Higher order duality in multiobjective fractional programming with square root term under...

$: Higher order duality in multiobjective fractional programming with square root term under generalized higher order (F,α,β,ρ,σ,d)-V-type I univex functions$
Applied Mathematics and Computation 247 (2014) 880–897

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Applied Mathematics and Computation

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Higher order duality in multiobjective fractional programmingwith square root term under generalized higher orderðF;a; b;q;r; dÞ-V-type I univex functions

http://dx.doi.org/10.1016/j.amc.2014.09.0360096-3003/� 2014 Elsevier Inc. All rights reserved.

E-mail address: [email protected]

A.K. TripathyDepartment of Mathematics, Trident Academy of Technology, F2/A, Chandaka Industrial Estate, Patia, Bhubaneswar 24, Odisha, India

a r t i c l e i n f o

Keywords:Multiobjective fractional programmingGeneralized higher order ðF;a; b;q;r; dÞ-V-type I univex functionEfficient solutionSchwartz inequality

a b s t r a c t

In this paper, a new generalized class of higher order ðF;a; b;q;r;dÞ-V-type I univex func-tion is introduced with some examples for a differentiable multiobjective programming(MP). Then multiobjective fractional programming problem (MFP) is considered in whichthe numerator and denominator of objective functions contain square root of positivesemidefinite quadratic form and the necessary and sufficient conditions for efficient solu-tion for (MFP) are established under generalized higher order ðF;a; b;q;r; dÞ-V-type I uni-vex functions. Again, higher order dual program is proposed for (MFP) and the dualityresults are established under generalized of higher order ðF;a; b;q;r; dÞ-V-type I univexfunctions. Also, some computational work has been done to substantiate the analysis.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

The fractional programming problem has been one of the most interesting topics in the past few years. Mond [26] andZhang and Mond [35] have considered fractional programming problems containing square root of positive semidefinitequadratic form. The popularity of this kind of problem lies in the fact that although the objective functions are nondifferen-tiable, a simple formulation of the dual may be given. Duality for various forms of mathematical problems involving squareroots of positive semidefinite quadratic forms has been discussed by Ahmad et al. [2], Mond [26], Zhang and Mond [35] andothers. Duality in fractional programming is an important class of duality theory and several contribution have been made inpast (reader can see [2–4,10,19–21,26,30,31,33,35]).

Second order and higher order duality provides a tighter bound for the value of the objective function when approxima-tions are used because there are more parameters involved. Mangasarian [23] first formulated a class of second order andhigher order duality for nonlinear programming problems. Mond and Zhang [28] obtained duality results for various higherorder dual problems under higher order invexity assumptions. Higher order duality in nonlinear programming under variousgeneralized convex functions has been studied by many researchers like Chen [6], Suneja et al. [32], Ahmad et al.[1], Gulatiet al.[11], Kim and Lee [17] and Gulati and Sani [10].

On the other hand, to relax convexity assumptions imposed on the functions in theorems on optimality conditions andduality, various generalized convexity notations have been introduced. Reader can see [5,9,13,14,16,22,24,25,29,33]. Kuk andTanino [18] obtained the duality results in nonsmooth optimization involving generalized type I functions. Gulati et al. [9]introduced generalized ðF;a;q; dÞ-V-type I functions and established sufficient optimality condition and duality for

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A.K. Tripathy / Applied Mathematics and Computation 247 (2014) 880–897 881

multiobjective programming under the aforesaid functions. Hachimi and Aghezzaf [12] obtained second order duality resultsin multiobjective programming under second order ðF;a;q; dÞ-type 1 function. Gulati and Agarwal [8] introduced secondorder ðF;a;q; dÞ-V-type I function for multiobjective programming problem and proved duality results involving aforesaidfunctions. Tripathy and Devi [33] introduced generalized ðd;q;g; hÞ-type 1 univex functions and obtained duality resultsof mixed type duality for nondifferentiable multiobjective fractional programming under these functions. Jayswal et al.[15] introduced a new class of generalized ðF;a;q; hÞ-d-V-univex function for nonsmooth multiobjective programmingproblems.

In this paper, motivated by Jayswal et al. [15,16], Gulati et al. [9], Gulati and Agarwal [7,8] and Suneja et al. [29], we haveintroduced a new generalized class of higher order ðF;a; b;q;r; dÞ-V-type I univex functions with examples. Then multiob-jective fractional programming problem (MFP) is considered in which the numerator and denominator of objective functionscontain square root of positive semidefinite quadratic form and the necessary and sufficient conditions for efficient solutionfor (MFP) are established under generalized higher order ðF;a; b;q;r; dÞ-V-type I univex functions. Again, higher order dualprogram is proposed for (MFP) and the duality results are established under generalized of higher order ðF;a; b;q;r; dÞ-V-type I univex functions. Some computational work has been done to substantiate the analysis.

2. Notations and preliminaries

Let Rn be n-dimensional Euclidean space and Rnþ be its nonnegative orthant. Throughout this paper, the following conven-

tion for vectors x and y in Rn will be adopted:

x < y() xi < yi for all i ¼ 1;2; . . . ;n; x5y() xi5yi for all i ¼ 1;2; . . . ;n;

x 6 y() xi 6 yi for all i ¼ 1;2; . . . ;n; but x – y:

Consider the multiobjective programming problem:

ðMPÞMinimize f ðxÞ ¼ ðf 1ðxÞ; f 2ðxÞ; . . . ; f kðxÞÞSubject to hðxÞ50; x�X;

where X # Rn is an open set and f : X ! Rk and h : X ! Rm are continuously differentiable functions.Let X0 ¼ fx�XjhðxÞ50g be the set of feasible solutions of (MP).Since the objectives in multiobjective problems generally conflict with one another, an optimal solution is chosen from

the set of efficient/weak efficient solutions.

Definition 2.1. A point x 2 X0 is an efficient (Pareto optimal) solution of (MP), if there does not exist x 2 X0 such thatf ðxÞ 6 f ðxÞ that is there does not exist any x 2 X0 such that

fiðxÞ5fiðxÞ for all i ¼ 1;2; . . . ; k and f rðxÞ < f rðxÞ for some r 2 f1;2; . . . ; kg:

Definition 2.2. A point x 2 X0 is an weak efficient (weak Pareto optimal) solution of (MP), if there does not exist x 2 X0 suchthat f ðxÞ < f ðxÞ .

Definition 2.3 (Schwartz inequality). Let x; y 2 Rn and A 2 Rn � Rn be a positive semidefinite matrix, thenxT Ay 6 ðxT AxÞ

12ðyT AyÞ

12. Equality holds if for some k P 0;Ax P kAy.

Definition 2.4. A function F : X � X � Rn ! R is said to be sub-linear in the third argument if,

(i) Fðx;u; a1 þ a2Þ 6 Fðx;u; a1Þ þ Fðx;u; a2Þ;8a1; a2 2 Rn;(ii) Fðx;u;aaÞ ¼ aFðx;u; aÞ;8a 2 R;a P 0.

Clearly Fðx;u; 0Þ ¼ 0.Let f ¼ ðf 1; f 2; . . . ; f kÞ : X ! Rk;h ¼ ðh1; h2; . . . ;hmÞ : X ! Rm,K ¼ ðK1;K2; . . . ;KkÞ : X � Rn ! Rk, H ¼ ðH1;H2; . . . ;HmÞ : X � Rn ! Rm be differentiable functions and p 2 Rn; q 2 Rn.Let b0 : X � X ! Rþ; b1 : X � X ! Rþ;w0 : R! R;w1 : R! R,a : X � X ! Rþ n f0g; b : X � X ! Rþ n f0g; k ¼ ðk1; k2; . . . ; kkÞ 2 Rk

þ; y ¼ ðy1; y2; . . . ; ymÞ 2 Rmþ ;q;r 2 R.

Suppose that d : X � X ! R is a pseudo metric, that is for all x; y; z 2 X;

(i) dðx; yÞP 0 and x ¼ y) dðx; yÞ ¼ 0, (ii) dðx; yÞ ¼ dðy; xÞ,(iii) dðx; yÞ 6 dðx; zÞ þ dðz; yÞ.

882 A.K. Tripathy / Applied Mathematics and Computation 247 (2014) 880–897

Definition 2.5. ðf ;hÞ is said to be higher order ðF;a; b;q;r; dÞ-V-type I univex function at u 2 X with respect to Kðu; pÞ andHðu; qÞ, if there exist functions b0

i ; b1j ;a0;a1 : X � X ! Rþ n f0g; b0; b1 : X � X : Rþ;w0;w1 : R! R and q;r 2 R such that for all

x 2 X and for all i ¼ 1;2; . . . ; k; j ¼ 1;2; . . . ;m,

b0ðx;uÞw0ðb0i ðx;uÞ½fiðxÞ � fiðuÞ � Kiðu; pÞ þ pTrpKiðu;pÞ�ÞP Fðx;u;a0ðx;uÞðrfiðuÞ þ rpKiðu; pÞÞÞ þ qd2ðx;uÞ

and b1ðx;uÞw1ðb1j ðx;uÞ½hjðuÞ þ Hjðu; qÞ � qTrqHjðu;pÞ�ÞP Fðx;u; a1ðx;uÞðrhjðuÞ þ rqHjðu; qÞÞÞ þ rd2ðx;uÞ:

Remark 2.1.

(i) If w0ðtÞ ¼ w1ðtÞ ¼ t; boðx;uÞ ¼ b1ðx;uÞ ¼ 1, b0i ðx;uÞ ¼ b1

j ðx;uÞ ¼ 1, i ¼ 1;2; . . . ; k; j ¼ 1;2; . . . ;m in Definition 2.5, we getthe definition of higher order ðF;a;q; dÞ-type I function introduced by Gulati and Agarwal [7].

(ii) If w0ðtÞ ¼ w1ðtÞ ¼ t; boðx;uÞ ¼ b1ðx;uÞ ¼ 1;a0ðx;uÞ ¼ a1ðx;uÞ ¼ 1, b0i ðx;uÞ ¼ b1

j ðx;uÞ ¼ 1; i ¼ 1;2; . . . ; k; j ¼ 1;2; . . . ;min Definition 2.5, we get the definition of higher order ðF;qi;rjÞ -type I function introduced by Suneja et al. [29].

(iii) If we take Kiðu; pÞ ¼ 0 and Hjðu; qÞ ¼ 0 for all i = 1,2,. . .,k; j = 1,2,. . .,m and w0ðtÞ ¼ w1ðtÞ ¼ t; boðx;uÞ ¼ b1ðx;uÞ ¼ 1 inDefinition 2.5, we get the definition of ðF;a;q; dÞ-V-type I function given by Gulati et al. [9].

Definition 2.6. ðf ;hÞ is said to be higher order ðF;a; b;q;r; dÞ-V-pseudo quasi type I univex function at u 2 X with respect toKðu; pÞ and Hðu; qÞ, if there exist functions b0

i ; b1j ;a0;a1 : X � X ! Rþ n f0g; b0; b1 : X � X : Rþ;w0;w1 : R! R and q;r 2 R such

that for all x 2 X,

F x;u; a0ðx;uÞXk

i¼1

rfiðuÞ þ rpKiðu;pÞ� � ! !

=� qd2ðx;uÞ

) b0ðx;uÞw0

Xk

i¼1

b0i ðx; uÞ fiðxÞ � fiðuÞ � Kiðu;pÞ þ pTrpKiðu;pÞ

� �� !=0

and

� b1ðx;uÞw1

Xm

j¼1

b1j ðx; uÞ hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ

� � !50

) F x;u; a1ðx;uÞXm

j¼1

rhjðuÞ þ rqHjðu; qÞ� � ! !

5� rd2ðx;uÞ:

Definition 2.7. ðf ;hÞ is said to be higher order ðF;a; b;q;r; dÞ-V-strictly pseudo quasi type I univex function at u 2 X withrespect to Kðu; pÞ and Hðu; qÞ, if there exist functions b0

i ; b1j ;a0;a1 : X � X ! Rþ n f0g; b0; b1 : X � X : Rþ;w0;w1 : R! R and

q;r 2 R such that for all x 2 X,

F x;u; a0ðx;uÞXk

i¼1

ðrfiðuÞ þ rpKiðu;pÞÞ ! !

=� qd2ðx;uÞ

) b0ðx;uÞw0

Xk

i¼1


� �� !> 0

and

� b1ðx;uÞw1

Xm

j¼1


� � !50


j¼1


5� rd2ðx;uÞ:

Definition 2.8. ðf ;hÞ is said to be higher order ðF;a; b;q;r; dÞ-V- quasi pseudo type I univex function at u 2 X with respect toKðu; pÞ and Hðu; qÞ, if there exist functions b0

i ; b1j ;a0;a1 : X � X ! Rþ n f0g; b0; b1 : X � X : Rþ;w0;w1 : R! R and q;r 2 R such

that for all x 2 X,

b0ðx;uÞw0

Xk

i¼1


� �� !50

) F x;u; a0ðx;uÞXk

i¼1


5� qd2ðx;uÞ


and

� b1ðx;uÞw1

Xm

j¼1


� � !=0


j¼1


=� rd2ðx;uÞ:

Definition 2.9. ðf ;hÞ is said to be higher order ðF;a; b;q;r; dÞ-V- quasi strictly pseudo type I univex function at u 2 X withrespect to Kðu; pÞ and Hðu; qÞ, if there exist functions b0

i ; b1j ;a0;a1 : X � X ! Rþ n f0g; b0; b1 : X � X : Rþ;w0;w1 : R! R and

q;r 2 R such that for all x 2 X,

b0ðx;uÞw0

Xk

i¼1


� �� !50

) F x;u; a0ðx;uÞXk

i¼1


5� qd2ðx;uÞ

and F x;u;a1ðx; uÞXm

j¼1

ðrhjðuÞ þ rqHjðu; qÞÞ ! !

=� rd2ðx;uÞ

) b1ðx;uÞw1

Xm

j¼1

b1j ðx;uÞ hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ

� � !> 0:

Example 2.1. For (MP) let k ¼ 2;X ¼ R; f ¼ ðf 1; f 2Þ : X ! R2;h : X ! R such that f 1ðxÞ ¼ x4 þ x2 þ 1; f 2ðxÞ ¼ 2x2 þ 8x;hðxÞ ¼ �x.

So the feasible region is X0 ¼ fx 2 Xjx P 0g.Let Fðx;u; aÞ ¼ jaj2 ðx2 þ u2Þ; b0ðx;uÞ ¼ b1ðx;uÞ ¼ 2;w0ðtÞ ¼ 2t;w1ðtÞ ¼ 4t, dðx;uÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ u2p

.Let K ¼ ðK1;K2Þ : X � R! R2; H : X � R! R defined by K1ðx; pÞ ¼ �2pðxþ 1Þ2; K2ðx; pÞ ¼ 3pðx2 þ 1Þ; Hðx; qÞ ¼

qðxþ 1Þ2.Also, let a0ðx;uÞ ¼ 2 ¼ a1ðx;uÞ; b0

1ðx;uÞ ¼ b02ðx;uÞ ¼ b1

1ðx;uÞ ¼ 1;q1 ¼ �2;q2 ¼ 1;r1 ¼ �6;u ¼ 0.Now, 8x 2 X and u ¼ 0; we have

b0ðx;uÞw0ðb01ðx;uÞ½f 1ðxÞ � f 1ðuÞ � K1ðu;pÞ þ pTrpK1ðu;pÞ�Þ ¼ 4ð2x2 þ 8xÞ=Fðx;u; a0ðx;uÞðrf 1ðuÞ

þ rpK1ðu;pÞÞÞ þ q1d2ðx;uÞ ¼ 4x2;

b0ðx;uÞw0ðb02ðx;uÞ½f 2ðxÞ � f 2ðuÞ � K2ðu;pÞ þ pTrpK2ðu;pÞ�Þ ¼ 4ðx4 þ x2Þ=Fðx;u; a0ðx;uÞðrf 1ðuÞ

þ rpK1ðu;pÞÞÞ þ q2d2ðx;uÞ ¼ 4x2

and

�b1ðx;uÞw1ðb11ðx;uÞ½hðuÞ þ Hðu; qÞ � qTrqHðu;pÞ�Þ ¼ 0=Fðx;u;a1ðx;uÞðrhðuÞ þ rqHðu; qÞÞÞ þ r1d2ðx;uÞ ¼ �6x2:

This shows that ðf ; hÞ is higher order ðF;a; b;q;r; dÞ-V-type I univex function at u ¼ 0;8x 2 X with respect to Kðu; pÞ andHðu; qÞ. However, for the above defined problem if we take

ðiÞ x ¼ 5; f 1ðxÞ � f 1ðuÞ � K1ðu; pÞ þ pTrpK1ðu; pÞ ¼ 2x2 þ 8x

< Fðx;u;a0ðx;uÞðrf 1ðuÞ þ rpK1ðu;pÞÞÞ þ q1d2ðx;uÞ ¼ 4x2:

ðiiÞ x ¼ 1; f 2ðxÞ � f 2ðuÞ � K2ðu;pÞ þ pTrpK2ðu;pÞ ¼ x4 þ x2

< Fðx; u;a0ðx;uÞðrf 2ðuÞ þ rpK2ðu;pÞÞÞ þ q2d2ðx;uÞ ¼ 4x2:

This shows that ðf ;hÞ is not higher order ðF;a;q; dÞ-type I at u ¼ 0;8x 2 X with respect to Kðu; pÞ and Hðu; qÞ as introduced byGulati and Agarwal [9] and Ahmad et al. [1].

Example 2.2. For (MP) let k ¼ 2;X ¼ R; f ¼ ðf 1; f 2Þ : X ! R2;h : X ! R such that f 1ðxÞ ¼ �x2 � 4xþ 8; f 2ðxÞ ¼ �x4 þ 2;hðxÞ ¼ �x.

So the feasible region is X0 ¼ fx 2 Xjx P 0g.Let Fðx;u; aÞ ¼ jajðx2 þ u2Þ; b0ðx;uÞ ¼ b1ðx;uÞ ¼ 2;w0ðtÞ ¼ 2t;w1ðtÞ ¼ 3t, dðx;uÞ ¼


.Let K ¼ ðK1;K2Þ : X � R! R2; H : X � R! R defined by K1ðx; pÞ ¼ 2pðx2 þ 1Þ; K2ðx; pÞ ¼ 2pðxþ 1Þ2; Hðx; qÞ ¼

2qðx2 þ 1Þ.


Also let a0ðx;uÞ ¼ 12 ;a1ðx;uÞ ¼ 1; b0

1ðx;uÞ ¼ b02ðx;uÞ ¼ b1

1ðx; uÞ ¼ 1;q ¼ �2;r ¼ 2;u ¼ 0.Now 8x 2 X and u ¼ 0; we have

b0ðx;uÞw0

X2

i¼1

b0i ðx;uÞ fiðxÞ � fiðuÞ � Kiðu;pÞ þ pTrpKiðu;pÞ

� �� !¼ 4ð�x4 � x2 � 4xÞ50

) F x;u;a0ðx;uÞX2

i¼1

ðrfiðuÞ þ rpKiðu; pÞÞ ! !

þ qd2ðx;uÞ ¼ �2x2 � 0

and

Fðx;u;a1ðx;uÞðrhðuÞþrqHðu;qÞÞÞþrd2ðx;uÞ ¼ 3x2 P 0)�b1ðx;uÞw1ðb11ðx;uÞ½hðuÞþHðu;qÞ�qTrqHðu;qÞ�Þ ¼ 0 P 0:

This shows that ðf ;hÞ is higher order ðF;a; b;q;r; dÞ-V-quasi pseudo type I univex function at u ¼ 0;8x 2 X with respect toKðu; pÞ and Hðu; qÞ. However, for the above defined problem if we take

ðiÞ x ¼ 1; b0ðx;uÞw0ðf 1ðxÞ � f 1ðuÞ � K1ðu; pÞ þ pTrpK1ðu;pÞÞ ¼ 4ð�x2 � 4xÞ< Fðx;u; a0ðx;uÞðrf 1ðuÞ þ rpK1ðu; pÞÞÞ þ q1d2ðx;uÞ ¼ �x2;

b0ðx;uÞw0ðf 2ðxÞ � f 2ðuÞ � K2ðu; pÞ þ pTrpK2ðu;pÞÞ ¼ �4x4

< Fðx;u; a0ðx;uÞðrf 2ðuÞ þ rpK2ðu; pÞÞÞ þ q2d2ðx;uÞ ¼ 0

and � b1ðx; uÞw1ðhðuÞ þ Hðu; qÞ � qTrqHðu; qÞÞ ¼ 0

< Fðx;u; a1ðx;uÞðrhðuÞ þ rqHðu; qÞÞÞ þ rd2ðx;uÞ ¼ 3x2:

This shows that ðf ;hÞ is not higher order ðF;a; b;q;r; dÞ-V-type I univex at u ¼ 0;8x 2 X with respect to Kðu; pÞ and Hðu; qÞ.

Example 2.3. For (MP) let k ¼ 2;X ¼ R; f ¼ ðf 1; f 2Þ : X ! R2;h ¼ ðh1;h2Þ : X ! R2 such that f 1ðxÞ ¼ x4 þ 2x2 þ 2;f 2ðxÞ ¼ x3 þ 4xþ 4;h1ðxÞ ¼ �x;h2ðxÞ ¼ �x2.

So the feasible region is X0 ¼ fx 2 X : x P 0g.Let Fðx;u; aÞ ¼ 2jajðx2 þ u2Þ; b0ðx;uÞ ¼ 4; b1ðx;uÞ ¼ 3;w0ðtÞ ¼ t

2 ;w1ðtÞ ¼ 3t, dðx;uÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ u2p

.Let K ¼ ðK1;K2Þ : X � R! R2; H ¼ ðH1;H2Þ : X � R! R2 defined by K1ðx; pÞ ¼ 2pðxþ 1Þ2; K2ðx; pÞ ¼ �8pðx3 þ 1Þ;

H1ðx; qÞ ¼ qðxþ 1Þ; H2ðx; qÞ ¼ 2qðxþ 1Þ.Also let a0ðx;uÞ ¼ 2;a1ðx;uÞ ¼ 1

3 ; b01ðx;uÞ ¼ b0

2ðx;uÞ ¼ 12 ; b

11ðx;uÞ ¼ b1

1ðx;uÞ ¼ 1;q ¼ 3;r ¼ �3;u ¼ 0.Now 8x 2 X and u ¼ 0; we have

F x;u;a0ðx;uÞX2

i¼1

rf iðuÞþrpKiðu;pÞ� � ! !

þqd2ðx;uÞ¼11x2=0

) b0ðx;uÞw0

X2

i¼1

b0i ðx;uÞ f iðxÞ� f iðuÞ�Kiðu;pÞþpTrpKiðu;pÞ

� �� !

¼ðx4þx3þ2x2þ4xÞ=0

and

�b1ðx;uÞw1

X2

j¼1


� � !¼ 0 � 0

) F x;u;a1ðx;uÞX2

j¼1

ðrhðuÞ þ rqHðu; qÞÞ ! !

þ rd2ðx;uÞ

¼ �x26 0:

This shows that ðf ;hÞ is higher order ðF;a; b;q;r; dÞ-V-quasi pseudo type I univex function at u ¼ 0;8x 2 X with respect toKðu; pÞ and Hðu; qÞ.

However, for the above defined problem if we take

ðiÞ x ¼ 1; b0ðx;uÞw0

X2

i¼1

ðb0i ðx;uÞ½fiðxÞ � fiðuÞ � Kiðu; pÞ þ pTrpKiðu; pÞ�Þ

!¼ ðx4 þ x3 þ 2x2 þ 4xÞ

< Fðx;u;a0ðx; uÞX2

i¼1

rfiðuÞ þ rpKiðu; pÞ� � !

þ q1d2ðx;uÞ ¼ 11x2:


This shows that ðf ;hÞ is not higher order ðF;a; b;q;r; dÞ-V-type I univex at u ¼ 0;8x 2 X with respect to Kðu; pÞ and Hðu; qÞ.Therefore the above examples clearly illustrate that the class of higher order ðF;a; b;q;r; dÞ-V- pseudo-quasi type I univex

and higher order ðF;a; b;q;r; dÞ-V- quasi- pseudo type I univex functions are more generalized than the cited in literature.

3. Optimality conditions

We consider the following multiobjective nondifferentiable fractional programming problems:

� (MFP) Minimize f ðxÞþðxT BxÞ12

gðxÞ�ðxT CxÞ12¼ f 1ðxÞþðxT B1xÞ

12

g1ðxÞ�ðxT C1xÞ12; f 2ðxÞþðxT B2xÞ

12

g2ðxÞ�ðxT C2xÞ12; . . . ; f kðxÞþðxT BkxÞ

12

gkðxÞ�ðxT CkxÞ12

� �

Subject to

hðxÞ � 0; x 2 X;

where X # Rn; f ¼ ðf 1; f 2; . . . ; f kÞ : Rn ! Rk; g ¼ ðg1; g2; . . . ; gkÞ : Rn ! Rk and h ¼ ðh1;h2; . . . ; hmÞ : Rn ! Rm are continuously dif-ferentiable function. Bi and Ci; i ¼ 1;2;3; . . . ; k are positive semidefinite matrices of order n. In the sequel, we assume that

fiðxÞ þ ðxT BixÞ12 P 0 and giðxÞ � ðxT CixÞ

12 > 0, for i ¼ 1;2; . . . ; k.

Let X0 ¼ fx 2 X # Rn : hjðxÞ � 0; j ¼ 1;2; . . . ;mg be the set of feasible solutions of problems (MFP).

Theorem 3.1 [1]Karush–Kuhn–Tucker type necessary optimality conditions. Suppose that x0 2 X is an efficient solution of(MFP) and a constraint qualification [27,34] is satisfied at x0, then there exist k > 0; k 2 Rk; y P 0; y 2 Rm;wi 2 Rn;

zi 2 Rn; i ¼ 1;2; . . . ; k and K : X � Rn ! Rk;H : X � Rn ! Rm such that

Xk

i¼1kir

fiðx0Þ þ xT0Biwi

giðx0Þ � xT0Cizi

� �þXm

j¼1

yjrhjðx0Þ ¼ 0; ð3:1Þ

Xm

j¼1yjhjðx0Þ ¼ 0; ð3:2Þ

ðxT0Bix0Þ

12 ¼ xT

0Biwi; ðxT0Cix0Þ

12 ¼ xT

0Cizi; i ¼ 1;2; . . . ; k; ð3:3ÞwT

i Biwi � 1; zTi Cizi � 1; i ¼ 1;2; . . . ; k: ð3:4Þ

Theorem 3.2 (Karush–Kuhn–Tucker type sufficient optimality conditions). Suppose that x0 2 X be feasible solution of (MFP).Let there exist k > 0; k 2 Rk; y P 0; y 2 Rm;wi 2 Rn; zi 2 Rn; i ¼ 1;2; . . . ; k and K : X � Rn ! Rk;H : X � Rn ! Rm such that

Xk

i¼1kir

fiðx0Þ þ xT0Biwi


� �þXm

j¼1

yjrhjðx0Þ þXk

i¼1

kirpKiðx0; pÞ þXm

j¼1

yjrqHjðx0; qÞ ¼ 0; ð3:5Þ

Kiðx0; pÞ � pTrpKiðx0;pÞ ¼ 0; i ¼ 1;2; . . . ; k; ð3:6ÞXm

j¼1yjðhjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞÞ ¼ 0; ð3:7Þ

ðxT0Bix0Þ

12 ¼ xT

0Biwi; ðxT0Cix0Þ

12 ¼ xT

0Cizi; i ¼ 1;2; . . . ; k; ð3:8ÞwT

i Biwi � 1; zTi Cizi � 1; i ¼ 1;2; . . . ; k: ð3:9Þ

Moreover, if

(i) f ð:Þþð:ÞT Bwgð:Þ�ð:ÞT Cz

;hð:Þ

is higher order ðF;a; b;q;r; dÞ-V-pseudo quasi type I univex function at x0 with respect to Kðx0; pÞ and

Hðx0; qÞ,(ii) for any t 2 R; w0ðtÞP 0 ) t P 0and t P 0) w1ðtÞP 0; b0ðx; x0Þ > 0; b1ðx; x0Þ > 0,(iii) b1

j ðx; x0Þ ¼ 1; 8j ¼ 1;2; . . . ;m;(iv) q

a0ðx;x0Þþ r

a1ðx;x0ÞP 0.

Then x0 is an efficient solution for (MFP).

Proof. Suppose x0 is not an efficient solution for (MFP). Then there exists x 2 X0 such that

fiðxÞ þ ðxT BixÞ12

giðxÞ � ðxT CixÞ126

fiðx0Þ þ ðxT0Bix0Þ

12

giðx0Þ � ðxT0Cix0Þ

12; 8i 2 f1;2; . . . ; kg

andf rðxÞ þ ðxT BrxÞ

12

grðxÞ � ðxT CrxÞ12<

f rðx0Þ þ ðxT0Brx0Þ

12

grðx0Þ � ðxT0Crx0Þ

12; for some r 2 f1;2; . . . ; kg:


As xT Biwi 6 ðxT BixÞ12; ðxT

0Bix0Þ12 ¼ xT

0Biwi; xT Cizi 6 ðxT CixÞ12; ðxT

0Cix0Þ12 ¼ xT

0Cizi, we can write

fiðxÞ þ xT Biwi

giðxÞ � xT Cizi6

fiðx0Þ þ xT0Biwi


; 8i 2 f1;2; . . . ; kg

andf rðxÞ þ xT Brwr

grðxÞ � xT Crzr<

f rðx0Þ þ xT0Brwr

grðx0Þ � xT0Crzr

; for some r 2 f1;2; . . . ; kg:

Since k > 0 and b0i ðx; x0Þ > 0; i ¼ 1;2; . . . ; k, from the above inequalities we get

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi

giðxÞ � xT Cizi� fiðx0Þ þ xT

0Biwi


� �< 0:

Again using (3.6) in above inequality, we get

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi


� Kiðx0; pÞ þ pTrpKiðx0;pÞ� �

< 0: ð3:10Þ

Also from (3.7) we havePm

j¼1yj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ� ¼ 0.Since b1

j ðx; x0Þ ¼ 1;8j ¼ 1;2; . . . ;m, we can write

Xm

j¼1

b1j ðx; x0Þyj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ� ¼ 0:

As hypothesis (ii) holds, therefore the above inequalities yields

b1ðx; x0Þw1

Xm

j¼1

b1j ðx; x0Þyj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ�

!P 0

) � b1ðx; x0Þw1

Xm

j¼1

b1j ðx; x0Þyj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ�

!6 0:

So by hypothesis (i), we getFðx; x0; a1ðx; x0Þ

Pmj¼1yjðrhjðx0Þ þ rqHjðx0; qÞÞÞ 6 �rd2ðx; x0Þ.

Since a1ðx; x0Þ > 0, the above inequality implies that

F x; x0;Xm

j¼1

yj rhjðx0Þ þ rqHjðx0; qÞ� � !

6 � ra1ðx; x0Þ

d2ðx; x0Þ: ð3:11Þ

Now (3.5) along with the Definition 2.4 gives

F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi


� �þrpKiðx0; pÞ

� � !þ F x; x0;

Xm

j¼1


P F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi


� �þrpKiðx0;pÞ

� �þXm

j¼1


¼ Fðx; x0; 0Þ ¼ 0

) F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� � !

P �F x; x0;Xm

j¼1


Pr

a1ðx; x0Þd2ðx; x0Þ ðusing ð3:11ÞÞ

P � qa0ðx; x0Þ

d2ðx; x0Þ ðusing hypothesis ðivÞÞ

) F x; x0;a0ðx; x0ÞXk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� � !P �qd2ðx; x0Þ:


So by hypothesis (i), we get

b0ðx; x0Þw0

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi


� Kiðx0;pÞ þ pTrpKiðx0; pÞ� �( )

P 0:

Since w0ðtÞP 0) t P 0 and b0ðx; x0Þ > 0, the above inequality implies that

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi



P 0:

This is a contradiction to (3.10). Hence we proved. h

Theorem 3.3 (Karush–Kuhn–Tucker type sufficient optimality conditions). Suppose that x0 2 X be the feasible solution ofðMFPÞ. Let there exist k > 0; k 2 Rk; y P 0; y 2 Rm;wi 2 Rn; zi 2 Rn; i ¼ 1;2; . . . ; k and K : X � Rn ! Rk;H : X � Rn ! Rm satisfyingthe conditions from (3.5) to (3.9). Moreover, if.

(i) f ð:Þþð:ÞT Bwgð:Þ�ð:ÞT Cz

;hð:Þ

is higher order ðF;a; b;q;r; dÞ-V-quasi strictly pseudo type I univex function at x0 with respect to Kðx0; pÞand Hðx0; qÞ,

(ii) for any t 2 R; t < 0) w0ðtÞ < 0 and w1ðtÞ < 0) t < 0; b0ðx; x0Þ > 0; b1ðx; x0Þ > 0,(iii) b1

j ðx; x0Þ ¼ 1;8j ¼ 1;2; . . . ;m;(iv) q

a0ðx;x0Þþ r

a1ðx;x0ÞP 0.

Then x0 is an efficient solution for ðMFPÞ.

Proof. Suppose x0 is not an efficient solution for (MFP). Then there exists x 2 X0 such that

fiðxÞ þ ðxT BixÞ12

giðxÞ � ðxT CixÞ126

fiðx0Þ þ ðxT0Bix0Þ

12

giðx0Þ � ðxT0Cix0Þ

12; 8i 2 f1;2; . . . ; kg

andf rðxÞ þ ðxT BrxÞ

12

grðxÞ � ðxT CrxÞ12<

f rðx0Þ þ ðxT0Brx0Þ

12

grðx0Þ � ðxT0Crx0Þ

12; for some r 2 f1;2; . . . ; kg:

As xT Biwi 6 ðxT BixÞ12; ðxT

0Bix0Þ12 ¼ xT

0Biwi; xT Cizi 6 ðxT CixÞ12; ðxT

0Cix0Þ12 ¼ xT

0Cizi, we can write

fiðxÞ þ xT Biwi

giðxÞ � xT Cizi6

fiðx0Þ þ xT0Biwi


; 8i 2 f1;2; . . . ; kg

andf rðxÞ þ xT Brwr

grðxÞ � xT Crzr<

f rðx0Þ þ xT0Brwr

grðx0Þ � xT0Crzr

; for some r 2 f1;2; . . . ; kg:

Since k > 0 and b0i ðx; x0Þ > 0; i ¼ 1;2; . . . ; k, from the above inequalities we get

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi


� �< 0:

Again using (3.6) in above inequality, we get

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi



< 0:

As hypothesis (ii) holds,

b0ðx; x0Þw0

Xk

i¼1

b0i ðx; x0Þki

fiðxÞ þ xT Biwi


0Biwi


� Kiðx0;pÞ þ pTrpKiðx0; pÞ� �( )

< 0: ð3:12Þ

By hypothesis (i), (3.12) implies

F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� � !6 � q

a0ðx; x0Þd2ðx; x0Þ: ð3:13Þ

Now (3.5) along with the Definition 2.4 gives


F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� � !þ F x; x0;

Xm

j¼1

yj½rhjðx0Þ þ rqHjðx0; qÞ� !

P F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� �þXm

j¼1

yjðrhjðx0Þ þ rqHjðx0; qÞÞ !

¼ Fðx; x0; 0Þ ¼ 0

) F x; x0;Xm

j¼1


P �F x; x0;Xk

i¼1

ki rfiðx0Þ þ xT

0Biwi



� � !

Pq

a0ðx; x0Þd2ðx; x0Þ ðusing ð3:13ÞÞ

P � ra1ðx; x0Þ

d2ðx; x0Þ ðusing hypothesis ðivÞÞ

) F x; x0;Xm

j¼1


P � ra1ðx; x0Þ

d2ðx; x0Þ

) F x; x0;a1ðx; x0ÞXm

j¼1


P �rd2ðx; x0Þ:

So by hypothesis (i), we get

� b1ðx; x0Þw1

Xm

j¼1

b1j ðx; x0Þyj hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ

� � !> 0

) b1ðx; x0Þw1

Xm

j¼1

b1j ðx; x0Þyj hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ

� � !< 0:

Since w1ðtÞ < 0) t < 0 and b1ðx; x0Þ > 0, the above inequality implies

Xm

j¼1

b1j ðx; x0Þyj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ� < 0:

Since b1j ðx; x0Þ ¼ 1;8j ¼ 1;2; . . . ;m,

Xm

j¼1

yj½hjðx0Þ þ Hjðx0; qÞ � qTrqHjðx0; qÞ� < 0:

This is a contradiction to (3.7). Hence we proved. h

4. Higher order multiobjective fractional duality

In this section, we have introduced higher order multiobjective fractional dual of (MFP) and established weak, strong andconverse duality theorems under generalized higher order ðF;a; b;q;r; dÞ-V-type I univex functions.

We consider the following higher order fractional dual for (MFP).

� (HMFD): Maximize v ¼ ðv1;v2; . . . ;vkÞ

Subject to

Xk

i¼1

ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ� þXm

i¼1

yjrhjðuÞ þXk

i¼1

ki½rpðKiðu; pÞ � v iGiðu; pÞÞ� þXm

i¼1

yjrqHjðu; qÞ ¼ 0; ð4:1Þ

Xk

i¼1

kifiðuÞ þ uT Biwi � v iðgiðuÞ � uT CiziÞ þ Kiðu;pÞ � v iGiðu;pÞ�pTrpðKiðu;pÞ � v iGiðu;pÞÞ

( )P 0; ð4:2Þ


Xm

i¼1

yjðhjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞÞP 0; ð4:3Þ

wTi Biwi � 1; zT

i Cizi � 1; i ¼ 1;2; . . . ; k; ð4:4Þ

u P 0;v i P 0; i ¼ 1;2; . . . ; k; ð4:5Þ

y P 0; ki > 0; i ¼ 1;2; . . . ; k; ð4:6Þ

where K;G : X � Rn ! Rk;H : X � Rn ! Rm are continuously differentiable functions, and p 2 Rn; q 2 Rn;w ¼ ðw1;w2; . . . ;wkÞ;z ¼ ðz1; z2; . . . ; zkÞ;wi; zi 2 Rn.

Theorem 4.1 (Weak Duality). Let x and ðu;v ; k; y;w; z; p; qÞ be the feasible solution for (MFP) and (HMFD) respectively. If

(i) ðf ð:Þ þ ð:ÞT Bw� vðgð:Þ � ð:ÞT CzÞ;hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V-pseudo quasi type 1 univex function at u with respectto Kðu; pÞ � vGðu; pÞ and Hðu; qÞ

(ii) for any t 2 R;w0ðtÞP 0) t P 0 and t P 0) w1ðtÞP 0; b0ðx;uÞ > 0; b1ðx;uÞ > 0,(iii) b1

j ðx;uÞ ¼ 1;8j ¼ 1;2; . . . ;m, and(iv) q

a0ðx;uÞþ r

a1ðx;uÞP 0.

Then

f 1ðxÞþðxT B1xÞ12


12


� �iðv1; . . . ;vkÞ:

Proof. Suppose that contradiction holds. That is



12


� �6 ðv1; . . . ;vkÞ:

) fiðxÞþðxT BixÞ12

giðxÞ�ðxT CixÞ126 v i;8i ¼ 1;2; . . . ; k and f r ðxÞþðxT Br xÞ

12

grðxÞ�ðxT Cr xÞ12< v r , for some r 2 f1;2; . . . ; kg.

) fiðxÞ þ ðxT BixÞ12 � v iðgiðxÞ � ðxT CixÞ

12Þ � 0;8i ¼ 1;2; . . . ; k and.

f rðxÞ þ ðxT BrxÞ12 � v rðgrðxÞ � ðxT CrxÞ

12Þ < 0, for some r 2 f1;2; . . . ; kg.

Since ki > 0; b0i ðx;uÞ > 0; i ¼ 1;2; . . . ; k; we obtain

Xk

i¼1

b0i ðx;uÞki½fiðxÞ þ ðxT BixÞ

12 � v iðgiðxÞ � ðxT CixÞ

12Þ� < 0: ð4:7Þ

Now by Schwartz inequality and (4.4), we obtain

xT Biwi 6 ðxT BixÞ12; xT Cizi 6 ðxT CixÞ

12; i ¼ 1;2; . . . ; k:

So, (4.7) becomes

Xk

i¼1

b0i ðx;uÞki½fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ� < 0: ð4:8Þ

As ðu; v; k; y;w; z; p; qÞ is feasible solution for (HMFD), from (4.3) and hypothesis (ii) and (iii) we get

b1ðx;uÞw1ðXm

j¼1

b1j ðx;uÞyj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�ÞP 0

)� b1ðx;uÞw1ðXm

j¼1

b1j ðx; uÞyj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�Þ � 0;

which by the hypothesis (i) and the fact that a1ðx;uÞ > 0 implies

F x;u;Xm

j¼1

yj½rhjðuÞ þ rqHjðu; qÞ� !

6 � ra1ðx;uÞ

d2ðx;uÞ: ð4:9Þ

Now using feasibility condition (4.1) along with sub-linearity of F, we get


Fðx;u;Xk

i¼1

ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu; pÞ � v iGiðu; pÞÞ�Þ þ Fðx;u;Xm

j¼1

yj½rhjðuÞ þ rqHjðu; qÞ�Þ

P F x;u;

Xk

i¼1

ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ

þXk

i¼1

kirpðKiðu;pÞ � v iGiðu;pÞÞ�

þXm

j¼1

yj½rhjðuÞ þ rqHjðu; qÞ�

0BBBBBBBBBBB@

1CCCCCCCCCCCA

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼ Fðx;u; 0Þ ¼ 0:

) Fðx;u;Xk

i¼1

ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu;pÞ � v iGiðu;pÞÞ�Þ

P �Fðx;u;Xm

j¼1


Pr

a1ðx;uÞd2ðx;uÞ ðusing ð4:9ÞÞ

P � qa0ðx;uÞ

d2ðx;uÞ: ðusing hypothesis ðivÞÞ

Since a0ðx;uÞ > 0, we get

F x;u;Xk

i¼1

ki rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu;pÞ � v iGiðu;pÞÞ� � !

P �qd2ðx;uÞ:

Using hypothesis (i), the above inequality gives

b0ðx;uÞw0Pk

i¼1b0i ðx; uÞki

fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ�ðfiðuÞ þ uT Biwi � v iðgiðuÞ � uT CiziÞÞ�ðKiðu; pÞ � v iGiðu; pÞÞþpTrpðKiðu;pÞ � v iGiðu;pÞÞ

0BBB@

1CCCA

0BBB@

1CCCAP 0:

Since w0ðtÞP 0) t P 0; b0ðx;uÞ > 0 and feasibility condition (4.2) holds, we have

Xk

i¼1

b0i ðx;uÞki½fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ�P 0:

� �
This is a contradiction to (4.8). So f 1ðxÞþðxT B1xÞ
12


12


iðv1; . . . ;vkÞ. h

Theorem 4.2 (Weak Duality). Let x and ðu;v ; k; y;w; z; p; qÞ be the feasible solution for ðMFPÞ and ðHMFDÞ respectively. If

(i) ðf ð:Þ þ ð:ÞT Bw� vðgð:Þ � ð:ÞT CzÞ; hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V-quasi strictly pseudo type I univex function at u withrespect to Kðu; pÞ � vGðu; pÞ and Hðu; qÞ

(ii) for any t 2 R; t 6 0) w0ðtÞ � 0 and w1ðtÞ < 0) t < 0; b0ðx;uÞ > 0; b1ðx;uÞ > 0,(iii) b1

j ðx;uÞ ¼ 1;8j ¼ 1;2; . . . ;m, and(iv) q

a0ðx;uÞþ r

a1ðx;uÞP 0.

Then



12


� �iðv1; . . . ;vkÞ:

Proof. Suppose that contradiction holds. That is



12


� �6 ðv1; . . . ;vkÞ:

) fiðxÞþðxT BixÞ12

giðxÞ�ðxT CixÞ126 v i;8i ¼ 1;2; . . . ; k and f r ðxÞþðxT Br xÞ

12

grðxÞ�ðxT Cr xÞ12< v r , for some r 2 f1;2; . . . ; kg.


) fiðxÞ þ ðxT BixÞ12 � v iðgiðxÞ � ðxT CixÞ

12Þ � 0;8i ¼ 1;2; . . . ; k and.

f rðxÞ þ ðxT BrxÞ12 � v rðgrðxÞ � ðxT CrxÞ

12Þ < 0, for some r 2 f1;2; . . . ; kg.

Since ki > 0; b0i ðx;uÞ > 0; i ¼ 1;2; . . . ; k; we obtain

Xk

i¼1

b0i ðx;uÞki½fiðxÞ þ ðxT BixÞ

12 � v iðgiðxÞ � ðxT CixÞ

12Þ� < 0: ð4:10Þ

Now by Schwartz inequality and (4.4), we obtain

xT Biwi 6 ðxT BixÞ12; xT Cizi 6 ðxT CixÞ

12; i ¼ 1;2; . . . ; k:

So, (4.10) becomes

Xk

i¼1

b0i ðx;uÞki½fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ� < 0: ð4:11Þ

Subtracting (4.2) from (4.11), we get

Xk

i¼1

b0i ðx;uÞki

fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ�ðfiðuÞ þ uT Biwi � v iðgiðuÞ � uT CiziÞÞ�ðKiðu;pÞ � v iGiðu;pÞÞ þ pTrpðKiðu;pÞ � v iGiðu;pÞÞ

0B@

1CA < 0:

Using hypothesis (ii), we get

b0ðx;uÞw0Pk

i¼1b0i ðx;uÞki

fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ�ðfiðuÞ þ uT Biwi � v iðgiðuÞ � uT CiziÞÞ�ðKiðu; pÞ � v iGiðu;pÞÞþpTrpðKiðu; pÞ � v iGiðu; pÞÞ

0BBB@

1CCCA

0BBB@

1CCCA � 0:

So by hypothesis (i) with the fact a0ðx;uÞ > 0, we get

F x;u;

Pki¼1ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ

þXk

i¼1

kirpðKiðu; pÞ � v iGiðu; pÞÞ

0BB@

1CCA

0BB@

1CCA 6 � q

a0ðx;uÞd2ðx;uÞ: ð4:13Þ

Now using feasibility condition (4.1) along with sub-linearity of F, we get

F x;u;Xk

i¼1

ki rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu; pÞ � v iGiðu; pÞÞ� � !

þ F x;u;Xm

j¼1


P F x;u;

Pki¼1ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ

þXk

i¼1

kirpðKiðu;pÞ � v iGiðu;pÞÞ

þXm

j¼1

yj½rhjðuÞ þ rqHjðu; qÞ�

0BBBBBBB@

1CCCCCCCA

0BBBBBBB@

1CCCCCCCA¼ Fðx;u; 0Þ ¼ 0:

) F x;u;Xm

j¼1


P �F x;u;Xk

i¼1

ki rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu; pÞ � v iGiðu; pÞÞ� � !

Pq

a0ðx;uÞd2ðx;uÞ ðusing ð4:13ÞÞ

P � ra1ðx;uÞ

d2ðx;uÞ ðusing hypothesisðivÞÞ:

Since a1ðx;uÞ > 0, we get

F x;u;Xm

j¼1

yi½rhjðuÞ þ rqðHjðu; qÞÞ� !

P �rd2ðx; uÞ;


which by hypothesis (i) gives

� b1ðx;uÞw1

Xk

j¼1

b1j ðx; uÞyj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�

!> 0

) b1ðx;uÞw1

Xk

j¼1


!< 0

) b1ðx;uÞw1ðXk

j¼1

yj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�Þ < 0: ðusing hypothesis ðiiiÞÞ

So by hypothesis (ii), we getPk

j¼1yj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�Þ < 0.This is a contradiction to the feasibility condition (4.3). Hence we proved. h

Theorem 4.3 (Strong Duality). Let x be an efficient solution of (MFP) at which a constraint qualification [27,34] is satisfied and

Kiðx;0Þ ¼ 0; rpKiðx;0Þ ¼ rfiðxÞ þ Biwi; rpKiðx;0Þ ¼ rgiðxÞ � Cizi;

Giðx;0Þ ¼ 0; Hjðx; 0Þ ¼ 0;rqHjðx; 0Þ ¼ rhjðxÞ; i ¼ 1;2; . . . ; k; j ¼ 1;2; . . . ;m:

ð4:13Þ

Then there exist k 2 Rk; y 2 Rm;wi; zi 2 Rn; i ¼ 1;2; . . . ; k such that ðx;w; z;v ; k; y; p ¼ 0; q ¼ 0Þ is a feasible solution of ðHMFDÞ.Further, if the conditions of weak duality Theorem 4.1 or Theorem 4.2 are satisfied for each feasible solution of ðMFPÞ and ðHMFDÞ,then ðx;w; z;v; k; y; p ¼ 0; q ¼ 0Þ is an efficient solution of ðHMFDÞ and the corresponding value of objective functions are equal.

Proof. Since x be an efficient solution of (MFD) at which a constraint qualification [27,34] is satisfied, then by Theorem 3.1,there exist k 2 Rk; y 2 Rm;wi; zi 2 Rn; i ¼ 1;2; ::; k such that

Xk

i¼1kir

fiðxÞ þ xT Biwi

giðxÞ � xT Cizi

� �þXm

j¼1

yjrhjðxÞ ¼ 0; ð4:14Þ

Xm

j¼1

yjhjðxÞ ¼ 0; ð4:15Þ

ðxT BixÞ12 ¼ xT Biwi; ðxT CixÞ

12 ¼ xT Cizi; i ¼ 1;2; . . . ; k; ð4:16Þ

wTi Biwi 6 1; zT

i Cizi 6 1; i ¼ 1;2; . . . ; k; ð4:17Þk > 0; yj P 0; j ¼ 1;2; . . . ;m: ð4:18Þ

Eq. (4.15) can be written as

Xk

i¼1

fgiðxÞ�xT Cizigr½fiðxÞþxT Biwi ��ffiðxÞþxT Biwigr giðxÞ�xT Cizi½ �ðgiðxÞ�xT CiziÞ

2

� �þXm

j¼1

yjrhjðxÞ ¼ 0

)Xk

i¼1

ki

giðxÞ � xT Cizi

� �r½fiðxÞ þ xT Biwi� � fiðxÞþxT Biwi

giðxÞ�xT Cizi

r½giðxÞ � xT Cizi�

n oþXm

j¼1

yjrhjðxÞ ¼ 0: ð4:19Þ

Since ki > 0; yj P 0; giðxÞ � xT Cizi > 0, let

ki

giðxÞ � xT Cizi¼ k̂i;

fiðxÞ þ xT Biwi

giðxÞ � xT Cizi¼ v i; i ¼ 1;2; . . . ; k: ð4:20Þ

Clearly

k̂i > 0; i ¼ 1;2; . . . ; k: ð4:21Þ

Thus using (4.20) in (4.19), we obtain

Xk

i¼1

k̂iðr½fiðxÞ þ xT Biwi� � v ir½giðxÞ � xT Cizi�Þ þXm

j¼1

yjrhjðxÞ ¼ 0: ð4:22Þ

Again fiðxÞþxT BiwigiðxÞ�xT Cizi

¼ v i

) fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ ¼ 0; i ¼ 1;2; . . . ; k

)Xk

i¼1

k̂i½fiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ� ¼ 0: ð4:23Þ


So the relation (4.15)–(4.17), (4.18), (4.21)–(4.23) along with (4.13) imply that ðx;w; z;v ; k; y; p ¼ 0; q ¼ 0Þ is feasible for(HMFD) and the corresponding value of objective functions are equal. The proof of the remaining part follows from the weakduality Theorem 4.1 and Theorem 4.2.

Theorem 4.4 (Strict Converse Duality). Let x and ðu;v ; k; y;w; z; p ¼ 0; q ¼ 0Þ be the feasible solution for (MFP) and (HMFD)respectively. If

(i) ðf ð:Þ þ ð:ÞT Bw� vðgð:Þ � ð:ÞT CzÞ;hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V- strictly pseudo quasi type I univex function at uwith respect to Kðu; pÞ � vGðu; pÞ and Hðu; qÞ,

(ii) for any t 2 R;w0ðtÞ > 0) t > 0 and t P 0) w1ðtÞP 0; b0ðx;uÞ > 0; b1ðx;uÞ > 0,(iii) b1

j ðx;uÞ ¼ 1;8j ¼ 1;2; . . . ;m, and(iv) q

a0ðx;uÞþ r

a1ðx;uÞP 0.

Then x ¼ u.

5. Computational work

Example 5.1.

Let X # Rþ and f ¼ ðf 1; f 2Þ : X ! R2; g ¼ ðg1; g2Þ : X ! R2;

h ¼ ðh1; h2Þ : X ! R2 defined by f 1ðxÞ ¼ x4 þ x3 þ 1; f 2ðxÞ ¼ x3 þ 2xþ 3;

g1ðxÞ ¼ x2 þ 1; g2ðxÞ ¼ xþ 1; h1ðxÞ ¼ �x; h2ðxÞ ¼ �x2 þ 1: Also; let B1 ¼ B2 ¼ C1 ¼ C2 ¼ 0:

Then the nondifferentiable multiobjective fractional programming problem (MFP) considered in Section 3 becomes:

� (MFP) Minimizef ðxÞ ¼ x4 þ x3 þ 1

2 ;x3 þ 2xþ 3

� �ð5:1Þ

gðxÞ x þ 1 xþ 2
Subject to h1ðxÞ ¼ �x 6 0;
h2ðxÞ ¼ �x2 þ 1 6 0; ð5:2Þx 2 X: ð5:3Þ

Here the feasible region is X0 ¼ fxjx P 1g. Clearly the minimum value of the objective functions is ð1:5;2Þ.

To find the corresponding higher order dual, let K ¼ ðK1;K2Þ : X � R! R2,G ¼ ðG1;G2Þ : X � R! R2;H ¼ ðH1;H2Þ : X � R! R2 defined by K1ðu; pÞ ¼ 2pðuþ 1Þ2;K2ðu; pÞ ¼ �6pðu3 þ 1Þ;G1ðu; pÞ ¼

pðuþ 1Þ;G2ðu; pÞ ¼ pðuþ 1Þ,H1ðu; qÞ ¼ qðuþ 1Þ;H2ðu; qÞ ¼ 2qðuþ 1Þ.Then our proposed dual reduces to.

� (HMFD):

Maximize v ¼ ðv1;v2ÞSubject to

k1ð4u3 þ 3u2 � 2v1uÞ þ k2ð3u2 þ 2� v2Þ þ k1½2ðuþ 1Þ2 � v1ðuþ 1Þ� þ k2½�6ðu3 þ 1Þ � v2ðuþ 1Þ�� y1 � 2y2uþ y1ðuþ 1Þ þ 2y2ðuþ 1Þ ¼ 0; ð5:4Þ

k1½ðu4 þ u3 þ 1Þ � v1ðu2 þ 1Þ� þ k2½ðu3 þ 2uþ 3Þ � v2ðuþ 2Þ�P 0; ð5:5Þ� y1u� y2ðu2 � 1ÞP 0; ð5:6Þ

u P 0; v1 P 0; v2 P 0; k1 > 0; k2 > 0; y1 P 0; y2 P 0: ð5:7Þ

Clearly ðu ¼ 12 ;v1 ¼ 1

2 ;v2 ¼ 2624 ; k1 ¼ 1

2 ; k2 ¼ 12 ; y1 ¼ 1

2 ; y2 ¼ 12Þ is a feasible solution of (HMFD). The value of the objective func-

tions of (HMFD) is v ¼ ðv1 ¼ 12 ;v2 ¼ 26

24Þ.So, for any x 2 X0;

f ðxÞgðxÞiv ¼ 1

2 ;2624

� �.

Now, let F : X � X � R! R defined by Fðx;u; aÞ ¼ jajðx2 þ u2Þ. Clearly F satisfies the properties in Definition 2.4.

Now from (5.6) and taking b1 ¼ 4;w1ðtÞ ¼ 2t;a1ðx;uÞ ¼ 45 ; b

11ðx;uÞ ¼ b1

2ðx;uÞ ¼ 1;r ¼ �2; dðx;uÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ u2p

, we see that fory1 ¼ 1

2 ; y2 ¼ 12 ; q1; q2 2 R at u ¼ 1

2


�b1ðx;uÞw1

X2

j¼1


!¼ �1 < 0

) Fðx;u;a1ðx;uÞX2

j¼1

yj½rhjðuÞ þ rqHjðu; qÞ�Þ ¼ ðx2 þ u2Þ 5a1ðx;uÞ4

þ r� �

6 0:

Again, let a ¼ a0ðx;uÞðP2

i¼1ki½rfiðuÞ � v irgiðuÞ þ rpðKiðu; pÞ � v iGiðu; pÞÞ�Þ.For b0 ¼ 2;w0ðtÞ ¼ t;a0ðx;uÞ > 0; b0

1ðx;uÞ ¼ b02ðx;uÞ ¼ 1;q ¼ 2; dðx;uÞ ¼


, we see that for v1 ¼ 12 ;v2 ¼ 26

24 ; k1 ¼ 12 ;

k2 ¼ 12 ; y1 ¼ 1

2 ; y2 ¼ 12 ; p1; p2 2 R at u ¼ 1

2

Fðx;u; aÞ þ qd2ðx; uÞ ¼ jajðx2 þ u2Þ þ qðx2 þ u2Þ ¼ ðx2 þ u2Þ 3a0ðx;uÞ4

þ q� �

P 0

) b0ðx;uÞw0

P2i¼1b

0i ðx;uÞkiðfiðxÞ � v igiðxÞÞ

�X2

i¼1

b0i ðx;uÞkiðfiðuÞ � v igiðuÞ

�X2

i¼1

b0i ðx;uÞkiðKiðu;pÞ � v iGiðu;pÞÞ

þX2

i¼1

b0i ðx;uÞkiðpTrpðKiðu;pÞ � v iGiðu;pÞÞÞ

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA:

¼ 212

x4 þ 2x3 þ 3x2� x2

2

� �� 63

32

� �P 0; 8x 2 X0:

So ðf ð:Þ � vgð:Þ;hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V-pseudo quasi type I univex function at u ¼ 12 with respect to

Kðu; pÞ � vGðu; pÞ and Hðu; qÞ along with the conditions (ii), (iii) and (iv) in Theorem 4.1. Hence the weak duality Theorem 4.1holds good.

Again we can see that at x ¼ 1 and ðu ¼ 1;v1 ¼ 1:5;v2 ¼ 2; k1 ¼ k2 ¼ 12 ; y1 ¼ 0; y2 ¼ 1Þ is a feasible solution of (MFP) and

(HMFD) respectable and x ¼ u.Also, (i) f ðxÞ

gðxÞ ¼ ð1:5;2Þ ¼ v ,(ii) ðf ð:Þ � vgð:Þ;hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V-strictly pseudo quasi type I univex function at u ¼ 1 with respect to

Kðu; pÞ � vGðu; pÞ and Hðu; qÞ along with the conditions (ii), (iii) and (iv) in Theorem 4.4 and for v1 ¼ 1:5;v2 ¼ 2; k1 ¼ 1

2 ; k2 ¼ 12 ; y1 ¼ 0; y2 ¼ 1; p ¼ 0 and b0 ¼ 1 ¼ b2;w0ðtÞ ¼ w1ðtÞ ¼ t;a0ðx;uÞ ¼ 1;a1ðx;uÞ ¼ 1

2 ; b01ðx;uÞ ¼ b0

2ðx;uÞ ¼b1

1ðx;uÞ ¼ b12ðx;uÞ ¼ 1;q ¼ 2;r ¼ � 3

2 ; dðx;uÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ u2p

. Hence Theorem 4.4 (Strictly Converse Duality) holds good.

Example 5.2.

Let X # R2þ; x ¼ ðx1; x2Þ 2 X and f ¼ ðf 1; f 2Þ : X ! R2; g ¼ ðg1; g2Þ : X ! R2;

h ¼ ðh1;h2Þ : X ! R2 defined by f 1ðxÞ ¼ x21 þ x2

2 þ 1; f 2ðxÞ ¼ x21 þ x1x2 þ 3;

g1ðxÞ ¼ x21 þ x2 þ 1; g2ðxÞ ¼ x1 þ x2 þ 2; h1ðxÞ ¼ �x1 � x2 þ 1; h2ðxÞ ¼ �x2

1 þ x2 þ 1:� �
Also, let B1 ¼ B2 ¼ C1 ¼ C2 ¼
1 00 1 .

Then the nondifferentiable multiobjective fractional programming problem (MFP) considered in Section 3 becomes:

� ðMFPÞ Minimize f ðxÞþðxT BxÞ121 ¼

x21þx2

2þ1þffiffiffiffiffiffiffiffiffix2

1þx22

p2

ffiffiffiffiffiffiffiffiffi2 2

p ;x2

1þx1x2þ3þffiffiffiffiffiffiffiffiffix2

1þx22

pffiffiffiffiffiffiffiffiffi2 2

p� �

gðxÞ�ðxT CxÞ2 x1þx2þ1� x

1þx

2x1þx2þ2� x

1þx

2 ð5:8Þ
Subject to h1ðxÞ ¼ �x1 � x2 þ 1 6 0;
h2ðxÞ ¼ �x21 þ x2 þ 1 6 0; ð5:9Þ

x 2 X: ð5:10Þ

Here the feasible solution is X0 ¼ fðx1; x2Þjx1 P 1; x2 6 x21 � 1g.

Clearly the minimum value of the first objective functions tends to 1 as x!1 and minimum value of second objectivefunction is 2.5 at x ¼ ð1;0Þ.

To find the corresponding higher order dual, let K ¼ ðK1;K2Þ : X � R! R2,G ¼ ðG1;G2Þ : X � R! R2;H ¼ ðH1;H2Þ : X � R! R2 defined by.


K1ðu; pÞ ¼ ðp1 þ p2Þðu21 � u2 � 1Þ; K2ðu; pÞ ¼ 2ðp1 þ p2Þðu1 þ u2

2 þ 12Þ;

G1ðu; pÞ ¼ ðp1 þ p2Þðu1 þ u2 þ 1Þ; G2ðu; pÞ ¼ ðp1 þ p2Þðu1 þ u2 þ 1Þ,H1ðu; qÞ ¼ ðq1 þ q2Þðu1 � u2 þ 1Þ; H2ðu; qÞ ¼ ðq1 þ q2Þðu1 þ u2 þ 1Þ. Also, let w1 ¼ w2 ¼

w0

w00

� �, z1 ¼ z2 ¼

z0

z00

� �,

Then our proposed dual reduces to.

� (HMFD):

Maximize v ¼ ðv1;v2ÞSubject to

k1ð2u1 þw0 � 2v1u1 þ v1z0 þ u21 � u2 � 1� v1u1 � v1u2 � v1Þ þ k2ð2u1 þ u2 þw0 � v2 þ v2z0 þ 2u1 þ 2u2

2

þ 1� v2u1 � v2u2 � v2Þ þ y1ðu1 � u2Þ þ y2ð�u1 þ u2 þ 1Þ ¼ 0; ð5:11Þ

k1ð2u2 þw00 � v1 þ v1z00 þ u21 � u2 � 1� v1u1 � v1u2 � v1Þ þ k2ðu1 þw00 � v2 þ v2z00 þ 2u1 þ 2u2

2 þ 1� v2u1 � v2u2 � v2Þ þ y1ðu1 � u2Þ þ y2ðu1 þ u2 þ 2Þ ¼ 0; ð5:12Þ

k1½u21 þ u2

2 þ 1þ u1w0 þ u2w00 � v1ðu21 þ 1� u1z0 � u2z00Þ� þ k2½u2

1 þ u1u2 þ 3þ u1w0 þ u2w00

� v2ðu22 þ 2� u1z0 � u2z00Þ�P 0; ð5:13Þ

y1ð�u1 � u2 þ 1Þ þ y2ð�u21 þ u2 þ 1ÞP 0; ð5:14Þ

ðw0Þ2 þ ðw00Þ2 6 1; ðz0Þ2 þ ðz00Þ2 6 1; ð5:15Þu P 0; v1 P 0; v2 P 0; k1 > 0; k2 > 0; y1 P 0; y2 P 0: ð5:16Þ

From the dual constraints (5.11) to (5.16) we observed that ðu1 ¼ 0;u2 ¼ 1;v1 ¼ 1;v2 ¼ 35 ; k1 ¼ 1

2 ; k2 ¼ 12 ; y1 ¼ 2

3 ; y2 ¼ 13 ;

w0 ¼ 12 ;w

00 ¼ 12 ; z

0 ¼ 12 ; z

00 ¼ 12 ; p1; p2; q1; q2 2 RÞ is a feasible solution of (HMFD) and the value of the objective functions of

(HMFD) is v ¼ ðv1 ¼ 1; v2 ¼ 35Þ.

From the above discussion, we observe that for any x 2 X0,

f ðxÞ þ ðxT BxÞ12

gðxÞ � ðxT CxÞ12iv ¼ 1;

23

� �:

Now, let F : X � X � R2 ! R defined by Fðx;u; aÞ ¼ jajððx1 � u1Þ2 þ ðx2 � u2Þ2Þ. Clearly F satisfies the properties in Definition 2.4.

Now from (5.13) and taking b1 ¼ 2;w1ðtÞ ¼ t;a1ðx; uÞ ¼ 9; b11ðx;uÞ ¼ b1

2ðx;uÞ ¼ 1;r ¼ �2; dðx; uÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � u1Þ2 þ ðx2 � u2Þ2

q,

we see that for y1 ¼ 12 ; y2 ¼ 1

2 ; q1; q2 2 R at u ¼ 12

�b1ðx;uÞw1

X2

j¼1

b1j ðx;uÞyj½hjðuÞ þ Hjðu; qÞ � qTrqHjðu; qÞ�

!¼ �8

3< 0) Fðx;u;a1ðx;uÞ

X2

j¼1


¼ ðx21 þ ðx2 � 1Þ2Þð1þ rÞ < 0:

Again, leta ¼ a0ðx;uÞ

P2i¼1ki½rfiðuÞ þ Biwi � v iðrgiðuÞ � CiziÞ þ rpðKiðu; pÞ � v iGiðu; pÞÞ�

.

For b0 ¼ 2;w0ðtÞ ¼ t;a0ðx;uÞ > 0; b01ðx;uÞ ¼ b0

2ðx; uÞ ¼ 1;q ¼ 2; dðx;uÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � u1Þ2 þ ðx2 � u2Þ2

q, we see that for

v1 ¼ 1; ;v2 ¼ 35 ; k1 ¼ 1

2 ; k2 ¼ 12 ; y1 ¼ 2

3 ; y2 ¼ 13 ;w

0 ¼ 12 ;w

00 ¼ 12 ; z

0 ¼ 12 ; z

00 ¼ 12 ; p1; p2 2 R at u ¼ ð0;1Þ

Fðx;u; aÞ þ qd2ðx;uÞ ¼ ðx21 þ ðx2 � 1Þ2Þ 9a0ðx; uÞ

100þ q

� �P 0) b0ðx;uÞw0P2

i¼1b0i ðx;uÞkiðfiðxÞ þ xT Biwi � v iðgiðxÞ � xT CiziÞ

�X2

i¼1

b0i ðx;uÞkiðfiðuÞ þ uT Biwi � v iðgiðuÞ � uT CiziÞÞ

�X2

i¼1

b0i ðx;uÞkiðKiðu;pÞ � v iGiðu;pÞÞ

þX2

i¼1

b0i ðx;uÞkiðpTrpðKiðu;pÞ � v iGiðu;pÞÞÞ

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA:

¼ 12

x21 þ

25

x22 þ x1x2 þ

3310

x1 þ3310

x2 þ245

� �� 2 P 0; 8x 2 X0:


So ðf ð:Þ þ ð:ÞT Bw� vðgð:Þ � ð:ÞT CzÞ;hð:ÞÞ is higher order ðF;a; b;q;r; dÞ-V-pseudo quasi type I univex function at u ¼ ð0;1Þwithrespect to Kðu; pÞ � vGðu; pÞ and Hðu; qÞ along with the conditions (ii), (iii) and (iv) in Theorem 4.1. Hence the weak dualityTheorem 4.1 holds good.

6. Conclusions

In this paper, a new generalized class of higher order ðF;a; b;q;r; dÞ-V-type I univex function is introduced with someexamples for a differentiable multiobjective programming (MP). The KKT necessary and sufficient conditions for efficientsolution for (MFP) are established for nondifferentiable multiobjective fractional programming problem (MFP) under gener-alized higher order ðF;a; b;q;r; dÞ-V-type I univex functions. Again, higher order dual program is proposed for (MFP) and theduality results are established under generalized of higher order ðF;a; b;q;r; dÞ-V-type I univex functions. Some computa-tional work has been done to substantiate the analysis. The results developed in this paper can be further extended to higherorder minimax mixed fractional programming.

Acknowledgements

The author is thankful to the referees for their valuable suggestions for the improvement of the paper.

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