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Dubey et al. Journal of Inequalities and Applications (2015) 2015:354 DOI 10.1186/s13660-015-0876-0

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Optimality and duality results for a nondifferentiable multiobjective fractional programming problem Ramu Dubey1 , Shiv K Gupta1* and Meraj Ali Khan2 *

Correspondence: [email protected] 1 Department of Mathematics, Indian Institute of Technology, Roorkee, 247 667, 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

1 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 : Rn × Rn → R and k : Rn × Rn → Rm , Mangasarian [] formulated a higher-order dual for a nonlinear optimization problem, {min f (x), subject to g(x) ≤ }. Inspired by this concept, many researchers have worked in this direction. Chen [] has formulated higherorder 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 []. In the last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems. In Chen [], multiobjective fractional problem and its duality theorems have been considered under higher-order (F, α, ρ, d)convexity. Later on, Suneja et al. [] 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 [] 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. © 2015 Dubey et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Yuan et al. [] 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. [] 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 × Rn −→ R (X ⊂ Rn ), p, s ∈ Rn 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 ¯ ¯ α, is also higher-order (C, ¯ 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 . A function C : X × X × Rn −→ R (X ⊂ Rn ) is said to be convex on Rn iff, for any fixed (x, u) ∈ X × X and for any x , x ∈ Rn ,   Cx,u λx + ( – λ)x ≤ λCx,u (x ) + ( – λ)Cx,u (x ),

∀λ ∈ (, ).

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 Cx,u () = , ∀(x, u) ∈ X × X. Let H : X × Rn → R, φ : X → R be differentiable functions on X. Assume that α, γ : X × X → R+ \{}, ρ ∈ R, d : X × X → R+ satisfying d(x, x ) =  ⇔ x = x and p, s ∈ Rn . Definition . 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,      φ(x) – φ(u) ≥ (>)Cx,u ∇φ(u) + ∇p H(u, p) α(x, u) +

 ρd(x, u)   H(u, s) – sT ∇s H(u, s) + . γ (x, u) α(x, u)

Remark . A differentiable function f = (f , f , . . . , fk ) : X → Rk is (C, α, γ , ρ, d)-convex if for all i = , , . . . , k, fi is (C, αi , γi , ρi , di )-convex. Remark . (i) If H(u, ·) = , then Definition . becomes that of a (C, α, ρ, d)-convex function as given in [, ]. Further if α(x, u) = , ρ = , and Cx,u (a) = ηT (x, u)a, for η : X × X → Rn , then (C, α, γ , ρ, d)-convexity reduces to invexity (see Hanson []). (ii) If H(u, ·) =  (·)T ∇  f (u)(·), α(x, u) = γ (x, u), and p = s, then Definition . reduces to the first expression of a second-order (C, α, ρ, d)-type I-convex function given in Gupta et al. []. (iii) If α(x, u) = γ (x, u) = , ρ = , H(u, ·) =  (·)T ∇  f (u)(·), p = s, and Cx,u (a) = ηT (x, u)a, where η : X × X → Rn , the above definition becomes that of η-bonvexity introduced by Pandey [].

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(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 []. Furthermore, if γ (x, u) = α(x, u), then Definition . reduces to higher-order (F, α, ρ, d)-convexity as in Ying []. Moreover, if H(u, ·) = , then Definition . becomes that of a (F, α, ρ, d)-convex function as introduced by Liang et al. []. Remark . 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 . 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) =

x + x +  , x+

Cx,u (a) = a (x – u) ,

H(u, ·) =

(·) , u+

d(x, u) = (x – u) .

Clearly, the function C (defined above) is convex with respect to the third variable satisfying Cx,u () = , ∀x, u ∈ X. Also, d(x, u) =  ⇔ x = u. Let ρ = – and α = γ =  . Then at u = , for all x ∈ X, we have       x – x   f (x) – f (u) = ≥ Cx,u ∇x f (u) + ∇p H(u, p) α(x, u)  x+ +

 ρd(x, u)    H(u, s) – sT ∇s H(u, s) + = – (x – ) . γ (x, u) α(x, u) 

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 . [] Let C be a compact convex set in Rn . The support function of C is defined by  S(x|C) = max xT y : y ∈ C .

3 Problem formulation and optimality conditions Consider the following nondifferentiable multiobjective programming problem:

(MFP)

Minimize subject to

f (x) + S(x|C ) f (x) + S(x|C ) fk (x) + S(x|Ck ) , ,..., g (x) – S(x|D ) g (x) – S(x|D ) gk (x) – S(x|Dk )  x ∈ X  = x ∈ X ⊂ Rn : hj (x) + S(x|Ej ) ≤ , j = , , . . . , m ,



F(x) =

where fi , gi : X → R (i = , , . . . , k) and hj : X → R (j = , , . . . , m) are continuously differentiable functions. Assume that fi (·) + S(·|Ci ) ≥  and gi (·) – S(·|Di ) > ; Ci , Di , and Ej are compact convex sets in Rn and S(x|Ci ), S(x|Di ), and S(x|Ej ) denote the support functions of compact convex sets, Ci , Di , and Ej , i = , , . . . , k, j = , , . . . , m, respectively. Definition . [] A point x ∈ X  is a weakly efficient solution of (MFP), if there exists  )+S(x |C ) i) i < gfi (x . no x ∈ X  such that for every i = , , . . . , k, gfii(x)+S(x|C (x)–S(x|Di ) (x )–S(x |D ) i

i

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Definition . [] A point x ∈ X  is an efficient solution (or a Pareto optimal solution)  )+S(x |C ) i) i of (MFP), if there exists no x ∈ X  such that for every i = , , . . . , k, gfii(x)+S(x|C ≤ gfi (x (x)–S(x|Di ) (x )–S(x |D ) and for some r = , , . . . , k,

fr (x)+S(x|Cr ) gr (x)–S(x|Dr )


 (λ¯ i ≥  and λ¯ = ), i = , , . . . , k, x¯ ∈ X is an optimal solution for the following single-objective problem:

(FPλ )

minimize G(x) = subject to

  k fi (x) + S(x|Ci ) λ¯ i gi (x) – S(x|Di ) i=

x ∈ X ⊂ Rn ,

then x¯ is an efficient solution (a weakly efficient solution) for (MFP). Theorem . For some t, assume ft (·) + (·)T zt and –(gt (·) – (·)T vt ) are (C, αt , γt , ρt , dt )Tz t convex at u ∈ X with respect to Ht , p, and s. Then ( gft (·)+(·) T ) is also higher-order t (·)–(·) vt ¯ α¯ t , γ¯t , ρ¯t , d¯ t )-convex at u ∈ X with respect to H ¯ t , p, and s, where (C, 

 gt (u) – uT vt α¯ t (x, u) = γ¯t (x, u) = γt (x, u), αt (x, u), gt (x) – xT vt     ft (u) + uT zt  ft (u) + uT zt ¯ ρ¯t = ρt  + , Ht (u, ·) = Ht (u, ·), + gt (u) – uT vt (gt (u) – uT vt ) (gt (u) – uT vt )   dt (x, u) , d¯ t (x, u) = gt (x) – xT vt and    (gt (u) – uT vt ) a ft (u) + uT zt + gt (u) – uT vt C , x,u (gt (u) – uT vt ) ft (u) + uT zt + gt (u) – uT vt   ft (u) + uT zt ¯ t (u, p). a=∇ + ∇p H gt (u) – uT vt

C¯ x,u (a) =



Proof For any x, u ∈ X, 

ft (x) + xT zt ft (u) + uT zt – gt (x) – xT vt gt (u) – uT vt





 ft (x) + xT zt – ft (u) – uT zt = gt (x) – xT vt    gt (x) – xT vt – gt (u) + uT vt  T – ft (u) + u zt . (gt (x) – xT vt )(gt (u) – uT vt )

Since ft (·) + (·)T zt and –(gt (·) – (·)T vt ) are (C, αt , γt , ρt , dt )-convex at u ∈ X with respect to Ht , p, and s, we have   ft (u) + uT zt  ft (x) + xT zt – αt (x, u) gt (x) – xT vt gt (u) – uT vt

     ≥ Cx,u ∇ ft (u) + uT zt + ∇p Ht (u, p) (gt (x) – xT vt )  ρt dt (x, u)   Ht (u, s) – sT ∇s Ht (u, s) + + γt (x, u) αt (x, u)

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

    ft (u) + uT zt Cx,u –∇ gt (u) – uT vt + ∇p Ht (u, p) T T (gt (x) – x vt )(gt (u) – u vt )  ρt dt (x, u)   T Ht (u, s) – s ∇s Ht (u, s) + , + γt (x, u) αt (x, u)

+

which implies   ft (u) + uT zt  ft (x) + xT zt – αt (x, u) gt (x) – xT vt gt (u) – uT vt    ft (u) + uT zt + gt (u) – uT vt gt (u) – uT vt ≥ (gt (x) – xT vt )(gt (u) – uT vt ) ft (u) + uT zt + gt (u) – uT vt

    × Cx,u ∇ ft (u) + uT zt + ∇p Ht (u, p)  ρt dt (x, u)   T Ht (u, s) – s ∇s Ht (u, s) + + γt (x, u) αt (x, u)    ft (u) + uT zt + gt (u) – uT vt ft (u) + uT zt + (gt (x) – xT vt )(gt (u) – uT vt ) ft (u) + uT zt + gt (u) – uT vt

    × Cx,u –∇ gt (u) – uT vt + ∇p Ht (u, p)  ρt dt (x, u)   T + Ht (u, s) – s ∇s Ht (u, s) + . γt (x, u) αt (x, u) This further yields   ft (x) + xT zt ft (u) + uT zt  – αt (x, u) gt (x) – xT vt gt (u) – uT vt   ft (u) + uT zt + gt (u) – uT vt ≥ (gt (x) – xT vt )(gt (u) – uT vt )    

(gt (u) – uT vt ) ft (u) + uT zt ∇ × Cx,u ft (u) + uT zt + gt (u) – uT vt gt (u) – uT vt     ft (u) + uT zt  + ∇ + H (u, p) + p t T T  gt (u) – u vt (gt (u) – u vt ) γt (x, u)(gt (x) – xT vt ) 

   ft (u) + uT zt  × + Ht (u, s) – sT ∇s Ht (u, s) T (gt (u) – u vt )   ρt dt (x, u) ft (u) + uT zt + + . αt (x, u)(gt (x) – xT vt ) (gt (u) – uT vt ) T

(x)–x vt Multiplying by ( ggt(u)–u T v ), the above inequality gives t

t

  ft (x) + xT zt ft (u) + uT zt (gt (x) – xT vt ) – αt (x, u)(gt (u) – uT vt ) gt (x) – xT vt gt (u) – uT vt   ft (u) + uT zt + gt (u) – uT vt ≥ (gt (u) – uT vt )

    (gt (u) – uT vt ) ft (u) + uT zt × Cx,u ∇ ft (u) + uT zt + gt (u) – uT vt gt (u) – uT vt

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   ft (u) + uT zt + H (u, p) ∇ p t gt (u) – uT vt (gt (u) – uT vt )

    ft (u) + uT zt   T + Ht (u, s) – s ∇s Ht (u, s) + γt (x, u)(gt (u) – uT vt ) (gt (u) – uT vt )   ft (u) + uT zt ρt dt (x, u) + . + αt (x, u)(gt (u) – uT vt ) (gt (u) – uT vt )

+

Setting 

 gt (u) – uT vt γ¯t (x, u) = γt (x, u), αt (x, u), gt (x) – xT vt   ft (u) + uT zt ρ¯t = ρt  + , gt (u) – uT vt     ft (u) + uT zt  dt (x, u) ¯ t (x, u) = ¯ t (u, ·) = + (u, ·), d H , H t (gt (u) – uT vt ) (gt (u) – uT vt ) gt (x) – xT vt     ft (u) + uT zt + gt (u) – uT vt (gt (u) – uT vt ) a ¯ C(x,u) (a) = Cx,u , (gt (u) – uT vt ) ft (u) + uT zt + gt (u) – uT vt α¯ t (x, u) =

and  a=∇

ft (u) + uT zt gt (u) – uT vt



¯ t (u, p). + ∇p H

It follows that    ft (x) + xT zt ft (u) + uT zt – α¯ t (x, u) gt (x) – xT vt gt (u) – uT vt   

 ft (u) + uT zt ¯ + ∇ ≥ C¯ (x,u) ∇ (u, p) H p t gt (u) – uT vt +

¯   ¯ ¯ t (u, s) + ρ¯t dt (x, u) . Ht (u, s) – sT ∇s H γ¯t (x, u) α¯ t (x, u)

zt ¯ ¯ t , γ¯t , ρ¯t , d¯ t )-convex at u ∈ X with respect to H ¯ t , p, Hence, ( gft (·)+(·) T ) is higher-order (C, α t (·)–(·) zt and s.  T

Theorem . Let X ⊂ Rn be an open convex set. Let ψi : X → R be higher-order (C, αi , γi , ρi , di )-convex at one point in X with respect to φi , ρi ≥ , and for all x ∈ X, φi (x, ·) = –(·)T ∇ψi (x), i = , , . . . , k; then only one of the following two cases holds: (i) there exists x ∈ X such that ψi (x) < , i = , , . . . , k;  (ii) there exists λ ∈ Rk+ \ {} such that ki= λi ψi (x) ≥ , for all x ∈ X. Proof If (i) has a solution, that is, there exists x ∈ X such that ψi (x) < , i = , , . . . , k, then  for every λ ∈ Rk+ \ {}, we have ki= λi ψi (x) < , which implies that (ii) does not hold. If (i) has no solution, let K = ψ(X) + int(Rk+ ), ψ = (ψ , ψ , . . . , ψk )T , then K ∩ (–Rk+ ) = φ. For any z , z ∈ K and β ∈ (, ), there exist x , x ∈ X and s , s ∈ int(Rk+ ) such that     βz + ( – β)z = βψ x + ( – β)ψ x + βs + ( – β)s .

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Since u = βx + ( – β)x ∈ X and φi (x, ·) = –(·)T ∇ψi (x), it follows from the higher-order (C, αi , γi , ρi , di )-convexity of ψi with respect to φi , p, and s that we have     j ψi x – ψi (u) αi (xj , u)   ≥ Cx,u ∇ψi (u) + ∇p φi (u, p) + =

   ρi d xj , u , αi (xj , u)

  ρi d(xj , u)   φi (u, s) – sT ∇s φi (u, s) + j γi (x , u) αi (xj , u)

j = , .

Thus, there exists t ≥  such that       βψi x + ( – β)ψi x = ψi βx + ( – β)x + t, which implies that     βz + ( – β)z = βψ x + ( – β)ψ x + βs + ( – β)s + (t, t, . . . , t)T . Since Rk+ is a closed convex cone, we have βz + ( – β)z ∈ K , that is, K is an open convex set. It follows from the convex separated theorem that there exists λ ∈ Rk+ \ {} such that,  for all x ∈ X, ki= λi ψi (x) ≥ .  We now discuss some optimality conditions for the problem (MFP). Theorem . If u ∈ X is a weakly efficient solution of (MFP), fi (·) + (·)T zi , and –(gi (·) – (·)T vi ) are higher-order (C, αi , γi , ρi , di )-convex functions at u ∈ X with respect to φi (u, ·) = (gi (u)–uT vi ) fi (u)+uT zi T ¯ – (f (u)+u T z +g (u)–uT v ) × (·) ∇( g (u)–uT v ), and ρi ≥ , i = , , . . . , k. Then there exists  ≤ λ ∈ i i i i i i Rk , λ¯ =  such that u is an optimal solution of (FPλ¯ ). Proof If u ∈ X is a weakly efficient solution of (MFP), then there does not exist any x ∈ X such that  ψj (x) =

fj (x) + xT zj fj (u) + uT zj – gj (x) – xT vj gj (u) – uT vj

 < ,

j = , , . . . , k.

Since fj (·) + (·)T zj and –(gj (·) – (·)T vj ) are higher-order (C, αj , γj , ρj , dj )-convex at u with f (x)+xT z

respect to φj , p and s, j = , , . . . , k, it follows from Theorem . that ψj (x) = ( gj (x)–xT vj – j

fj (u)+uT zj ) gj (u)–uT vj

j

¯ α¯ j , γ¯j , ρ¯j , d¯ j )-convex at u with respect to φ¯ j , p, and s, j = is higher-order (C, , , . . . , k, where  gj (x) – xT vj αj (x, u), γ¯j (x, u) = γj (x, u), gj (u) – uT vj   fj (u) + uT zj , ρ¯j = ρj  + gj (u) – uT vj     fj (u) + uT zj fj (u) + uT zj  T φ + (u, ·) = –(·) ∇ φ¯ j (u, ·) = j gj (u) – uT vj (gj (u) – uT vj ) gj (u) – uT vj 

α¯ j (x, u) =

= –(·)T ∇ψj (u),

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

d¯ j (x, u) =



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 dj (x, u) , gj (x) – xT vj

and   fj (u) + uT zj + gj (u) – uT vj (gj (u) – uT vj ) a Cx,u , (gj (u) – uT vj ) fj (u) + uT zj + gj (u) – uT vj   fj (u) + uT zj + ∇p φ¯ j (u, p). a=∇ gj (u) – uT vj C¯ x,u (a) =

Using Theorem ., there exists  ≤ λ¯ ∈ Rk , λ¯ =  such that

k

¯

j= λj ψj (x) ≥ ,

that is,

    k k f (x) + xT zj fj (u) + uT zj ¯λj j ¯ ≥ , λj gj (x) – xT vj gj (u) – uT vj j= j= which implies that u is an optimal solution of (FPλ¯ ).



Theorem . (Necessary condition) [] Assume that x¯ is an efficient solution of (MFP) and the Slater constraint qualification is satisfied on X. Then there exist λ¯ ∈ Rk , μ¯ ∈ Rm , ¯ j ∈ Rn , i = , , . . . , k, j = , , . . . , m, such that z¯ i ∈ Rn , v¯ i ∈ Rn , and w k

λ¯ i ∇



i= m

fi (¯x) + x¯ T z¯ i gi (¯x) – x¯ T v¯ i

 m   ¯ j = , + μ¯ j ∇ hj (¯x) + x¯ T w j=

  ¯ j = , μ¯ j hj (¯x) + x¯ T w

j=

x¯ T z¯ i = S(¯x|Ci ),

i = , , . . . , k,

x¯ T v¯ i = S(¯x|Di ),

i = , , . . . , k,

¯ j = S(¯x|Ej ), x¯ T w

j = , , . . . , m,

z¯ i ∈ Ci ,

v¯ i ∈ Di ,

λ¯ i > ,

μ¯ j ≥ ,

¯ j ∈ Ej , w

i = , , . . . , k, j = , , . . . , m,

i = , , . . . , k, j = , , . . . , m.

Theorem . (Sufficient condition) Let u be a feasible solution of (MFP). Assume that there exist λi > , i = , , . . . , k and μj ≥ , j = , , . . . , m, such that k i= m

 λi ∇

fi (u) + uT zi gi (u) – uT vi

 m   μj ∇ hj (u) + uT wj = , + j=

  μj hj (u) + uT wj = ,

j=

uT zi = S(u|Ci ),

i = , , . . . , k,

uT vi = S(u|Di ),

i = , , . . . , k,

uT wj = S(u|Ej ),

j = , , . . . , m,

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

zi ∈ Ci ,

vi ∈ Di ,

wj ∈ Ej ,

Page 9 of 18

i = , , . . . , k, j = , , . . . , m.

Let for any i = , , . . . , k, j = , , . . . , m, (i) (fi (·) + (·)T zi ) and –(gi (·) – (·)T vi ) be higher-order (C, αi , γi , ρi , di )-convex at u, with respect to Hi , p, and s, (ii) (hj (·) + (·)T wj ) be higher-order (C, ξj , δj , ηj , cj )-convex at u, with respect to Kj , q, and r, k m cj (x,u) d¯ i (x,u) (iii) i= λi ρ¯i α¯ i (x,u) + j= μj ηj ξj (x,u) ≥ , (iv) γ¯i (x, u) = ζ (x, u), ξj (x, u) = σ (x, u), and δj (x, u) = σ (x, u), k m k T ¯ ¯ ¯ (v) i= λi (∇p Hi (u, p)) + j= μj (∇q Kj (u, q)) = , i= λi (Hi (u, s) – s ∇s Hi (u, s)) ≥  m T and j= μj (Kj (u, r) – r ∇r Kj (u, r)) ≥ , where 

 gi (u) – uT vi α¯ i (x, u) = αi (x, u), γ¯i (x, u) = γi (x, u), gi (x) – xT vi    fi (u) + uT zi ¯ + Hi (u, ·), Hi (u, ·) = gi (u) – uT vi (gi (u) – uT vi )

  fi (u) + uT zi ρ¯i = ρi  + , gi (u) – uT vi   di (x, u) ¯di (x, u) = . gi (x) – xT vi

Then u is an efficient solution of (MFP). Proof Suppose u is not an efficient solution of (MFP), then there exists x ∈ X  such that fi (x) + S(x|Ci ) fi (u) + S(u|Ci ) ≤ , gi (x) – S(x|Di ) gi (u) – S(u|Di )

for all i = , , . . . , k,

fr (x) + S(x|Cr ) fr (u) + S(u|Cr ) < , gr (x) – S(x|Dr ) gr (u) – S(u|Dr )

for some r = , , . . . , k,

and

which implies fi (x) + S(x|Ci ) fi (x) + xT zi fi (u) + S(u|Ci ) fi (u) + uT zi ≤ , ≤ = T gi (x) – x vi gi (x) – S(x|Di ) gi (u) – S(u|Di ) gi (u) – uT vi for all i = , , . . . , k,

()

and fr (x) + S(x|Cr ) fr (u) + S(u|Cr ) fr (u) + uT zr fr (x) + xT zr ≤ , < = gr (x) – xT vr gr (x) – S(x|Dr ) gr (u) – S(u|Dr ) gr (u) – uT vr for some r = , , . . . , k. Since

λi α¯ i (x,u)

k i=

()

> , i = , , . . . , k, the inequalities () and () give

  λi fi (x) + xT zi fi (u) + uT zi < . – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

Page 10 of 18

zi ¯ ¯ i , γ¯i , ρ¯i , d¯ i )-convex From Theorem ., for each i,  ≤ i ≤ k, ( gfi (·)+(·) T ) is higher-order (C, α i (·)–(·) vi ¯ i , p, and s, therefore at u ∈ X  with respect to H T

  fi (x) + xT zi fi (u) + uT zi  – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi     fi (u) + uT zi ¯ ¯ ≥ Cx,u ∇ + ∇p Hi (u, p) gi (u) – uT vi +

¯   ¯ ¯ i (u, s) + ρ¯i di (x, u) , Hi (u, s) – sT ∇s H γ¯i (x, u) α¯ i (x, u)

()

where  gi (u) – uT vi αi (x, u), γ¯i (x, u) = γi (x, u), gi (x) – xT vi   fi (u) + uT zi , ρ¯i = ρi  + gi (u) – uT vi     fi (u) + uT zi  di (x, u) ¯ i (x, u) = ¯ i (u, ·) = H , + (u, ·), d H i gi (u) – uT vi (gi (u) – uT vi ) gi (x) – xT vi 

α¯ i (x, u) =

and    (gi (u) – uT vi ) a fi (u) + uT zi + gi (u) – uT vi C , x,u (gi (u) – uT vi ) fi (u) + uT zi + gi (u) – uT vi   fi (u) + uT zi ¯ i (u, p). a=∇ + ∇p H gi (u) – uT vi C¯ x,u (a) =



Also, by the higher-order (C, ξj , δj , ηj , cj )-convexity of (hj (·) + (·)T wj ) at u with respect to Kj , q, and r, j = , , . . . , m, we have    hj (x) + xT wj – hj (u) – uT wj ξj (x, u)     ≥ Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q) +

Let τ =

 ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . δj (x, u) ξj (x, u)

k

i= λi

(fi (u)+uT zi +gi (u)–uT vi ) (gi (u)–uT vi )

+

m

j= μj

> .

Adding the inequalities obtained by multiplying () by k i=

()

λi τ

and () by

μj , τ

we get

  λi fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi +

m j=



 μj  hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u)

k λi (fi (u) + uT zi + gi (u) – uT vi ) i=

τ

(gi (u) – uT vi )

 Cx,u

(gi (u) – uT vi ) fi (u) + uT zi + gi (u) – uT vi



Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

Page 11 of 18

      fi (u) + uT zi fi (u) + uT zi  × ∇ + H (u, p) + ∇ p i gi (u) – uT vi gi (u) – uT vi (gi (u) – uT vi )    k  λi   fi (u) + uT zi  + Hi (u, s) – sT ∇s Hi (u, s) + τ γ¯i (x, u) gi (u) – uT vi (gi (u) – uT vi ) i= +

 k λi ρi di (x, u) i=

+

m μj j=

+

τ αi (x, u)

τ

fi (u) + uT zi  + gi (u) – uT vi (gi (u) – uT vi )

    Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q)

m μj j=



 μj ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . τ δj (x, u) τ ξj (x, u) j= m

Further, using the convexity of C, we have k i=

  λi fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi +

m j=

 μj  hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u)

  k   k  fi (u) + uT zi ¯ i (u, p) ≥ Cx,u λi ∇ λi ∇p H + τ i= gi (u) – uT vi i=  m m   T μj ∇ hj (u) + u wj + μj ∇q Kj (u, q) + j=

+

k λi i=

+

j= k   ¯ λi ρ¯i d¯ i (x, u) ¯ i (u, s) + Hi (u, s) – sT ∇s H τ γ¯i (x, u) τ α¯ i (x, u) i=

m μj j=

 μj ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . τ δj (x, u) τ ξj (x, u) j= m

It follows from hypotheses (iii)-(v) that k i=

  λi fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi +

m j=

 μj  hj (x) + xT wj – hj (u) – uT wj τ σ (x, u)

   k   m    fi (u) + uT zi ≥ Cx,u . λi ∇ μj ∇ hj (u) + uT wj + τ i= gi (u) – uT vi j= Using the fact Cx,u () =  and k i=

m

T j= μj (hj (u) + u wj ) = ,

we get

  m  μj  λi fi (x) + xT zi fi (u) + uT zi + hj (x) + xT wj ≥ . – T T τ α¯ i (x, u) gi (x) – x vi gi (u) – u vi τ σ (x, u) i=

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

Page 12 of 18

Finally, using feasibility of the primal problem (MFP), we have k i=

  λi fi (x) + xT zi fi (u) + uT zi – ≥ , α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi 

which contradicts (). Therefore, u is an efficient solution of (MFP).

4 Duality model Consider the following higher-order Mond-Weir type dual (MFD) of (MFP):

(MFD) maximize

f (u) + uT z f (u) + uT z fk (u) + uT zk G(u) = , , . . . , g (u) – uT v g (u) – uT v gk (u) – uT vk

subject to   k m   fi (u) + uT zi λi ∇ μj ∇ hj (u) + uT wj + T gi (u) – u vi i= j= +

k

¯ i (u, p) + λi ∇p H

i= m

m



()

μj ∇q Kj (u, q) = ,

j=

  μj hj (u) + uT wj + Kj (u, r) – rT ∇r Kj (u, r) ≥ ,

()

j= k   ¯ i (u, s) ≥ , ¯ i (u, s) – sT ∇s H λi H

()

i=

zi ∈ Ci ,

vi ∈ Di , μj ≥ ,

λi > ,

wj ∈ Ej ,

i = , , . . . , k, j = , , . . . , m,

i = , , . . . , k, j = , , . . . , m.

We now discuss the duality results for the primal-dual pair (MFP) and (MFD). Theorem . (Weak duality theorem) Let x ∈ X  and (u, z, v, μ, λ, w, p, q, r, s) be feasible for (MFD). Suppose that: (i) (fi (·) + (·)T zi ) and –(gi (·) – (·)T vi ) are higher-order (C, αi , γi , ρi , di )-convex at u, with respect to Hi , p, and s, i = , , . . . , k, (ii) (hj (·) + (·)T wj ) is higher-order (C, ξj , δj , ηj , cj )-convex at u, with respect to Kj , q, and r, j = , , . . . , m, m k cj (x,u) d¯ i (x,u) (iii) i= λi ρ¯i α¯ i (x,u) + j= μj ηj ξj (x,u) ≥ , (iv) γ¯i (x, u) = ζ (x, u) and ξj (x, u) = δj (x, u) = σ (x, u), i = , , . . . , k, j = , , . . . , m, where 

 gi (x) – xT vi αi (x, u), gi (u) – uT vi   di (x, u) ¯di (x, u) = . gi (x) – xT vi

α¯ i (x, u) =

  fi (u) + uT zi ρ¯i = ρi  + , gi (u) – uT vi

Then the following cannot hold: fi (x) + S(x|Ci ) fi (u) + uT zi ≤ , gi (x) – S(x|Di ) gi (u) – uT vi

for all i = , , . . . , k,

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

Page 13 of 18

and fr (x) + S(x|Cr ) fr (u) + uT zr < , gr (x) – S(x|Dr ) gr (u) – uT vr

for some r = , , . . . , k.

Proof Suppose that () and () hold, then using i = , , . . . , k, we have k i=

λi α¯ i (x,u)

()

> , xT zi ≤ S(x|Ci ), xT vi ≤ S(x|Di ),

  λi fi (x) + xT zi fi (u) + uT zi < . – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi

()

zi ¯ ¯ i , γ¯i , ρ¯i , d¯ i )-convex at From hypothesis (i) and Theorem ., ( gfi (·)+(·) T ) is higher-order (C, α i (·)–(·) vi ¯ i , p, and s, i = , , . . . , k, therefore u with respect to H T

   fi (x) + xT zi fi (u) + uT zi – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi     fi (u) + uT zi ¯ i (u, p) ≥ C¯ x,u ∇ + ∇ H p gi (u) – uT vi +

¯   ¯ ¯ i (u, s) + ρ¯i di (x, u) , Hi (u, s) – sT ∇s H γ¯i (x, u) α¯ i (x, u)

()

and by the higher-order (C, ξj , δj , ηj , cj )-convex of (hj (·) + (·)T wj ) at u with respect to Kj , q, and r, j = , , . . . , m, we have    hj (x) + xT wj – hj (u) – uT wj ξj (x, u)     ≥ Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q) +

Let τ =

k

 ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . δj (x, u) ξj (x, u)

 (fi (u)+uT zi +gi (u)–uT vi ) + m j= μj > . (gi (u)–uT vi ) μj λi () by τ and () by τ and add them,

i= λi

Multiply k i=

to get

  λi fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi +

m j=



 μj  hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u) 

 (gi (u) – uT vi ) τ (gi (u) – uT vi ) fi (u) + uT zi + gi (u) – uT vi i=       fi (u) + uT zi  fi (u) + uT zi + + ∇p Hi (u, p) × ∇ gi (u) – uT vi gi (u) – uT vi (gi (u) – uT vi )    k  λi   fi (u) + uT zi  T Hi (u, s) – s ∇s Hi (u, s) + + τ γ¯i (x, u) gi (u) – uT vi (gi (u) – uT vi ) i=

k λi (fi (u) + uT zi + gi (u) – uT vi )

Cx,u

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

+

 k λi ρi di (x, u) i=

+

m μj j=

+

τ αi (x, u)

τ

 fi (u) + uT zi + gi (u) – uT vi (gi (u) – uT vi )



    Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q)

m μj j=

Page 14 of 18

 μj ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . τ δj (x, u) τ ξj (x, u) j= m

Further, using convexity on C, ()-(), and hypotheses (iii)-(iv), we have k i=

  m  μj  λi fi (x) + xT zi fi (u) + uT zi + – hj (x) + xT wj ≥ . T T τ α¯ i (x, u) gi (x) – x vi gi (u) – u vi τ σ (x, u) i=

Since x is a feasible solution of (MFP), it follows that k i=

  λi fi (x) + xT zi fi (u) + uT zi ≥ . – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi 

This contradicts (). Hence we have the result.

Theorem . (Strong duality theorem) Assume u¯ is an efficient solution of (MFP) and let the Slater constraint qualification be satisfied on X. Also, if ¯ u, ¯ ) = , H( ¯ u, ¯ ) = , ∇s H(

¯ ) = , K(u,

¯ u, ¯ ) = , ∇p H(

¯ ) = , ∇q K(u,

¯ ) = , ∇r K(u,

¯ j ∈ Rn , i = , , . . . , k, j = , , . . . , m, then there exist  < λ¯ ∈ Rk , μ¯ ∈ Rm , z¯ i ∈ Rn , v¯ i , and w ¯ z¯ , v¯ , μ, ¯ p¯ = , q¯ = , s¯ = , r¯ = ) is a feasible solution of (MFD) and such that (u, ¯ λ¯ , w, the objective function values of (MFP) and (MFD) are equal. Furthermore, if the conditions of Theorem . hold for all feasible solutions of (MFP) and each feasible solution ¯ z¯ , v¯ , μ, ¯ p¯ = , q¯ = , ¯ λ¯ , w, (u , z , v , μ , λ , w , p = , q = , r = , s = ) of (MFD), then (u, r¯ = , s¯ = ) is an efficient solution of (MFD). Proof Assume u¯ is an efficient solution of (MFP) and the Slater constraint qualification is satisfied on X. Then, from Theorem ., there exist λ¯ i > , μ¯ ∈ Rm , z¯ i ∈ Rn , v¯ i ∈ Rn , and ¯ j ∈ Rn , i = , , . . . , k, j = , , . . . , m, such that w k i= m

λ¯ i ∇



¯ + u¯ T z¯ i fi (u) ¯ – u¯ T v¯ i gi (u)

 m   ¯ j = , ¯ + u¯ T w + μ¯ j ∇ hj (u)

()

j=

  ¯ j = , ¯ + u¯ T w μ¯ j ∇ hj (u)

()

j=

¯ i ), u¯ T z¯ i = S(u|C

i = , , . . . , k,

()

¯ i ), u¯ T v¯ i = S(u|D

i = , , . . . , k,

()

¯ j = S(u|E ¯ j ), u¯ T w

j = , , . . . , m,

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

z¯ i ∈ Ci ,

v¯ i ∈ Di ,

λ¯ i > ,

¯ j ∈ Ej , w

μ¯ j ≥ ,

Page 15 of 18

i = , , . . . , k, j = , , . . . , m,

i = , , . . . , k, j = , , . . . , m.

() ()

¯ z¯ , v¯ , μ, ¯ p¯ = , q¯ = , r¯ = , s¯ = ) is feasible for (MFD) and from ()-(), the Thus, (u, ¯ λ¯ , w, objective function values of (MFP) and (MFD) are equal. ¯ w, ¯ z¯ , v¯ , μ, ¯ p¯ = , q¯ = , r¯ = , s¯ = ) is an efficient solution of We now show that (u, ¯ λ, (MFD). If not, then there exists (u , z , v , μ , λ , w , p , q , r , s ) of (MFD) such that ¯ + u¯ T z¯ i fi (u ) + u T zi fi (u) ≤ , T ¯ – u¯ v¯ i gi (u ) – u T v i gi (u)

i = , , . . . , k,

¯ + u¯ T z¯ r fr (u ) + u T zr fr (u) < , ¯ – u¯ T v¯ r gr (u ) – u T v r gr (u)

for some r = , , . . . , k.

and

This contradicts the weak duality theorem. Hence we have the result.



Theorem . (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) (fi (·) + (·)T zi ) and –(gi (·) – (·)T vi ) are higher-order strictly (C, αi , γi , ρi , di )-convex at u, with respect to Hi , p, and s, i = , , . . . , k, (ii) (hj (·) + (·)T wj ) is higher-order (C, ξj , δj , ηj , cj )-convex at u, with respect to Kj , q, and r, j = , , . . . , m, (iii) γ¯i (x, u) = ζ (x, u) and ξj (x, u) = δj (x, u) = σ (x, u), i = , , . . . , k, j = , , . . . , m, k m cj (x,u) d¯ i (x,u) (iv) i= λi ρ¯i α¯ i (x,u) + j= μj ηj ξj (x,u) ≥ , k fi (x)+xT zi fi (u)+uT zi λi (v) i= α¯ i (x,u) ( g (x)–xT v – g (u)–uT v ) ≤ , i

i

i

i

where  gi (x) – xT vi αi (x, u), α¯ i (x, u) = gi (u) – uT vi   di (x, u) d¯ i (x, u) = . gi (x) – xT vi

  fi (u) + uT zi , ρ¯i = ρi  + gi (u) – uT vi



Then x = u. Proof Suppose that x = u and exhibit a contradiction. Let (u, z, v, μ, λ, w, p, q, r, s) be feasible for (MFD), then  Cx,u () = Cx,u

k i=

+

k i=

 λi ∇

fi (u) + uT zi gi (u) – uT vi

¯ i (u, p) + λi ∇p H

m j=

 m   + μj ∇ hj (u) + uT wj j=

 μj ∇q Kj (u, q) = .

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

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zi ¯ ¯ i , γ¯i , ρ¯i , d¯ i )-convex at u with respect to H ¯ i , p, Since ( gfi (·)+(·) T ) is higher-order strictly (C, α i (·)–(·) vi and s, we have    fi (x) + xT zi fi (u) + uT zi – α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi     fi (u) + uT zi ¯ ¯ > Cx,u ∇ + ∇p Hi (u, p) gi (u) – uT vi T

+ Let τ =

¯   ¯ ¯ i (u, s) + ρ¯i di (x, u) . Hi (u, s) – sT ∇s H γ¯i (x, u) α¯ i (x, u)

k

 (fi (u)+uT zi +gi (u)–uT vi ) + m j= μj > . (gi (u)–uT vi ) λi by τ in the above inequality, we

i= λi

Multiplying

get



 λi  fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi     fi (u) + uT zi ¯ i (u, p) > C¯ x,u ∇ + ∇ H p gi (u) – uT vi +

 ¯   ¯ ¯ i (u, s) + ρ¯i di (x, u) . Hi (u, s) – sT ∇s H γ¯i (x, u) α¯ i (x, u)

()

By the higher-order (C, ξj , δj , ηj , cj )-convex of (hj (·) + (·)T wj ) at u with respect to Kj , q, and r, j = , , . . . , m, we obtain        hj (x) + xT wj – hj (u) – uT wj ≥ Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q) ξj (x, u) +

 ηj cj (x, u)   Kj (u, r) – rT ∇r Kj (u, r) + . δj (x, u) ξj (x, u)

It follows that

 μj   hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u)     ≥ Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q)

  ηj cj (x, u)   T Kj (u, r) – r ∇r Kj (u, r) + . + δj (x, u) ξj (x, u)

Taking summation over i in () and over j in (), we get k i=

  λi fi (x) + xT zi fi (u) + uT zi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi +

m j=

>

 μj  hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u)

    fi (u) + uT zi ¯ ¯ + ∇p Hi (u, p) Cx,u ∇ τ gi (u) – uT vi

k λi i=

+

k λi i=

  ¯ ¯ i (u, s) Hi (u, s) – sT ∇s H τ γ¯i (x, u)

()

Dubey et al. Journal of Inequalities and Applications (2015) 2015:354

+

k λi ρ¯i d¯ i (x, u) i=

+

m j=

τ α¯ i (x, u)

+

m μj j=

τ

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    Cx,u ∇ hj (u) + uT wj + ∇q Kj (u, q)

 μj ηj cj (x, u) μj   Kj (u, r) – rT ∇r Kj (u, r) + . τ δj (x, u) τ ξj (x, u) j= m

Further, using convexity on C and (), we obtain   fi (x) + xT zi fi (u) + uT zi λi – τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi

k i=

+

m j=

>

k i=

+

 μj  hj (x) + xT wj – hj (u) – uT wj τ ξj (x, u)

 μj    λi   ¯ ¯ i (u, s) + Hi (u, s) – sT ∇s H Kj (u, r) – rT ∇r Kj (u, r) τ γ¯i (x, u) τ δ (x, u) j j= m

k λi ρ¯i d¯ i (x, u) i=

τ α¯ i (x, u)

+

m μj ηj cj (x, u) j=

τ ξj (x, u)

.

This, together with ()-() and hypotheses (iii)-(iv), shows k i=

  m  μj  fi (x) + xT zi fi (u) + uT zi λi hj (x) + xT wj > . – + τ α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi τ σ (x, u) i=

Since x is a feasible solution of (MFP), it follows that k i=

  fi (x) + xT zi fi (u) + uT zi λi – > . α¯ i (x, u) gi (x) – xT vi gi (u) – uT vi

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 ¯ ¯ α, ratio of higher-order (C, α, γ , ρ, d)-convex function is also a higher-order (C, ¯ 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 1 Department of Mathematics, Indian Institute of Technology, Roorkee, 247 667, India. 2 Department of Mathematics, University of Tabuk, Tabuk, Saudi Arabia.

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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-0131-1436, for their support to carry out this work. Received: 9 June 2015 Accepted: 29 October 2015 References 1. Mangasarian, OL: Second and higher-order duality in nonlinear programming. J. Math. Anal. Appl. 51, 607-620 (1975) 2. Chen, X: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J. Math. Anal. Appl. 290, 423-435 (2004) 3. Kassem, MA: Higher-order symmetric duality in vector optimization problem involving generalized cone-invex functions. Appl. Math. Comput. 209, 405-409 (2009) 4. Suneja, SK, Srivastava, MK, Bhatia, M: Higher order duality in multiobjective fractional programming with support functions. J. Math. Anal. Appl. 347, 8-17 (2008) 5. Ying, G: Higher-order symmetric duality for a class of multiobjective fractional programming problems. J. Inequal. Appl. 2012, 142 (2012) 6. Yuan, DH, Chinchuluun, A, Pardalos, PM: Nondifferentiable minimax fractional programming problems with (C, α , ρ , d)-convexity. J. Optim. Theory Appl. 185, 185-199 (2006) 7. Chinchuluun, A, Yuan, D, Pardalos, PM: Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154, 133-147 (2007) 8. Long, XJ: Optimality conditions and duality for nondifferentiable multiobjective programming problems with (C, α , ρ , d)-convexity. J. Optim. Theory Appl. 148, 197-208 (2011) 9. Hanson, MA: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80, 545-550 (1981) 10. Gupta, SK, Danger, D, Ahmad, I: On second order duality for nondifferentiable minimax fractional programming problems involving type-I functions. ANZIAM J. 55, 479-494 (2014) 11. Pandey, S: Duality for multiobjective fractional programming involving generalized η -convex functions. Opsearch 28, 31-43 (1991) 12. Gulati, TR, Saini, H: Higher-order (F, α , β , ρ , d)-convexity and its application in fractional programming. Eur. J. Pure Appl. Math. 4, 266-275 (2011) 13. Liang, ZA, Huang, HX, Pardalos, PM: Optimality conditions and duality for a class of nonlinear fractional programming problem. J. Optim. Theory Appl. 110, 611-619 (2001) 14. Gupta, SK, Kailey, N, Sharma, MK: Multiobjective second-order nondifferentiable symmetric duality involving (F, α , ρ , d)-convex function. J. Appl. Math. Inform. 28, 1395-1408 (2010) 15. 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. 28, 107-121 (2008) 16. Egudo, RR: Multiobjective fractional duality. Bull. Aust. Math. Soc. 37, 367-378 (1988) 17. Geoffrion, AM: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618-630 (1968) 18. Gulati, TR, Geeta: Duality in nondifferentiable multiobjective fractional programming problem with generalized invexity. J. Appl. Math. Comput. 35, 103-118 (2011)