (b) in the above corollary might be especially useful when
the function f is piecewise smooth in which case @B f(x) consists of a finite number
of matrices.
APPLICATIONS OF H-DIFFERENTIABILITY 17
7. Minimizing the merit function under P -conditions
The following theorem is similar to Theorem 6.
Theorem 9 Suppose f : R n is H-differentiable at
x with an H-differential
is an NCP function of f: Assume that \Psi := 1jj\Phijj 2 is H-differentiable
at
x with an H-differential given by
Further suppose that T (x) consists of P-matrices. Then
Proof. Suppose 0: Then by description of T \Psi (x); we have T \Psi
Conversely, suppose that 0 2 T \Psi (x), so that for some \Phi(x) T [V A +W
We claim that 0: Suppose, if possible,
that \Phi(x) 6= 0: If which leads to
a contradiction since for some
Hence y 6= 0 and
contradicting the P-property of A. Hence
Theorem is H-differentiable at
x with an H-differential
T (x). Suppose that \Psi is H-differentiable at
x with an H-differential given by
Further suppose that T (x) has the row-P-property. Then
Proof. The proof is similar to that of Theorem 7. To show that 0 2 co T \Psi (x) )
proceed as in the proof of Theorem 7. We have statements (22)
and (23) in our new setting where we may assume (as before) that Y i and Z i
are nonnegative for all i. Since
taking
we see that the matrix in (23) is nonsingular. It follows that
Remark We note that Theorems 9 and 10 are applicable to the min-function \Phi of
Example 8.
8. Minimizing the merit function under regularity (strict regularity)
conditions
We now generalize the concept of a regular (strictly regular) point [14] in order to
weaken the hypotheses in the Theorems 6 and 7.
For a given H-differentiable function f and
we define the following
subsets of I = ng.
Definition 4. Consider f ,
x, and the index sets as above. Let T (x) be an H-
differential of f at
x. Then the vector  x 2 R n is called a regular (strictly regular)
point of f with respect to T (x) if for every nonzero vector z 2 R n such that
z
there exists a vector s 2 R n such that
Theorem is H-differentiable at
x with an H-differential
\Phi be an NCP function satisfying the following conditions:
Suppose \Psi is H-differentiable with an H-differential given by
x is a regular point if and only if  x solves NCP(f).
Proof. Suppose that 0 2 T \Psi (x) and
x is a regular point. Then for some
APPLICATIONS OF H-DIFFERENTIABILITY 19
We claim that 0: Assume the contrary that
x is not a solution of NCP(f ).
x is a regular point, and y and
z have the same sign, by taking a vector s 2 R n satisfying (25) and (26), we have
and
contradict (30). Hence  x is a solution to NCP(f ). The 'if'
part of the theorem follows easily from the definitions.
Remark Theorem 11 is applicable to the NCP functions of Examples 5, 6 and 7.
A slight modification of the above theorem leads to the following result.
Theorem is H-differentiable at
x with an H-differential
\Phi be an NCP function satisfying the following conditions:
Suppose \Psi is H-differentiable with an H-differential given by
x is a strictly regular point if and only if
x solves NCP(f).
Proof. The proof is similar to that of Theorem 11.
Concluding Remarks
In this paper, we considered two applications of H-differentiability. The first
application dealt with the necessary optimality condition in H-differentiable op-
timization. In the second application, for a nonlinear complementarity problem
corresponding to an H-differentiable function, with an associated NCP function
\Phi and a merit function described conditions under which every
global/local minimum or a stationary point of \Psi is a solution of NCP(f ). We
would like to note here that similar methodologies can be carried out for other
merit functions. For example, we can consider the Implicit Lagrangian function of
Mangasarian and Solodov [16]:
\Theta
fixed parameter and x   y is the Hadamard (=componentwise)
product of vectors x and y. (In [16], it is shown that
By defining the merit function
and formulating the concept of strictly regular point, we can extend the results of
[4] for H-differentiable functions.
Our results recover/extend various well known results stated for continuously
differentiable (locally Lipschitzian, semismooth, C-differentiable) functions.

Acknowledgements

We thank the referees for their constructive comments.



--R

"A Penalized Fischer-Burmeister NCP-Function: Theoretical Investigation and Numerical Results,"
Optimization and Nonsmooth Analysis
The Linear Complementarity Problem
"On Unconstrained and Constrained Stationary Points of the Implicit Lagrangian,"
"A New Merit Function for Nonlinear Complementarity Problems and Related Algorithm,"
"Regularity Properties of a Semismooth Reformulation of Variational Inequalities,"
"A Special Newton-Type Optimization Method,"
"Solution of Monotone Complementarity Problems with Locally Lipschitzian Functions,"
"On the Resolution of Monotone Complementarity Problems,"
"Algebraic Univalence Theorems for Nonsmooth Functions,"
"A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems,"
"Unconstrained Minimization Approaches to Nonlinear Complementarity Prob- lems,"
"A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems,"
"A Semismooth Equation Approach to the Solution of Nonlinear Complementarity Problems,"

"Nonlinear Complementarity as Unconstrained and Constrained Minimization,"
"Semismooth and Semiconvex Functions in Constrained Optimization,"
"On P- and S- Functions and Related Classes of N-Dimensional Nonlinear Mappings,"
"A Smoothing Function and its Applications,"
"Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations,"
"C-differentiability, C-differential Operators and Generalized Newton Methods,"

"Regularization of P0 -functions in Box Variational Inequality Problems,"
Variational Analysis
"On Characterizations of P- and P 0 - Properties in Nonsmooth Functions,"
"On Some Properties of P-matrix Sets,"
--TR

--CTR
M. A. Tawhid, On the local uniqueness of solutions of variational inequalities under H-differentiability, Journal of Optimization Theory and Applications, v.113 n.1, p.149-164, April 2002
Jong-Shi Pang , Defeng Sun , Jie Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Mathematics of Operations Research, v.28 n.1, p.39-63, February
merit function;nonlinear complementarity problem;locally Lipschitzian function;NCP function;h-differentiability;generalized Jacobian