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Michel-Penot subdifferential and Lagrange multiplier rule
Triloki Nath and S. R. Singh*
Department of Mathematics, Faculty of Science
Banaras Hindu University, VARANASI-221005
INDIA.
*Corresponding author email: singh_shivaraj@rediffmail.com
Abstract:-In this paper, we investigate some properties of Michel Penot subdifferentials of locally
Lipschitz functions and establish Lagrange multiplier rule in terms of Michel-Penot subdifferentials
for nonsmooth mathematical programming problem.
Key-Words: Nonsmooth optimization; approximate subdifferentials; generalized gradient; Michel
Penot subdifferential; Banach space.
1 Introduction
In this paper, we consider a mathematical
programming problem on a Banach space and
derive necessary conditions in Lagrange
multiplier form. The main tool in this paper is
Michel-Penot subdifferential of Locally
Lipschitz function defined on Banach space.
Most of extensions of Lagrange multiplier
rules for various problems of nonsmooth
optimization are given in terms of generalized
gradient of Clarke or certain approximate
subdifferentials (see [5], [2]). The extensions
involving smaller, particularly convex valued,
subdifferentials with certain calculus rules such
as those introduced by Michel and Penot [14],
Treiman [13], Dolecki [15] and Frankowaska [6]
were obtained only for problems containing
finitely many inequality constraints with no
equality constraints at all. The past studies
reveal that to obtain more precise and more
selective first order necessary conditions, the
size of subdifferential must be smaller.
The reason for this disparity is quite obvious
that the small convex-valued subdifferentials
lack upper semi-continuity, which is needed to
handle equality constraints. Further the
approximate subdifferentials and generalized
gradients are smallest among upper
semicontinuous and convex-valued upper
semicontinuous subdifferentials with calculus
(see [8]). Since the subdifferentials of Michel-
Penot and Treiman are naturally connected with
the Gateaux and Frechet derivatives (in the sense
that the function is differentiable in the
corresponding sense if the subdifferential is a
singleton), it is reasonable to ask whether it is
possible to obtain Lagrange multiplier rule
involving these subdifferentials for problems
with finitely many equality constraints. Ioffe [3]
has given affirmative answer of this question
which is a particular case of the Theorem 4 in
[2], involving weak prederivative, a concept
associated with Ioffe's fan theory (see [1]).
Ioffe [3] has shown that a Lagrange
multiplier rule involving the Michel-Penot
subdifferential is valid for the problem (P):
Cx
njxh
mixgtosubject
xfMin
j
i,,..2,1;0)(
,,..2,1;0)(
(P)
where
X
is a Banach space, .),..2,1(, migf i
and .),..2,1( njhj are functions from
X
to
R
and Cis a closed convex subset of
X
.
We consider the above problem (P) with
C closed but not necessarily convex subset. The
Lagrangian function
L
for this problem is given
by
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).(),,()(,
)(,)(
),,,,(
xdKxh
xgxf
KxL
C
Where (.)
C
ddenotes the distance function of
closed setC, and denotes the Euclidean
norm on nm
R
1.
We state the following Multiplier rule in terms
of Michel-Penot subdifferential.
Theorem 1.1 Assume that ,..,2,1(,, ihgf ji
.),..2,1; njm
are locally Lipschitz at
x
. If
x
is a solution of (P), then for every
K
sufficiently
large, there exist 0,0 i
and R
j
not all
zero such that 0)( xgii
and
)(),,(
)()()(0 11
xdK
xhxgxf
C
n
jjj
m
iii
where
denotes the Michel-Penot
subdifferential.
The Michel-Penot subdifferential of a locally
Lipschitzian function is the principal part of
Clarke subdifferential. It coincides with the
Gateaux derivative at differentiable (in Gateaux
sense) points. Here, it is interesting to note that a
locally Lipschitz function can be determined by
its Michel-Penot subdifferential uniquely up to
an additive constant, though this can not be done
by its Clarke subdifferential, if the set of
abnormal point
)()(.. xfxfei
is not
negligible.
The Michel-Penot subdifferential of Gateaux-
differentiable function is singleton whereas
Clarke subdifferential may have more than one
point, e.g. the function RRf
:defined by
.0,0
,0,
1
sin
)( 2
xif
xif
x
x
xf
Then f is globally Lipschitz and differentiable
with derivative
.0,0
,0,
1
cos
1
sin2
)( xif
xif
xx
x
xDf
here
0)0()0( Dff
and
1,0)0( f
.
Analogous to Clarke's subdifferential was
motivated in stochastic programming (see Birge
and Qi [12]). As the objective function of a
stochastic programming problem is generally a
multi integral of several variables (Birge and Qi
[11]), and thus to avoid the computation of
unnecessary extraneous subgradients, the study
of Michel-Penot subdifferential of integral
functionals is fruitful.
In some way one may consider the present
work analogous to the existing general schemes
for Mathematical Programming; (see e.g. Clarke
[4], Halkin [7]) but the treatment of the problem
is completely different. In fact in our present
study not only results and realms of applicability
are different but there is a fundamental
difference in the approach too. Here, instead of
postulating the existence of certain convex
and/or linear approximations or use of Ioffe's fan
theory [3], we need the Michel-Penot
subdifferential which is intrinsic to the problem.
Thus our work is closer in spirit to the classical
theory (with derivatives) and to the convex
analysis treatment (with subgradients).
The paper is organized as follows: In section
2, we discuss some fundamental notions of
nonsmooth analysis and reproduce the basic
properties of Michel-Penot directional derivative
and Subdifferential to make the study self
contained. In section 3, we investigate M-P
regularity (semiregularity for locally
Lipschitzian) and various calculus rules. Various
inclusion relationship has been established,
which in turn as equality under M-P regularity.
It is noteworthy that by the generalized
Rademacher's theorem equality holds almost
everywhere in finite dimensions and in separable
Banach space, relative to Haar measure. In
sections 4 and 5, we discuss Michel-Penot
subdifferential of integral functionals and a
general formula for point wise maxima of
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locally Lipschitz function on infinite index set.
In the last section, we employed these results to
establish multiplier rule in terms of Michel-
Penot subdifferential.
2 Preliminaries
In view of making the study self contained, we
need to reproduce the following notions of
nonsmooth analysis:
Definition2.1 Let RX
:
be a locally
Lipschitz function at,
x
then the Clarke
generalized directional derivative of
at a point
x
and in the direction Xd
, denoted by
);(
0dx
is given as:
tytdy
dx txy
)(- )(
suplim);( 0,
0
and the Clarke generalized gradient of
at
x
is
given by
XdxxdxXxx ,,);(:)( *0**
where *
X
denotes the topological dual of
X
.,. denotes the dual pairing between *
X
and
.X
Let C be a non empty subset of
X
and
consider its distance functions ,:
RXdC
defined by
.:inf)( CccxxdC
This function is not everywhere differentiable
but it is (globally) Lipschitz, with the Lipschitz
constant equal to 1.
Let Cx
be given, we say that Xv
is a
tangent vector to C at
x
if .0);(
0vxdC
The set of tangent vectors to C at
x
is a closed
convex cone in
X
called Clarke tangent cone of
Cat
x
and denoted by ).,( CxT That is
0);(|),( 0 vxdXvCxT C
and the Clarke normal cone is polar to ),,( CxT
defined as
.),(,0,|),( *** CxTvvxXxCxN
Let Xx
and RX
:
be a locally Lipschitz
function at
x
. Then the Michel-Penot
directional derivative of
at the point
x
in the
direction
x
, denoted by ),;( vx
is given by
ttwxtwtvx
vx tXw
)(- )(
suplimsup);( 0
and the Michel-Penot subdifferential of
at
x
is given by
XvxxvxXxx ,,);(:)( ***
It is known (see [14]) that when a function f is
Gateaux differentiable at
x
,
)()( xfxf .
The following properties of the Michel-Penot
directional derivatives and Michel-Penot
subdifferentials will be useful in the sequel.
Proposition2.1 Let Xx
and RXgf
:, be
locally Lipschitz functions at
x
, and the local
Lipschitz constant of f be f
L ,then the
following hold:
(i)The function );( vxfv
is finite,
positively homogeneous and subadditive on .X
(ii) As a function of );(, vxfv is Lipschitz
continuous with Lipschitz constant .
f
L
(iii) )(xf
is a nonempty, convex, weak -
compact subset of
X
and f
Lx
for
every )(xfx
, one has
.)(|,max);( xfvvxf
(iv) For every scalar ),())((, xfxf
and for every Xv
,
).;()();( vxfvxf
(v) )()())(( xgxfxgf
and ,);();();()( vxgvxfvxgf
the equalities hold if both f and
g
are MP-
regular.
(vi) If
x
is a local minima of f, then
)(0 xf
and .0);( Xvvxf
(vii) );( vxf is upper semi continuous as a
function of ).;( vx
Proof. The proof of (i) to (vi) may be seen in
Michel and Penot [14] and Birge and Qi [12].
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We prove (vii): Let i
x and i
v be arbitrary
sequences converging to
x
and
v
respectively.
For each
i
by definition of upper limit and
supremum, there exist i
y in
X
and
0
i
t, such
that
1
, iiii zztxy and
i
txy iii
1
,
Then
i
iiiiiii
i
iiiiii
i
iiiiii
ii
tztvtxfztvtxf
tztxfztvtxf
tztxfztvtxf
i
vxf
)()(
)()(
)()(1
);(
As the last term on right hand side of the above
expression is bounded in magnitude by vvK i,
thus taking upper limit, we get
i
iiiii
i
ii
i
tztxfztvtxf
vxf
)()(
suplim
);(suplim
Therefore
);(
)()(
suplimsup
);(suplim
vxf
tztxfztvtxf
vxf
i
iii
iXz
ii
i
This shows that );( vxf is upper semi-
Continuous as a function of ),( vx .
Lemma 2.1 Let i
x and i
be sequences in
X
and
X
such that .)( ii xf
Suppose that
i
x Converges to
x
, and that
is a cluster point
of i
in the weak-topology. Then )(xf
(i.e. the multifunction f
is weak-closed).
Proof. Let Xv
be given and there is a
subsequence v
i,
which converges to ,,v
since ,)( ii xf
then ,,);( vvxf ii
which implies, by upper semi continuity of
.),(.;
fthat
.,);( vvxf
3 M-P regularity and calculus rules
Analogous to Clarke regularity [5], semi-
regularity, which is a weaker notion, was
introduced by Birge and Qi [12] for locally
Lipschitz function. Ye [9] extended this notion
as M-P regularity to any function.
Definition 3.1 ([12], [9]) A function
RXf
:is said to be M-P regular (or
semiregular if f is locally Lipschitz) at
x
if (i)
the usual directional derivative );( vxf
exists finitely for all
v
in ,X
(ii) );();( vxfvxf
for all
v
in .X
Remark 3.1 If there is no ambiguity between
the two notions M-P regularity and
semiregularity. We use the word M-P regularity
for semiregularity (signifies local Lipschitzian
property is present) also. Note that every convex
function as well as every Gateaux differentiable
function is M-P regular.
We can prove the following Mean Value
Theorem, already proved by Borwein et. al [10],
analogously as Theorem 2.3.7 in Clarke
[5].Which is stronger than that of Clarke [5].
Theorem 3.1 (Mean-Value Theorem [10]).
Suppose RXUf
:be locally Lipschitz
on the open set U. Let
yx,be a line segment
inU. Then there exists a point
u
in
yx, such
that .,)()()( xyufxfyf
Chain Rules
We now intend to provide chain rules, for
Michel-Penot subdifferentials. Let n
RUh :
and RRg n:are locally Lipschitz function at
x
, so that RUhgf
:is also locally
Lipschitz at
x
. Then we have the following
chain rules:
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Theorem 3.2 (see [12]) Chain rule 1
))((
,)(|
co)( 1
___
xhg
xh
xf ii
n
iii
,
(where ___
co denotes weak- closed convex hull)
and equality holds under any one of the
following additional hypotheses:
(i)
g
is regular at ),(xh each i
h is M-P regular at
x
and every element of )(( xhg
has
nonnegative component. In this case it follows
that f is M-P regular.
(ii)
g
is M-P regular at ),(xh and h is Gateaux
differentiable. In this case it follows that f is
regular at
x
and ___
co is superfluous.
(iii)
g
is Gateaux differentiable at ),(xh and
1
n (in this case the ___
co is superfluous). )(
i
g
is M-P regular at ),(xh each i
his M-P
regular at
x
and every element of ))(( xhg
has nonnegative components, in this case also, it
follows that f is M-P regular.
Remark 3.2 The condition )(
iis weaker than
(i), an easy consequence of Theorem 2.3.9 in
Clarke [5] and the upper semicontinuity of
multifunction g
results in equality in this
case.
Theorem 3.3 (see [12]) Chain rule 2.
Assume that RRg n:is differentiable (for
locally Lipschitz functions on a finite
dimensional spaces, Gateaux differentiability is
the same as Frechet differentiability) and
n
RXh :is locally Lipschitz function then
:hgf
is Lipschitz near
x
, and one has:
)1(),())(()( xhxhgxf i
i.e.
)2(,
))((
,)(|
)( 1
xhg
xh
xf ii
n
iii
and equality holds, if each )(xhiis M-P regular
at
x
, and ))(( xhg
has nonnegative
components. In this case, it follows that f is M-
P regular at
x
.
(b) Let YXF
: (
Y
is another Banach space),
and let RYg
:. Suppose that
F
is Gateaux
differentiable at
x
and
g
is Lipschitz near
)(xF . Then Fgf
is Lipschitz near
x
, and
one has ),())(()( xDFxFgxf
Equality holds if
g
or
g
is MP-regular at
)(xF , in which case f or f
is also MP-
regular at
x
. Equality also holds if
F
maps
every neighborhood of
x
to a set which is dense
in a neighborhood of )(xF ( e. g. if )(xDF is
onto ).
Corollary 3.1 Let RYg
:be Lipschitz near
x
, and suppose that the space
X
, is
continuously embedded in
Y
, is dense in
Y
and
contains the point
x
. Then the restriction f of
g
to
X
is Lipschitz near
x
and
)()( xgxf
Proposition 3.1 (point wise maxima)
Suppose
nifi,...2,1| is a finite collection of
functions, each of which is Lipschitz near
x
.
The function RUf
:is defined as
.,...,2,1|)(max)( niufuf i
Then for any Ux
,
,)(|)()( xIixfcoxf i
where
.)()(|)( xfxfixI i If i
fis M-P
regular at
x
for each )(xIi
, then equality
holds and fis regular at
x
.
Proof. The proof follows from proposition
2.3.12 in [5]. The assertion regarding equality
and regularity follows from Theorem (i) of chain
rule 1.
Proposition 3.2 (Basic Calculus)
Suppose that 21,ff be Lipschitz near
x
. Then
21 ff
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and )0( 2
2
1f
f
f are Lipschitz near
x
, and
)()()(
)()())((
12
2121
productsxfxf
xfxfxff
)(0)(
)( )()()()(
)(
2
2
2
2112
2
1
quotientsxfif
xf xfxfxfxf
x
f
f
Also, if in addition 0)(,0)( 21 xfxf and if
21 ,ff are both M-P regular at
x
, then equality
holds in Product rule and 21 ff is M-P regular
at
x
. If in addition and 21,ff are M-P regular
at
x
, then equality holds
in Quotient rule and
2
1
f
f is M-P regular at
.
x
4 Michel-Penot subdifferential of
Integral Functionals
Suppose that U is an open subset of Banach
space
X
. Let ),,(
Tbe a positive measure
space. We consider a family of functions
TtRUft |: under following hypotheses:
(i) For each
x
in U, the map )(xft t
is
measurable;
(ii) For some integrable function RTk
:, for
all
x
and
y
in U and
t
in
T
, one has
.)()()( yxtkyfxf tt
We now invoke the Michel-Penot subdifferential
of the integral functional f on
X
given by
Ttdtxfxf )()()(
as
Ttdtxfxf )3()()()(
as given in [5], expression (3) (see [12]) has the
following interpretation:
For each
in )(xf
there is a map t
t
from
T
to
X
with )(xftt
a.e. (almost
everywhere relative to
); such that for every
v
in
X
, the function vt t,
is in ),(
1RXL
and .)(,,
Ttdtvv
Now, consider three cases:
(a)
T
is countable.
(b)
X
is separable.
(c)
T
is a separable metric space,
is a regular
measure, and the map )(uft t
is weak -
upper semicontinuous for each
u
in U.
Theorem 4.1. Let f be defined at some point
x
in U. Then f is defined and Lipschitz in U.
If at least one of (a),(b) or (c) is satisfied, then
formula (3) holds. Further, if )(
t
fis
M-P regular at
x
for each
t
, then fis M-P
regular at
x
and equality holds in expression (3).
Proof. The Lipschitz condition on Ufollows
from the hypotheses (i) and (ii) given above in
this section. In case (b) the formula (3) holds has
been shown by Birge and Qi [12], in case (a) or
(c) proof is similar to the proof of Theorem 2.7.2
of [5]. The proof for equality part is identical to
the corresponding part of Theorem 2.7.2 of [5]
using M-P regularity and M-P subdifferential
instead of regularity and Clarke subdifferential.
The advantage of Theorem 4.1 is that under
moderate conditions on
X
and
T
, the right
hand side of (3) is singleton almost everywhere
in .UAlmost everywhere means except a set of
Haar measure zero, that is there exists a Borel
probability measure,
on
X
such that
0)(
xN
for all Xx
, and we say N is a
Haar zero set.
In practical situations, for instance in control
theory, a different type of integral functional
occurs frequently is that
X
is space of functions
on
T
. When
T
is countable or
X
is restricted to
space of continuous functions on
T
(e.g.
1,0CX ) or finite dimensional, Theorem 4.1
will apply directly. However, not generally,
when
X
is a p
L
space. We state the following
Theorem for
p1.
Theorem 4.2 Let ),,(
Tbe a
- finite
positive measure space and
Y
a separable
Banach space, and ),( YTLpbe the space of
p
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integrable ( for 1
p, essentially bounded
)functions from
T
to
Y
. Suppose
X
is a closed
subspace of ),( YTLpand a family of functions
RYft: such that
(i) )(yft t
is measurable for each
y
in
Y
,
)(
ii there exists 0
, a function ),( RTLk q
(
q
is conjugate index to
p
defined as
,1
11 qp 1
qif
p
) such that for all
Tt
, for all ,)(, 21 Y
Btxyy
.)()()( 2121 yytkyfyf tt
Then, for any
x
satisfying the conditions,
Ttdtxfxf )()()(
is Lipschitz near
x
and
(3) holds. Further, if t
fis M-P regular at )(tx
for all Tt
, then equality holds in (3).
Proof. Similar to Theorems 2.7.3 and 2.7.5 of
Clarke [5], the measurability of
));(( vtxft t
for all )(tx and
v
follows
since the supremum over Xw
in the
definition of ));(( vtxft can be replaced by
supremum over countable dense subset of
Y
for
each fixed
t
. We use the fact that ));(( vtxft is
sup of countable limsup of measurable functions
over a countable set (see Birge and Qi [12],
Theorem 5.3, have considered n
R
Y
).
5 Pointwise Maxima-A General
Formula
We have already studied functionals which are
pointwise maxima of some finite indexed family
of functions. Now, we study much more
complex situation in which indexed set is
infinite.
Let
t
f be a family of functions on
TtX
, and
T
is a topological space. Suppose
there is some point
x
in
X
such that t
f is
locally Lipschitz at
x
for each Tt
. We
define a new type of partial MP-subdifferential,
in which it has also been taken care of variations
in parameters
.
t
We denote by
)(uftT
the set
.
ofpoint cluster
,,,
,)(|
co)( ___
i
iii
itii
wais
Ttttxx
xfX
xf
i
Definition 5.1
The multifunction )(),( yfy
is said to
be (weak ) closed at ),( xt provided
)()( xfuf ttT .
Clearly, in view of Lemma 2.1, if
t
is isolated
in
T
, then condition certainly holds.
Lemma 5.1 We consider the following
hypotheses
(i)
T
is sequentially compact.
(ii) There exists a neighborhood U of
x
such
that the map )(yft t
is upper semicontinuous
for each Uy
.
(iii) For each Tt
, t
f is Lipschitz of given
rank (i.e. Lipschitz constant)
K
on U, and
Ttxft|)( is bounded. Then a function
RXf
:given by
Ttyfyf t |)(max)(
is defined and finite (with the maximum
defining fattained) on U, and f is Lipschitz
on U of rank
K
.
Let
.)()(|max:)( yfyfTtyM t
Observe that )(yM is nonempty and closed for
each
y
in U. Let us denote by ][SP the
collection of probability radon measures on S,
for any subset S of
T
.
Theorem 5.1 Under the hypotheses given in
above lemma 5.1, suppose that either
(a)
X
is separable, or
(b)
T
is metrizable.
Then one has
)4()(|)()(
)(
TtT XMPdtxf
xf
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Further, if the multifunction )(),( xfxt t
is
closed at ),( xt for each ),(xMt
and if t
f is
M-P regular at
x
for each
t
in ),(xM and
equality holds in (4) with
).()( xfuf ttT
Proof. The proof is similar to the proof of
Theorem 2.8.2 of Clarke [5], except we use
Michel-Penot directional derivative instead of
Clarke directional derivative, and use the Mean
value Theorem stated in Borwein et. al [10].
Remark 5.1 The interpretation of the set
occurring on the right-hand side
expression (4) is Completely analogous to that
of (1) in Theorem 2.7.2 of Clarke [5]. In
particular, each element
of that set is an
element of
X
, corresponding to which there is
a mapping vt t,
from
T
to
X
and an
element
, of
)(XMP such that, for every
v
in
X
, vt t,
is
-integrable, and
.)(,,
Ttdtvv
6 Proof of the multiplier Theorem
The following results are pivotal to establish the
multiplier rule:
Theorem 6.1 (Ekeland's Theorem)
Let ),( dV be a complete metric space with
metric d, and let
RVF:be a l. s. c.
function which is bounded below. If
u
is a point
in V satisfying
FuF inf)(
for some 0
, then, for every 0
there
exists a point
v
in V such that
(i) ),()( uFvF
(ii) ,),(
vud
(iii) for all ,Vinvw
).(),()( vFvwdwF
Proposition 6.1 (see [5] Proposition(2.4.3)) Let
f be Lipschitz of rank
K
on a set S. Let
SCx
and suppose that fattains a
minimum over C at
x
. Then for any
K
K
ˆ,
the function )(
ˆ
)()( ydKyfyg C
attains a
minimum over S at
x
. If
K
K
ˆand C is
closed, then any other point minimizing
g
over
S must also lie in C.
Now, we have sufficient machinery to establish
the stated multiplier rule as the main theorem.
Proof. Proof of the theorem
Let
,
1),,(,0,0
|),,(
nm RRRt
T
let any 0
be given and define RXF
:by
.
))(),(,)()((
),,(
max)(
yhygxfyf
yF T
Note that Fis locally Lipschitz at
x
and that
.)(
xF We claim that 0
F onC, if
0)(
yF then it can be easily shown that
0)(,0)( yhyg ji (i.e.
y
is a feasible point
for
P
) and ,)()(
xfyf a contradiction.
Therefore
x
satisfies ,inf)(
FuF C
and by Theorem 6.1, there is a point
u
in
Bx
such that for all
)),(( BxCCy
we have ).()( uFuyyF
If
K
ˆ is the Lipschitz constant then any
K
K
ˆ
is local Lipschitz constant (
is sufficiently
small) for the function uyyF
)( near
the point
.
u
y
By proposition 6.1,
u
therefore also minimizes
over some neighborhood of
u
,
the function ),()( yKduyyFy C
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,)(
)5(,
)(),,,,(max
uyyG
uy
xfKyL
T
where
)(),,,,(max)( xfKyLyG T.
For
sufficiently small, then, we have )6(.)(0 BuG
Now, using Theorem 5.1, we estimate ).(uG
First, we claim that mapping )7(),,(),( KtyLyt
is closed in the sense of definition 5.1. Observe
that for any pair 21,tt of points in
T
, the function
))(,,()(
),,(),,(
21
21 yhgftt
KtyLKtyLy
is Lipschitz near x of rank 21 ttK , thus
BttKKtyLKtyL xx 2121 ),,(),,(
by proposition 2.1 (v) and 2.1 (iii). The
closure property, Lemma 2.1, of the M-P
subdifferential implies that the map (7) is closed.
Since )(uF is positive, there is a Unique u
t in
T
at which
F
(and hence G) attains
maximum.
Now Theorem 5.1, is applied to estimate
:)(uG
),,(
)(),,()(
KtyL
dtKtyLuG
ux
Tux
Using the above estimation in equation (6), we
get )8(),,(0 BKtyL ux
Proceeding in the same manner as above for a
sequence ,0
i
then the corresponding
sequence xuiand a subsequence
.Ttt i
u since the map (7) is closed; hence
theorem follows from equation (8).
Remark 6.1 We can prove above multiplier rule
similar to proof of Theorem 1 in [4], we use
proposition 2.1 (v), Proposition 3.1 and
Proposition 2.1 (iii) in stead of Propositions 8, 9
and 1 respectively in [4].
Remark 6.2 when C is convex also then
)()( xNxd CC (normal cone in the sense of
convex analysis), and multiplier rule coincides
with that of given in Ioffe [3].
7 Vector Optimization Problem
(VOP):
Let p
pRffff ),...,,( 21 be the objective
function, and then the problem (P) is called
(VOP). The feasible point
x
is termed as an
efficient (weak efficient) or Pareto optimal for
the VOP, if there is no feasible point
y
for
which prxfyf rr 1);()(
.1);()( prxfyf rr
The Lagrangian
L
for the VOP is given by
)(),,()(,
)(,)(,),,,,(
xdKxh
xgxfKxL
C
where .),...,,( 21 p
pR
Theorem7.1. (Multiplier rule for VOP)
Assume that .),..,2,1;,..2,1;,..,2,1(,, njmiprhgf jir
are locally Lipschitz at
x
. If
x
is a weak
efficient point of VOP, then for every
K
sufficiently large there exist 0,),...,,( 21 i
p
pR
and Rvj not all zero such that
mixgii ,..,2,1;0)(
and
)(),,()(
)()(0
1
11
xdKxh
xgxf
C
n
jjj
m
iii
p
rrr
Proof. Proof is analogous to the proof of
Theorem 1.1, we will consider
as
p
vector
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p
R),...,,(
appearing in the definition of
F
.
Acknowledgment: Authors are highly thankful
to Council of Scientific and Industrial Research,
Govt. of India, New Delhi-110012, INDIA, for
providing financial support to carry out the
present research work.
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