Research Journal of Applied Science, Engineering and Technology 2(6): 580-588, 2010
ISSN: 2040-7467
© M axwell Scientific Organization, 2010
Submitted Date: June 13, 2010
Accepted Date: July 18, 2010
Published Date: September 10, 2010
Second-Order Duality for Nondifferentiable Multiobjective Programming
Involving (N, D)-Univexity
Deo Brat Ojha
Department of Mathematics, R.K.G.I.T., Uttar Pradesh Technical University,
Ghaziabad, U .P., India
Abs tract: The co ncepts of ( N, D)-invex ity have been introd uced by C arsiti, Ferrara and Stefan escu. W e
consider a second-order dual model asso ciated to a multiobjective p rogramm ing problem involving support
functions and a weak du ality result is established under appropriate sec ond-orde r (N, D)-univexity conditions.
Key w ords: Duality theore m, multiobjective programm ing, second -order ( N, D)-(pseudo/quasi)-convexity,
second -order ( N, D)-(pseu do/qu asi)-univexity
established some duality results under generalized secondorder F-convexity assumptions.
For N(x, a,(y, r)) = F(x, a; y) + rd 2 (x, a), where F(x,
a) is sublinear on R n, the definition of ( N, D) - invex ity
reduces to the definition of (F, D) -convexity introduced
by Preda (1992), which in turn Jeyakumar (1985)
generalizes the concepts of F-convexity and D-invex ity
(Vial, 1983).
The more recent literature, X u (1996), Ojha (2005)
and O jha a nd M ukherjee (2006) for duality under
generalized (F, D)-convexity, Mishra (2000) and
Yang et al. (2003) for duality under second order
F-convexity. Liang et al. (2003) and Hachimi (2004) for
optimality criteria and duality involving (F, ", D, d) convexity or generalized (F, ", D, d) -type functions. The
(F, D) -convexity was recently generalized to (N, D)invex ity by C aristi et al. (2006 ), and here w e will use this
concept to extend some theoretical results of
multiobjective programming.
W henever the objective function and all active
restriction functions satisfy simultaneously the same
generalized invex ity at a Kuhn-Tucker point which is an
optimum condition, then all these functions shou ld satisfy
the usual invexity, too . This is n ot the case in
multiobjective
programm ing;
Ferrara
and
Stefanescu (2008) showed that sufficiency Kuhn-Tucker
condition can be proved under ( N, D)-invex ity, even if
Han son’s invexity is not satisfied, (Puglisi, 2009). The
interested
reader
may consult
Ahmad and
Husain (2005), Chandra et al. (1985, 1998), Chandra and
Husain (1981), Chandra and Prasad (1993), Craven
(1977), Dorn (1960), Kim et al. (1997), M ishra (2000),
Mond et al. (1991), Unger and Hunter (1974), Weir and
Mond (1988) and Y ang and Hou (2001).
Therefore, the results of this study are real extensions
of the similar results known in the literature.
INTRODUCTION
For nonlinear programming problems, a number of
duals have been suggested among which the Wolfe dual
(Dorn, 1960; Hanson and Mond, 1982) are well known.
W hile studying duality under generalized convexity,
Mon d and Weir (1981) proposed a number of deferent
duals for nonlinear programm ing pro blems with
nonnegative variables and proved va rious duality
theorems under appropriate pseudo-convexity/quasiconvexity assumptions.
The study of second order duality is signiWcant due to
the comp utational advan tage over Wrst order duality as it
provides tighter bounds for the value of the objective
function when approximations are used by G ulati
et al. (1997), Suneja et al. (2003) and Yang et al. (2003).
Mangasarian (1975) considered a nonlinear programming
problem and discussed second order duality under
inclusion condition. Mond (1974) was the Wrst who
present second order convexity. He also gave simpler
conditions than Mangasarian using a generalized form of
convexity, which was later called bonvexity by Bector
and Chandra (1987). Further, Jeyakumar (1985, 1986) and
Yang et al. (2003) discussed also second order
Mangasarian type dual formulation under D-convexity and
generalized representation conditions respectively. Zhang
and Mond (1996) established some duality theorems for
second-order duality in nonlinear programming under
generalized second-order B-invexity, de Wned in their
study. Mond (1974) has show n that se cond order dua lity
can be useful from computational point of view, since one
may obtain better lower bounds for the primal problem
than otherwise. The case of some optimization problems
that involve n-set functions was studied by P reda (1998).
Recently, Yang et al. (2003) proposed four second-order
dual models for nonlinear programming problems and
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
In this study, we de Wne the seco nd-order ( N, D)invexity, we also consider a class of Multiobjective
programming problems and for the dual model we prove
a weak d uality result.
Definition: A a , X 0 point is said to be a weak efficient
solution of problem (VP) if there is no x , X such that
f (x) < f (a) .
Definition: A po int a , X 0 is said to be a properly
efficient solution of (VP) if it is efficient and there exist
a positive constant K such that for each x , X 0 and for
each i , {1, 2, ,.,.,., p} satisfying fi (x) < fi (a), there exist
at least one i , {1, 2, .,.,., p} such that f j (x) < f j (a) and
f i (a) < f i (x) # K (f j (x) - f j (a)).
Denoting by WE (P), E(P) and PE(P ) the sets of all
weakly efficient, efficient and properly efficient solutions
of (VP), w e have P E(P) f E(P) f W E (P).
W e denote by Lf (a) the gradient vector at a of a
MATERIALS AND METHODS
we denote by R n the n-dimensional Euclidean space,
and by R n+ its nonnegative orthant. Further,
For any vector x , R n, y , R n we
denote:
, and by L2f (a) the
differen tiable function
Hessian matrix of f at a. For a real valued twice
differen tiable function R(x, y) defined on an op en set in
R p × R q, we denote by Lx R(a, b) the gradient vector of R
with respect to x at (a, b), and by Lxx R(a, b) the Hessian
matrix with respect to x at (a, b). Similarly, we may
define:
Let C d R n be a compact convex set. The support function
of C is defined by s(x|C) = max {x T y | y , C}. Being
convex and every where finite, it has a subdiferential, that
is, there exist z , R n such that s(y|C) $ s(x|C) + z T (y - x)
for all y , C.
The subdifferantials of s(x|C) is given by
Ly R(a, b), Lxy R(a, b) and Lyy R(a, b)
s(x|C) =
For convenience, let us write the definitions of ( N, D)-
{z , C | z T = s(x|C)}.
For any set D d R n, the normal cone to D at a point x
, D is defined by:
unive xity from (1 ), Let
be a differentiable
function (X 0 f R n), X f X 0, and a , X 0. An element of all
(n+1)- dimensional Euclidean Space R n+1 is represented as
the ordered pair (z, r) with z , R n and r , R, D is a real
number and N is real valued function defined on X 0 × X 0 ×
R n+1, such that N (x,a) is convex on R n+1 and N(x, a (0, r))
$ 0, for every (x, a) , X 0 × X 0 and r , R +. b 0, b 1: X × X ×
[0,1] ÷R + b(x, a) =
For a com pact convex set C. We obviously have y , N C(x)
if and only if s(y|C) = x T y, or equivalently, if
, and b does not
depend upon 8 if the corresponding functions are
.
differentiable.
W e consider
is an n-dimensional
vector- valued function and h: X × R n ÷ R be
differentiable function.
W e assum e that R0, R1: R ÷R satisfying
, are
n
differential functions and x d R is an open set. We define
the following multiobjective programming problem:
, and b 0
and
(P) minimize f (x) = (f 1(x)....f p(x)) subject to g(x) $ 0, x
,X
(x,a) > 0 and b 1 (x, a)
Let X 0 be the set of all feasible solutions of (P) that
is, X 0 = {x , X | g(x) $ 0}. We quote some definitions and
also give some new on es.
Examp le:
0 and R0( ") = - R0( ") and R1(-")
= - R1( ").
min f(x) = x-1
g(x) = -x-1 # 0, x , X 0 , [1,4)
N (x,a;(y,r)) = 2(2 r-1) |x-a| +y, x-a ,
Definition: A vector a , X 0 is said to be an efficient
solution of problem (P) if there exit no x , X 0 such that f
(a) - f (x) , R 0+\{0} i.e., f i (x) # f i (a) for all i , {1,.,., p}
and for at least one j , {1,.,., p} we have f i (x) # f i (a).
for R2(x) = x, R1(x) = -x, D =
(for f ) and D = 1 (for
g), then this is (N, D)-univex bu t it is not ( N, D)-invex.
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
f : X ÷R, f(x) = x log x, h: X ×R ÷R, h(u,y) = -y log u
Definition: A real-valued twice differentiable function
f (., y): X × X ÷R is said to be se cond-order ( N, D)-univex
at u , X with respect to p , R n, if for all b: X × X ÷R +, N:
X × X × R n+1÷R, D is a real number, we have:
W e have:
(1)
K: X × X × R n+1÷R, taking:
D = 0 N (x, y; b) = |b| + |b|2
Definition: A rea l-valued twice differentiable function f
(., y): X × X ÷R is said to be seco nd-order ( N,D)pseu dounivex at a , X with respect to p , R n, if for all b:
X × X ÷R +, N: X × X × R n+1÷R, D is a real number, we
have:
It is obvious our mapping is more generalized rather than
previous ones.
Hence f(x) = x log x is second-order ( N,D)-univex at
u , X, with respect to h(u,y) = -y logu.
A real valued twice differentiable function g is
second order F-pseudoc oncave if !g is second order Fpseudoconvex.
W e shall make use of the following generalized
Schwartz inequality:
(2)
whe re x, y , R n and A , R n × R n is a positive sem idefinite
matrix. Equality holds if for some 8 $ 0, Ax = 8 Ay.
Definition: A real-valued twice differentiable function f
(., y): X × X ÷R is said to be seco nd-order ( N,D)quasiunivex at a , X with respect to p , R n, if for all b: X
× X ÷R +, N: X × X × R n+1÷R, D is a real number, we
have:
RESULTS AND DISCUSSION
M ond -W eir type second order sym metric duality: W e
consider here the follow ing pair of second order
nondifferentiable multiobjective with r-objectives and
establish weak, strong and converse duality theorems.
(3)
(MP) Minimize:
H(x, y, w, p) =
{H 1(x, y, w, p), H 2(x, y, w, p) ,.,., H r(x, y, w, p)}
Remark:
C If we consider the case b = 1, N (x, u; (Lf(u), D)) = F
(x, u; Lf(u)) (with F is sublinear in third argu men t,
then the above definition reduce to Definition 4 of
Chen [4].
C
W hen
subject to:
(4)
and N (x, u;
( Lf(u), D)) = F (x, u; Lf(u)) = 0(x, u) T Lf (u) where
0: X×X ÷R n, the above definition reduce to 0(pseudo/quasi)-bonvexity.
(5)
Examp le:
W e present here a function w hich is second-order ( N, D)univex for b = 1. Let us consider X = (0, 4) and
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
Theorem (Weak du ality):
Let (x, y, 8, w, p) be a feasible solution of (M P) and
(u, v, 8, a, q) a feasible solution of (MD ). Then the
inequalities can not hold simultaneously:
(6)
8>0
x$0
(7)
(8)
(MD) M aximize:
C
J(u, v, a, q) = { J 1(u, v, a, q), J 2(u, v, a, q),.,., J r(u, v, a, q)}
is second order ( N, D)pseu dounivex at u,
subject to:
C
is second order (N, D)pseu dounicav e at y
(9)
C
N (x,u; (>, D)) + u T >
0, for > , R n, and
C
N (v, y; (., D)) + y T .
0, for . , R n, then
H(x, y, w, p) Ü J(u, v, a, q).
(10)
Proof:
W ith the help of:
(11)
8>0
(12)
(13)
we have:
where;
(By hypo thesis (iii) and (10), also by the second
order( N, D)-pseudou nivexity of:
8i , R, p i , R n, q i , R n, I = 1,2,.,.,r and fi, i =1, 2,.,., r are
thrice differentiable function from R n × R n ÷R, E i and C i,
i = 1, 2,.,., r are positive semi definite matrices. Also, we
mean here:
at u, with property o f b and R, provides:
b i = R n × R m × R n × R m ÷R + p = (p 1, p 2,.,., p r),
q = (q 1, q 2,.,., q r), w = (w 1, w 2,.,., w r), a = (a 1, a 2,.,., a r)
Remark:
Since th e o bje ctiv e fu nctio ns o f (M P ) and (M D) contain
the support functions s(x | C i) and s(v | D i), i = 1, 2,.,.,p,
these problems are nondifferentiable multiobjective
programming problems.
(14)
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
Now,
Hence, H(x, y, w, p) Ü J(u, v, a, q).
we have:
Theorem (Strong d uality): Let f be thrice differentiable
on R n × R n and
solution for (MP), and
(by hypothesis (iv),(5) and by the second order ( N, D)-
C
pseu dounicav ity
be a weak efficient
, assum e that:
is nonsingular for all i = 1, 2,.,., r;
at y, with
property of b and R, gives:
C
the matrix
C
definite
the set
is positive or negative
(15)
are linearly independent
where;
Combin ing (14 ) and (1 5), we get:
i
=
1 , 2 , . , . ,r .
Then
is a feasible solution of (M D),
, i = 1,2,.,.,r and the two objectives
have the sam e values. Also, if the hypothesis of previous
theorem are satisfied for all feasible solutions of (MP) and
(MD ), then
is an efficient
solution for (M D).
Proof: Since
is a weak efficient
solution of (MP), by Fritz-John co ndition, there ex ist " ,
R r, $ , R n, ( , R, v , R r and > , R n such that:
Applying Schwartz inequality, (6) and (1 1), we get:
(16)
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
Conseque ntly, (18) gives:
(30)
(17)
Since
is nonsingular for i = 1, 2,.,., r, from (20),
it follows that:
from (17), w e get:
(18)
(19)
(20)
using (31), w e get:
(21)
(22)
(31)
(23)
Premultiplying (32) by
and using (30), we
get:
(24)
(25)
(26)
by hypothesis (ii) implies:
(32)
(27)
Therefore, fro m (32 ), we g et:
(28)
(29)
Since
and
, (25) implies * = 0.
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
which by hyp othesis (iii) gives:
(38)
(33)
Thus in either case:
If ( = 0, then "i = 0,1 = 1,2,.,., r and fro m (33 ), $ = 0.
Also from (16) and (1 9), we get, >i = 0, v i = 0, i = 1, 2,.,.,
r. Thus ( ", $, (, v, *, >) = 0, a contradiction to (29).
Hence ( > 0, since
"i > 0,1 = 1, 2,.,., r. Using (33 ) in (31),
2,.,., r, hence
(39)
i = 1, 2,.,., r, (34) implies
Hence:
i = 1,
i = 1, 2,.,., r. Using (33) and
i = 1, 2,.,., r in (16), it gives
,
which by (34) gives:
(34)
(using (21) and (39)).
Now follows from
Previous
Theorem
that
is an efficient solu tion for (MD ).
(35)
A converse duality theorem may be merely stated as its
proof would run analogously to this Theorem.
Also , from (33), we get:
Theorem
(Converse
n
duality):
Let
f
be
n
differen tiable on R × R and
(36)
be a
weak efficient solution for (MD), and
(MP). Assum e that:
Hence from (27) an d (35-37),
. Then
C
and from (19) and (33 ):
is nonsingular for all i = 1, 2,.,., r
C
the matrix
C
negative definite
the set
(37)
which is the condition in the Schwartz
Therefore:
fixed in
is
feasible for (M D).
Let
thrice
is positive or
inequality.
are linearly independent
In case, v i > 0, (24) gives
and so
where,
i
=
1,2,.,.,r.
Then
is a feasible solution of (M P),
. In case v i = 0,
i = 1,2,.,.,r, and the two
gives
and so:
objectives have the same values. Also, if the hypothesis of
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Res. J. Appl. Sci. Eng. Technol., 5(6): 580-588, 2010
Theorem (W eak d uality) are satisfied for all feasible
s o l u ti o n s
of
(MP)
and
(MD),
then
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is an efficient solution for (MP ).
CONCLUSION
C
C
C
If b = 1, R / 1, E i = C i = 0, i = 1, 2,.,., r, and N (x, u;
( Lf(u), D)) = F (x, u; Lf (u)) for D = 0 then (MP) and
(MD) reduce to the second order multiobjective
symmetric dual programstudied by Suneja et al.
(2003) with omission o f non-negativity co nstraints
from (MP) and (MD). If in addition p = q = 0, and
r = 1, then we get the first order symmetric dual
programs of Chandra et al. (1998).
If b = 1, R / 1, we set p = q = 0, and N (x, u; (Lf(u),
D)) = F (x, u; Lf (u)) for D = 0 in (M P) and (M D),
then we obtain a pair of first order symmetric dual
nondifferentiable multiobjective programs considered
by Mond et al. (1991).
If we set, b = 1, R / 1, N (x, u; (Lf(u), D)) = F (x, u;
Lf (u)) for D = 0 in (MP) and (MD), then we obtain a
pair of second order symmetric dual
nondifferentiable multiobjective programs considered
by A hmad and H usain (2005).
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