# Optimality and Duality of Approximate Quasi Weakly Efficient Solution for Nonsmooth Vector Optimization Problems

Authors: Wenjing Li; Guolin Yu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper aims at studying optimality conditions and duality theorems of an approximate quasi weakly efficient solution for a class of nonsmooth vector optimization problems (VOP). First, a necessary optimality condition to the problem (VOP) is established by using the Clarke subdifferential. Second, the concept of approximate pseudo quasi type-I function is introduced, and under its hypothesis, a sufficient optimality condition to the problem (VOP) is also obtained. Finally, the approximate Mond–Weir dual model of the problem (VOP) is presented, and then, weak, strong, and converse duality theorems are established.


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## Body

## 1. Introduction

Optimality and duality are two core contents of vector optimization theory and applications. Subdifferential and subgradient (see [1]) as the powerful tools to characterize nonsmooth vector optimization problems have attracted much attention from many scholars (see [2–6]). Clarke subdifferential (see [1, 7, 8]) is an important nonsmooth analytical tool with many good properties. This paper will employ Clarke subdifferential to study the optimality conditions and duality theorems for a class of nonsmooth vector optimization problem (VOP).In practical problems, most solutions obtained by numerical algorithms are approximate ones. Therefore, it is of great theoretical value and practical significance to study approximate solutions of vector optimization problems. In addition, the convexity of functions and its generalization play a critical role in establishing sufficient conditions for vector optimization problems. Until now, the concepts of invexity (see [9]), pseudo-convexity, and quasi-convexity (see [10]) had been introduced, and the optimality conditions for different efficient solutions to vector optimization problems had been discussed. For more references dealing with approximate solutions and approximate convexity, please refer to [11–14]. In this paper, we will propose a kind of generalized convexity, termed as the approximate pseudo quasi type-I functions, and use it to explore optimality to the problem (VOP) with respect to the approximate quasi weakly efficiency.It is known that duality assertions allow to study a minimization problem through a maximization problem and to know what one can expect in the best case. There are many dual models in the literature, for instance, Lagrange dual (see [15, 16]), Mond–Weir duality (see [17–19]), Wolfe duality (see [17, 20]), conjugate duality (see [21]), and symmetric duality (see [22]). In this paper, we will introduce an approximate Mond–Weir dual model for the problem (VOP) and examine duality theorems between it and primal problem involving approximate quasi weakly efficient solution.The article is organized as follows: in Section2, we give some symbols, concepts, and lemmas, which will be used in the subsequent sections; in Section 3, the necessary and sufficient optimality conditions are established; in Section 4, an approximate Mond–Weir dual model of the problem (VOP) is presented, and then, some duality theorems are obtained.


## 2. Preliminaries

Letℝn be the n-dimensional Euclidean space, K is a nonempty subset of ℝn, intK, clK, and coK denote its interior, closure, and convex hull, respectively. Bx¯,r stands for the open ball of radius r>0 around x¯∈ℝn. The inner product between x and y is denoted by x,y for any x,y∈ℝn, and the norm of x be defined by x. Let(1)ℝ+n=x=x1,x2,…,xn∈ℝn:xi≥0,i=1,…,n,ℝ++n=x=x1,x2,…,xn∈ℝn:xi>0,i=1,…,n.Without other specifications, we always suppose thatD⊂ℝm be pointed closed convex cone with intD≠∅. The dual cone of D be denoted by (see [1])(2)D∗=ξ∈ℝm:ξ,y≥0,∀y∈D.Definition 1 (see [3, 23]).
An vector-valued mappingF:ℝn⟶ℝm is said to be locally Lipschitz at x¯∈ℝn, if there exists a constant L>0 and r>0 such that(3)Fx1−Fx2≤Lx1−x2,∀x1,x2∈Bx¯,r.
IfF is locally Lipschitz at x, for any x∈ℝn, then F is called locally Lipschitz mapping on ℝn.
It had been pointed out in [24] that the sum of two locally Lipschitz mappings is locally Lipschitz in finite-dimensional spaces.Definition 2 (see [1]).
Letφ:ℝn⟶ℝ∪+∞ be a locally Lipschitz function, the Clarke directional derivative of φ at x¯∈ℝn in the direction v∈ℝn be defined as(4)φ°x¯;v≔limx⟶x¯supt⟶0φx+tv−φxt.
TheClarke subdifferential of φ at x¯∈ℝn is defined as(5)∂φx¯≔ξ∈ℝn:ξ,v≤φ°x¯;v,∀v∈ℝn.Definition 3 (see [1]).
LetK be a nonempty subset of ℝn, the Clarke contingent cone and Clarke normal cone associated with a set K at a point x¯∈K are defined by(6)Tx¯;K≔y∈ℝn:∀xn∈K,xn⟶x¯,∀tn⟶0,∃yn⟶y,s.txn+tnyn∈K,∀n,Nx¯;K≔ξ∈ℝn:ξ,y≤0,∀y∈Tx¯;K.Lemma 1 (see [24]).
LetF:ℝn⟶ℝm be a locally Lipschitz at x¯∈ℝn and the function f:ℝm⟶ℝ be a locally Lipschitz at Fx¯, then(7)∂f∘Fx¯⊂∪η∈∂fFx¯∂η∘Fx¯.Definition 4 (see [25]).
LetK be a nonempty subset of ℝn, the indicator function δK on K is defined as(8)δKx=0,x∈K,+∞,x∉K.Remark 1.
IfK is a closed set, then δK is a lower semicontinuous function.
We collect some properties of Clarke subdifferential in the following Lemma2, which will be used in the later subsection, and their proofs can be found in [1] pp. 25–109.Lemma 2 (see [1]).
Letx∈ℝn, φ:ℝn⟶ℝ, and ψ:ℝn⟶ℝ be a locally Lipschitz function. Then,(i)
for allt>0, ∂φtx⊂t∂φx(ii)
∂φ+ψx⊂∂φx+∂ψx(iii)
ifx is local minimum point of ψ, then 0∈∂ψx(iv)
ifK is closed set, then Nx;K=∂δKxGerstewitz’s function is an important nonlinear scalarization function (see [26]), which has been proved to be an effective tool for characterizing vector optimization problem. We shall use some properties of Gerstewitz’s function (see pp.302–310 in [26]) to prove optimality conditions in Section 3.Lemma 3 (see [26]).
Lete∈intD, Gerstewitz’s function Θe:ℝm⟶ℝ be defined as(9)Θey=inft∈ℝ:y∈te−D,y∈ℝm,and Θe is continuous and locally Lipschitz.The next Lemma4 collects some properties of Gerstewitz’s function (see Lemma 2.3 and Lemma 3.1 in [25]), which will be used to prove the main conclusions of this article.Lemma 4 (see [25]).
Lete∈intD and t∈ℝ. Then,(i)
Θey≤t⇔y∈te−D(ii)
Θey≥t⇔y∉te−intD(iii)
∂Θey⊂D∗/0(iv)
∂Θey=ξ∈D∗:ξ,y=ΘeyLetfx=f1x,…,fpx:ℝn⟶ℝp and gx=g1x,…,gqx:ℝn⟶ℝq be vector-valued mappings, K be a nonempty closed subset of ℝn. We consider the following vector optimization problem (VOP):(10)VOPminfx=f1x,…,fpxs.t.gjx≤0,j=1,…,q,x∈K.LetΩ≔x∈K:gjx≤0,j=1,…,q be the set of feasible solutions for problem (VOP).The notion of approximate quasi weakly efficiency was introduced in literature [27] (see Definition 3.2). Now, let’s present this concept.Definition 5 (see [27]).
Letϵ≥0, e∈ℝ++p. x¯∈Ω is called an ϵe-quasi weakly efficient solution to problem (VOP), if(11)fx−fx¯+ϵx−x¯e∉−ℝ++p,∀x∈Ω.
Now, we give an example ofϵe-quasi weakly efficient solution for problem (VOP).Example 1.
LetK=ℝ+, x∈K, the vector-valued mapping f:ℝ⟶ℝ2 be defined as(12)fx=x2−x−3,−2x+1,and the mapping g:ℝ⟶ℝ be defined by(13)gx=−x2.
Takingϵ=1, x¯=0, and e=1,2∈ℝ++2, we get Ω=ℝ+, and for all x∈K,(14)fx−fx¯+ϵx−x¯e=x2−x−3,−2x+1−−3,1+x,2|x|=x2−x−3+3+x,−2x+1−1+2x=x2,0∉−ℝ+2.
Hence,x¯ is an ϵe-quasi weakly efficient solution of problem (VOP).


## 3. Optimality Conditions

In this section, we first give a necessary optimality condition using Clarke subdifferential for problem (VOP) with respect to quasi weakly efficiency. Furthermore, a kind of generalized convexity is introduced and is applied to establish a sufficient condition to problem (VOP).Theorem 1.
In problem (VOP), letϵ≥0, e¯×e^∈ℝ++p×ℝ++q, x¯∈Ω, f and g be locally Lipschitz on ℝn. If x¯ is an εe¯-quasi weakly efficient solution of problem (VOP), then there exists α¯,β¯∈ℝ+p×ℝ+q/0,0. such that(15)0∈∂α¯∘fx¯+∂β¯∘gx¯+α¯,e¯ϵB+Nx¯;K,(16)β¯,gx¯=0,where α¯°fx≔α¯,fx, and B is the closed unit ball of ℝn.Proof.
Sincex¯ is an εe¯-quasi weakly efficient solution of problem (VOP), we get(17)fx−fx¯+ϵx−x¯e¯∉−ℝ++p,∀x∈Ω.
This means that(18)fx−fx¯+ϵx−x¯e¯,gx∉−ℝ++p×ℝ++q,∀x∈K.
By Lemma4 (ii), we obtain(19)Θe¯×e^fx−fx¯+ϵx−x¯e¯,gx≥0,∀x∈K.
LetHx=fx−fx¯+ϵx−x¯e¯,gx, then(20)Θe¯×e^Hx≥0,∀x∈K.
Particularly, it holds(21)Θe¯×e^Hx¯≥0.
Noticing thatHx¯=0,gx¯ and 0,gx¯∈−ℝ+p,ℝ+q, we get from Lemma 4 (i) that(22)Θe¯×e^Hx¯=Θe¯×e^0,gx¯≤0.
By (21), we arrive at(23)Θe¯×e^Hx¯=0.
Together with (20), we derive(24)Θe¯×e^Hx≥0=Θe¯×e^Hx¯,∀x∈K.
It means thatx¯ is a minimal solution of Θe¯×e^°Hx on K. Furthermore, this leads to that x¯ is a minimal solution of the function Θe¯×e^°Hx+δKx on ℝn. It yields from Lemma 2 that(25)0∈∂Θe¯×e^∘H+δKx¯⊂∂Θe¯×e^∘Hx¯+∂δKx¯.
SinceK is closed, it follows from Lemma 2 (iv) that(26)∂δKx¯=Nx¯;K.
Combining with (25), we get(27)0∈∂Θe¯×e^°Hx¯+Nx¯;K.
Becausef, g, Gerstewitz’s funtion Θe¯×e^, and ⋅−x¯ are locally Lipschitz, it follows from Lemma 1 that H is also locally Lipschitz and(28)∂Θe¯×e^°Hx¯⊂∪η∈∂Θe¯×e^Hx¯∂η°Hx¯.
Thus, there existsη¯=α¯,β¯∈∂Θe¯×e^Hx¯ with α¯,β¯∈ℝp×ℝq such that(29)0∈∂η¯°Hx¯+Nx¯;K.
By Lemma4 (iii), we obtain(30)∂Θe¯×e^Hx¯⊂ℝ+p×ℝ+q∗0,0=ℝ+p×ℝ+q0,0.
Hence,α¯,β¯∈ℝ+p×ℝ+q/0,0. Noticing Hx=fx−fx¯+ϵx−x¯e¯,gx, we derive from (29) that(31)0∈∂α¯,β¯°Hx¯+Nx¯;K=∂α¯,β¯°f−fx¯+ϵ⋅−x¯e¯,gx¯+Nx¯;K⊂⊂∂α¯°fx¯+α¯,e¯ϵ∂⋅−x¯+∂β¯°gx¯+Nx¯;K⊂∂α¯°fx¯+∂β¯°gx¯+α¯,e¯ϵB+Nx¯;K,
which implies that(32)0∈∂α¯∘fx¯+∂β¯∘gx¯+α¯,e¯ϵB+Nx¯;K.
We now prove that (16) holds. It is clear that(33)α¯,β¯∈∂Θe¯×e^Hx¯=∂Θe¯×e^0,gx¯.
By Lemma4 (iv), we obtain(34)α¯,β¯,0,gx¯=Θe¯×e^0,gx¯=Θe¯×e^Hx¯=0,that is,(35)β¯,gx¯=0.
Therefore, (16) is true.
Motivated by the concepts of approximate pseudoconvex of type I and type II given in [10] (see Definition 8 and Definition 9 in [10]), we present the following generalized convexity, which will be utilized to establish a sufficient condition for problem (VOP).Definition 6.
Letϵ≥0, φ,ψ:ℝn⟶ℝ be locally Lipschitz functions. It is said that φ,ψ is ϵ-pseudo quasi type-I function at x¯∈ℝn, if for any ξ∈∂φx¯, ζ∈∂ψx¯, it holds(36)ξ,x−x¯+ϵx−x¯≥0⇒φx−φx¯+ϵx−x¯≥0,ψx−ψx¯≤0⇒ζ,x−x¯≤0,∀x∈ℝn.Example 2.
LetK=ℝ+, the real-valued functions φ, ψ:K⟶ℝ be defined by(37)φx=x2−2x+1,x∈0,1,x−1,x∈1,+∞,ψx=−x−8,∀x∈K.
Letϵ=3, x¯=0, because functions φ and ψ are differentiable, we derive ∂φ0=0, ∂ψ0=−1, and for any ξ∈∂φ0, ζ∈∂ψ0 and x∈K, it holds(38)ξ,x−x¯+ϵx−x¯=ξx+3x≥0,ζ,x−x¯≤0,φx−φx¯+ϵx−x¯=x2+x≥0,x∈0,1,4x≥0,x∈1,+∞,ψx−ψx¯=−x≤0.
Hence,φ,ψ is 3-pseudo quasi type-I at x¯=0.Theorem 2.
In problem (VOP), letϵ≥0, e¯×e^∈ℝ++p×ℝ++q, and x¯∈K. Suppose that f and g are locally Lipschitz on ℝn, and there exist α¯∈ℝ+p\0 and β¯∈ℝ+p/0 such that (15) and (16) hold. If α¯°f,β¯°g is α¯,e¯ϵ-pseudo quasi type-I at x¯, then x¯ is an ϵe¯-quasi weakly efficient solution of problem (VOP).Proof.
It follows from (15) that there exist ξ¯∈∂α¯°fx¯, ζ¯∈∂β¯°gx¯, ρ∈Nx¯;K, b∈B such that(39)ξ¯+ζ¯+ρ+α¯,e¯ϵb=0.
This means that(40)ξ¯,x−x¯+ζ¯,x−x¯+ρ,x−x¯+α¯,e¯ϵb,x−x¯=0,∀x∈K.
It is obvious that(41)ρ,x−x¯≤0,∀x∈K.
Together with (40), we obtain(42)ξ¯,x−x¯+ζ¯,x−x¯+α¯,e¯ϵb,x−x¯≥0,∀x∈K.
Noticing thatb=1, we arrive at(43)b,x−x¯≤x−x¯,∀x∈K.
By (42), one gets(44)ξ¯,x−x¯+ζ¯,x−x¯+α¯,e¯ϵx−x¯≥0,∀x∈K.
Suppose thatx¯ is not ϵe¯-quasi weakly efficient solution of problem (VOP), there exists x^∈K such that(45)fx^−fx¯+ϵx^−x¯e¯∈−ℝ++p.
Becauseα¯∈ℝ+p/0, one has(46)α¯,fx^−fx¯+ϵx^−x¯e¯<0,that is,(47)α¯,fx^−α¯,fx¯+α¯,e¯ϵx^−x¯<0.
Sinceβ¯∈ℝ+q/0 and gx^∈−ℝ+q, we get from (16) that(48)β¯,gx^−β¯,gx¯≤0.
In addition, becauseα¯°f,β¯°g is α¯,e¯ϵ-pseudo quasi type-I at x¯, it yields(49)ξ¯,x^−x¯+α¯,e¯ϵx^−x¯<0,ζ¯,x^−x¯≤0.
Hence,(50)ξ¯,x^−x¯+ζ¯,x^−x¯+α¯,e¯ϵx^−x¯<0
which contradicts (43). Thus, x¯ is an ϵe¯-quasi weakly efficient solution of problem (VOP).


## 4. Approximate Mond-Weir Dual

In this section, we propose an approximate Mond–Weir dual model for problem (VOP) and establish duality theorems with respect to quasi weakly efficient solution. Letϵ≥0 and e¯∈ℝ++p, for problem (VOP), we consider the following approximate Mond–Weir dual problem MWDϵ:(51)MWDϵmaxfys.t.0∈∂α∘fy+∂β∘gy+Ny;K+α,e¯ϵB,giy≥0,i=1,…,q,y∈K,α∈ℝ+p\0,β∈ℝ+q\0.The feasible set of problemMWDϵ is denoted by(52)SM=y,α,β∈K×ℝ+p\0×ℝ+q0:0∈∂α∘fy+∂β∘gy+α,e¯ϵB+Ny;K,giy≥0,i=1,…,q,y∈K.Definition 7.
y¯,α¯,β¯∈SM is called an ϵe¯-quasi weakly efficient solution to problem MWDϵ, if for any y,α,β∈SM, it holds(53)fy−fy¯−ϵy−y¯e¯∉ℝ++p.
Feng and Qiu ([25]) proposed a kind of generalized invex functions (see Definition 2.2 in [25]) and used it to derive a sufficient condition for an vector equilibrium problem with respect to weakly efficient solution. We extend the generalized invexity introduced in [25] to the following two class concepts of generalized convex functions and employ them to establish duality theorems between problem (VOP) and MWDϵ.Definition 8.
Letτ∈ℝ and τ≠0, σ:ℝn⟶ℝ be a locally Lipschitz function.(1)
σ is said to be quasi-τ-convex at x¯∈ℝn, if for any η∈∂σx¯, it holds that(54)1τeτσx−σx¯−1≤0⇒η,x−x¯≤0,∀x∈ℝn.(2)
σ is said to be ϵ-strictly τ-quasi convex at x¯∈ℝn, if for any η∈∂σx¯, it holds that(55)1τeτσx−σx¯+ϵx−x¯−1<0⇒η,x−x¯+ϵx−x¯<0,∀x∈ℝn.
The next Example3 is an approximate strictly τ-quasi convex function.Example 3.
LetK=ℝ+, τ∈ℝ, and τ≠0, f:K⟶ℝ be defined as(56)fx=−13x+2−ln3,x∈0,1,lne−x3,x∈1,+∞..
Settingϵ=1/4, x¯=0, because f is differentiable, we derive ∂f0=−1/3, and for any x∈K, x≠x¯, it holds(57)fx−f0+14x−0=−112x<0,x∈0,1,−34x<0,x∈1,+∞..
Since(58)1τeτfx−f0+ϵx−0−1<0,∀x∈K,and for any ξ∈∂f0,(59)ξ,x−x¯+ϵx−x¯=ξx+14x<0,we get that f is 1/4-strictly τ-quasi convex at x¯=0.Theorem 3 (Weak duality).
Letτ∈ℝ and τ≠0, y¯,α,β∈SM. If α°f is α,e¯ϵ-strictly quasi-τ-convex at y¯∈K, β°g is quasi-τ-convex at y¯∈K, then(60)fy¯−fx−ϵx−y¯e¯∉ℝ++p,∀x∈Ω.Proof.
Sincey¯,α,β∈SM, we have(61)0∈∂α°fy¯+∂β°gy¯+α,e¯ϵB+Ny¯;K.
It yields that there existξ∈∂α°fy¯, ζ∈∂β°gy¯, ρ∈Ny¯;K, and b∈B such that(62)ξ+ζ+α,e¯ϵb+ρ=0.
This means that(63)ξ,x−y¯+ζ,x−y¯+α,e¯ϵb,x−y¯+ρ,x−y¯=0,∀x∈K.
It is clear that(64)ρ,x−y¯≤0,∀x∈K.
This leads to(65)ξ,x−y¯+ζ,x−y¯+α,e¯ϵb,x−y¯≥0,∀x∈K.
Noticing thatb=1, we obtain(66)b,x−y¯≤x−y¯,∀x∈K,which implies that(67)ξ,x−y¯+ζ,x−y¯+α,e¯ϵx−y¯≥0,∀x∈K.
Suppose to the contrary, that is, (60) is not true, and there exists x^∈Ω such that(68)fy¯−fx^−ϵx^−y¯e¯∈ℝ++p.
Becauseα∈ℝ+p/0, one gets(69)α,fy¯−fx^−ϵx^−y¯e¯0,which means that(70)α,fx^−α,fy¯+α,e¯ϵx^−y¯<0.
In view ofτ∈ℝ and τ≠0, we get(71)1τeτα,fx^−α,fy¯+α,e¯ϵx^−y¯‖−1<0.
Sinceα°f is α,e¯ϵ-strictly quasi-τ-convex at y¯∈K, and ξ∈∂α°fy¯, it yields that(72)ξ,x^−y¯+α,e¯ϵx^−y¯<0.
Noticing thatx^∈Ω, giy¯≥0, i=1,…,q, and β∈ℝ+q/0, one has(73)gix^≤0,i=1,…,q,β,gx^−β,gy¯≤0.
Fromτ∈ℝ and τ≠0, we get(74)1τeτβ,gx^−β,gy¯−1≤0.
Becauseζ∈∂β°gy¯ and β°g is quasi-τ-convex at y¯∈K, it follows that(75)ζ,x^−y¯≤0.
Combining with (72) and (75), we arrive at(76)ξ,x^−y¯+ζ,x^−y¯+α,e¯ϵx^−y¯<0,
which contradicts (67).Theorem 4 (Strong duality).
Letτ∈ℝ, τ≠0. Suppose that y¯∈Ω is an ϵe¯-quasi weakly efficient solution of (VOP), and there exists α¯,β¯∈ℝ+p×ℝ+q/0,0 such that (15) and (16) hold. If α¯°f is α¯,e¯ϵ-strictly-τ-convex at y¯, β¯°g is quasi-τ-convex at y¯, then y¯,α¯,β¯ is an ϵe¯-quasi weakly efficient solution of dual problem MWDϵ.Proof.
Sincey¯∈Ω is an ϵe¯-quasi weakly efficient solution of (VOP), it follows from Theorem 1 that there exists α¯,β¯∈ℝ+p×ℝ+q/0,0 such that(77)0∈∂α¯°fy¯+∂β¯°gy¯+Ny¯;K+α¯,e¯ϵB,β¯,gy¯=0.
We claim thatgiy¯≥0, i=1,…,q. In fact, if giy¯<0i=1,…,q, then(78)β¯,gy¯<0,due to β¯,gy¯=0. This is a contradiction. Thus, y¯,α¯,β¯∈SM. In addition, since α¯°f is α¯,e¯ϵ-strictly quasi-τ-convex at y¯ and β¯°g is quasi-τ-convex at y¯, we get from Theorem 3 that(79)fy−fy¯−ϵy−y¯e¯∉ℝ++p,∀y,α,β∈SM.
Therefore,y¯,α¯,β¯ is an ϵe¯-quasi weakly efficient solution of dual problem MWDϵ.Theorem 5 (Converse duality).
Letτ∈ℝ, τ≠0, y¯∈K, and there exists α¯,β¯∈ℝ+p×ℝ+q/0,0 such that y¯,α¯,β¯∈SM. If α¯°f is α¯,e¯ϵ-strictly quasi-τ-convex at y¯, β¯°g and γ¯°h are quasi-τ-convex at y¯, then y¯ is an ϵe¯-quasi weakly efficient solution of problem (VOP).Proof.
Supposing thaty¯ is not an ϵe¯-quasi weakly efficient solution of problem (VOP), then there exists x^∈Ω such that(80)fx^−fy¯+ϵx^−y¯e¯∈−ℝ++p,gjx^≤0,j=1,…,q.
Sinceα¯∈ℝ+p/0, it follows that(81)α¯,fx^−α¯,fy¯+α¯,e¯ϵx^−y¯<0.
Becauseτ∈ℝ and τ≠0, we get(82)1τeτα¯,fx^−α¯,fy¯+α¯,e¯ϵx^−y¯−1<0.
By the similar arguments as in Theorem3, we conclude(83)ξ,x^−y¯+ζ,x^−y¯+α¯,e¯ϵx^−y¯<0,which contradicts to the following equation:(84)ξ¯,x^−y¯+ζ¯,x^−y¯+α¯,e¯ϵx^−y¯≥0.
Thus,y¯ is an ϵe¯-quasi weakly efficient solution of problem (VOP).


## 5. Conclusions

We have derived a necessary optimality condition for a class of nonsmooth vector optimization problem (VOP) by using the properties of Gerstewitz’s function and Clarke subdifferential. The concept of approximate pseudo quasi type-I function is introduced, and it is utilized to establish a sufficient optimality condition for an approximate quasi weakly efficient solution to problem (VOP). We have also proposed an approximate Mond–Weir dual modelMWDϵ for problem (VOP) and established the weak, strong, and converse duality theorems with respect to approximate quasi weakly efficiency.The optimality of vector optimization problems with respect to approximate solution is an interesting topic. There were some results related to approximate convexities and quasi efficiency in literature [12, 28, 29], and the readers can refer to them and references [30] therein.



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