Optimality and duality results for a nondifferentiable multiobjective fractional programming problem

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R E S E A R C H

Open Access

Optimality and duality results for a

nondifferentiable multiobjective fractional

programming problem

Ramu Dubey

1

_{, Shiv K Gupta}

1*

_{and Meraj Ali Khan}

2

*_{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 func-tions,h:Rn_×_Rn_→_R_and_k_:_Rn_×_Rn_→_Rm_{, Mangasarian [] formulated a higher-order} dual for a nonlinear optimization problem,{minf(x), subject tog(x)≤}. Inspired by this concept, many researchers have worked in this direction. Chen [] has formulated higher-order multiobjective symmetric dual programs and established duality relations under higher-orderF-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, Sunejaet al.[] discussed higher-order Mond-Weir and Schaible type nondiﬀerentiable dual programs and their duality theorems under higher-order (F,ρ,σ )-typeI-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.

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Yuanet al.[] introduced a class of functions called (C,α,ρ,d)-convex functions and de-rived duality theorems for a nondiﬀerentiable minimax fractional programming problem under (C,α,ρ,d)-convexity. Chinchuluunet al.[] later studied nonsmooth multiobjec-tive 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 functionH :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 is also higher-order (C¯,α¯,γ¯,ρ¯,d¯)-convex. A sufficient optimality condition related to the efficient solution of a multiobjective nondifferentiable fractional problem has been estab-lished. 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 functionC:X×X×Rn−→R(X⊂Rn) is said to be convex onRniff, for any fixed (x,u)∈X×Xand for anyx,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. LetCbe a convex function with respect to the third variable such thatCx,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+satisfyingd(x,x) = ⇔x=xandp,s∈Rn.

Deﬁnition . The functionφis said to be higher-order (strictly) (C,α,γ,ρ,d)-convex at

uwith respect toH,pandsif for eachx∈X, 

α(x,u)

φ(x) –φ(u)≥(>)Cx,u

∇φ(u) +∇pH(u,p)

+ 

γ(x,u)

H(u,s) –sT∇sH(u,s)

+ρd(x,u)

α(x,u) .

Remark . A diﬀerentiable functionf = (f,f, . . . ,fk) :X→Rkis (C,α,γ,ρ,d)-convex if for alli= , , . . . ,k,fiis (C,αi,γi,ρi,di)-convex.

Remark .

(i) IfH(u,·) = , then Deﬁnition . becomes that of a(C,α,ρ,d)-convex function as given in [, ]. Further ifα(x,u) = ,ρ= , andCx,u(a) =ηT(x,u)a, for

η:X×X→Rn_{, then}₍_C_,_α_,_γ_,_ρ_,_d₎_{-convexity reduces to invexity (see Hanson []).}

(ii) IfH(u,·) =_(·)T∇f(u)(·),α(x,u) =γ(x,u), andp=s, then Deﬁnition . reduces to the ﬁrst expression of a second-order(C,α,ρ,d)-type I-convex function given in Guptaet al.[].

(iii) Ifα(x,u) =γ(x,u) = ,ρ= ,H(u,·) = _(·)T_∇_f₍_u₎₍_·₎_,_p₌_{s, and}_C

x,u(a) =ηT(x,u)a,

whereη:X×X→Rn_{, the above deﬁnition becomes that of}_η_-bonvexity

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(iv) IfCis sublinear with respect to third variable andp=s, then the above deﬁnition reduces to higher-order(F,α,γ,ρ,d)-convexity as given in Gulati and Saini []. Furthermore, ifγ(x,u) =α(x,u), then Deﬁnition . reduces to higher-order

(F,α,ρ,d)-convexity as in Ying []. Moreover, ifH(u,·) = , then Deﬁnition . becomes that of a(F,α,ρ,d)-convex function as introduced by Lianget 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 . LetX={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 functionC(deﬁned above) is convex with respect to the third variable satis-fyingCx,u() = ,∀x,u∈X. Also,d(x,u) = ⇔x=u.

Letρ= – andα=γ =_. Then atu= , for allx∈X, we have 

α(x,u)

f(x) –f(u)= 

x_–_x x+ 

≥Cx,u

∇xf(u) +∇pH(u,p)

+ 

γ(x,u)

H(u,s) –sT∇sH(u,s)

+ρd(x,u)

α(x,u) = –  (x– )

_.

Hence,f is a higher-order (C,α,γ,ρ,d)-convex function with respect toH,p, ands. But considering the sameC,f is not higher-order (F,α,γ,ρ,d)-convex becauseCis not a sub-linear functional with respect to the third variable.

Deﬁnition .[] LetCbe a compact convex set inRn_{. The support function of}_C_is deﬁned by

S(x|C) =maxxT_y_:_y_∈_C _.

3 Problem formulation and optimality conditions

Consider the following nondiﬀerentiable multiobjective programming problem:

(MFP) Minimize F(x) =

f(x) +S(x|C) g(x) –S(x|D)

, f(x) +S(x|C)

g(x) –S(x|D)

, . . . ,fk(x) +S(x|Ck)

gk(x) –S(x|Dk)

subject to x∈X₌_x_∈_X_⊂_Rn_:_h

j(x) +S(x|Ej)≤,j= , , . . . ,m , wherefi,gi:X→R(i= , , . . . ,k) andhj:X→R(j= , , . . . ,m) are continuously diﬀer-entiable functions. Assume thatfi(·) +S(·|Ci)≥ andgi(·) –S(·|Di) > ;Ci,Di, andEjare compact convex sets inRn_and_S₍_x_|_C

i),S(x|Di), andS(x|Ej) denote the support functions of compact convex sets,Ci,Di, andEj,i= , , . . . ,k,j= , , . . . ,m, respectively.

Deﬁnition .[] A pointx∈Xis a weakly eﬃcient solution of (MFP), if there exists nox∈X_{such that for every}_i_{= , , . . . ,}_k_, fi(x)+S(x|Ci)

gi(x)–S(x|Di)<

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Deﬁnition .[] A pointx_∈_X_{is an eﬃcient solution (or a Pareto optimal solution)}

of (MFP), if there exists nox∈X_{such that for every}_i_{= , , . . . ,}_k_, fi(x)+S(x|Ci) gi(x)–S(x|Di)≤

fi(x)+S(x|Ci) gi(x)–S(x|Di) and for somer= , , . . . ,k, fr(x)+S(x|Cr)

gr(x)–S(x|Dr) <

fr(x)+S(x|Cr) gr(x)–S(x|Dr).

Lemma .[] If for a givenλ¯i>  (λ¯i≥andλ¯= ),i= , , . . . ,k,x¯∈X is an optimal

solution for the following single-objective problem:

(FPλ) minimize G(x) = k

i=

¯

λi

fi(x) +S(x|Ci)

gi(x) –S(x|Di)

subject to x∈X⊂Rn_,

thenx is an eﬃcient solution¯ (a weakly eﬃcient solution)for(MFP).

Theorem . For some t, assume ft(·) + (·)Tzt and–(gt(·) – (·)Tvt)are(C,αt,γt,ρt,dt)

-convex at u ∈ X with respect to Ht, p, and s. Then (ft(·)+(·) T_z

t

gt(·)–(·)T_v_t) is also higher-order (C¯,α¯t,γ¯t,ρ¯t,d¯t)-convex at u∈X with respect toH¯t,p,and s,where

¯

αt(x,u) =

gt(u) –uTvt

gt(x) –xTvt

αt(x,u), γ¯t(x,u) =γt(x,u),

¯

ρt=ρt

 +ft(u) +u T_z

t

gt(u) –uTvt

, H¯t(u,·) =

 (gt(u) –uTvt)

+ ft(u) +u T_z

t (gt(u) –uTvt)

Ht(u,·),

¯

dt(x,u) =

dt(x,u)

gt(x) –xTvt

,

and

¯

Cx,u(a) =

ft(u) +uTzt+gt(u) –uTvt (gt(u) –uTvt)

Cx,u

(gt(u) –uTvt)a

ft(u) +uTzt+gt(u) –uTvt

,

a=∇

ft(u) +uTzt

gt(u) –uTvt

+∇pH¯t(u,p).

Proof For anyx,u∈X,

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

=

ft(x) +xTzt–ft(u) –uTzt

gt(x) –xTvt

–ft(u) +uTzt

gt(x) –xTvt–gt(u) +uTvt (gt(x) –xTvt)(gt(u) –uTvt)

.

Sinceft(·) + (·)Tzt and –(gt(·) – (·)Tvt) are (C,αt,γt,ρt,dt)-convex atu∈Xwith respect to

Ht,p, ands, we have



αt(x,u)

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

≥ 

(gt(x) –xTvt)

Cx,u

∇ft(u) +uTzt

+∇pHt(u,p)

+ 

γt(x,u)

Ht(u,s) –sT∇sHt(u,s)

+ρtdt(x,u)

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+ ft(u) +u T_z

t (gt(x) –xTvt)(gt(u) –uTvt)

Cx,u

–∇gt(u) –uTvt

+∇pHt(u,p)

+ 

γt(x,u)

+ρtdt(x,u)

αt(x,u)

,

which implies



αt(x,u)

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

≥

ft(u) +uTzt+gt(u) –uTvt (gt(x) –xTvt)(gt(u) –uTvt)

gt(u) –uTvt

×

Cx,u

∇ft(u) +uTzt

+∇pHt(u,p)

+ 

γt(x,u)

+ρtdt(x,u)

αt(x,u)

+

ft(u) +uTzt

×

Cx,u

–∇gt(u) –uTvt

+∇pHt(u,p)

+ 

γt(x,u)

+ρtdt(x,u)

αt(x,u)

.

This further yields



αt(x,u)

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

≥

×

Cx,u

(gt(u) –uTvt)

∇

ft(u) +uTzt

gt(u) –uTvt

+



gt(u) –uTvt

+ ft(u) +u T_z

∇pHt(u,p)

+ 

γt(x,u)(gt(x) –xTvt)

×

 + ft(u) +u T_z

t (gt(u) –uTvt)

+ ρtdt(x,u)

αt(x,u)(gt(x) –xTvt)

 + ft(u) +u T_z

t (gt(u) –uTvt)

.

Multiplying by (gt(x)–xTvt

gt(u)–uT_v_t), the above inequality gives

(gt(x) –xTvt)

αt(x,u)(gt(u) –uTvt)

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

≥

×

Cx,u

(gt(u) –uTvt)

∇

ft(u) +uTzt

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+



gt(u) –uTvt

+ ft(u) +u T_z

∇pHt(u,p)

+ 

γt(x,u)(gt(u) –uTvt)

 + ft(u) +u T_z

t (gt(u) –uTvt)

+ ρtdt(x,u)

αt(x,u)(gt(u) –uTvt)

 + ft(u) +u T_z

t (gt(u) –uTvt)

.

Setting

¯

αt(x,u) =

gt(u) –uTvt

gt(x) –xTvt

αt(x,u), γ¯t(x,u) =γt(x,u),

¯

ρt=ρt

 +ft(u) +u T_z

t

gt(u) –uTvt

,

¯

Ht(u,·) =

 (gt(u) –uTvt)

+ ft(u) +u T_z

Ht(u,·), d¯t(x,u) =

dt(x,u)

gt(x) –xTvt

,

¯

C(x,u)(a) =

Cx,u

(gt(u) –uTvt)a

,

and

a=∇

ft(u) +uTzt

gt(u) –uTvt

+∇pH¯t(u,p).

It follows that



¯

αt(x,u)

ft(x) +xTzt

gt(x) –xTvt

– ft(u) +u T_z

t

gt(u) –uTvt

≥

¯

C(x,u)

∇

ft(u) +uTzt

gt(u) –uTvt

+∇pH¯t(u,p)

+ 

¯

γt(x,u) _¯

Ht(u,s) –sT∇sH¯t(u,s)

+ρ¯td¯t(x,u)

¯

αt(x,u)

.

Hence, (ft(·)+(·)Tzt

gt(·)–(·)T_z_t) is higher-order (C¯,α¯t,γ¯t,ρ¯t,d¯t)-convex atu∈Xwith respect toH¯t,p,

ands.

Theorem . Let X ⊂ Rn _{be an open convex set}_. _Let _ψ

i : 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 existsx∈Xsuch thatψi(x) < ,i= , , . . . ,k;

(ii) there existsλ∈Rk

+\ {}such that

k

i=λiψi(x)≥,for allx∈X.

Proof If (i) has a solution, that is, there existsx∈Xsuch thatψi(x) < ,i= , , . . . ,k, then for everyλ∈Rk₊\ {}, we havek_i_=λiψi(x) < , which implies that (ii) does not hold.

If (i) has no solution, letK=ψ(X) +int(Rk

+),ψ= (ψ,ψ, . . . ,ψk)T, thenK∩(–Rk+) =φ.

For anyz_,_z_∈_K_and_β_∈_{(, ), there exist}_x_,_x_∈_X_and_s_,_s_∈_int₍_Rk

+) such that

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Sinceu=βx_{+ ( –}_β₎_x_∈_X_and_φ

i(x,·) = –(·)T∇ψi(x), it follows from the higher-order (C,αi,γi,ρi,di)-convexity ofψiwith respect toφi,p, andsthat we have



αi(xj,u)

ψi

xj–ψi(u)

≥Cx,u

∇ψi(u) +∇pφi(u,p)

+ 

γi(xj,u)

φi(u,s) –sT

∇sφi(u,s) +

ρid(xj,u)

αi(xj,u)

= 

αi(xj,u)

ρid

xj,u, j= , .

Thus, there existst≥ such that

βψi

x+ ( –β)ψi

x=ψi

βx+ ( –β)x+t,

which implies that

βz+ ( –β)z=βψx+ ( –β)ψx+βs+ ( –β)s+ (t,t, . . . ,t)T.

SinceRk

+is a closed convex cone, we haveβz+ ( –β)z∈K, that is,Kis an open convex

set. It follows from the convex separated theorem that there existsλ∈Rk

+\ {}such that,

for allx∈X,k_i_=λiψi(x)≥.

We now discuss some optimality conditions for the problem (MFP).

Theorem . If u∈X is a weakly eﬃcient solution of(MFP),fi(·) + (·)Tzi,and–(gi(·) – (·)T_v

i)are higher-order(C,αi,γi,ρi,di)-convex functions at u∈X with respect toφi(u,·) = – (gi(u)–uTvi)

(fi(u)+uTzi+gi(u)–uTvi)×(·)

T_∇₍fi(u)+uTzi

gi(u)–uTvi),andρi≥,i= , , . . . ,k.Then there exists≤ ¯λ∈

Rk_,_λ¯_{= }_{such that u is an optimal solution of}_(FP_¯ λ).

Proof Ifu∈Xis a weakly eﬃcient solution of (MFP), then there does not exist anyx∈X

such that

ψj(x) =

fj(x) +xTzj

gj(x) –xTvj

– fj(u) +u T_z

j

gj(u) –uTvj

< , j= , , . . . ,k.

Sincefj(·) + (·)Tzjand –(gj(·) – (·)Tvj) are higher-order (C,αj,γj,ρj,dj)-convex atu with respect toφj,pands,j= , , . . . ,k, it follows from Theorem . thatψj(x) = (

fj(x)+xTzj gj(x)–xTvj – fj(u)+uTzj

gj(u)–uTvj) is higher-order (C¯,α¯j,γ¯j,ρ¯j,d¯j)-convex at u with respect to φ¯j, p, and s,j= , , . . . ,k, where

¯

αj(x,u) =

gj(x) –xTvj

gj(u) –uTvj

αj(x,u), γ¯j(x,u) =γj(x,u),

¯

ρj=ρj

 + fj(u) +u T_z

j

gj(u) –uTvj

,

¯

φj(u,·) =



gj(u) –uTvj

+ fj(u) +u T_z

j (gj(u) –uTvj)

φj(u,·) = –(·)T∇

fj(u) +uTzj

gj(u) –uTvj

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¯

dj(x,u) =

dj(x,u)

gj(x) –xTvj

,

and

¯

Cx,u(a) =

fj(u) +uTzj+gj(u) –uTvj (gj(u) –uTvj)

Cx,u

(gj(u) –uTvj)a

fj(u) +uTzj+gj(u) –uTvj

,

a=∇

fj(u) +uTzj

gj(u) –uTvj

+∇pφ¯j(u,p).

Using Theorem ., there exists ≤ ¯λ∈Rk_,_λ¯_{=  such that}k

j=λ¯jψj(x)≥, that is, k

j=

¯

λj

fj(x) +xTzj

gj(x) –xTvj

≥

k

j=

¯

λj

fj(u) +uTzj

gj(u) –uTvj

,

which implies thatuis an optimal solution of (FPλ¯).

Theorem .(Necessary condition) [] Assume thatx is an eﬃcient solution of¯ (MFP)

and the Slater constraint qualiﬁcation is satisﬁed on X.Then there existλ¯ ∈Rk,μ¯ ∈Rm,

¯

zi∈Rn,v¯i∈Rn,andw¯j∈Rn,i= , , . . . ,k,j= , , . . . ,m,such that k

i=

¯

λi∇

fi(x¯) +x¯Tz¯i

gi(x¯) –x¯Tv¯i

+ m

j=

¯

μj∇

hj(x¯) +x¯Tw¯j

= ,

m

j=

¯

μj

hj(x¯) +x¯Tw¯j

= ,

¯

xT¯zi=S(x¯|Ci), i= , , . . . ,k,

¯

xT¯vi=S(x¯|Di), i= , , . . . ,k,

¯

xTw¯j=S(x¯|Ej), j= , , . . . ,m,

¯

zi∈Ci, v¯i∈Di, w¯j∈Ej, i= , , . . . ,k,j= , , . . . ,m,

¯

λi> , μ¯j≥, i= , , . . . ,k,j= , , . . . ,m.

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

k

i= λi∇

fi(u) +uTzi

gi(u) –uTvi

+ m

j= μj∇

hj(u) +uTwj

= ,

m

j= μj

hj(u) +uTwj

= ,

uTzi=S(u|Ci), i= , , . . . ,k,

uT_v

i=S(u|Di), i= , , . . . ,k,

(9)

zi∈Ci, vi∈Di, wj∈Ej, i= , , . . . ,k,j= , , . . . ,m.

Let for any i= , , . . . ,k,j= , , . . . ,m,

(i) (fi(·) + (·)Tzi)and–(gi(·) – (·)Tvi)be higher-order(C,αi,γi,ρi,di)-convex atu,with respect toHi,p,ands,

(ii) (hj(·) + (·)Twj)be higher-order(C,ξj,δj,ηj,cj)-convex atu,with respect toKj,q,andr,

(iii) k_i_=λiρ¯i

¯

di(x,u)

¯ αi(x,u)+

m j=μjηj

cj(x,u) ξj(x,u)≥,

(iv) γ¯i(x,u) =ζ(x,u),ξj(x,u) =σ(x,u),andδj(x,u) =σ(x,u),

(v) k_i_=λi(∇pH¯i(u,p)) + m

j=μj(∇qKj(u,q)) = , k

i=λi(H¯i(u,s) –sT∇sH¯i(u,s))≥ andm_j_=μj(Kj(u,r) –rT∇rKj(u,r))≥,

where

¯

αi(x,u) =

gi(u) –uTvi

gi(x) –xTvi

αi(x,u),γ¯i(x,u) =γi(x,u), ρ¯i=ρi

 + fi(u) +u T_z

i

gi(u) –uTvi

,

¯

Hi(u,·) =



gi(u) –uTvi

+ fi(u) +u T_z

i (gi(u) –uTvi)

Hi(u,·), d¯i(x,u) =

di(x,u)

gi(x) –xTvi

.

Then u is an eﬃcient solution of(MFP).

Proof Supposeuis not an eﬃcient solution of (MFP), then there existsx∈Xsuch that

fi(x) +S(x|Ci)

gi(x) –S(x|Di)≤

fi(u) +S(u|Ci)

gi(u) –S(u|Di)

, for alli= , , . . . ,k,

and

fr(x) +S(x|Cr)

gr(x) –S(x|Dr)

< fr(u) +S(u|Cr)

gr(u) –S(u|Dr)

, for somer= , , . . . ,k,

which implies

fi(x) +xTzi

gi(x) –xTvi ≤

fi(x) +S(x|Ci)

gi(x) –S(x|Di)≤

fi(u) +S(u|Ci)

gi(u) –S(u|Di)

= fi(u) +u T_z

i

gi(u) –uTvi ,

for alli= , , . . . ,k, ()

and

fr(x) +xTzr

gr(x) –xTvr ≤

fr(x) +S(x|Cr)

gr(x) –S(x|Dr)

< fr(u) +S(u|Cr)

gr(u) –S(u|Dr)

= fr(u) +u T_z

r

gr(u) –uTvr ,

for somer= , , . . . ,k. ()

Since λi

¯

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

k

i= λi

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

(10)

From Theorem ., for eachi, ≤i≤k, (fi(·)+(·)Tzi

gi(·)–(·)Tvi) is higher-order (C¯,α¯i,γ¯i,ρ¯i,d¯i)-convex atu∈Xwith respect toH¯i,p, ands, therefore



¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

≥ ¯Cx,u

∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p)

+ 

¯

γi(x,u) _¯

Hi(u,s) –sT∇sH¯i(u,s)

+ρ¯id¯i(x,u)

¯

αi(x,u)

, ()

where

¯

αi(x,u) =

gi(u) –uTvi

gi(x) –xTvi

αi(x,u), γ¯i(x,u) =γi(x,u),

¯

ρi=ρi

 + fi(u) +u T_z

i

gi(u) –uTvi

,

¯

Hi(u,·) =



gi(u) –uTvi

+ fi(u) +u T_z

Hi(u,·), d¯i(x,u) =

di(x,u)

gi(x) –xTvi

,

and

¯

Cx,u(a) =

fi(u) +uTzi+gi(u) –uTvi (gi(u) –uTvi)

Cx,u

(gi(u) –uTvi)a

fi(u) +uTzi+gi(u) –uTvi

,

a=∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p).

Also, by the higher-order (C,ξj,δj,ηj,cj)-convexity of (hj(·) + (·)Twj) atuwith respect toKj,

q, andr,j= , , . . . ,m, we have



ξj(x,u)

hj(x) +xTwj–hj(u) –uTwj

≥Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ 

δj(x,u)

Kj(u,r) –rT∇rKj(u,r)

+ηjcj(x,u)

ξj(x,u)

. ()

Letτ=k_i_=λi(fi(u)+u T_z

i+gi(u)–uTvi)

(gi(u)–uTvi) + m

j=μj> .

Adding the inequalities obtained by multiplying () byλi

τ and () by μj

τ , we get

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

j= μj

τ ξj(x,u)

≥

k

i= λi

τ

(fi(u) +uTzi+gi(u) –uTvi) (gi(u) –uTvi)

Cx,u

(gi(u) –uTvi)

(11)

×

∇

fi(u) +uTzi

gi(u) –uTvi

+



gi(u) –uTvi

+ fi(u) +u T_z

∇pHi(u,p) + k i= λi τ  ¯

γi(x,u)



gi(u) –uTvi

+ fi(u) +u T_z

Hi(u,s) –sT∇sHi(u,s) + k i= λi τ

ρidi(x,u)

αi(x,u)



gi(u) –uTvi

+ fi(u) +u T_z

+ m j= μj

τ Cx,u

∇hj(u) +uTwj

+∇qKj(u,q) + m j= μj τ 

δj(x,u)

Kj(u,r) –rT∇rKj(u,r) + m j= μj τ

ηjcj(x,u)

ξj(x,u) .

Further, using the convexity ofC, we have

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi + m j= μj

τ ξj(x,u)

≥Cx,u  τ _k i= λi∇

fi(u) +uTzi

gi(u) –uTvi

+ k

i=

λi∇pH¯i(u,p)

+ m

j= μj∇

hj(u) +uTwj

+ m

j=

μj∇qKj(u,q) + k i= λi τ  ¯

γi(x,u) _¯

Hi(u,s) –sT∇sH¯i(u,s) + k i= λi τ ¯

ρid¯i(x,u)

¯

αi(x,u)

+ m j= μj τ 

δj(x,u)

Kj(u,r) –rT∇rKj(u,r) + m j= μj τ

ηjcj(x,u)

ξj(x,u) .

It follows from hypotheses (iii)-(v) that

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi + m j= μj

τ σ(x,u)

≥Cx,u  τ _k i= λi∇

fi(u) +uTzi

gi(u) –uTvi

+ m

j= μj∇

hj(u) +uTwj

.

Using the factCx,u() =  and m

j=μj(hj(u) +uTwj) = , we get

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi + m i= μj

τ σ(x,u)

hj(x) +xTwj

(12)

Finally, using feasibility of the primal problem (MFP), we have

k

i= λi

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

≥,

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

4 Duality model

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

(MFD) maximize G(u) =

f(u) +uTz g(u) –uTv

,f(u) +u T_z

 g(u) –uTv

, . . . ,fk(u) +u T_z

k

gk(u) –uTvk

subject to k

i= λi∇

fi(u) +uTzi

gi(u) –uTvi

+ m

j= μj∇

hj(u) +uTwj

+ k

i=

λi∇pH¯i(u,p) + m

j=

μj∇qKj(u,q) = ,

()

m

j= μj

hj(u) +uTwj+Kj(u,r) –rT∇rKj(u,r)

≥, ()

k

i= λi

_¯

≥, ()

zi∈Ci, vi∈Di, wj∈Ej, i= , , . . . ,k,j= , , . . . ,m,

λi> , μj≥, 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(·) + (·)Tzi)and–(gi(·) – (·)Tvi)are higher-order(C,αi,γi,ρi,di)-convex atu,with respect toHi,p,ands,i= , , . . . ,k,

(ii) (hj(·) + (·)Twj)is higher-order(C,ξj,δj,ηj,cj)-convex atu,with respect toKj,q,and

r,j= , , . . . ,m, (iii) k_i_=λiρ¯i

¯

di(x,u)

¯ αi(x,u)+

m j=μjηj

(iv) γ¯i(x,u) =ζ(x,u)andξj(x,u) =δj(x,u) =σ(x,u),i= , , . . . ,k,j= , , . . . ,m,

where

¯

αi(x,u) =

gi(x) –xTvi

gi(u) –uTvi

αi(x,u), ρ¯i=ρi

 +fi(u) +u T_z

i

gi(u) –uTvi

,

¯

di(x,u) =

di(x,u)

gi(x) –xTvi

.

Then the following cannot hold:

fi(x) +S(x|Ci)

gi(x) –S(x|Di)≤

fi(u) +uTzi

gi(u) –uTvi

(13)

and

fr(x) +S(x|Cr)

gr(x) –S(x|Dr)

< fr(u) +u T_z

r

gr(u) –uTvr

, for some r= , , . . . ,k. ()

Proof Suppose that () and () hold, then using λi

¯

αi(x,u)> ,x T_z

i≤S(x|Ci),xTvi≤S(x|Di),

i= , , . . . ,k, we have

k

i= λi

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

< . ()

From hypothesis (i) and Theorem ., (fi(·)+(·)Tzi

gi(·)–(·)Tvi) is higher-order (C¯,α¯i,γ¯i,ρ¯i,d¯i)-convex at

uwith respect toH¯i,p, ands,i= , , . . . ,k, therefore 

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

≥ ¯Cx,u

∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p)

+ 

¯

γi(x,u) _¯

+ρ¯id¯i(x,u)

¯

αi(x,u)

, ()

and by the higher-order (C,ξj,δj,ηj,cj)-convex of (hj(·) + (·)Twj) atuwith respect toKj,q, andr,j= , , . . . ,m, we have



ξj(x,u)

≥Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ 

δj(x,u)

+ηjcj(x,u)

ξj(x,u)

. ()

i+gi(u)–uTvi)

j=μj> . Multiply () by λi

τ and () by μj

τ and add them, to get

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

j= μj

τ ξj(x,u)

≥

k

i= λi

τ

(fi(u) +uTzi+gi(u) –uTvi) (gi(u) –uTvi)

Cx,u

(gi(u) –uTvi)

fi(u) +uTzi+gi(u) –uTvi

×

∇

fi(u) +uTzi

gi(u) –uTvi

+



gi(u) –uTvi

+ fi(u) +u T_z

∇pHi(u,p)

+ k

i= λi

τ



¯

γi(x,u)



gi(u) –uTvi

+ fi(u) +u T_z

(14)

+ k

i= λi

τ

ρidi(x,u)

αi(x,u)



gi(u) –uTvi

+ fi(u) +u T_z

+ m

j= μj

τ Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ m

j= μj

τ



δj(x,u)

+ m

j= μj

τ

ηjcj(x,u)

ξj(x,u) .

Further, using convexity onC, ()-(), and hypotheses (iii)-(iv), we have

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

i= μj

τ σ(x,u)

hj(x) +xTwj

≥.

Sincexis a feasible solution of (MFP), it follows that

k

i= λi

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

≥.

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

Theorem .(Strong duality theorem) Assumeu is an eﬃcient solution of¯ (MFP)and let

the Slater constraint qualiﬁcation be satisﬁed on X.Also,if

¯

H(u¯, ) = , K(u¯, ) = , ∇pH¯(u¯, ) = , ∇qK(u¯, ) = ,

∇sH¯(u¯, ) = , ∇rK(u¯, ) = ,

then there exist  <λ¯∈Rk_,_μ_¯ _∈_Rm_,_¯_z

i∈Rn,v¯i,andw¯j∈Rn,i= , , . . . ,k,j= , , . . . ,m,

such that (u¯,z¯,¯v,μ¯,λ¯,w¯,p¯ = ,q¯ = ,¯s= ,r¯ = ) is a feasible solution of (MFD) and

the objective function values of(MFP) and(MFD)are equal.Furthermore,if the

condi-tions of Theorem .hold for all feasible solutions of(MFP) and each feasible solution

(u,z,v,μ,λ,w,p= ,q= ,r = ,s= ) of (MFD),then (u¯,z¯,v¯,μ¯,λ¯,w¯,p¯= ,q¯ = ,

¯

r= ,¯s= )is an eﬃcient solution of(MFD).

Proof Assumeu¯is an eﬃcient solution of (MFP) and the Slater constraint qualiﬁcation is

satisﬁed onX. Then, from Theorem ., there existλ¯i> ,μ¯∈Rm,z¯i∈Rn,¯vi∈Rn, and

¯

wj∈Rn,i= , , . . . ,k,j= , , . . . ,m, such that k

i=

¯

λi∇

fi(u¯) +u¯Tz¯i

gi(u¯) –u¯Tv¯i

+ m

j=

¯

μj∇

hj(u¯) +u¯Tw¯j

= , ()

m

j=

¯

μj∇

hj(u¯) +u¯Tw¯j

= , ()

¯

uT¯zi=S(u¯|Ci), i= , , . . . ,k, ()

¯

uTv¯i=S(u¯|Di), i= , , . . . ,k, ()

¯

(15)

¯

zi∈Ci, v¯i∈Di, w¯j∈Ej, i= , , . . . ,k,j= , , . . . ,m, ()

¯

λi> , μ¯j≥, i= , , . . . ,k,j= , , . . . ,m. ()

Thus, (u¯,¯z,v¯,μ¯,λ¯,w¯,p¯= ,q¯= ,r¯= ,¯s= ) is feasible for (MFD) and from ()-(), the objective function values of (MFP) and (MFD) are equal.

We now show that (u¯,z¯,v¯,μ¯,λ¯,w¯,p¯ = ,q¯= ,r¯= ,¯s= ) is an eﬃcient solution of (MFD). If not, then there exists (u,z,v,μ,λ,w,p,q,r,s) of (MFD) such that

fi(u¯) +u¯T¯zi

gi(u¯) –u¯Tv¯i ≤

fi(u) +uTzi

gi(u) –uTvi

, i= , , . . . ,k,

and

fr(u¯) +u¯Tz¯r

gr(u¯) –u¯Tv¯r < fr(u

_{) +}_uT_z r

gr(u) –uTvr

, for somer= , , . . . ,k.

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(·) + (·)Tzi)and–(gi(·) – (·)Tvi)are higher-order strictly(C,αi,γi,ρi,di)-convex at

u,with respect toHi,p,ands,i= , , . . . ,k,

(ii) (hj(·) + (·)Twj)is higher-order(C,ξj,δj,ηj,cj)-convex atu,with respect toKj,q,and

r,j= , , . . . ,m,

(iii) γ¯i(x,u) =ζ(x,u)andξj(x,u) =δj(x,u) =σ(x,u),i= , , . . . ,k,j= , , . . . ,m,

(iv) k_i_=λiρ¯i

¯

di(x,u)

¯ αi(x,u)+

m j=μjηj

(v) k_i_= λi

¯ αi(x,u)(

fi(x)+xTzi gi(x)–xTvi–

fi(u)+uTzi gi(u)–uTvi)≤,

where

¯

αi(x,u) =

gi(x) –xTvi

gi(u) –uTvi

αi(x,u), ρ¯i=ρi

 +fi(u) +u T_z

i

gi(u) –uTvi

,

¯

di(x,u) =

di(x,u)

gi(x) –xTvi

.

Then x=u.

Proof Suppose thatx=uand exhibit a contradiction. Let (u,z,v,μ,λ,w,p,q,r,s) be feasi-ble for (MFD), then

Cx,u() =Cx,u _k

i= λi∇

fi(u) +uTzi

gi(u) –uTvi

+ m

j= μj∇

hj(u) +uTwj

+ k

i=

λi∇pH¯i(u,p) + m

j=

μj∇qKj(u,q)

(16)

Since (fi(·)+(·)Tzi

gi(·)–(·)Tvi) is higher-order strictly (C¯,α¯i,γ¯i,ρ¯i,d¯i)-convex atuwith respect toH¯i,p, ands, we have



¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

>C¯x,u

∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p)

+ 

¯

γi(x,u) _¯

+ρ¯id¯i(x,u)

¯

αi(x,u) .

i+gi(u)–uTvi)

j=μj> . Multiplying byλi

τ in the above inequality, we get

λi

τ



¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

>C¯x,u

∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p)

+ 

¯

γi(x,u) _¯

+ρ¯id¯i(x,u)

¯

αi(x,u)

. ()

By the higher-order (C,ξj,δj,ηj,cj)-convex of (hj(·) + (·)Twj) atuwith respect toKj,q, and

r,j= , , . . . ,m, we obtain



ξj(x,u)

≥Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ 

δj(x,u)

+ηjcj(x,u)

ξj(x,u) .

It follows that

μj

τ



ξj(x,u)

≥Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ 

δj(x,u)

+ηjcj(x,u)

ξj(x,u)

. ()

Taking summation overiin () and overjin (), we get

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

j= μj

τ ξj(x,u)

> k

i= λi

τ C¯x,u

∇

fi(u) +uTzi

gi(u) –uTvi

+∇pH¯i(u,p)

+ k

i= λi

τ



¯

γi(x,u) _¯

(17)

+ k

i= λi

τ

¯

ρid¯i(x,u)

¯

αi(x,u) +

m

j= μj

τ Cx,u

∇hj(u) +uTwj

+∇qKj(u,q)

+ m

j= μj

τ



δj(x,u)

+ m

j= μj

τ

ηjcj(x,u)

ξj(x,u) .

Further, using convexity onCand (), we obtain

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

j= μj

τ ξj(x,u)

> k

i= λi

τ



¯

γi(x,u) _¯

+ m

j= μj

τ



δj(x,u)

+ k

i= λi

τ

¯

ρid¯i(x,u)

¯

αi(x,u) +

m

j= μj

τ

ηjcj(x,u)

ξj(x,u) .

This, together with ()-() and hypotheses (iii)-(iv), shows

k

i= λi

τα¯i(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

+ m

i= μj

τ σ(x,u)

hj(x) +xTwj

> .

Sincexis a feasible solution of (MFP), it follows that

k

i= λi

¯

αi(x,u)

fi(x) +xTzi

gi(x) –xTvi

– fi(u) +u T_z

i

gi(u) –uTvi

> .

This contradicts the hypothesis (v). Hence the proof is completed.

5 Conclusions

In this article, we consider a class of fractional programming problem havingk-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¯,α¯,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.

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