# Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

Authors: Pallavi Kharbanda; Divya Agarwal; Deepa Sinha
Journal: ISRN Applied Mathematics
 (2014)

---

## Abstract

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of(φ,α,ρ,σ)-dI-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.


---

## Body

## 1. Introduction

Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and thus explored the extent of optimality conditions and duality applicability in mathematical programming problems. Consequently, various generalizations of convex functions have been introduced in the literature (see Hanson [1], Vial [2], Hanson and Mond [3], Jeyakumar and Mond [4], Hanson et al. [5], Liang et al. [6], and Gulati et al. [7]).Nonsmooth optimization provides analytical tools for studying optimization problems involving functions that are not differentiable in the usual sense. Several nonlinear analysis problems arise from areas of optimization theory, game theory, differential equations, mathematical physics, convex analysis, and nonlinear functional analysis. For a nondifferentiable multiobjective programming problem, there exists a generalization of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized subgradient. Instead of Clarke generalized subgradient, Ye [8] used the concept of directional derivative to define the class of d invex functions. Also, he derived necessary and sufficient optimality conditions taking functions f′(xo;y) and gJ′(xo;y) to be convex. However, Antczak [9] considered the directional derivatives of objective and constraint functions to be preinvex and derived duality results for Wolfe type, Mond-Weir type, and mixed type dual programs. Mishra and Noor [10] extended the class of functions to d-V-type I functions and obtained sufficient optimality and duality results for Mond-Weir type multiobjective dual program. Nahak and Mohapatra [11] obtained duality results for multiobjective programming problem under (d-ρ-η-θ) invexity assumptions. Slimani and Radjef [12] introduced a far wider class of nondifferentiable functions called dI-V-type I functions in which each component is directionally differentiable in its own direction instead of the same direction and established sufficient optimality and duality results.On the other hand, Bector et al. [13] generalized the notion of convexity to univex functions. Rueda et al. [14] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [15] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions. Generalizing the functions, Mishra et al. [16] obtained duality results for a nondifferentiable multiobjective programming problem under generalized d-univexity. As an extension, Ahmad [17] introduced a new class of dI-V-type I univex functions which was generalized to a class of (dI-ρ-σ)-V-type I univex functions by Kharbanda et al. [18].In this paper, we introduce a new generalized class of(φ,α,ρ,σ)-dI-V-type I univex functions which generalizes the class of functions introduced by Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8]. Further, we establish weak, strong, converse, and strict converse duality results for Mond-Weir type multiobjective dual program.


## 2. Preliminaries and Definitions

The following convention of vectors inRn will be followed throughout this paper: x≧y⇔xi≧yi, i=1,2,…,n; x≥y⇔x≧y, x≠y; x>y⇔xi>yi, i=1,2,…,n. Let D be a nonempty subset of Rn,η:D×D→Rn, and let xo be an arbitrary point of D and h:D→R,f:D→Rm, ϕ:R→R, b:D×D→R+. Also, we denote R≥m={y:y∈Rm and y≥0},  R≧(>)k={y:y∈Rk and y≧0   (y>0)} and i=1,m¯={1,2,…,m},  j=1,k¯={1,2,…,k}.Definition 1 (Weir and Mond [20] and Weir and Jeyakumar [21]).
The functionh is called preinvex on D if, for all x,xo∈D, there exists a vector function η such that ∀λ∈[0,1],  xo+λη(x,xo)∈D, one has
(1)λh(x)+(1-λ)h(xo)≧h(xo+λη(x,xo)).Definition 2 (Mititelu [22]).
The setD is said to be invex at xo with respect to η, if, for each x∈D,
(2)xo+λη(x,xo)∈D,∀λ∈[0,1].Dis said to be an invex set with respect to η, if D is invex at each xo∈D with respect to same η.Definition 3 (Antczak [9]).
LetD⊆Rn be an invex set. An m-dimensional vector-valued function ψ:D→Rm is said to be preinvex with respect to η, if each of its components is preinvex on D with respect to the same function η.Definition 4 (Clarke [23]).
The functionh is said to be locally Lipschitz at xo∈D, if there exist a neighbourhood v(xo) of xo and a constant k>0 such that
(3)|h(y)-h(x)|≦k∥y-x∥,∀x,y∈v(xo),
where ∥·∥ denotes the Euclidean norm. Also, one says that h is locally Lipschitz on D if it is locally Lipschitz at every point of D.Definition 5 (Bector et al. [13]).
A differentiable functionh is said to be univex at xo with respect to ϕ,η,b if, ∀x∈D, one has
(4)b(x,xo)ϕ(h(x)-h(xo))≧[∇h(xo)]Tη(x,xo).Definition 6 (Mishra et al. [16]).
LetD⊆Rn be a nonempty open set. The function f is called d-univex at xo∈D with respect to ϕ,η,b if it is directionally differentiable at xo such that, for any x∈D,
(5)b(x,xo)ϕ(fi(x)-fi(xo))≧fi′(xo;η(x,xo)),hhhhhhhhhhhhhhhhhhhhhhhhhh∀i=1,m¯,
where fi′(xo;η(x,xo)) denotes the directional derivative of fi at xo in the direction η(x,xo):
(6)fi′(xo;η(x,xo))=lim⁡λ→0+⁡fi(xo+λη(x,xo))-fi(xo)λ.
If the above inequalities are satisfied at any point xo∈D, then f is said to be d-univex on D with respect to η.Definition 7 (Slimani and Radjef [12]).
The functionh is said to be semidirectionally differentiable at xo∈D in the direction η(x,xo) if its directional derivative h′(xo;η(x,xo)) exists finite for all x∈D.Definition 8 (Slimani and Radjef [12]).
LetD⊆Rn be a nonempty open set. The function f is called dI-invex at xo∈D with respect to (ηi)i=1,m¯, if, for any x∈D,
(7)fi(x)-fi(xo)≧fi′(xo;ηi(x,xo)),∀i=1,m¯,
where fi is semidirectionally differentiable at xo in direction ηi:D×D→Rn, for i=1,m¯.We consider the following nonlinear multiobjective programming problem:(8)(MP)hhhMinimizef(x)=(f1(x),f2(x),…,fm(x))hhhsubject tog(x)≦0,
where x∈D and the functions f:D→Rm, g:D→Rk, and D is a nonempty open subset of Rn. Let X={x∈D:g(x)≦0} be the set of feasible solutions of (MP). For xo∈D, if we denote
(9)J(xo)={j∈{1,2,…,k}:gj(xo)=0},J~(xo)={j∈{1,2,…,k}:gj(xo)<0},J-(xo)={j∈{1,2,…,k}:gj(xo)>0},
then
(10)J(xo)∪J~(xo)∪J-(xo)={1,2,…,k}.Now we define a new class of(φ,α,ρ,σ)-dI-V-type I univex functions where φ:D×D×R→R is a functional which for any x,xo∈D satisfies the following properties:(i)
φ(x,xo;a1+a2)≦φ(x,xo;a1)+φ(x,xo;a2), for all a1,a2∈R, (subadditive in third argument);(ii)
φ(x,xo;αa)=αφ(x,xo;a), for all α∈R,  α≧0 and for all a∈R, (positive homogeneous in third argument);(iii)
φ(x,xo;a)>0, for all a>0.Let the functionsf:D→Rm  and  g:D→Rk where fi and gj are semidirectionally differentiable functions in the directions ηi:X×D→Rn and θj:X×D→Rn for i=1,m¯ and j=1,k¯. Also, let α,α~ be the vectors in Rm+k whose components are the functions αi1,αj2:X×D→R+∖{0},α~i1,α~j2:X×D→R+∖{0}, respectively, for i=1,m¯, j=1,k¯, while ρ∈Rm+k and ρ~∈R2 whose components are in R and σ:X×D→Rn;  bo and b1 are nonnegative functions defined on X×D, ϕo:R→R, and ϕ1:R→R.Definition 9.
(
f
,
g
) is said to be (φ,α,ρ,σ)-dI-V-type I univex at xo∈D with respect to (ηi)i=1,m¯ and (θj)j=1,k¯ if for all x∈X(11)bo(x,xo)ϕo(fi(x)-fi(xo))≧φ(x,xo;αi1(x,xo)fi′(xo;ηi(x,xo)))+ρi1∥σ(x,xo)∥2,∀i=1,m¯,-b1(x,xo)ϕ1(gj(xo))≧φ(x,xo;αj2(x,xo)gj′(xo;θj(x,xo)))+ρj2∥σ(x,xo)∥2,∀j=1,k¯.If the inequalities inf are strict (whenever x≠xo), then (f,g) is said to be semistrictly (φ,α,ρ,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯.Remark 10.
(i) If in the above definition,φ(x,xo;αi1(x,xo)fi′(xo;ηi(x,xo))) = αi1(x,xo)fi′(xo;ηi(x,xo)), φ(x,xo;αj2(x,xo)gj′(xo;θj(x,xo))) = αj2(x,xo)gj′(xo;θj(x,xo)), for all i=1,m¯ and j=1,k¯, then we obtain the definition of (dI-ρ-σ)-V-type I univex function given by Kharbanda et al. [18]. Also, if in addition, we take ρi1,ρj2=0, then the above definition reduces to definition of dI-V-type I univex function introduced by Ahmad [17].
(ii) Ifφ is same as in (i) and ρi1,ρj2=0,  αi1(x,xo)=αj2(x,xo)=1, for all i=1,m¯ and  j=1,k¯ and bo(x,xo)=b1(x,xo)=1, ϕo(t)=t, and ϕ1(t)=t, then the above definition becomes definition of dI-V-type I function introduced by Slimani and Radjef [12].
(iii) Iff and g are differentiable functions and φ(x,xo;αi1(x,xo)fi′(xo;ηi(x,xo))) = αi1(x,xo)[∇fi(xo)]Tη(x,xo), φ(x,xo;αj2(x,xo)gj′(xo;θj(x,xo))) = αj2(x,xo)[∇gj(xo)]Tη(x,xo), and ρi1,ρj2=0,  i=1,m¯ and  j=1,k¯  and  bo(x,xo)=b1(x,xo)=1,  ϕo(t)=t,  ϕ1(t)=t, then above definition reduces to V-type I functions given by Hanson et al. [5]. Also, if, in addition, we take αi1(x,xo)=αj2(x,xo)=1, then we get the definition of type I function defined by Hanson and Mond [3].
(iv) If, in the above definition,φ(x,xo;αi1(x,xo)fi′(xo;ηi(x,xo))) = αi1(x,xo)fi′(xo;η(x,xo)), φ(x,xo;αj2(x,xo)gj′(xo;θj(x,xo))) = αj2(x,xo)gj′(xo;η(x,xo)), and ρi1,ρj2=0, i=1,m¯, and j=1,k¯ and bo(x,xo)=b1(x,xo)=1, ϕo(t)=t, and ϕ1(t)=t, then we obtain the definition of d-type I function introduced by Suneja and Srivastava [19].Definition 11.
(
f
,
g
) is said to be quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo∈D with respect to (ηi)i=1,m¯ and (θj)j=1,k¯, if for some vectors μ∈R≧m,  λ∈R≧k and for all x∈X(12)bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))≦0⟹φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))≦-ρ~1∥σ(x,xo)∥2,b1(x,xo)ϕ1(∑j=1k‍λjα~j2(x,xo)gj(xo))≧0⟹φ(x,xo;∑j=1k‍λjgj′(xo;θj(x,xo)))≦-ρ~2∥σ(x,xo)∥2.If the second (implied) inequality inf is strict (x≠xo), then (f,g) is said to be semistrictly quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯.Definition 12.
(
f
,
g
) is said to be pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo∈D with respect to (ηi)i=1,m¯ and (θj)j=1,k¯, if for some vectors μ∈R≧m,  λ∈R≧k and for all x∈X(13)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))≧-ρ~1∥σ(x,xo)∥2⟹bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))≧0,φ(x,xo;∑j=1k‍λjgj′(xo;θj(x,xo)))≧-ρ~2∥σ(x,xo)∥2⟹b1(x,xo)ϕ1(∑j=1k‍λjα~j2(x,xo)gj(xo))≦0.If the second (implied) inequality inf (resp., g) is strict (x≠xo), then (f,g) is said to be semistrictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex in f (resp., g) and if the second (implied) inequalities in f and g are both strict, then (f,g) is said to be strictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯.Definition 13.
(
f
,
g
) is said to be quasi-pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo∈D with respect to (ηi)i=1,m¯ and (θj)j=1,k¯, if for some vectors μ∈R≧m,  λ∈R≧k and for all x∈X(14)bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))≦0⟹φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))≦-ρ~1∥σ(x,xo)∥2,φ(x,xo;∑j=1k‍λjgj′(xo;θj(x,xo)))≧-ρ~2∥σ(x,xo)∥2⟹b1(x,xo)ϕ1(∑j=1k‍λjα~j2(x,xo)gj(xo))≦0.If the second (implied) inequality ing is strict (x≠xo), then (f,g) is said to be quasistrictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯.Definition 14.
(
f
,
g
) is said to be pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo∈D with respect to (ηi)i=1,m¯ and (θj)j=1,k¯, if for some vectors μ∈R≧m,  λ∈R≧k and for all x∈X(15)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))≧-ρ~1∥σ(x,xo)∥2⟹bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))≧0,(16)b1(x,xo)ϕ1(∑j=1k‍λjα~j2(x,xo)gj(xo))≧0⟹φ(x,xo;∑j=1k‍λjgj′(xo;θj(x,xo)))≦-ρ~2∥σ(x,xo)∥2.If the second (implied) inequality inf is strict (x≠xo), then (f,g) is said to be strictly pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯.


## 3. Illustration

In this section, we give some nontrivial examples which illustrate that the class of functions introduced in this paper is nonempty.Example 15.
Letf:R2→R and g:R2→R be defined by
(17)f(x1,x2)={x12+x22+|x1|+4ifx1≠0,x2≠04ifx1=0,x2=08+x12+x22else,g(x1,x2)={-2-4x12-6x22ifx1≠0,x2≠00else.
Let η(x,y)=(1+x12,2+y22), θ(x,y)=(0,x2), φ(x,y;a)=(|a|/2)(1+x12y22), σ(x,y)=(x1+y1,1+y2), and α1(x,y)=8,  α2(x,y)=4 where x=(x1,x2), y=(y1,y2).
Also, letbo(x,y)=1/2, b1(x,y)=4, ϕo(t)=4t, ϕ1(t)=4t, ρ1=-4, and ρ2=-2.
The setX of feasible solutions of problem is nonempty. Clearly, f and g are semidirectionally differentiable at xo=(0,0) with f′(xo;η(x,xo))=1+x12 and g′(xo;θ(x,xo))=0.
It is easy to see that for allx∈X(18)bo(x,xo)ϕo(f(x)-f(xo))≧φ(x,xo;α1(x,xo)f′(xo;η(x,xo)))+ρ1∥σ(x,xo)∥2,-b1(x,xo)ϕ1(g(xo))≧φ(x,xo;α2(x,xo)g′(xo;θ(x,xo)))+ρ2∥σ(x,xo)∥2.
Therefore (f,g) is (φ,α,ρ,σ)-dI-V-type I function at xo.
However, if we takex=(1,1), then
(19)(i)bo(x,x)ϕo(f(x)-f(xo))<α1(x,xo)f′(xo;η(x,xo))+ρ1∥σ(x,xo)∥2,bo(x,x)ϕo(f(x)-f(xo))<α1(x,xo)f′(xo;η(x,xo)).
Thus(f,g) is neither (dI-ρ-σ)-V-type I univex function given by Kharbanda et al. [18] nor dI-V-type I univex function at xo as given by Ahmad [17].
Hence the above example clearly illustrates that the class of(φ,α,ρ,σ)-dI-V-type I univex functions is more generalized than the class of (dI-ρ-σ)-V-type I univex functions and the class of dI-V-type I univex functions.
Next we show that(f,g) is pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex function but not (φ,α,ρ,σ)-dI-V-type I univex function.Example 16.
Letf:R2→R3 and g:R2→R2 be defined by
(20)f1(x1,x2)={4x12-2x2+4x22ifx1≠0,x2≠09+9x22ifx1=0,x2≠00else,f2(x1,x2)={2x22x12+6+x2ifx1≠0,x2≠00else,f3(x1,x2)={1+6x12ifx1≠0,x2=02+3x22ifx1=0,x2≠00else,g1(x1,x2)={0ifx1=0orx2=0-1+x12else,g2(x1,x2)={2-x12ifx1≠0,x2=02-6x22ifx1=0,x2≠0-1else.
Let η1(x,y)=(1+x12+y12,1+x22), η2(x,y)=(0,x1+x2), η3(x,y)=(4+x14,2+x22+y22), θ1(x,y)=(1+x1,0), θ2(x,y)=(2+x14,2+2x22), φ(x,y;a)=|a|(x12+y12), bo(x,y)=4, b1(x,y)=4, ϕo(t)=t/2,  ϕ1(t)=4t, and σ(x,y)=(x1+y1,x2+y2).
Also, letα11(x,y)=2, α21(x,y)=1, α31(x,y)=1, α12(x,y)=1, α22(x,y)=3, ρ~1=1, ρ~2=-4, ρ11=4, ρ21=1, ρ31=-6, ρ12=-4, ρ22=-6, μ1=μ2=1/2,  μ3=0, and λ1=1,  λ2=0 where x=(x1,x2), y=(y1,y2), α~i1(x,y)=1/αi1(x,y), and α~j2(x,y)=1/αj2(x,y), i=1,2,3, j=1,2.
The setX of feasible solutions of problem is nonempty. Clearly, f1,f2,f3 and g1,g2 are semidirectionally differentiable at xo=(0,0) with f1′(xo;η1(x,xo))=-2(1+x22),  f2′(xo;η2(x,xo))=0, f3′(xo;η3(x,xo))=0, g1′(xo;θ1(x,xo))=0, and g2′(xo;θ2(x,xo))=0.
It is easy to see that for allx∈X(21)φ(x,xo;∑i=13‍μifi′(xo;ηi(x,xo)))+ρ~1∥σ(x,xo)∥2=x22x12+x22+2x12≧0⟹bo(x,xo)ϕo(∑i=13‍μiα~i1(x,xo)(fi(x)-fi(xo)))≧0,b1(x,xo)ϕ1(∑j=12‍λjα~j2(x,xo)gj(xo))=0≧0⟹φ(x,xo;∑j=12‍λjgj′(xo;θj(x,xo)))+ρ~2∥σ(x,xo)∥2=-4(x12+x22)≦0.
Therefore (f,g) is pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex function at xo.
However, for the above defined problem, if we take(i)
x=(1,1),
(22)bo(x,xo)ϕo(f1(x)-f1(xo))<φ(x,xo;α11(x,xo)f1′(xo;η1(x,xo)))+ρ11∥σ(x,xo)∥2;(ii)
x=(1,0),
(23)bo(x,xo)ϕo(f2(x)-f2(xo))<φ(x,xo;α21(x,xo)f2′(xo;η2(x,xo)))+ρ21∥σ(x,xo)∥2.So (f,g) is not (φ,α,ρ,σ)-dI-V-type I univex function at xo.


## 4. Sufficient Optimality Conditions

In this section, we discuss some sufficient optimality conditions for a point to be an efficient solution of (MP) under newly defined class of(φ,α,ρ,σ)-dI-V-type I univex functions.Theorem 17.
Suppose there exist a feasible solutionxo of (MP) and vector functions ηi:X×D→Rn,  i=1,m¯,  θj:X×D→Rn,  j∈J(xo), and scalars μi>0,  i=1,m¯, and λj≧0,  j∈J(xo) such that(i)
∑i=1mμifi′(xo;ηi(x,xo))+∑j∈J(xo)λjgj′(xo;θj(x,xo))≧0, ∀x∈X,(ii)
for anyu∈R, u<0⇒ϕo(u)<0 and u≧0⇒ϕ1(u)≧0, bo(x,xo)>0, b1(x,xo)≧0,(iii)
(f,gJ(xo)) is pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j∈J(xo), and(iv)
ρ~1+ρ~2≧0,then xo is an efficient solution of (MP).Proof.
Suppose thatxo is not an efficient solution of (MP). Then there exists an x∈X of (MP) such that f(x)≤f(xo).
Asμi>0,  α~i1(x,xo)>0,  i=1,m¯, therefore
(24)∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo))<0.
Also gj(xo)=0,  λj≧0,  α~j2(x,xo)>0,  j∈J(xo) imply
(25)∑j∈J(xo)‍λjα~j2(x,xo)gj(xo)=0.
Since hypothesis (ii) holds, therefore inequality (24) and equality (25) become
(26)bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))<0,b1(x,xo)ϕ1(∑j∈J(xo)‍λjα~j2(x,xo)gj(xo))≧0.
Using hypothesis (iii), we obtain
(27)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))<-ρ~1∥σ(x,xo)∥2,φ(x,xo;∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))≦-ρ~2∥σ(x,xo)∥2.
The above inequalities along with subadditivity of φ yield
(28)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo))+∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))<-(ρ~1+ρ~2)∥σ(x,xo)∥2≦0(using hypothesis (iv)).
But as hypothesis (i) holds and φ(x,xo;0)=0 and φ(x,xo;a)>0 for a>0, therefore
(29)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo))+∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))≧0.
Thus we get a contradiction and hence the proof.Theorem 18.
Suppose there exist a feasible solutionxo of (MP) and vector functions ηi:X×D→Rn,  i=1,m¯, and θj:X×D→Rn,  j∈J(xo), and scalars μi≧0, i=1,m¯, and λj≧0,  j∈J(xo) satisfying(i)
∑i=1mμifi′(xo;ηi(x,xo))+∑j∈J(xo)λjgj′(xo;θj(x,xo))≧0, for all x∈X,(ii)
for anyu∈R, u≦0⇒ϕo(u)≦0 and u≧0⇒ϕ1(u)≧0, bo(x,xo)≧0, b1(x,xo)≧0,(iii)
(f,gJ(xo)) is strictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j∈J(xo), and(iv)
ρ~1+ρ~2≧0,then xo is an efficient solution of (MP).Proof.
Suppose thatxo is not an efficient solution of (MP). Then there exists x∈X of (MP) such that f(x)≤f(xo).
Asμi≧0, i=1,m¯, α~i1(x,xo)>0, i=1,m¯, and gj(xo)=0, λj≧0,  α~j2(x,xo)>0, j∈J(xo), therefore
(30)∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo))≦0,∑j∈J(xo)‍λjα~j2(x,xo)gj(xo)=0.
Using hypothesis (ii), we obtain
(31)bo(x,xo)ϕo(∑i=1m‍μiα~i1(x,xo)(fi(x)-fi(xo)))≦0,b1(x,xo)ϕ1(∑j∈J(xo)‍λjα~j2(x,xo)gj(xo))≧0.
Since (f,gJ(xo)) is strictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo, therefore, the above inequalities yield
(32)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))<-ρ~1∥σ(x,xo)∥2,φ(x,xo;∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))<-ρ~2∥σ(x,xo)∥2.
Using subadditivity of φ and hypothesis (iv), we get
(33)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo))+∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))<-(ρ~1+ρ~2)∥σ(x,xo)∥2≦0.
But hypothesis (i) and properties of φ imply
(34)φ(∑j∈J(xo)‍λjgj′(xo;θj(x,xo))x,xo;∑i=1m‍μifi′(xo;ηi(x,xo))+∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))≧0,
which leads to a contradiction. Hence xo is an efficient solution of (MP).In order to illustrate the result obtained, we will give an example of a multiobjective optimization problem in which the efficient solution will be obtained by the application of Theorem18.Example 19.
Let(35)Minimizef(x)=(f1(x),f2(x))subject tog(x)=(g1(x),g2(x))≦0,
where f:R2→R2 and g:R2→R2 be defined as
(36)f1(x1,x2)={2+x22+8|x1|ifx1≠0,x2≠00ifx1=0,x2=0x12+x22else,f2(x1,x2)={x12+x22-4|x1|ifx1≠0,x2≠00ifx1=0,x2=04+2x12+3x24else,g1(x1,x2)={-x14+1ifx1≠0,x2=00ifx1=0,x2=04x1-x12-3x22else,g2(x1,x2)={2x1-3x14ifx1≠0,x2=00ifx1=0,x2=0-x22-x12+2else.
Let η1(x,y)=(1+x12+y12,0), η2(x,y)=(1+x12,2+2x22), θ1(x,y)=(1+x12,2+x24), θ2(x,y)=(2+2x12,0), φ(x,y;a)=|a|(1+x12y12), bo(x,y)=4, b1(x,y)=4, ϕo(t)=t/2, ϕ1(t)=2t, and σ(x,y)=(x1+y1,1+y2).
Also, letα11(x,y)=4, α21(x,y)=2, α12(x,y)=2, α22(x,y)=1, ρ~1=4, ρ~2=-4, ρ11=0, ρ21=4, ρ12=-2, ρ22=-5, μ1=μ2=1/2, λ1=1/2, and λ2=0 where x=(x1,x2),  y=(y1,y2),α~i1(x,y)=1/αi1(x,y), and  α~j2(x,y)=1/αj2(x,y), i=1,2,  j=1,2.
The setX of feasible solutions of problem is nonempty. Clearly, f1,f2 and g1,g2 are semidirectionally differentiable at xo=(0,0) with f1′(xo;η1(x,xo))=0,  f2′(xo;η2(x,xo))=-4(1+x12),  g1′(xo;θ1(x,xo))=4(1+x12), and g2′(xo;θ2(x,xo))=4(1+x12).
It is easy to see that for allx∈X∖{xo}(37)φ(x,xo;∑i=12‍μifi′(xo;ηi(x,xo)))≧-ρ~1∥σ(x,xo)∥2⟹bo(x,xo)ϕo(∑i=12‍μiα~i1(x,xo)(fi(x)-fi(xo)))>0,b1(x,xo)ϕ1(∑j=12‍λjα~j2(x,xo)gj(xo))≧0⟹φ(x,xo;∑j=12‍λjgj′(xo;θj(x,xo)))<-ρ~2∥σ(x,xo)∥2.
Hence (f,gJ(xo)) is strictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex function at xo.
Also, hypotheses (i), (ii), and (iv) of Theorem18 are clearly satisfied and it follows that xo is an efficient solution of the above defined multiobjective optimization problem, whereas it will be impossible to apply for this purpose the sufficient optimality conditions given in Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8].Theorem 20.
Suppose there exist a feasible solutionxo of (MP) and vector functions ηi:X×D→Rn,  i=1,m¯, and θj:X×D→Rn,  j∈J(xo), and scalars μi≧0,  i=1,m¯, and λj≧0,  j∈J(xo), satisfying(i)
∑i=1mμifi′(xo;ηi(x,xo))+∑j∈J(xo)λjgj′(xo;θj(x,xo))≧0, for all x∈X,(ii)
for anyu∈R, u≦0⇒ϕo(u)≦0 and u≧0⇒ϕ1(u)≧0,  bo(x,xo)≧0, b1(x,xo)≧0, and(iii)
ρ~1+ρ~2≧0.Also, if either the fact that(a)
(f,gJ(xo)) is semistrictly quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j∈J(xo), or(b)
(f,gJ(xo)) is quasistrictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j∈J(xo), or(c)
(f,gJ(xo)) is strictly pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at xo with respect to (ηi)i=1,m¯ and (θj)j∈J(xo) holds,then xo is an efficient solution of (MP).Proof.
If (a) or (c) holds, then proceeding as in previous theorem we get(38)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))<-ρ~1∥σ(x,xo)∥2,φ(x,xo;∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))≦-ρ~2∥σ(x,xo)∥2.
And if (b) holds, we get
(39)φ(x,xo;∑i=1m‍μifi′(xo;ηi(x,xo)))≦-ρ~1∥σ(x,xo)∥2,φ(x,xo;∑j∈J(xo)‍λjgj′(xo;θj(x,xo)))<-ρ~2∥σ(x,xo)∥2.
The remaining part of proof runs on the lines of the proof of Theorem 18.Now, following Antczak [9] and Slimani and Radjef [12], we state the following necessary optimality conditions.Theorem 21 (Karush-Kuhn-Tucker type necessary optimality conditions).
If(i)
xo is a weakly efficient solution of (MP),(ii)
gj is continuous at xo for j∈J~(xo),(iii)
there exist vector functionsηi:X×D→Rn, i=1,m¯, and θj:X×D→Rn,  j∈J(xo), such that at xo∈D the following inequalities are satisfied with respect to η:X×D→Rn:
(40)fi′(xo;η(x,xo))≦fi′(xo;ηi(x,xo)),fi′xo;ηi(x,xo)ii∀x∈X,∀i=1,m¯,gj′(xo;η(x,xo))≦gj′(xo;θj(x,xo)),iiiiiiiiiiiiiiiiiiiiiiiiii∀x∈X,∀j∈J(xo),(iv)
for alli=1,m¯ and j∈J(xo), fi and gj are semidirectionally differentiable at xo and the functions fi′(xo;ηi(x,xo)), i=1,m¯, and gj′(xo;θj(x,xo)), j∈J(xo), are preinvex functions of x on X,(v)
the functiong satisfies dI-constraint qualification at xo with respect to (θj)j∈J(xo), then there exist μ∈R⩾m and λ∈R≧k such that
(41)∑i=1m‍μifi′(xo;ηi(x,xo))+∑j=1k‍λjgj′(xo;θj(x,xo))≧0,mmmmmmmmmmmimmmmmmmmm∀x∈X,λjgj(xo)=0,∀j=1,k¯.


## 5. Mond-Weir Type Duality

In this section, we consider Mond-Weir type dual of (MP) and establish weak, strong, converse, and strict converse duality theorems. Consider(42)(MWD)hhhhMax⁡hhhhf(y)hhhhsubjectto∑i=1m‍μifi′(y;ηi(x,y))hhhhhhsubjectto+∑j=1k‍λjgj′(y;θj(x,y))≧0,hhhhhhhhhhhhhhhhhhhhhiihhhhhhhh∀x∈X,(43)λjgj(y)≧0,j=1,k¯,
where y∈D, μ∈R⩾m, λ∈R≧k, ηi:X×D→Rn, for all i=1,m¯, and θj:X×D→Rn, for all j=1,k¯. Let W be the set of feasible points of (MWD).Theorem 22 (weak duality).
Letx and (y,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) be the feasible solutions of (MP) and (MWD), respectively, with μ>0. If (i)
(f,g) is pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at y with respect to (ηi)i=1,m¯ and (θj)j=1,k¯,(ii)
for anyu∈R,  u<0⇒ϕo(u)<0 and u≧0⇒ϕ1(u)≧0, bo(x,y)>0, b1(x,y)≧0, and(iii)
ρ~1+ρ~2≧0,then
(44)f(x)≰f(y).Proof.
Suppose to the contrary that(45)f(x)⩽f(y).
As μi>0, α~i1(x,y)>0, i=1,m¯, we get
(46)∑i=1m‍μiα~i1(x,y)(fi(x)-fi(y))<0.
Since λjgj(y)≧0, α~j2(x,y)>0, j=1,k¯, therefore
(47)∑j=1k‍λjα~j2(x,y)gj(y)≧0.
Using hypothesis (ii), we get
(48)bo(x,y)ϕo(∑i=1m‍μiα~i1(x,y)(fi(x)-fi(y)))<0,b1(x,y)ϕ1(∑j=1k‍λjα~j2(x,y)gj(y))≧0.
Since hypothesis (i) holds, therefore the above inequalities yield
(49)φ(x,y;∑i=1m‍μifi′(y;ηi(x,y)))<-ρ~1∥σ(x,y)∥2,φ(x,y;∑j=1k‍λjgj′(y;θj(x,y)))≦-ρ~2∥σ(x,y)∥2.
Using subadditivity of φ and hypothesis (iii), we get
(50)φ(x,y;∑i=1m‍μifi′(y;ηi(x,y))+∑j=1k‍λjgj′(y;θj(x,y)))<-(ρ~1+ρ~2)∥σ(x,y)∥2≦0.
However, the feasible condition (42) and properties of φ imply
(51)φ(x,y;∑i=1m‍μifi′(y;ηi(x,y))+∑j=1k‍λjgj′(y;θj(x,y)))≧0.
Thus we get a contradiction and hence the proof.The proof of the following theorems runs on the lines of the proof of Theorem22.Theorem 23 (weak duality).
Letx and (y,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) be the feasible solutions of (MP) and (MWD), respectively. If(i)
(f,g) is pseudo-quasi (φ,α~,ρ~,σ)-dI-V-type I univex at y with respect to (ηi)i=1,m¯ and (θj)j=1,k¯,(ii)
for anyu∈R, u<0⇒ϕo(u)<0 and u≧0⇒ϕ1(u)≧0, bo(x,y)>0, b1(x,y)≧0, and(iii)
ρ~1+ρ~2≧0,then
(52)f(x)≮f(y).Theorem 24 (weak duality).
Letx and (y,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) be the feasible solutions of (MP) and (MWD), respectively. If(i)
(f,g) is quasistrictly pseudo (φ,α~,ρ~,σ)-dI-V-type I univex at y with respect to (ηi)i=1,m¯ and (θj)j=1,k¯,(ii)
for anyu∈R, u≦0⇒ϕo(u)≦0 and u≧0⇒ϕ1(u)≧0,  bo(x,y)≧0, b1(x,y)≧0, and(iii)
ρ~1+ρ~2≧0,then
(53)f(x)≰f(y).Theorem 25 (strong duality).
Letxo be a weakly efficient solution of (MP) and gj is continuous at xo for j∈J~(xo). Also, the vector functions ηi,  i=1,m¯ and θj,  j∈J(xo) exist for which
(54)fi′(xo;η(x,xo))≦fi′(xo;ηi(x,xo)),hhhhhhhhhhhhh∀x∈X,∀i=1,m¯,gj′(xo;η(x,xo))≦gj′(xo;θj(x,xo)),hhhhhhhhhhhh∀x∈X,∀j∈J(xo)
at xo and fi,gj are semidirectionally differentiable at xo with fi′(xo;ηi(x,xo)) and gj′(xo;θj(x,xo)) as preinvex functions on X. Also if g satisfies dI-constraint qualification at xo, then ∃μ∈R≥m and λ∈R≧k such that (xo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) is feasible for (MWD) and the objective function values of (MP) and (MWD) are equal. Moreover, if any weak duality holds, then (xo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) is a weakly efficient solution of (MWD).Proof.
Sincexo is a weakly efficient solution of (MP), therefore, by Theorem 21, there exist μ∈R⩾m and λ∈R≧k such that
(55)∑i=1m‍μifi′(xo;ηi(x,xo))+∑j=1k‍λjgj′(xo;θj(x,xo))≧0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x∈X,λjgj(xo)=0,∀j=1,k¯.
It follows that(xo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯)∈W and therefore feasible for (MWD). Clearly, objective function values of (MP) and (MWD) are equal at these points.
Suppose(xo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) is not a weakly efficient solution for (MWD). Then ∃(y~,μ~,λ~,(η~i)i=1,m¯,(θ~j)j=1,k¯)∈W such that f(xo)<f(y~) which contradicts weak duality theorems. Therefore (xo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) is a weakly efficient solution of (MWD).Theorem 26 (converse duality).
Let(yo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) be a feasible solution of (MWD). Suppose hypotheses of Theorem 24 hold at yo, then yo is an efficient solution of (MP).Proof.
Suppose thatyo is not an efficient solution of (MP). Then ∃xo∈X such that
(56)f(xo)⩽f(yo).
Hence, by Theorem 24 (weak duality), we obtain a contradiction. Therefore yo is an efficient solution of (MP).Theorem 27 (strict converse duality).
Letxo and (yo,μ,λ,(ηi)i=1,m¯,(θj)j=1,k¯) be the feasible solutions of (MP) and (MWD), respectively. If (i)
f(xo)≦f(yo),(ii)
(f,g) is semistrictly quasi (φ,α~,ρ~,σ)-dI-V-type I univex at yo with respect to (ηi)i=1,m¯ and (θj)j=1,k¯,(iii)
for anyu∈R,  u≦0⇒ϕo(u)≦0 and u≧0⇒ϕ1(u)≧0, bo(xo,yo)≧0, b1(xo,yo)≧0, and(iv)
ρ~1+ρ~2≧0,then xo=yo.Proof.
Supposexo≠yo.
Sinceμ∈R≥m, α~i1(xo,yo)>0, i=1,m¯, and hypothesis (i) holds, therefore
(57)∑i=1m‍μiα~i1(xo,yo)fi(xo)≦∑i=1m‍μiα~i1(xo,yo)fi(yo).
As α~j2(xo,yo)>0,  j=1,k¯, and yo is feasible solution of (MWD), therefore
(58)∑j=1k‍λjα~j2(xo,yo)gj(yo)≧0.
Using hypothesis (iii), we obtain
(59)bo(xo,yo)ϕo(∑i=im‍μiα~i1(xo,yo)(fi(xo)-fi(yo)))≦0,b1(xo,yo)ϕ1(∑j=1k‍λjα~j2(xo,yo)gj(yo))≧0.
Applying hypothesis (ii) to the above inequalities, we get
(60)φ(xo,yo;∑i=1m‍μifi′(yo;ηi(xo,yo)))<-ρ~1∥σ(xo,yo)∥2,φ(xo,yo;∑j=1k‍λjgj′(yo;θj(xo,yo)))≦-ρ~2∥σ(xo,yo)∥2.
Using subadditivity of φ and hypothesis (iv), we get
(61)φ(∑j=1k‍xo,yo;∑i=1m‍μifi′(yo;ηi(xo,yo))+∑j=1k‍λjgj′(yo;θj(xo,yo)))<0
which is a contradiction as feasible condition (42) holds and φ(xo,yo;0)=0 and φ(xo,yo;a)>0 for a>0 yield
(62)φ(∑j=1k‍xo,yo;∑i=1m‍μifi′(yo;ηi(xo,yo))+∑j=1k‍λjgj′(yo;θj(xo,yo)))≧0.
Hence xo=yo.



---

*Source: 904640-2014-05-22.xml*