Research ArticleSecond Order Duality in Multiobjective Fractional Programmingwith Square Root Term under Generalized Univex Function
Arun Kumar Tripathy
Department of Mathematics Trident Academy of Technology F2A Chandaka Industrial EstateBhubaneswar Odisha 751024 India
Correspondence should be addressed to Arun Kumar Tripathy arun tripathy06rediffmailcom
Received 12 March 2014 Accepted 10 April 2014 Published 7 July 2014
Academic Editor Majid Soleimani-damaneh
Copyright copy 2014 Arun Kumar Tripathy This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programmingproblem inwhich the numerator and denominator of objective function contain square root of positive semidefinite quadratic formAlso the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterizationtechnique is used to establish duality results under generalized second order 120588-univexity assumption
1 Introduction
A fractional programming problem arises in many typesof optimization problems such as portfolio selection pro-duction information theory and numerous decision mak-ing problems in management science More specifically itcan be used in engineering and economics to minimize aratio of physical or economical function or both such ascosttime costvolume and costbenefit in order to measurethe efficiency or productivity of the system Many economicnoneconomic and indirect applications of fractional pro-gramming problem have also been given by Bector [1] Bectorand Chandra [2] Craven [3] Mond and Weir [4] Stancu-Minasian [5] Schaible and Ibaraki [6] Ahmad et al [7]Ahmad and Sharma [8] and Gulati et al [9]
The central concept in optimization is known as theduality theory which asserts that given a (primal) mini-mization problem the infimum value of the primal problemcannot be smaller than the supermom value of the associated(dual) maximization problem and the optimal values ofprimal and dual problems are equal Duality in fractionalprogramming is an important class of duality theory andseveral contributions have been made in the past [1 5 8 10ndash14] Second order duality provides a tighter bound for thevalue of the objective function when approximations areused For more details one can consult [15 page 93] Another
advantage of second order duality when applicable is that ifa feasible point in the primal is given and first order dualitydoes not apply then we can use second order duality toprovide a lower bound of the value of the primal problem (see[4])
Multiobjective fractional programming duality has beenofmuch interest in the recent past Schaible [16] and Bector etal [11] derived Fritz John and Karush-Kuhn Tucker necessaryand sufficient optimality condition for a class of nondifferen-tiable convex multiobjective fractional programming prob-lems and established some duality theorems Liang et al [1718] discussed the optimality condition and duality for non-linear fractional programming Santos et al [19] discussedthe optimality and duality for nonsmooth multiobjectivefractional programming with generalized convexity Bectoret al [20] and Xu [21] gave a mixed type duality for fractionalprogramming established some sufficient conditions andobtained various duality results between the mixed dual andprimal problem Zhou and Wang [22] introduced a classof mixed type dual for nonsmooth multiobjective fractionalprogramming and established the duality results under (119881 120588)invexity assumption
Duality for various forms of mathematical problemsinvolving square roots of positive semidefinite quadraticforms has been discussed bymany authors [10 23ndash25] Mond[25] considered a nonlinear fractional programming problem
Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 541524 8 pageshttpdxdoiorg1011552014541524
2 International Scholarly Research Notices
involving square roots of positive semidefinite quadratic formin the numerator and denominator and proved the neces-sary and sufficient condition for optimality Kim et al [2627] formulated a nondifferentiable multiobjective fractionalproblem in which numerators contain support function Oneof the most known approaches used for solving nonlin-ear fractional programming problem is called parametricapproach Dinklebaeh [28] and Jagannathan [12] introducedthis approach that was used later byOsuna-Gomez et al [13]to characterize solution of a multiobjective fractional prob-lem under generalized convexity Tripathy [14] introducedthree approaches given by Dinklebaeh [28] and Jagannathan[12] for both primal and mixed type dual of a nondifferen-tiablemultiobjective fractional programming and establishedthe duality results under generalized 120588-univexity
To relax convexity assumption imposed on the functionin theorems on optimality and duality various generalizedconvexity concepts have been proposed Hanson [29] intro-duced the class of invex functions Bector et al [30] intro-duced univex function Mishra [31] derived the optimalitycondition for multiobjective programming with generalizedunivexity Jayswal [32] presented minimax fractional pro-gramming under generalized 120588-univexity assumption
Motivated by the earlier authors in this paper we haveintroduced three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem in which the numerator and denom-inator of objective function contain square root of positivesemidefinite quadratic form Also we have established thenecessary and sufficient optimality condition and used aparameterization technique to establish duality results undergeneralized 120588-univexity assumption
2 Notations and Preliminaries
Let R119899 be the 119899-dimensional Euclidean space and R119899+its
nonnegative orthant The following conventions for inequal-ity will be used throughout this paper For any 119909 =
(1199091 1199092 119909
119899) 119910 = (119910
1 1199102 119910
119899) we denote the following
(i) 119909 gt 119910 hArr 119909119894gt 119910119894 for all 119894 = 1 2 119899
(ii) 119909 ge 119910 hArr 119909119894ge 119910119894 for all 119894 = 1 2 119899
Throughout the paper let 119883 be a nonempty open subset ofR119899
Consider the following nondifferentiable multiobjectivefractional programming problem
21 Multiobjective Fractional Primal Problem
(i) MFP0 Minimize
119891 (119909) + (119909119879
119861119909)12
119892 (119909) minus (119909119879119862119909)12
= (1198701(119909) 119870
2(119909) 119870
119896(119909)) (1)
where
119870119894(119909) =
119891119894(119909) + (119909
119879
119861119894119909)12
119892119894(119909) minus (119909119879119862
119894119909)12
119894 = 1 2 119896 (2)
(ii) MFP1 Minimize
119865 (119909) = (1198651(119909) 119865
2(119909) 119865
119896(119909)) (3)
where
119865119894(119909) = 119891
119894(119909) + (119909
119879
119861119894119909)12
minus ]119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(4)
]119894are fixed parameters
(iii) 119872119865119875120582 Minimize 120582119865(119909) 120582 is 119896-dimensional strictlypositive vector
all subject to same constraint
ℎ (119909) le 0 119909 isin 119883 sube R119899
(5)
where 119891119894 R119899 rarr R 119892
119894 R119899 rarr R 119894 = 1 2 119896 and ℎ =
(ℎ1 ℎ
119898) ℎ119895 R119899 rarr R 119895 = 1 2 119898 are differentiable
functions 119861119894and 119862
119894 119894 = 1 2 119896 are positive semidefinite
matrices of order 119899 In the sequel we assume that 119891119894(119909) ge 0
and 119892119894(119909) gt 0 on R119899 for 119894 = 1 2 119896
Let 1198830= 119909 isin 119883 sube R119899 ℎ
119895(119909) le 0 119895 = 1 2 119898 for all
feasible solutions of MFP0 MFP1 and MFP120582 and denote 119868 =1 2 3 119896119872 = 1 2 3 119898 119869
1= 119895 isin 119872 ℎ
119895(119909) = 0
and 1198692= 119895 isin 119872 ℎ
119895(119909) lt 0 It is obvious that 119869
1cup 1198692= 119872
Throughout the paper consider 119891119894 119883 rarr R 120578 119883 times
119883 rarr R119899 119901 isin R119899 120588 isin RAssume that 120595 R rarr R satisfying 119886 le 0 rArr 120595(119886) le 0 or
120595(119886) le 0 rArr 119886 le 0 and 120595(minus119886) = minus120595(119886) 119870 119883 times 119883 rarr R+
For 119909 119909 isin 119883 we can write119870(119909 119909) = lim120582rarr0119887(119909 119909 120582) ge 0
Definition 1 The real differentiable function 119891119894is said to be
second order 120588-univex at 119909 isin 119883 with respect to 120578 120595 and 119870if
119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)]
ge 120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
forall119909 isin 119883
(6)
Definition 2 The real differentiable function 119891119894is said to be
second order 120588-pseudounivex at 119909 isin 119883 with respect to 120578 120595and119870 if
120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
ge 0
997904rArr 119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)] ge 0
forall119909 isin 119883
(7)
International Scholarly Research Notices 3
Definition 3 The real differentiable function 119891119894is said to be
second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if
119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)] le 0
997904rArr 120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
le 0
forall119909 isin 119883
(8)
Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]
Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865
119894(119909) 119894 = 1 2 119896 and 119865
119903(119909) lt 119865
119903(119909) for some
119903 isin 1 2 119896
Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0
such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865
119894(119909) lt 119865
119894(119909) we have 119865
119894(119909) minus 119865
119894(119909) le
119872(119865119903(119909) minus 119865
119903(119909)) for some 119903 isin 1 2 119896 such that 119865
119903(119909) gt
119865119903(119909)
We assume that 119891119894(119909) + (119909
119879
119861119894119909)12
ge 0 119892119894(119909) minus
(119909119879
119862119894119909)12
gt 0 119894 = 1 2 119896 for all 119909 isin 119883
Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12
The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ
119895(119909) = 0 and
V119894= (119891119894(119909) + (119909
119879
119861119894119909)12
)(119892119894(119909) minus (119909
119879
119862119894119909)12
)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ
119895(119909) le 0 119895 isin
119869(119909) satisfying any one of the following conditions
(a) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) + 119861
119894119909
(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909)) + (119908
119879
(V2119894119862119894)119908)12
ge 0 119908 isin119882(119909) 119894 = 1 2 119896
(b) 119909119879119861119894119909 = 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909) minus
119862119894119909(119909119879
119862119894119909)12
) + (119908119879
119861119894119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(c) 119909119879119861119894119909 = 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909)) +
(119908119879
119861119894119908)12
+ (119908119879
(V2119894119862119894)119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(d) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) +
119861119894119909(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909) minus 119862
119894119909(119909119879
119862119894119909)12
) ge 0119908 isin 119882(119909) 119894 = 1 2 119896
Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of
119872119865119875120582 then 1199090 is properly efficient for MFP1
Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0
if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0
Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then
there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that
nabla120582119865 (119909) + 119910119879
nablaℎ (119909)
=
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119895=1
119910119895nablaℎ119895(119909) = 0
(9)
119865119894(119909) = 119891
119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
(10)
119910119879
ℎ (119909) = 0 (11)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (12)
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
(13)
119910 ge 0 (14)
V119894ge 0 119894 = 1 2 119896 (15)
Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119875(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] + sum119898
119895=1119910119895ℎ119895(119909) is second order 120588-
pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830
with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R
satisfying 120595(119886) le 0 rArr 119886 le 0
(ii) 120588 ge 0
Then 119909 is an efficient solution of MFP1
Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)
and (13) we have
nabla119875 (119909)
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909))
4 International Scholarly Research Notices
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
+
119898
sum
119895=1
119910119895ℎ119895(119909)) = 0
(16)
Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +
nabla2
119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +
120588119909 minus 1199092
ge 0Since 119875(119909) is second order 120588-pseudounivex with respect
to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +
(12)119901119879
(nabla2
119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives
119875 (119909) minus 119875 (119909) +1
2119901119879
nabla2
119875 (119909) 119901
ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1
2119901119879
nabla2
119875 (119909) 119901
(17)
Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
(18)
Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883
0such that
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
le 119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(19)
and 119891119905(119909) + (119909
119879
119861119905119909)12
minus V119905119892119905(119909) minus (119909
119879
119862119905119909)12
le 119891119905(119909) +
(119909119879
119861119905119909)12
minus V119905119892119905(119909)minus (119909
119879
119862119905119909)12
for some 119905 isin 1 2 119896
The above relation together with the relation 120582119894gt 0
implies that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(20)
From the relations (5) (11) and (14) we get
119898
sum
119895=1
119910119895ℎ119895(119909) le
119898
sum
119895=1
119910119895ℎ119895(119909) (21)
Consequently (20) and (21) yield
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
(22)
This contradicts (18) Hence 119909 is an efficient solution forMFP1
Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119876(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] is second order 120588-pseudounivex withrespect to 120578 120595
0 and 119870 at 119909 isin 119883
0and 119867(119909) =
sum119898
119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with
respect to 120578 1205951 and 119870 at 119909 isin 119883
0with (nabla2119876(119909))119901 = 0
and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and
1205950 1205951 R rarr R satisfying 120595
0(119886) ge 0 rArr 119886 ge 0 and
1205951(119886) le 0 rArr 119886 le 0
(ii) 120588 + 120590 ge 0
Then 119909 is an efficient solution of1198721198651198751
Proof Suppose hypothesis holds
International Scholarly Research Notices 5
From the relations (5) (11) and (14) we get119898
sum
119895=1
119910119895ℎ119895(119909)
le
119898
sum
119895=1
119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0
(23)
Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following
119867(119909) minus 119867 (119909) +1
2119901119879
(nabla2
119867(119909)) 119901 le 0
997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +
1
2119901119879
(nabla2
119867(119909)) 119901 le 0
(24)
Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951
and119870 implies the following
120578(119909 119909)119879
nabla119867 (119909) + (nabla2
119867(119909)) 119901 + 120590119909 minus 1199092
le 0
997904rArr 120578(119909 119909)119879
nabla119867 (119909) + 120590119909 minus 1199092
le 0
(25)
From (9) we get
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119894=1
119910119894nablaℎ119894(119909) = 0
997904rArr nabla119876 (119909) + nabla119867 (119909) = 0
997904rArr 120578(119909 119909)119879
[nabla119876 (119909) + nabla119867 (119909)] = 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + 120578(119909 119909)119879
nabla119867 (119909)
+ 120590119909 minus 1199092
minus 120590119909 minus 1199092
= 0
(26)
Using (25) in (26) we get
120578(119909 119909)119879
nabla119876 (119909) minus 120590119909 minus 1199092
ge 0 (27)
Since 120588 + 120590 ge 0 we get
120588119909 minus 1199092
ge minus120590119909 minus 1199092
(28)
So we have
120578(119909 119909)119879
nabla119876 (119909) + 120588119909 minus 1199092
ge 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + nabla2
119876 (119909) 119901 + 120588119909 minus 1199092
ge 0
(29)
Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870
we obtained
119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +
1
2119901119879
nabla2
119876 (119909) 119901] ge 0 (30)
Using the property of119870 and 1205950 we get
119876 (119909) minus 119876 (119909) +1
2119901119879
nabla2
119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(31)
If 119909 were not an efficient solution to MFP1 then from (20)we have
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(32)
This contradicts (31)Therefore 119909 is an efficient solution for MFP1
3 Second Order Mixed Type MultiobjectiveFractional Duality
(i) MMFD0 Maximize
119871 (119906)
= (1198711(119906) minus
1
2119901119879
nabla2
1198711(119906) 119901 119871
2(119906)
minus1
2119901119879
nabla2
1198712(119906) 119901 119871
119896(119906) minus
1
2119901119879
nabla2
119871119896(119906) 119901)
(33)
where
119871119894(119906) =
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
119892119894(119906) minus 119906119879119862
119894119911
119894 = 1 2 119896 (34)
(ii) MMFD1 Maximize
119866 (119906)
= (1198661(119906) minus
1
2119901119879
nabla2
1198661(119906) 119901 119866
2(119906)
minus1
2119901119879
nabla2
1198662(119906) 119901 119866
119896(119906) minus
1
2119901119879
nabla2
119866119896(119906) 119901)
(35)
where 119866119894(119906) = 119891
119894(119906) + 119910
119879
1198691
ℎ1198691
(119906) + 119906119879
119861119894119908 minus ]119894119892119894(119906) minus
119906119879
119862119894119911 119894 = 1 2 119896 ]
119894are fixed parameters
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Scholarly Research Notices
involving square roots of positive semidefinite quadratic formin the numerator and denominator and proved the neces-sary and sufficient condition for optimality Kim et al [2627] formulated a nondifferentiable multiobjective fractionalproblem in which numerators contain support function Oneof the most known approaches used for solving nonlin-ear fractional programming problem is called parametricapproach Dinklebaeh [28] and Jagannathan [12] introducedthis approach that was used later byOsuna-Gomez et al [13]to characterize solution of a multiobjective fractional prob-lem under generalized convexity Tripathy [14] introducedthree approaches given by Dinklebaeh [28] and Jagannathan[12] for both primal and mixed type dual of a nondifferen-tiablemultiobjective fractional programming and establishedthe duality results under generalized 120588-univexity
To relax convexity assumption imposed on the functionin theorems on optimality and duality various generalizedconvexity concepts have been proposed Hanson [29] intro-duced the class of invex functions Bector et al [30] intro-duced univex function Mishra [31] derived the optimalitycondition for multiobjective programming with generalizedunivexity Jayswal [32] presented minimax fractional pro-gramming under generalized 120588-univexity assumption
Motivated by the earlier authors in this paper we haveintroduced three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem in which the numerator and denom-inator of objective function contain square root of positivesemidefinite quadratic form Also we have established thenecessary and sufficient optimality condition and used aparameterization technique to establish duality results undergeneralized 120588-univexity assumption
2 Notations and Preliminaries
Let R119899 be the 119899-dimensional Euclidean space and R119899+its
nonnegative orthant The following conventions for inequal-ity will be used throughout this paper For any 119909 =
(1199091 1199092 119909
119899) 119910 = (119910
1 1199102 119910
119899) we denote the following
(i) 119909 gt 119910 hArr 119909119894gt 119910119894 for all 119894 = 1 2 119899
(ii) 119909 ge 119910 hArr 119909119894ge 119910119894 for all 119894 = 1 2 119899
Throughout the paper let 119883 be a nonempty open subset ofR119899
Consider the following nondifferentiable multiobjectivefractional programming problem
21 Multiobjective Fractional Primal Problem
(i) MFP0 Minimize
119891 (119909) + (119909119879
119861119909)12
119892 (119909) minus (119909119879119862119909)12
= (1198701(119909) 119870
2(119909) 119870
119896(119909)) (1)
where
119870119894(119909) =
119891119894(119909) + (119909
119879
119861119894119909)12
119892119894(119909) minus (119909119879119862
119894119909)12
119894 = 1 2 119896 (2)
(ii) MFP1 Minimize
119865 (119909) = (1198651(119909) 119865
2(119909) 119865
119896(119909)) (3)
where
119865119894(119909) = 119891
119894(119909) + (119909
119879
119861119894119909)12
minus ]119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(4)
]119894are fixed parameters
(iii) 119872119865119875120582 Minimize 120582119865(119909) 120582 is 119896-dimensional strictlypositive vector
all subject to same constraint
ℎ (119909) le 0 119909 isin 119883 sube R119899
(5)
where 119891119894 R119899 rarr R 119892
119894 R119899 rarr R 119894 = 1 2 119896 and ℎ =
(ℎ1 ℎ
119898) ℎ119895 R119899 rarr R 119895 = 1 2 119898 are differentiable
functions 119861119894and 119862
119894 119894 = 1 2 119896 are positive semidefinite
matrices of order 119899 In the sequel we assume that 119891119894(119909) ge 0
and 119892119894(119909) gt 0 on R119899 for 119894 = 1 2 119896
Let 1198830= 119909 isin 119883 sube R119899 ℎ
119895(119909) le 0 119895 = 1 2 119898 for all
feasible solutions of MFP0 MFP1 and MFP120582 and denote 119868 =1 2 3 119896119872 = 1 2 3 119898 119869
1= 119895 isin 119872 ℎ
119895(119909) = 0
and 1198692= 119895 isin 119872 ℎ
119895(119909) lt 0 It is obvious that 119869
1cup 1198692= 119872
Throughout the paper consider 119891119894 119883 rarr R 120578 119883 times
119883 rarr R119899 119901 isin R119899 120588 isin RAssume that 120595 R rarr R satisfying 119886 le 0 rArr 120595(119886) le 0 or
120595(119886) le 0 rArr 119886 le 0 and 120595(minus119886) = minus120595(119886) 119870 119883 times 119883 rarr R+
For 119909 119909 isin 119883 we can write119870(119909 119909) = lim120582rarr0119887(119909 119909 120582) ge 0
Definition 1 The real differentiable function 119891119894is said to be
second order 120588-univex at 119909 isin 119883 with respect to 120578 120595 and 119870if
119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)]
ge 120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
forall119909 isin 119883
(6)
Definition 2 The real differentiable function 119891119894is said to be
second order 120588-pseudounivex at 119909 isin 119883 with respect to 120578 120595and119870 if
120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
ge 0
997904rArr 119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)] ge 0
forall119909 isin 119883
(7)
International Scholarly Research Notices 3
Definition 3 The real differentiable function 119891119894is said to be
second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if
119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)] le 0
997904rArr 120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
le 0
forall119909 isin 119883
(8)
Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]
Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865
119894(119909) 119894 = 1 2 119896 and 119865
119903(119909) lt 119865
119903(119909) for some
119903 isin 1 2 119896
Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0
such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865
119894(119909) lt 119865
119894(119909) we have 119865
119894(119909) minus 119865
119894(119909) le
119872(119865119903(119909) minus 119865
119903(119909)) for some 119903 isin 1 2 119896 such that 119865
119903(119909) gt
119865119903(119909)
We assume that 119891119894(119909) + (119909
119879
119861119894119909)12
ge 0 119892119894(119909) minus
(119909119879
119862119894119909)12
gt 0 119894 = 1 2 119896 for all 119909 isin 119883
Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12
The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ
119895(119909) = 0 and
V119894= (119891119894(119909) + (119909
119879
119861119894119909)12
)(119892119894(119909) minus (119909
119879
119862119894119909)12
)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ
119895(119909) le 0 119895 isin
119869(119909) satisfying any one of the following conditions
(a) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) + 119861
119894119909
(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909)) + (119908
119879
(V2119894119862119894)119908)12
ge 0 119908 isin119882(119909) 119894 = 1 2 119896
(b) 119909119879119861119894119909 = 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909) minus
119862119894119909(119909119879
119862119894119909)12
) + (119908119879
119861119894119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(c) 119909119879119861119894119909 = 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909)) +
(119908119879
119861119894119908)12
+ (119908119879
(V2119894119862119894)119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(d) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) +
119861119894119909(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909) minus 119862
119894119909(119909119879
119862119894119909)12
) ge 0119908 isin 119882(119909) 119894 = 1 2 119896
Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of
119872119865119875120582 then 1199090 is properly efficient for MFP1
Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0
if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0
Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then
there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that
nabla120582119865 (119909) + 119910119879
nablaℎ (119909)
=
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119895=1
119910119895nablaℎ119895(119909) = 0
(9)
119865119894(119909) = 119891
119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
(10)
119910119879
ℎ (119909) = 0 (11)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (12)
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
(13)
119910 ge 0 (14)
V119894ge 0 119894 = 1 2 119896 (15)
Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119875(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] + sum119898
119895=1119910119895ℎ119895(119909) is second order 120588-
pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830
with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R
satisfying 120595(119886) le 0 rArr 119886 le 0
(ii) 120588 ge 0
Then 119909 is an efficient solution of MFP1
Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)
and (13) we have
nabla119875 (119909)
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909))
4 International Scholarly Research Notices
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
+
119898
sum
119895=1
119910119895ℎ119895(119909)) = 0
(16)
Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +
nabla2
119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +
120588119909 minus 1199092
ge 0Since 119875(119909) is second order 120588-pseudounivex with respect
to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +
(12)119901119879
(nabla2
119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives
119875 (119909) minus 119875 (119909) +1
2119901119879
nabla2
119875 (119909) 119901
ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1
2119901119879
nabla2
119875 (119909) 119901
(17)
Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
(18)
Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883
0such that
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
le 119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(19)
and 119891119905(119909) + (119909
119879
119861119905119909)12
minus V119905119892119905(119909) minus (119909
119879
119862119905119909)12
le 119891119905(119909) +
(119909119879
119861119905119909)12
minus V119905119892119905(119909)minus (119909
119879
119862119905119909)12
for some 119905 isin 1 2 119896
The above relation together with the relation 120582119894gt 0
implies that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(20)
From the relations (5) (11) and (14) we get
119898
sum
119895=1
119910119895ℎ119895(119909) le
119898
sum
119895=1
119910119895ℎ119895(119909) (21)
Consequently (20) and (21) yield
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
(22)
This contradicts (18) Hence 119909 is an efficient solution forMFP1
Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119876(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] is second order 120588-pseudounivex withrespect to 120578 120595
0 and 119870 at 119909 isin 119883
0and 119867(119909) =
sum119898
119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with
respect to 120578 1205951 and 119870 at 119909 isin 119883
0with (nabla2119876(119909))119901 = 0
and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and
1205950 1205951 R rarr R satisfying 120595
0(119886) ge 0 rArr 119886 ge 0 and
1205951(119886) le 0 rArr 119886 le 0
(ii) 120588 + 120590 ge 0
Then 119909 is an efficient solution of1198721198651198751
Proof Suppose hypothesis holds
International Scholarly Research Notices 5
From the relations (5) (11) and (14) we get119898
sum
119895=1
119910119895ℎ119895(119909)
le
119898
sum
119895=1
119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0
(23)
Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following
119867(119909) minus 119867 (119909) +1
2119901119879
(nabla2
119867(119909)) 119901 le 0
997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +
1
2119901119879
(nabla2
119867(119909)) 119901 le 0
(24)
Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951
and119870 implies the following
120578(119909 119909)119879
nabla119867 (119909) + (nabla2
119867(119909)) 119901 + 120590119909 minus 1199092
le 0
997904rArr 120578(119909 119909)119879
nabla119867 (119909) + 120590119909 minus 1199092
le 0
(25)
From (9) we get
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119894=1
119910119894nablaℎ119894(119909) = 0
997904rArr nabla119876 (119909) + nabla119867 (119909) = 0
997904rArr 120578(119909 119909)119879
[nabla119876 (119909) + nabla119867 (119909)] = 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + 120578(119909 119909)119879
nabla119867 (119909)
+ 120590119909 minus 1199092
minus 120590119909 minus 1199092
= 0
(26)
Using (25) in (26) we get
120578(119909 119909)119879
nabla119876 (119909) minus 120590119909 minus 1199092
ge 0 (27)
Since 120588 + 120590 ge 0 we get
120588119909 minus 1199092
ge minus120590119909 minus 1199092
(28)
So we have
120578(119909 119909)119879
nabla119876 (119909) + 120588119909 minus 1199092
ge 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + nabla2
119876 (119909) 119901 + 120588119909 minus 1199092
ge 0
(29)
Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870
we obtained
119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +
1
2119901119879
nabla2
119876 (119909) 119901] ge 0 (30)
Using the property of119870 and 1205950 we get
119876 (119909) minus 119876 (119909) +1
2119901119879
nabla2
119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(31)
If 119909 were not an efficient solution to MFP1 then from (20)we have
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(32)
This contradicts (31)Therefore 119909 is an efficient solution for MFP1
3 Second Order Mixed Type MultiobjectiveFractional Duality
(i) MMFD0 Maximize
119871 (119906)
= (1198711(119906) minus
1
2119901119879
nabla2
1198711(119906) 119901 119871
2(119906)
minus1
2119901119879
nabla2
1198712(119906) 119901 119871
119896(119906) minus
1
2119901119879
nabla2
119871119896(119906) 119901)
(33)
where
119871119894(119906) =
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
119892119894(119906) minus 119906119879119862
119894119911
119894 = 1 2 119896 (34)
(ii) MMFD1 Maximize
119866 (119906)
= (1198661(119906) minus
1
2119901119879
nabla2
1198661(119906) 119901 119866
2(119906)
minus1
2119901119879
nabla2
1198662(119906) 119901 119866
119896(119906) minus
1
2119901119879
nabla2
119866119896(119906) 119901)
(35)
where 119866119894(119906) = 119891
119894(119906) + 119910
119879
1198691
ℎ1198691
(119906) + 119906119879
119861119894119908 minus ]119894119892119894(119906) minus
119906119879
119862119894119911 119894 = 1 2 119896 ]
119894are fixed parameters
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Scholarly Research Notices 3
Definition 3 The real differentiable function 119891119894is said to be
second order 120588-quasiunivex at 119909 isin 119883 with respect to 120578 120595and119870 if
119870 (119909 119909) 120595 [119891119894(119909) minus 119891
119894(119909) +
1
2119901119879
(nabla2
119891119894(119909) 119901)] le 0
997904rArr 120578(119909 119909)119879
[nabla119891119894(119909) + nabla
2
119891119894(119909) 119901] + 120588119909 minus 119909
2
le 0
forall119909 isin 119883
(8)
Remark 4 If 119901 = 0 the above definitions reduce to thedefinitions of 120588-univex 120588-pseudounivex and 120588-quasiunivexas introduced in [14]
Definition 5 A feasible point119909 is said to be efficient forMFP1if there exists no other feasible point 119909 in MFP1 such that119865119894(119909) le 119865
119894(119909) 119894 = 1 2 119896 and 119865
119903(119909) lt 119865
119903(119909) for some
119903 isin 1 2 119896
Definition 6 (see [33]) A feasible point119909 is said to be properlyefficient for MFP1 if it is efficient and there exist 119872 gt 0
such that for each 119894 isin 1 2 119896 and for all feasible point119909 in MFP1 satisfying 119865
119894(119909) lt 119865
119894(119909) we have 119865
119894(119909) minus 119865
119894(119909) le
119872(119865119903(119909) minus 119865
119903(119909)) for some 119903 isin 1 2 119896 such that 119865
119903(119909) gt
119865119903(119909)
We assume that 119891119894(119909) + (119909
119879
119861119894119909)12
ge 0 119892119894(119909) minus
(119909119879
119862119894119909)12
gt 0 119894 = 1 2 119896 for all 119909 isin 119883
Definition 7 (generalized Schwarz Inequality) Let 119861 be apositive semidefinite matrix of order 119899 Then for all 119909 119908 isinR119899 119909119879119861119908 le (119909119879119861119909)12(119908119879119861119908)12
The equality holds if 119861119909 = 120582119861119908 for some 120582 ge 0Let 1198691(119909) = 119895 isin 119872 = 1 2 119898 ℎ
119895(119909) = 0 and
V119894= (119891119894(119909) + (119909
119879
119861119894119909)12
)(119892119894(119909) minus (119909
119879
119862119894119909)12
)Then define the set119882(119909) = 119908 isin R119899 119908119879nablaℎ
119895(119909) le 0 119895 isin
119869(119909) satisfying any one of the following conditions
(a) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) + 119861
119894119909
(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909)) + (119908
119879
(V2119894119862119894)119908)12
ge 0 119908 isin119882(119909) 119894 = 1 2 119896
(b) 119909119879119861119894119909 = 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909) minus
119862119894119909(119909119879
119862119894119909)12
) + (119908119879
119861119894119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(c) 119909119879119861119894119909 = 0 119909119879119862
119894119909 = 0 rArr 119908
119879
(nabla119891119894(119909) minus V
119894nabla119892119894(119909)) +
(119908119879
119861119894119908)12
+ (119908119879
(V2119894119862119894)119908)12
ge 0 119908 isin 119882(119909) 119894 =1 2 119896
(d) 119909119879119861119894119909 gt 0 119909119879119862
119894119909 gt 0 rArr 119908
119879
(nabla119891119894(119909) +
119861119894119909(119909119879
119861119894119909)12
minus V119894nabla119892119894(119909) minus 119862
119894119909(119909119879
119862119894119909)12
) ge 0119908 isin 119882(119909) 119894 = 1 2 119896
Lemma 8 (see [33]) If 1199090 isin 1198830is an optimal solution of
119872119865119875120582 then 1199090 is properly efficient for MFP1
Lemma 9 (see [12]) 1199090 isin 1198830is an efficient solution for MFP0
if and only if it is an efficient solution of MFP1 with 119865(1199090) = 0
Lemma 10 (see [10] necessary optimality condition) If 119909 isin1198830is an optimal solution of (119872119865119875120582) such that119882(119909) = 120601 then
there exist V119894isin R+ 119908 119911 isin R119899 and 119910 isin R119898 such that
nabla120582119865 (119909) + 119910119879
nablaℎ (119909)
=
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119895=1
119910119895nablaℎ119895(119909) = 0
(9)
119865119894(119909) = 119891
119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
(10)
119910119879
ℎ (119909) = 0 (11)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (12)
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
(13)
119910 ge 0 (14)
V119894ge 0 119894 = 1 2 119896 (15)
Theorem 11 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119875(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] + sum119898
119895=1119910119895ℎ119895(119909) is second order 120588-
pseudounivex with respect to 120578 120595 and 119870 at 119909 isin 1198830
with (nabla2119875(119909))119901 = 0 where119891119894 119883 rarr R119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883times119883 rarr R119899119870 119883times119883 rarr R+ and 120595 R rarr R
satisfying 120595(119886) le 0 rArr 119886 le 0
(ii) 120588 ge 0
Then 119909 is an efficient solution of MFP1
Proof Suppose that the hypothesis holdsSince the conditions of Lemma 10 are satisfied from (9)
and (13) we have
nabla119875 (119909)
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909))
4 International Scholarly Research Notices
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
+
119898
sum
119895=1
119910119895ℎ119895(119909)) = 0
(16)
Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +
nabla2
119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +
120588119909 minus 1199092
ge 0Since 119875(119909) is second order 120588-pseudounivex with respect
to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +
(12)119901119879
(nabla2
119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives
119875 (119909) minus 119875 (119909) +1
2119901119879
nabla2
119875 (119909) 119901
ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1
2119901119879
nabla2
119875 (119909) 119901
(17)
Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
(18)
Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883
0such that
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
le 119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(19)
and 119891119905(119909) + (119909
119879
119861119905119909)12
minus V119905119892119905(119909) minus (119909
119879
119862119905119909)12
le 119891119905(119909) +
(119909119879
119861119905119909)12
minus V119905119892119905(119909)minus (119909
119879
119862119905119909)12
for some 119905 isin 1 2 119896
The above relation together with the relation 120582119894gt 0
implies that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(20)
From the relations (5) (11) and (14) we get
119898
sum
119895=1
119910119895ℎ119895(119909) le
119898
sum
119895=1
119910119895ℎ119895(119909) (21)
Consequently (20) and (21) yield
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
(22)
This contradicts (18) Hence 119909 is an efficient solution forMFP1
Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119876(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] is second order 120588-pseudounivex withrespect to 120578 120595
0 and 119870 at 119909 isin 119883
0and 119867(119909) =
sum119898
119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with
respect to 120578 1205951 and 119870 at 119909 isin 119883
0with (nabla2119876(119909))119901 = 0
and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and
1205950 1205951 R rarr R satisfying 120595
0(119886) ge 0 rArr 119886 ge 0 and
1205951(119886) le 0 rArr 119886 le 0
(ii) 120588 + 120590 ge 0
Then 119909 is an efficient solution of1198721198651198751
Proof Suppose hypothesis holds
International Scholarly Research Notices 5
From the relations (5) (11) and (14) we get119898
sum
119895=1
119910119895ℎ119895(119909)
le
119898
sum
119895=1
119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0
(23)
Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following
119867(119909) minus 119867 (119909) +1
2119901119879
(nabla2
119867(119909)) 119901 le 0
997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +
1
2119901119879
(nabla2
119867(119909)) 119901 le 0
(24)
Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951
and119870 implies the following
120578(119909 119909)119879
nabla119867 (119909) + (nabla2
119867(119909)) 119901 + 120590119909 minus 1199092
le 0
997904rArr 120578(119909 119909)119879
nabla119867 (119909) + 120590119909 minus 1199092
le 0
(25)
From (9) we get
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119894=1
119910119894nablaℎ119894(119909) = 0
997904rArr nabla119876 (119909) + nabla119867 (119909) = 0
997904rArr 120578(119909 119909)119879
[nabla119876 (119909) + nabla119867 (119909)] = 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + 120578(119909 119909)119879
nabla119867 (119909)
+ 120590119909 minus 1199092
minus 120590119909 minus 1199092
= 0
(26)
Using (25) in (26) we get
120578(119909 119909)119879
nabla119876 (119909) minus 120590119909 minus 1199092
ge 0 (27)
Since 120588 + 120590 ge 0 we get
120588119909 minus 1199092
ge minus120590119909 minus 1199092
(28)
So we have
120578(119909 119909)119879
nabla119876 (119909) + 120588119909 minus 1199092
ge 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + nabla2
119876 (119909) 119901 + 120588119909 minus 1199092
ge 0
(29)
Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870
we obtained
119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +
1
2119901119879
nabla2
119876 (119909) 119901] ge 0 (30)
Using the property of119870 and 1205950 we get
119876 (119909) minus 119876 (119909) +1
2119901119879
nabla2
119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(31)
If 119909 were not an efficient solution to MFP1 then from (20)we have
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(32)
This contradicts (31)Therefore 119909 is an efficient solution for MFP1
3 Second Order Mixed Type MultiobjectiveFractional Duality
(i) MMFD0 Maximize
119871 (119906)
= (1198711(119906) minus
1
2119901119879
nabla2
1198711(119906) 119901 119871
2(119906)
minus1
2119901119879
nabla2
1198712(119906) 119901 119871
119896(119906) minus
1
2119901119879
nabla2
119871119896(119906) 119901)
(33)
where
119871119894(119906) =
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
119892119894(119906) minus 119906119879119862
119894119911
119894 = 1 2 119896 (34)
(ii) MMFD1 Maximize
119866 (119906)
= (1198661(119906) minus
1
2119901119879
nabla2
1198661(119906) 119901 119866
2(119906)
minus1
2119901119879
nabla2
1198662(119906) 119901 119866
119896(119906) minus
1
2119901119879
nabla2
119866119896(119906) 119901)
(35)
where 119866119894(119906) = 119891
119894(119906) + 119910
119879
1198691
ℎ1198691
(119906) + 119906119879
119861119894119908 minus ]119894119892119894(119906) minus
119906119879
119862119894119911 119894 = 1 2 119896 ]
119894are fixed parameters
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Scholarly Research Notices
= nabla(
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
+
119898
sum
119895=1
119910119895ℎ119895(119909)) = 0
(16)
Also from hypothesis (i) we have nabla2119875(119909)119901 = 0So we can write nabla119875(119909) + nabla2119875(119909)119901 = 0Now for 120578(119909 119909) isin R119899 we can write 120578(119909 119909)119879(nabla119875(119909) +
nabla2
119875(119909)119901) = 0For 120588 ge 0 we have 120578(119909 119909)119879(nabla119875(119909) + nabla2119875(119909)119901) +
120588119909 minus 1199092
ge 0Since 119875(119909) is second order 120588-pseudounivex with respect
to 120578 120595 and 119870 at 119909 isin 1198830 we have 119870(119909 119909)120595119875(119909) minus 119875(119909) +
(12)119901119879
(nabla2
119875(119909))119901 ge 0 and using the properties of 119870 120595 itgives
119875 (119909) minus 119875 (119909) +1
2119901119879
nabla2
119875 (119909) 119901
ge 0 997904rArr 119875 (119909) ge 119875 (119909) minus1
2119901119879
nabla2
119875 (119909) 119901
(17)
Since nabla2119875(119909)119901 = 0 the above inequality implies 119875(119909) ge 119875(119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
] +
119898
sum
119895=1
119910119895ℎ119895(119909)
(18)
Suppose that 119909 is not efficient solution of MFP1 then thereexist 119909 isin 119883
0such that
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
le 119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
119894 = 1 2 119896
(19)
and 119891119905(119909) + (119909
119879
119861119905119909)12
minus V119905119892119905(119909) minus (119909
119879
119862119905119909)12
le 119891119905(119909) +
(119909119879
119861119905119909)12
minus V119905119892119905(119909)minus (119909
119879
119862119905119909)12
for some 119905 isin 1 2 119896
The above relation together with the relation 120582119894gt 0
implies that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(20)
From the relations (5) (11) and (14) we get
119898
sum
119895=1
119910119895ℎ119895(119909) le
119898
sum
119895=1
119910119895ℎ119895(119909) (21)
Consequently (20) and (21) yield
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
+
119898
sum
119895=1
119910119895ℎ119895(119909)
(22)
This contradicts (18) Hence 119909 is an efficient solution forMFP1
Theorem 12 (sufficient optimality condition) Let 119909 isin 1198830be
a feasible solution of MFP1 and there exist 120582119894isin R+119908 119911 isin R119899
V119894isin R+ and 119910 isin R119898 satisfying the condition in Lemma 10 at
119909 Furthermore suppose that the following conditions hold
(i) 119876(119909) = sum119896
119894=1120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus
(119909119879
119862119894119909)12
] is second order 120588-pseudounivex withrespect to 120578 120595
0 and 119870 at 119909 isin 119883
0and 119867(119909) =
sum119898
119895=1119910119895ℎ119895(119909) is second order 120590-quasiunivex with
respect to 120578 1205951 and 119870 at 119909 isin 119883
0with (nabla2119876(119909))119901 = 0
and (nabla2119867(119909))119901 = 0 where 119891119894 119883 rarr R 119892
119894 119883 rarr R
ℎ119895 119883 rarr R 119894 = 1 2 119896 119895 = 1 2 119898 119901 isin R119899
120578 119883 times 119883 rarr R119899 119870 119883 times 119883 rarr R+ and
1205950 1205951 R rarr R satisfying 120595
0(119886) ge 0 rArr 119886 ge 0 and
1205951(119886) le 0 rArr 119886 le 0
(ii) 120588 + 120590 ge 0
Then 119909 is an efficient solution of1198721198651198751
Proof Suppose hypothesis holds
International Scholarly Research Notices 5
From the relations (5) (11) and (14) we get119898
sum
119895=1
119910119895ℎ119895(119909)
le
119898
sum
119895=1
119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0
(23)
Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following
119867(119909) minus 119867 (119909) +1
2119901119879
(nabla2
119867(119909)) 119901 le 0
997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +
1
2119901119879
(nabla2
119867(119909)) 119901 le 0
(24)
Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951
and119870 implies the following
120578(119909 119909)119879
nabla119867 (119909) + (nabla2
119867(119909)) 119901 + 120590119909 minus 1199092
le 0
997904rArr 120578(119909 119909)119879
nabla119867 (119909) + 120590119909 minus 1199092
le 0
(25)
From (9) we get
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119894=1
119910119894nablaℎ119894(119909) = 0
997904rArr nabla119876 (119909) + nabla119867 (119909) = 0
997904rArr 120578(119909 119909)119879
[nabla119876 (119909) + nabla119867 (119909)] = 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + 120578(119909 119909)119879
nabla119867 (119909)
+ 120590119909 minus 1199092
minus 120590119909 minus 1199092
= 0
(26)
Using (25) in (26) we get
120578(119909 119909)119879
nabla119876 (119909) minus 120590119909 minus 1199092
ge 0 (27)
Since 120588 + 120590 ge 0 we get
120588119909 minus 1199092
ge minus120590119909 minus 1199092
(28)
So we have
120578(119909 119909)119879
nabla119876 (119909) + 120588119909 minus 1199092
ge 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + nabla2
119876 (119909) 119901 + 120588119909 minus 1199092
ge 0
(29)
Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870
we obtained
119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +
1
2119901119879
nabla2
119876 (119909) 119901] ge 0 (30)
Using the property of119870 and 1205950 we get
119876 (119909) minus 119876 (119909) +1
2119901119879
nabla2
119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(31)
If 119909 were not an efficient solution to MFP1 then from (20)we have
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(32)
This contradicts (31)Therefore 119909 is an efficient solution for MFP1
3 Second Order Mixed Type MultiobjectiveFractional Duality
(i) MMFD0 Maximize
119871 (119906)
= (1198711(119906) minus
1
2119901119879
nabla2
1198711(119906) 119901 119871
2(119906)
minus1
2119901119879
nabla2
1198712(119906) 119901 119871
119896(119906) minus
1
2119901119879
nabla2
119871119896(119906) 119901)
(33)
where
119871119894(119906) =
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
119892119894(119906) minus 119906119879119862
119894119911
119894 = 1 2 119896 (34)
(ii) MMFD1 Maximize
119866 (119906)
= (1198661(119906) minus
1
2119901119879
nabla2
1198661(119906) 119901 119866
2(119906)
minus1
2119901119879
nabla2
1198662(119906) 119901 119866
119896(119906) minus
1
2119901119879
nabla2
119866119896(119906) 119901)
(35)
where 119866119894(119906) = 119891
119894(119906) + 119910
119879
1198691
ℎ1198691
(119906) + 119906119879
119861119894119908 minus ]119894119892119894(119906) minus
119906119879
119862119894119911 119894 = 1 2 119896 ]
119894are fixed parameters
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Scholarly Research Notices 5
From the relations (5) (11) and (14) we get119898
sum
119895=1
119910119895ℎ119895(119909)
le
119898
sum
119895=1
119910119895ℎ119895(119909) 997904rArr 119867(119909) le 119867 (119909) 997904rArr 119867(119909) minus 119867 (119909) le 0
(23)
Also from hypothesis (i) we get (nabla2119867(119909))119901 = 0So we have the following
119867(119909) minus 119867 (119909) +1
2119901119879
(nabla2
119867(119909)) 119901 le 0
997904rArr 119870 (119909 119909) 1205951119867 (119909) minus 119867 (119909) +
1
2119901119879
(nabla2
119867(119909)) 119901 le 0
(24)
Hence the 120590-quasiunivexity of 119867(119909) with respect to 120578 1205951
and119870 implies the following
120578(119909 119909)119879
nabla119867 (119909) + (nabla2
119867(119909)) 119901 + 120590119909 minus 1199092
le 0
997904rArr 120578(119909 119909)119879
nabla119867 (119909) + 120590119909 minus 1199092
le 0
(25)
From (9) we get
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+
119898
sum
119894=1
119910119894nablaℎ119894(119909) = 0
997904rArr nabla119876 (119909) + nabla119867 (119909) = 0
997904rArr 120578(119909 119909)119879
[nabla119876 (119909) + nabla119867 (119909)] = 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + 120578(119909 119909)119879
nabla119867 (119909)
+ 120590119909 minus 1199092
minus 120590119909 minus 1199092
= 0
(26)
Using (25) in (26) we get
120578(119909 119909)119879
nabla119876 (119909) minus 120590119909 minus 1199092
ge 0 (27)
Since 120588 + 120590 ge 0 we get
120588119909 minus 1199092
ge minus120590119909 minus 1199092
(28)
So we have
120578(119909 119909)119879
nabla119876 (119909) + 120588119909 minus 1199092
ge 0
997904rArr 120578(119909 119909)119879
nabla119876 (119909) + nabla2
119876 (119909) 119901 + 120588119909 minus 1199092
ge 0
(29)
Since 119876(119909) is 120588-pseudounivex with respect to 120578 1205950 and 119870
we obtained
119870 (119909 119909) 1205950[119876 (119909) minus 119876 (119909) +
1
2119901119879
nabla2
119876 (119909) 119901] ge 0 (30)
Using the property of119870 and 1205950 we get
119876 (119909) minus 119876 (119909) +1
2119901119879
nabla2
119876 (119909) 119901 ge 0 997904rArr 119876 (119909) ge 119876 (119909)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(31)
If 119909 were not an efficient solution to MFP1 then from (20)we have
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
lt
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minusV119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
(32)
This contradicts (31)Therefore 119909 is an efficient solution for MFP1
3 Second Order Mixed Type MultiobjectiveFractional Duality
(i) MMFD0 Maximize
119871 (119906)
= (1198711(119906) minus
1
2119901119879
nabla2
1198711(119906) 119901 119871
2(119906)
minus1
2119901119879
nabla2
1198712(119906) 119901 119871
119896(119906) minus
1
2119901119879
nabla2
119871119896(119906) 119901)
(33)
where
119871119894(119906) =
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
119892119894(119906) minus 119906119879119862
119894119911
119894 = 1 2 119896 (34)
(ii) MMFD1 Maximize
119866 (119906)
= (1198661(119906) minus
1
2119901119879
nabla2
1198661(119906) 119901 119866
2(119906)
minus1
2119901119879
nabla2
1198662(119906) 119901 119866
119896(119906) minus
1
2119901119879
nabla2
119866119896(119906) 119901)
(35)
where 119866119894(119906) = 119891
119894(119906) + 119910
119879
1198691
ℎ1198691
(119906) + 119906119879
119861119894119908 minus ]119894119892119894(119906) minus
119906119879
119862119894119911 119894 = 1 2 119896 ]
119894are fixed parameters
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Scholarly Research Notices
(iii) 119872119872119865119863120582 Maximize 120582119866(119906) 120582 is 119896-dimensionalstrictly positive vector
all subject to same constraints
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 119910
119879
1198692
[nablaℎ1198692(119906) + nabla
2
nablaℎ1198692(119906) 119901]
= 0
(36)
119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908 minus V119894119892119894(119906) minus 119906
119879
119862119894119911 ge 0
for 119894 = 1 2 119896(37)
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
1199101198692
isin R119898minus|1198691|
(38)
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896 (39)
119910 ge 0 V119894ge 0 119894 = 1 2 119896 (40)
where119891119894 119883 rarr R119892
119894 119883 rarr R ℎ
119895 119883 rarr R 119894 = 1 2 119896
119895 = 1 2 119898 are differentiable functions119908 119911 isin R119899119901 isin R119899119861119894 and 119862
119894 119894 = 1 2 119896 are positive semidefinite matrices of
order 119899For the following theorems we assume that 120578 119883 times119883 rarr
R119899 119870 119883 times 119883 rarr R+ and 120595
0 1205951 R rarr R satisfying
1205950(119886) ge 0 rArr 119886 ge 0 and 119887 le 0 rArr 120595
1(119887) le 0 and 120588 120590 isin R
Theorem 13 (weak duality) Let 119909 be a feasible solution for theprimalMFP120582 and let (119906 119910 V 119908) be feasible for dual SMMFD120582If
(i) sum119896119894=1120582119894119866119894(sdot) is second order 120588-pseudounivex with
respect to 120578 1205950 119870 and for 119910
1198692
isin R119898minus|1198691| 1199101198791198692
ℎ1198692
(sdot) issecond order 120590-quasiunivex with respect to 120578 120595
1 and
119870 along with
(ii) 120588 + 120590 ge 0 then Inf(120582119865(119909)) ge Sup(120582119866(119906))
Proof Now from the primal and dual constraints we have
ℎ (119909) le 0
119910119879
1198692
ℎ1198692(119906) minus
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 ge 0
(41)
So
119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906) +
1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901 le 0
997904rArr 119870 (119909 119906) 1205951[119910119879
1198692
ℎ1198692(119909) minus 119910
119879
1198692
ℎ1198692(119906)
+1
2119901119879
nabla2
(119910119879
1198692
ℎ1198692(119906)) 119901] le 0
(42)
Since 1199101198791198692
ℎ1198692
is second order 120590-quasiunivex with respect to120578 1205951 and119870 and in view of (42) for 119909 119906 isin R119899 we have
120578(119909 119906)119879
nabla [119910119879
1198692
ℎ1198692(119906)] + nabla
2
[119910119879
1198692
ℎ1198692(119906)] 119901 + 120590119909 minus 119906
2
le 0
(43)
Again from the dual constraint (36) we have119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 119910119879
1198692
119879
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
(44)
Since 120578(119909 119906) isin R119899 we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901] = 0
997904rArr 120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901]
+ 120578(119909 119906)119879
119910119879
1198692
[nablaℎ1198692(119906) + nabla
2
ℎ1198692(119906) 119901]
+ 120590119909 minus 1199062
minus 120590119909 minus 1199062
= 0
(45)
Using (43) in above equation we get 120578(119909 119906)119879sum119896119894=1120582119894
[nabla119866119894(119906) + nabla
2
119866119894(119906)119901] minus 120590119909 minus 119906
2
ge 0Since 120588 + 120590 ge 0 we get 120588119909 minus 1199062 ge minus120590119909 minus 1199062So we have
120578(119909 119906)119879
119896
sum
119894=1
120582119894[nabla119866119894(119906) + nabla
2
119866119894(119906) 119901] + 120588119909 minus 119906
2
ge 0
(46)
Since sum119896
119894=1120582119894119866119894(119906) is second order 120588-pseudounivex
with respect to 120578 1205950 and 119870 by Definition 2 and
(46) we get 119870(119909 119906)1205950sum119896
119894=1120582119894119866119894(119909) minus sum
119896
119894=1120582119894119866119894(119906) +
(12)119901119879
sum119896
119894=1120582119894119866119894(119906)119901 ge 0
Using the property of 1205950and119870 we get
119896
sum
119894=1
120582119894119866119894(119909) minus
119896
sum
119894=1
120582119894119866119894(119906) +
1
2119901119879
119896
sum
119894=1
120582119894119866119894(119906) 119901 ge 0
997904rArr
119896
sum
119894=1
120582119894119866119894(119909) ge
119896
sum
119894=1
120582119894119866119894(119906)
997904rArr
119896
sum
119894=1
120582119894[119891119894(119909) + 119910
119879
1198691
ℎ1198691(119909)
+119909119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906)
+119906119879
119861119894119908 minus V119894119892119894(119906) minus 119909
119879
119862119894119911]
(47)
Equation (5) gives ℎ(119909) le 0 rArr 1199101198791198691
ℎ1198691
(119909) le 0 for 1199101198691
ge 0
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Scholarly Research Notices 7
So (47) implies that
119896
sum
119894=1
120582119894[119891119894(119909) + 119909
119879
119861119894119908 minus V119894119892119894(119909) minus 119909
119879
119862119894119911]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minusV119894119892119894(119906) minus 119906
119879
119862119894119911]
(48)
Now by Schwarz Inequality and (39) we have
119909119879
119861119894119908 le (119909
119879
119861119894119909)12
(119908119879
119861119894119908)12
le (119909119879
119861119894119909)12
119909119879
119862119894119911 le (119909
119879
119862119894119909)12
(119911119879
119862119894119911)12
le (119909119879
119862119894119909)12
119894 = 1 2 119896
(49)
So both (48) and (49) imply that
119896
sum
119894=1
120582119894[119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
]
ge
119896
sum
119894=1
120582119894[119891119894(119906) + 119910
119879
1198691
ℎ1198691(119906) + 119906
119879
119861119894119908
minus V119894119892119894(119906) minus 119906
119879
119862119894119911]
997904rArr Inf (120582119865 (119909)) ge Sup (120582119865 (119906))
(50)
Theorem 14 (strong duality) Let 119909 be optimal solution forMFP120582 and let 119882(119909) = 120601 Then there exist 120582
119894isin R+ 119908 119911 isin
R119899 V119894isin R+ and 119910 isin R119898 such that (119906 120582 119910 V 119908 119911 119901 =
0) is a feasible solution for dual and the objective values ofboth primal and dual are equal to zero Furthermore if (i)(119906 120582 119910 V 119908 119911) is feasible for dual (ii) sum119896
119894=1120582119894119866119894(sdot) is second
order 120588-pseudounivex with respect to 120578 1205950 and 119870 and for
1199101198692
isin R119898minus|1198691| 1199101198791198691
ℎ1198692
(sdot) is second order 120590-quasiunivex withrespect to 120578 120595
1 and 119870 along with (iii) 120588 + 120590 ge 0 then
(119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for SMMFD0
Proof Since 119909 is optimal solution for (MFP120582) by Lemma 10there exist 120582
119894isin R+119908 119911 isin R119899 V
119894isin R+ and 119910 isin R119898 such that
119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911] + 119910
119879
nablaℎ (119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894(119892119894(119909) minus (119909
119879
119862119894119909)12
) = 0
119894 = 1 2 119896
119910119879
ℎ (119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(51)
which can be written as119896
sum
119894=1
120582119894[nabla119891119894(119909) + 119910
119879
1198691
nablaℎ1198691(119909) + 119861
119894119908 minus V119894nabla119892119894(119909) minus 119862
119894119911]
+ 119910119879
1198691
nablaℎ1198692(119909) = 0
119891119894(119909) + (119909
119879
119861119894119909)12
minus V119894119892119894(119909) minus (119909
119879
119862119894119909)12
+ 119910119879
1198691
ℎ1198691(119909) = 0
119910119879
1198691
ℎ1198692(119909) = 0
119908119879
119861119894119908 le 1 119911
119879
119862119894119911 le 1 119894 = 1 2 119896
(119909119879
119861119894119909)12
= 119909119879
119861119894119908 (119909
119879
119862119894119909)12
= 119909119879
119862119894119911
119894 = 1 2 119896
119910 ge 0
V119894ge 0 119894 = 1 2 119896
(52)
These are nothing but the dual constraintsSo (119906 120582 119910 V 119908 119911 119901 = 0) is feasible solution for dual
problemAnd the objective values of MFP120582 and SMMFD120582 are
equal to zero It follows fromTheorem 13 and for any feasiblesolution (119906 120582 119910 V 119908 119911 119901 = 0) of dual that 120582119866(119906) le 120582119866(119909)
So (119906 120582 119910 V 119908 119911 119901 = 0) is optimal solution ofSMMFD120582 Then applying Lemmas 8 and 9 we concludethat (119906 120582 119910 V 119908 119911 119901 = 0) is properly efficient for(SMMFD0)
4 Special Case
If 119901 = 0 119862119894= 0 119894 = 1 2 119896 then our dual programming
reduces to the dual programming proposed by Tripathy [14]
5 Conclusion
In this paper three approaches given by Dinklebaeh [28] andJagannathan [12] for both primal and second order mixedtype dual of a nondifferentiable multiobjective fractionalprogramming problem are introduced and the necessaryand sufficient optimality conditions are established and aparameterization technique is used to establish duality results
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Scholarly Research Notices
under generalized second order 120588-univexity assumptionTheresults developed in this paper can be further extendedto higher order mixed type fractional problem containingsquare root term Also the present work can be furtherextended to a class of nondifferentiable minimax mixedfractional programming problems
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] C R Bector ldquoDuality in nonlinear fractional programmingrdquoZeitschrift fur Operations Research vol 17 no 5 pp 183ndash1931973
[2] C R Bector and S Chandra ldquoFirst and Second order duality fora class of non differentiable fractional programmingrdquo Journal ofInformation amp Optimization Sciences vol 7 pp 335ndash348 1986
[3] B D Craven ldquoFractional programmingrdquo in Sig Ma Series inApplied Mathematics vol 4 pp 145ndash152 Heldermann BerlinGermany 1998
[4] B Mond and T Weir ldquoDuality for fractional programmingwith generalized convexity conditionrdquo Journal of Information ampOptimization Sciences vol 3 pp 105ndash124 1982
[5] I M Stancu-Minasian Fractional Programming TheoryMethod and Application Kluwer Academic PublishersDordrecht The Netherland 1997
[6] S Schaible and T Ibaraki ldquoFractional programmingrdquo EuropeanJournal of Operational Research vol 12 no 4 pp 325ndash338 1983
[7] I Ahmad S K Gupta N Kailey and R P Agarwal ldquoDualityin nondifferentiable minimax fractional programming with B-(pr)-invexityrdquo Journal of Inequalities and Applications vol 2011article 75 15 pages 2011
[8] I Ahmad and S Sharma ldquoSecond-order duality for nondif-ferentiable multiobjective programming problemsrdquo NumericalFunctional Analysis and Optimization vol 28 no 9-10 pp 975ndash988 2007
[9] T R Gulati I Ahmad and D Agarwal ldquoSufficiency andduality in multiobjective programming under generalized typei functionsrdquo Journal of Optimization Theory and Applicationsvol 135 no 3 pp 411ndash427 2007
[10] I Ahmad Z Husain and S Al-Homidan ldquoSecond-order dual-ity in nondifferentiable fractional programmingrdquo NonlinearAnalysis Real World Applications vol 12 no 2 pp 1103ndash11102011
[11] C R Bector S Chandra and I Husain ldquoOptimality conditionsand duality in subdifferentiable multiobjective fractional pro-grammingrdquo Journal of Optimization Theory and Applicationsvol 79 no 1 pp 105ndash125 1993
[12] R Jagannathan ldquoDuality for nonlinear fractional programsrdquoZeitschrift fur Operations Research vol 17 no 1 pp 1ndash3 1973
[13] R Osuna-Gomez A Ruız-Canales and P Rufian-Lizana ldquoMul-tiobjective fractional programmingwith generalized convexityrdquoTrabajos de Investigacion Operativa vol 8 no 1 pp 97ndash1102000
[14] A K Tripathy ldquoMixed type duality in multiobjective fractionalprogramming under generalized 120588-univex functionrdquo Journal ofMathematical Modelling and Algorithms in Operations Research2013
[15] S K Mishra S Wang and K K Lai ldquoRole of 120572-pseudo-univex functions in vector variational-like inequality problemsrdquoJournal of Systems Science and Complexity vol 20 no 3 pp344ndash349 2007
[16] S Schaible ldquoFractional programmingrdquo in Handbook of GlobalOptimization R Horst and P M Pardalos Eds pp 495ndash608 KluwerAcademic Publishers DordrechtTheNetherlands1995
[17] Z A Liang H X Huang and P M Pardalos ldquoOptimalityconditions and duality for a class of nonlinear fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 110 no 3 pp 611ndash619 2001
[18] Z-A Liang H-X Huang and P M Pardalos ldquoEfficiencyconditions and duality for a class of multiobjective fractionalprogramming problemsrdquo Journal of Global Optimization vol27 no 4 pp 447ndash471 2003
[19] L B Santos R Osuna-Gomez and A Rojas-Medar ldquoNons-moothmultiobjective fractional programmingwith generalizedconvexityrdquo Revista Integracion vol 26 no 1 pp 1ndash12 2008
[20] C R Bector S Chandra and A Abha ldquoOn mixed duality inmathematical programmingrdquo Journal of Mathematical Analysisand Applications vol 259 no 1 pp 346ndash356 2001
[21] Z Xu ldquoMixed type duality in multiobjective programmingproblemsrdquo Journal of Mathematical Analysis and Applicationsvol 198 no 3 pp 621ndash635 1996
[22] H Zhou and Y Wang ldquoOptimality and mixed duality for non-smooth multi-objective fractional programmingrdquo Pure Mathe-matics and Applications vol 14 no 3 pp 263ndash274 2003
[23] I Ahmad andZHusain ldquoDuality in nondifferentiableminimaxfractional programming with generalized convexityrdquo AppliedMathematics and Computation vol 176 no 2 pp 545ndash5512006
[24] I Ahmad and Z Husain ldquoOptimality conditions and dualityin nondifferentiable minimax fractional programming withgeneralized convexityrdquo Journal of Optimization Theory andApplications vol 129 no 2 pp 255ndash275 2006
[25] B Mond ldquoA class of nondifferentiable fractional program-ming problemrdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 58 no 8 pp 337ndash341 1978
[26] D S Kim S J Kim and M H Kim ldquoOptimality andduality for a class of nondifferentiable multiobjective fractionalprogramming problemsrdquo Journal of Optimization Theory andApplications vol 129 no 1 pp 131ndash146 2006
[27] D S Kim Y J Lee and K D Bae ldquoDuality in nondifferentiablemultiobjective fractional programs involving conesrdquo TaiwaneseJournal of Mathematics vol 13 no 6 A pp 1811ndash1821 2009
[28] W Dinklebaeh ldquoOn nonlinear fractional programmingrdquoMan-agement Science vol 13 no 7 pp 492ndash498 1967
[29] M A Hanson ldquoOn sufficiency of the Kuhn-Tucker conditionsrdquoJournal of Mathematical Analysis and Applications vol 80 no2 pp 545ndash550 1981
[30] C R Bector S K Suneja and S Gupta ldquoUnivex functionsand univex nonlinear programmingrdquo in Proceeding of theAdministrative Science Association of Canada pp 115ndash124 1992
[31] S K Mishra ldquoOn multiple-objective optimization with gener-alized univexityrdquo Journal of Mathematical Analysis and Applica-tions vol 224 no 1 pp 131ndash148 1998
[32] A Jayswal ldquoNon-differentiable minimax fractional program-ming with generalized 120572-univexityrdquo Journal of Computationaland Applied Mathematics vol 214 no 1 pp 121ndash135 2008
[33] A M Geoffrion ldquoProper efficiency and the theory of vectormaximizationrdquo Journal of Mathematical Analysis and Applica-tions vol 22 no 3 pp 618ndash630 1968
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of