Optimality and duality results for a nondifferentiable multiobjective fractional programming problem

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1 DubeyetalJournal of Inequalities and Applications :354 DOI 086/s R E S E A R C H Open Access Optimality and duality results for a nondifferentiable multiobjective fractional programming problem Ramu Dubey ShivKGupta * and Meraj Ali Khan 2 * Correspondence: skgiitr@gmailcom Department of Mathematics Indian Institute of Technology Roorkee India Full list of author information is available at the end of the article Abstract The purpose of this paper is to study a class of nondifferentiable multiobjective fractional programming problems in which every component of objective functions contains a term involving the support function of a compact convex set For a differentiable function we introduce the definition of higher-order C α γ ρ d-convex function A nontrivial example is also constructed which is in this class but not F α γ ρ d-convex Based on the C α γ ρ d-convexity sufficient optimality conditions for an efficient solution of the nondifferentiable multiobjective fractional programming problem are established Further a higher-order Mond-Weir type dual is formulated for this problem and appropriate duality results are proved under higher-order C α γ ρ d-assumptions MSC: 90C26; 90C30; 90C32; 90C46 Keywords: duality results; multiobjective programming problem; support function; KKT conditions; efficient solution Introduction Higher-order duality is significant due to its computational importance as it provides higher bounds whenever an approximation is used By introducing two different functions h : R n R n R and k : R n R n R m Mangasarian [] formulated a higher-order dual for a nonlinear optimization problem {min f x subject to gx 0}Inspiredbythis concept many researchers have worked in this direction Chen [2]hasformulatedhigherorder multiobjective symmetric dual programs and established duality relations under higher-order F-convexity assumptions A higher-order vector optimization problem and its dual have been studied by Kassem [3] In the last several years various optimality and duality results have been obtained for multiobjective fractional programming problems In Chen [2] multiobjective fractional problem and its duality theorems have been considered under higher-order F α ρ d- convexity Later on Suneja et al [4] discussed higher-order Mond-Weir and Schaible type nondifferentiable dual programs and their duality theorems under higher-order F ρ σ - type I-assumptions Recently Ying [5] has studied higher-order multiobjective symmetric fractional problem and formulated its Mond-Weir type dual Further duality results are obtained under higher-order F α ρ d-convexity 205 Dubey et al This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made

2 DubeyetalJournal of Inequalities and Applications :354 Page 2 of 8 Yuan et al [6] introduced a class of functions called C α ρ d-convex functions and derived duality theorems for a nondifferentiable minimax fractional programming problem under C α ρ d-convexity Chinchuluun et al [7] later studied nonsmooth multiobjective fractional programming problems in the framework of C α ρ d-convexity In this paper we first introduce the definition of higher-order C α γ ρ d-convex with respect to a differentiable function H : X R n R X R n p s R n and construct a nontrivial example which is higher-order C α γ ρ d-convex but not a F α γ ρ d- convex function We prove that the ratio of higher-order C α γ ρ d-convex functions is also higher-order C ᾱ γ ρ d-convex A sufficient optimality condition related to the efficient solution of a multiobjective nondifferentiable fractional problem has been established Finally we formulate a higher-order Mond-Weir type dual problem corresponding to the multiobjective nondifferentiable fractional programming problem and established usual duality relations under the aforesaid assumptions 2 Preliminaries Definition 2 AfunctionC : X X R n R X R n is said to be convex on R n iff for any fixed x u X X and for any x x 2 R n C xu λx λx 2 λcxu x λc xu x 2 λ 0 We now introduce the definition of higher-order C α γ ρ d-convex function Let C be a convex function with respect to the third variable such that C xu 0 = 0 x u X XLet H : X R n R φ : X R be differentiable functions on XAssumethatα γ : X X R \{0} ρ R d : X X R satisfying dx x 0 =0 x = x 0 and p s R n Definition 22 The function φ is said to be higher-order strictly C α γ ρ d-convex at u with respect to H p and s if for each x X ] φx φu >Cxu φu p Hu p αx u Hu s s T s Hu s ] ρdx u γ x u αx u Remark 2 A differentiable function f =f f 2 f k :X R k is C α γ ρ d-convex if for all i =2k f i is C α i γ i ρ i d i -convex Remark 22 i If Hu =0thenDefinition22 becomes that of a C α ρ d-convex function as givenin[6 8] Further if αx u= ρ =0andC xu a=η T x uafor η : X X R n thenc α γ ρ d-convexity reduces to invexity see Hanson [9] ii If Hu = 2 T 2 f u αx u=γ x uandp = sthendefinition22 reduces to the first expression of a second-order C α ρ d-type I-convex function given in Gupta et al [0] iii If αx u=γ x u= ρ =0 Hu = 2 T 2 f u p = sandc xu a=η T x ua where η : X X R n the above definition becomes that of η-bonvexity introduced by Pandey []

3 DubeyetalJournal of Inequalities and Applications :354 Page 3 of 8 iv If C is sublinear with respect to third variable and p = s then the above definition reduces to higher-order F α γ ρ d-convexity as given in Gulati and Saini [2] Furthermore if γ x u =αx uthendefinition 22 reduces to higher-order F α ρ d-convexity as in Ying [5] Moreover if Hu =0then Definition 22 becomes that of a F α ρ d-convex function as introduced by Liang et al [3] Remark 23 Every F α γ ρ d-convex function is C α γ ρ d-convex However the converse need not be true This is illustrated by the following example Example 2 Let X = {x : x } R f : X R C : X X R R H : X R R and d : X X R be defined as f x= x2 x 2 C xu a=a 2 x u 2 Hu = x u dx u=x u2 Clearly the function C defined above is convex with respect to the third variable satisfying C xu 0 = 0 x u XAlsodx u=0 x = u Let ρ = andα = γ = 3 Thenatu = for all x Xwehave 4 αx u [ f x f u ] = 4 3 x 2 x C xu x f u p Hu p x Hu s s T s Hu s ] ρdx u γ x u αx u = 3 x 2 Hence f is a higher-order C α γ ρ d-convex function with respect to H p and s But considering the same C f is not higher-order F α γ ρ d-convex because C is not a sublinear functional with respect to the third variable Definition 23 [4] LetC be a compact convex set in R n ThesupportfunctionofC is defined by Sx C=max { x T y : y C } 3 Problem formulation and optimality conditions Consider the following nondifferentiable multiobjective programming problem: { f xsx C MFP Minimize Fx= g x Sx D f 2xSx C 2 g 2 x Sx D 2 f } kxsx C k g k x Sx D k subject to x X 0 = { x X R n : h j xsx E j 0 j =2m } where f i g i : X R i =2k andh j : X R j =2m are continuously differentiable functions Assume that f i S C i 0andg i S D i >0;C i D i ande j are compact convex sets in R n and Sx C i Sx D i and Sx E j denote the support functions of compact convex sets C i D i ande j i =2k j =2mrespectively Definition 35] Apointx 0 X 0 is a weakly efficient solution of MFP if there exists no x X 0 such that for every i =2k f ixsx C i g i x Sx D i < f ix 0 Sx 0 C i g i x 0 Sx 0 D i

4 DubeyetalJournal of Inequalities and Applications :354 Page 4 of 8 Definition 32 [6] Apointx 0 X 0 is an efficient solution or a Pareto optimal solution of MFP if there exists no x X 0 such that for every i =2k f ixsx C i g i x Sx D i f ix 0 Sx 0 C i g i x 0 Sx 0 D i and for some r =2k f rxsx C r g r x Sx D r < f rx 0 Sx 0 C r g r x 0 Sx 0 D r Lemma 37] If for a given >0 0 and λ 0i =2k x Xisanoptimal solution for the following single-objective problem: FP λ minimize Gx= i= subject to x X R n fi xsx C i g i x Sx D i then x is an efficient solution a weakly efficient solution for MFP Theorem 3 For some t assume f t T and g t T v t are C α t γ t ρ t d t - convex at u X with respect to H t p and s Then f t T g t T v t is also higher-order C ᾱ t γ t ρ t d t -convex at u X with respect to H t p and s where gt u u T v t ᾱ t x u= g t x x T v t ρ t = ρ t f tuu T g t u u T v t dt x u d t x u= g t x x T v t α t x u H t u = γ t x u=γ t x u g t u u T v t f tuu T g t u u T v t 2 H t u and ft uu C xu a= T g t u u T v t g t u u C T v t 2 a g t u u T v t 2 xu f t uu T g t u u T v t ft uu T a = g t u u T p H t u p v t Proof For any x u X ft xx T f tuu T = g t x x T v t g t u u T v t ft xx T f t u u T g t x x T v t [ f t uu T ] g t x x T v t g t uu T v t g t x x T v t g t u u T v t Since f t T and g t T v t arec α t γ t ρ t d t -convex at u X with respect to H t pandswehave f tuu T g t x x T v t g t u u T v t ft xx T α t x u g t x x T v t { C xu ft uu T p H t u p Ht u s s T s H t u s ] ρ td t x u γ t x u α t x u }

5 DubeyetalJournal of Inequalities and Applications :354 Page 5 of 8 f t uu T { C g t x x T v t g t u u T xu gt u u T v t p H t u p v t γ t x u [ Ht u s s T s H t u s ] ρ td t x u α t x u which implies ft xx T f tuu T α t x u g t x x T v t g t u u T v t ft uu T g t u u T v t g t u u T v t g t x x T v t g t u u T v t f t uu T g t u u T v t { C xu ft uu T p H t u p Ht u s s T s H t u s ] ρ } td t x u γ t x u α t x u ft uu T g t u u T v t f t uu T g t x x T v t g t u u T v t f t uu T g t u u T v t { C xu gt u u T v t p H t u p } Ht u s s T s H t u s ] ρ } td t x u γ t x u α t x u This further yields ft xx T f tuu T α t x u g t x x T v t g t u u T v t ft uu T g t u u T v t g t x x T v t g t u u T v t g t u u {C T v t 2 ft uu T xu f t uu T g t u u T v t g t u u T v t f tuu T } g t u u T v t g t u u T v t 2 p H t u p γ t x ug t x x T v t [ f tuu T Ht u s s T g t u u T s H t u s ] v t ρ t d t x u f tuu T α t x ug t x x T v t g t u u T v t Multiplying by g tx x T v t g t u u T v t the above inequality gives g t x x T v t ft xx T f tuu T α t x ug t u u T v t g t x x T v t g t u u T v t ft uu T g t u u T v t g t u u T v t 2 g t u u {C T v t 2 ft uu T xu f t uu T g t u u T v t g t u u T v t

6 DubeyetalJournal of Inequalities and Applications :354 Page 6 of 8 Setting g t u u T v t γ t x ug t u u T v t ρ t d t x u α t x ug t u u T v t gt u u T v t ᾱ t x u= g t x x T v t ρ t = ρ t f tuu T g t u u T v t H t u = f tuu T g t u u T v t 2 } p H t u p [ f tuu T g t u u T v t f tuu T g t u u T v t α t x u g t u u T v t f tuu T g t u u T v t 2 C xu a= ft uu T g t u u T v t g t u u T v t 2 Ht u s s T s H t u s ] γ t x u=γ t x u C xu H t u dt x u d t x u= g t x x T v t g t u u T v t 2 a f t uu T g t u u T v t and ft uu T a = g t u u T p H t u p v t It follows that ᾱ t x u ft xx T f tuu T g t x x T v t g t u u T v t { C xu ft uu T g t u u T v t γ t x u p H t u p [ H t u s s T s H t u s ] ρ t d t x u ᾱ t x u Hence f t T is higher-order C ᾱ g t T γ t ρ t d t -convex at u X with respect to H t p t and s Theorem 32 Let X R n be an open convex set Let ψ i : X R be higher-order C α i γ i ρ i d i -convex at one point in X with respect to φ i ρ i 0 and for all x X φ i x = T ψ i x i =2k; then only one of the following two cases holds: i there exists x X such that ψ i x<0 i =2k; ii there exists λ R k \{0} such that k i= ψ i x 0 for all x X Proof If i has a solution that is there exists x X such that ψ i x<0i =2kthen for every λ R k \{0}wehave k i= ψ i x < 0 which implies that ii does not hold If i has no solution let K = ψxintr k ψ =ψ ψ 2 ψ k T thenk R k =φ For any z z 2 K and β 0 there exist x x 2 X and s s 2 intr k suchthat } βz βz 2 = βψ x βψ x 2 βs βs 2

7 DubeyetalJournal of Inequalities and Applications :354 Page 7 of 8 Since u = βx βx 2 X and φ i x = T ψ i x it follows from the higher-order C α i γ i ρ i d i -convexity of ψ i with respect to φ i pands that we have ψi x j ψ α i x j i u ] u C xu [ ψi u p φ i u p ] = α i x j u ρ id x j u j =2 Thus there exists t 0 such that { φi u s s T[ γ i x j s φ i u s ]} ρ idx j u u α i x j u βψ i x βψ i x 2 = ψ i βx βx 2 t which implies that βz βz 2 = βψ x βψ x 2 βs βs 2 t tt T Since R k is a closed convex cone we have βz βz 2 KthatisK is an open convex set It follows from the convex separated theorem that there exists λ R k \{0} such that for all x X k i= ψ i x 0 We now discuss some optimality conditions for the problem MFP Theorem 33 If u X is a weakly efficient solution of MFP f i T and g i T v i are higher-order C α i γ i ρ i d i -convex functions at u X with respect to φ i u = g i u u T v i 2 f i uu T g i u u T v i T f iuu T and ρ g i u u T v i 0 i =2k Then there exists 0 λ i R k λ 0such that u is an optimal solution of FP λ Proof If u X is a weakly efficient solution of MFP then there does not exist any x X such that fj xx T z j ψ j x= f juu T z j <0 j =2k g j x x T v j g j u u T v j Since f j T z j and g j T v j are higher-order C α j γ j ρ j d j -convex at u with respect to φ j p and s j =2k it follows from Theorem 3 that ψ j x = f jxx T z j g j x x T v j f j uu T z j g j u u T v j is higher-order C ᾱ j γ j ρ j d j -convex at u with respect to φ j p ands j = 2kwhere gj x x T v j ᾱ j x u= g j u u T v j ρ j = ρ j f juu T z j g j u u T v j φ j u = α j x u g j u u T v j = T ψ j u γ j x u=γ j x u f juu T z j φ g j u u T v j 2 j u = T fj uu T z j g j u u T v j

8 DubeyetalJournal of Inequalities and Applications :354 Page 8 of 8 dj x u d j x u= g j x x T v j and C xu a= f juu T z j g j u u T v j g j u u T v j 2 a C g j u u T v j 2 xu f j uu T z j g j u u T v j fj uu T z j a = g j u u T p φ j u p v j Using Theorem 32thereexists0 λ R k λ 0suchthat k j= λ j ψ j x 0 that is j= fj xx λ T z j j g j x x T v j j= fj uu λ T z j j g j u u T v j which implies that u is an optimal solution of FP λ Theorem 34 Necessary condition [8] Assume that x is an efficient solution of MFP and the Slater constraint qualification is satisfied on X Then there exist λ R k μ R m R n v i R n and R n i =2k j =2m such that i= fi x x T g i x x T v i hj x x T =0 j= h j x x T =0 j= x T = S x C i x T v i = S x D i x T = S x E j i =2k i =2k j =2m C i v i D i E j i =2k j =2m >0 0 i =2k j =2m Theorem 35 Sufficient condition Let u be a feasible solution of MFP Assume that there exist >0i =2kand 0 j =2m such that i= fi uu T g i u u T v i hj uu T =0 j= h j uu T =0 j= u T = Su C i u T v i = Su D i u T = Su E j i =2k i =2k j =2m

9 DubeyetalJournal of Inequalities and Applications :354 Page 9 of 8 C i v i D i E j i =2k j =2m Let for any i =2k j =2m i f i T and g i T v i be higher-order C α i γ i ρ i d i -convex at u with respect to H i p and s ii h j T be higher-order C ξ j δ j η j c j -convex at u with respect to K j q and r iii k i= λ d i ρ i xu i ᾱ i xu m j= μ c j xu jη j ξ j xu 0 iv γ i x u=ζ x u ξ j x u=σ x u and δ j x u=σ x u v k i= p H i u p m j= q K j u q = 0 k i= H i u s s T s H i u s 0 and m j= K j u r r T r K j u r 0 where gi u u T v i ᾱ i x u= g i x x T v i H i u = g i u u T v i Then u is an efficient solution of MFP α i x u γ i x u=γ i x u ρ i = ρ i f iuu T g i u u T v i f iuu T H g i u u T v i 2 i u di x u d i x u= g i x x T v i Proof Suppose u is not an efficient solution of MFP then there exists x X 0 such that f i xsx C i g i x Sx D i f iusu C i g i u Su D i for all i =2k and f r xsx C r g r x Sx D r < f rusu C r g r u Su D r forsomer =2k which implies f i xx T f ixsx C i g i x x T v i g i x Sx D i f iusu C i g i u Su D i = f iuu T g i u u T v i for all i =2k and f r xx T z r f rxsx C r g r x x T v r g r x Sx D r < f rusu C r g r u Su D r = f ruu T z r g r u u T v r for some r =2k 2 Since ᾱ i >0i =2ktheinequalitiesand2give xu i= fi xx T f iuu T <0 3 ᾱ i x u g i x x T v i g i u u T v i

10 DubeyetalJournal of Inequalities and Applications :354 Page 0 of 8 From Theorem 3foreachi i k f i T g i T v i is higher-order C ᾱ i γ i ρ i d i -convex at u X 0 with respect to H i pands therefore where fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i fi uu C xu T g i u u T p H i u p v i H i u s s T s H i u s ] ρ i d i x u γ i x u ᾱ i x u 4 gi u u T v i ᾱ i x u= g i x x T v i ρ i = ρ i f iuu T g i u u T v i H i u = α i x u g i u u T v i γ i x u=γ i x u f iuu T H g i u u T v i 2 i u di x u d i x u= g i x x T v i and fi uu C xu a= T g i u u T v i g i u u C T v i 2 a g i u u T v i 2 xu f i uu T g i u u T v i fi uu T a = g i u u T p H i u p v i Also by the higher-order C ξ j δ j η j c j -convexity of h j T atuwith respect to K j qandr j =2mwehave hj xx T h j u u T ] ξ j x u C xu hj uu T q K j u q Kj u r r T r K j u r ] η jc j x u δ j x u ξ j x u 5 Let = k i= f iuu T g i u u T v i m g i u u T v i 2 j= >0 Adding the inequalities obtained by multiplying 4by and 5by weget i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i i= j= [ hj xx T h j u u T ] ξ j x u [ f i uu T g i u u T v i g i u u T v i 2 C g i u u T v i 2 xu f i uu T g i u u T v i

11 DubeyetalJournal of Inequalities and Applications :354 Page of 8 fi uu T g i u u T v i i= i= j= j= γ i x u ρ i d i x u α i x u g i u u T v i g i u u T v i g i u u T v i f iuu T g i u u T v i 2 p H i u p f iuu T Hi u s s T g i u u T v i 2 s H i u s ] f iuu T g i u u T v i 2 C xu hj uu T q K j u q Kj u r r T r K j u r ] δ j x u Further using the convexity of C wehave i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i j= [ hj xx T h j u u T ] ξ j x u fi uu T C xu g i u u T v i i= j= p H i u p i= h j uu T m q K j u q j= j= η j c j x u ξ j x u i= j= H i u s s T s H i u s γ i x u Kj u r r T r K j u r ] δ j x u i= j= ρ i d i x u ᾱ i x u η j c j x u ξ j x u It follows from hypotheses iii-v that i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i j= [ hj xx T h j u u T ] σx u fi uu T C xu g i u u T v i i= h j uu T Using the fact C xu 0 = 0 and m j= h j uu T =0weget i= j= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i i= [ hj xx T ] 0 σx u

12 DubeyetalJournal of Inequalities and Applications :354 Page 2 of 8 Finally using feasibility of the primal problem MFP we have i= fi xx T f iuu T 0 ᾱ i x u g i x x T v i g i u u T v i which contradicts 3 Therefore u is an efficient solution of MFP 4 Duality model Consider the following higher-order Mond-Weir type dual MFD of MFP: { f uu T z MFD maximize Gu= f 2uu T z 2 f kuu T } z k g u u T v g 2 u u T v 2 g k u u T v k subject to fi uu T μ g i= i u u T j h j uu T v i j= p H i u p q K j u q=0 i= j= [ hj uu T K j u r r T r K j u r ] 0 7 j= [ H i u s s T s H i u s ] 0 8 i= C i v i D i E j i =2k j =2m >0 0 i =2k j =2m 6 We now discuss the duality results for the primal-dual pair MFP and MFD Theorem 4 Weak duality theorem Let x X 0 and u z v μ λ w p q r s be feasible for MFD Suppose that: i f i T and g i T v i are higher-order C α i γ i ρ i d i -convex at u with respect to H i p and s i =2k ii h j T is higher-order C ξ j δ j η j c j -convex at u with respect to K j q and r j =2m iii k i= λ d i ρ i xu i ᾱ i xu m j= μ c j xu jη j ξ j xu 0 iv γ i x u=ζ x u and ξ j x u=δ j x u=σ x u i =2k j =2m where gi x x T v i ᾱ i x u= g i u u T v i di x u d i x u= g i x x T v i α i x u ρ i = ρ i f iuu T g i u u T v i Then the following cannot hold: f i xsx C i g i x Sx D i f iuu T g i u u T v i for all i =2k 9

13 DubeyetalJournal of Inequalities and Applications :354 Page 3 of 8 and f r xsx C r g r x Sx D r < f ruu T z r g r u u T v r for some r =2k 0 λ Proof Suppose that 9 and0 hold then using i ᾱ i xu >0xT Sx C i x T v i Sx D i i =2kwehave i= fi xx T f iuu T <0 ᾱ i x u g i x x T v i g i u u T v i From hypothesis i and Theorem 3 f i T is higher-order C ᾱ g i T v i γ i ρ i d i -convex at i u with respect to H i pands i =2k therefore fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i fi uu C xu T g i u u T p H i u p v i H i u s s T s H i u s ] ρ i d i x u γ i x u ᾱ i x u 2 and by the higher-order C ξ j δ j η j c j -convex of h j T atu with respect to K j q and r j =2mwehave hj xx T h j u u T ] ξ j x u C xu hj uu T q K j u q Kj u r r T r K j u r ] η jc j x u δ j x u ξ j x u 3 Let = k i= f iuu T g i u u T v i m g i u u T v i 2 j= >0 Multiply 2by i= and 3by and add them to get fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i i= j= [ hj xx T h j u u T ] ξ j x u [ f i uu T g i u u T v i g i u u T v i 2 C g i u u T v i 2 xu f i uu T g i u u T v i fi uu T i= g i u u T v i γ i x u g i u u T v i g i u u T v i f iuu T g i u u T v i 2 p H i u p f iuu T Hi u s s T g i u u T v i 2 s H i u s ]

14 DubeyetalJournal of Inequalities and Applications :354 Page 4 of 8 i= j= j= ρ i d i x u α i x u g i u u T v i f iuu T g i u u T v i 2 C xu hj uu T q K j u q Kj u r r T r K j u r ] δ j x u j= η j c j x u ξ j x u Further using convexity on C6-8 and hypotheses iii-iv we have i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i Since x is a feasible solution of MFP it follows that i= fi xx T f iuu T 0 ᾱ i x u g i x x T v i g i u u T v i i= [ hj xx T ] 0 σx u This contradicts Hence we have the result Theorem 42 Strong duality theorem Assume ū is an efficient solution of MFP and let the Slater constraint qualification be satisfied on X Also if Hū0=0 Kū0=0 p Hū0=0 q Kū0=0 s Hū0=0 r Kū0=0 then there exist 0< λ R k μ R m R n v i and R n i = 2k j = 2m such that ū z v μ λ w p =0 q =0 s =0 r =0is a feasible solution of MFD and the objective function values of MFP and MFD are equal Furthermore if the conditions of Theorem 4 hold for all feasible solutions of MFP and each feasible solution u z v μ λ w p =0q =0r =0s =0of MFD then ū z v μ λ w p =0 q =0 r =0 s =0is an efficient solution of MFD Proof Assume ū is an efficient solution of MFP and the Slater constraint qualification is satisfied on X ThenfromTheorem34 thereexist >0 μ R m R n v i R n and R n i =2k j =2msuchthat i= fi ūū T g i ū ū T v i h j ūū T =0 4 j= h j ūū T = 0 5 j= ū T = Sū C i i =2k 6 ū T v i = Sū D i i =2k 7 ū T = Sū E j j =2m 8

15 DubeyetalJournal of Inequalities and Applications :354 Page 5 of 8 C i v i D i E j i =2k j =2m 9 >0 0 i =2k j =2m 20 Thus ū z v μ λ w p =0 q =0 r =0 s = 0 is feasible for MFD and from 6-8 the objective function values of MFP and MFD are equal Wenowshowthatū z v μ λ w p =0 q =0 r =0 s = 0 is an efficient solution of MFD If not then there exists u z v μ λ w p q r s ofmfdsuchthat f i ūū T f iu u T g i ū ū T v i g i u u T v i =2k i and f r ūū T z r < f ru u T z r forsomer =2k g r ū ū T v r g r u u T v r This contradicts the weak duality theorem Hence we have the result Theorem 43 Strict converse duality theorem Let x be a feasible solution for MFP and u z v μ λ w p q r s be feasible for MFD Suppose that: i f i T and g i T v i are higher-order strictly C α i γ i ρ i d i -convex at u with respect to H i p and s i =2k ii h j T is higher-order C ξ j δ j η j c j -convex at u with respect to K j q and r j =2m iii γ i x u=ζ x u and ξ j x u=δ j x u=σ x u i =2k j =2m iv k i= λ d i ρ i xu i ᾱ i xu m j= μ c j xu jη j ξ j xu 0 v k i= ᾱ i xu f ixx T f iuu T 0 g i x x T v i g i u u T v i where gi x x T v i ᾱ i x u= g i u u T v i di x u d i x u= g i x x T v i α i x u ρ i = ρ i f iuu T g i u u T v i Then x = u Proof Suppose that x u and exhibit a contradiction Let u z v μ λ w p q r sbefeasible for MFD then fi uu T C xu 0 = C xu μ g i= i u u T j h j uu T v i j= p H i u p q K j u q =0 2 i= j=

16 DubeyetalJournal of Inequalities and Applications :354 Page 6 of 8 Since f i T is higher-order strictly C ᾱ g i T v i γ i ρ i d i -convex at u with respect to H i p i and swehave fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i fi uu > C xu T g i u u T p H i u p v i H i u s s T s H i u s ] ρ i d i x u γ i x u ᾱ i x u Let = k i= f iuu T g i u u T v i m g i u u T v i 2 j= >0 Multiplying by in the above inequality we get [ fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i fi uu > C xu T g i u u T p H i u p v i γ i x u [ H i u s s T s H i u s ] ρ i d i x u ᾱ i x u ] 22 By the higher-order C ξ j δ j η j c j -convex of h j T atu with respect to K j qand r j =2mweobtain hj xx T h j u u T ] Cxu hj uu T q K j u q ξ j x u It follows that [ hj xx T h j u u T ] ξ j x u C xu hj uu T q K j u q δ j x u [ Kj u r r T r K j u r ] η jc j x u ξ j x u Taking summation over i in 22and over j in 23 we get i= > fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i i= j= [ hj xx T h j u u T ] ξ j x u C fi uu T xu g i u u T p H i u p v i i= H i u s s T s H i u s ] γ i x u Kj u r r T r K j u r ] η jc j x u δ j x u ξ j x u ] 23

17 DubeyetalJournal of Inequalities and Applications :354 Page 7 of 8 i= j= ρ i d i x u ᾱ i x u j= C xu hj uu T q K j u q Kj u r r T r K j u r ] δ j x u j= η j c j x u ξ j x u Further using convexity on C and 2 we obtain i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i j= [ hj xx T h j u u T ] ξ j x u > i= H i u s s T s H i u s ] γ i x u j= Kj u r r T r K j u r ] δ j x u i= ρ i d i x u ᾱ i x u j= η j c j x u ξ j x u This together with 7-8 and hypotheses iii-iv shows i= fi xx T f iuu T ᾱ i x u g i x x T v i g i u u T v i Since x is a feasible solution of MFP it follows that i= fi xx T f iuu T >0 ᾱ i x u g i x x T v i g i u u T v i i= [ hj xx T ] >0 σx u This contradicts the hypothesis v Hence the proof is completed 5 Conclusions In this article we consider a class of fractional programming problem having k-objectives in which each numerator and denominator of the objective function is nondifferentiable in terms of the support function of a compact convex set The important property that the ratio of higher-order C α γ ρ d-convex function is also a higher-order C ᾱ γ ρ d- convex function is obtained We also derive sufficient optimality conditions for an efficient solution for this problem Furthermore a higher-order Mond-Weir type dual is formulated and appropriate duality relations are obtained under higher-order C α γ ρ d-convexity assumptions Competing interests The authors declare that they have no competing interests Authors contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Author details Department of Mathematics Indian Institute of Technology Roorkee India 2 Department of Mathematics University of Tabuk Tabuk Saudi Arabia

18 DubeyetalJournal of Inequalities and Applications :354 Page 8 of 8 Acknowledgements The authors are thankful to the reviewers for their valuable and constructive suggestions The authors are also grateful to Deanship of Scientific Research DSR University of Tabuk Saudi Arabia under Grant No S for their support to carry out this work Received: 9 June 205 Accepted: 29 October 205 References Mangasarian OL: Second and higher-order duality in nonlinear programming J Math Anal Appl Chen X: Higher-order symmetric duality in nondifferentiable multiobjective programming problems J Math Anal Appl Kassem MA: Higher-order symmetric duality in vector optimization problem involving generalized cone-invex functions Appl Math Comput Suneja SK Srivastava MK Bhatia M: Higher order duality in multiobjective fractional programming with support functions J Math Anal Appl Ying G: Higher-order symmetric duality for a class of multiobjective fractional programming problems J Inequal Appl Yuan DH Chinchuluun A Pardalos PM: Nondifferentiable minimax fractional programming problems with C α ρ d-convexity J Optim Theory Appl Chinchuluun A Yuan D Pardalos PM: Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity Ann Oper Res Long XJ: Optimality conditions and duality for nondifferentiable multiobjective programming problems with C α ρ d-convexity J Optim Theory Appl Hanson MA: On sufficiency of the Kuhn-Tucker conditions J Math Anal Appl Gupta SK Danger D Ahmad I: On second order duality for nondifferentiable minimax fractional programming problems involving type-i functions ANZIAM J Pandey S: Duality for multiobjective fractional programming involving generalized η-convex functions Opsearch Gulati TR Saini H: Higher-order F α β ρ d-convexity and its application in fractional programming Eur J Pure Appl Math Liang ZA Huang HX Pardalos PM: Optimality conditions and duality for a class of nonlinear fractional programming problem J Optim Theory Appl Gupta SK Kailey N Sharma MK: Multiobjective second-order nondifferentiable symmetric duality involving F α ρ d-convex function J Appl Math Inform Chen X: Sufficiency conditions and duality for a class of multiobjective fractional programming problems with higher-order F α ρ d-convexity J Appl Math Comput Egudo RR: Multiobjective fractional duality Bull Aust Math Soc Geoffrion AM: Proper efficiency and the theory of vector maximization J Math Anal Appl Gulati TR Geeta: Duality in nondifferentiable multiobjective fractional programming problem with generalized invexityjapplmathcomput