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In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient optimality condition for a class of nonsmooth multiobjective programming problems. Utilizing the sufficient optimality conditions, weak and strong duality theorems are established for Wolfe type duality model.

The field of multiobjective programming, also called vector programming, has grown remarkably in different directions in the settings of optimality conditions and duality theory since the 1980s. It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions. The Clarke subdifferential [

In this paper, we consider the nonsmooth multiobjective programming including the locally Lipschitz functions. The new concepts of invex of order σ ( B , φ ) − V − type II functions are introduced. Then, a sufficient optimality condition is obtained for the nondifferentiable multiobjective programming problem under the new functions and the Wolfe type duality results are obtained.

Let R n be the n-dimensional Euclidean space and let X be a nonempty open subset of R n . For x = ( x 1 , x 2 , ⋯ , x n ) T , y = ( y 1 , y 2 , ⋯ , y n ) T ∈ R n , we denote:

x = y ⇔ x i = y i , i = 1 , 2 , ⋯ , n ; x ≦ y ⇔ x i ≤ y i , i = 1 , 2 , ⋯ , n ; x ≤ y ⇔ x i ≦ y i , i = 1 , 2 , ⋯ , n and x ≠ y ; x < y ⇔ x i < y i , i = 1 , 2 , ⋯ , n ; x ∈ R + n ⇔ x ≧ 0.

Definition 2.1. [

| f ( y ) − f ( z ) | ≦ k ‖ y − z ‖ , forall y , z ∈ B ( x , ε ) . (1)

where B ( x , ε ) is the open ball of radius ε about x.

Definition 2.2. [

f 0 ( x ; d ) = lim y → x sup λ → 0 + f ( y + λ d ) − f ( y ) λ . (2)

Definition 2.3. [

∂ f ( x ) = { ξ ∈ R n : f 0 ( x ; d ) ≥ 〈 ξ , d 〉 , ∀ d ∈ R n } . (3)

where 〈 ⋅ , ⋅ 〉 is the inner product in R n .

Consider the following nonsmooth multiobjective programming problem:

(MP) Minimize f ( x ) = ( f 1 ( x ) , f 2 ( x ) , ⋯ , f k ( x ) ) , s . t . g j ( x ) ≦ 0 , j = 1 , 2 , ⋯ , m , x ∈ X .

where f i : X → R ( i ∈ K = { 1 , 2 , ⋯ , k } ) and g j : X → R ( j ∈ M = { 1 , 2 , ⋯ , m } ) are locally Lipschitz functions and X is a convex set in R n .

Let X 0 = { x | g j ( x ) ≦ 0 , j ∈ M } be the set of feasible solutions of (MP), and

J ( x ) = { j | g j ( x ) = 0 , x ∈ X 0 , j ∈ M } .

Definition 2.4. A point x ¯ ∈ X 0 is a strict minimizer of order σ for (MP) with respect to a nonlinear function ψ : X × X → R n , if for a constant ρ ∈ int R + k , such that

f ( x ¯ ) ≦ f ( x ) + ρ ‖ ψ ( x , x ¯ ) ‖ σ ,forall x ∈ X 0 . (4)

Throughout the paper, we suppose that η : X × X → R n ; b 0 , b 1 : X × X → R + ; φ 0 , φ 1 : R → R ; α , β : X × X → R + \ { 0 } ; ρ i , τ ∈ R + , i ∈ K .

Definition 2.5. ( f , g ) is said to be invex of order σ ( B , φ ) − V − type II at x ¯ ∈ X , if there exist η , b 0 , b 1 , φ 0 , φ 1 , α , β , ρ i ( i ∈ K ) , τ and some vectors λ ∈ R + k and μ ∈ R + m such that for all x ∈ X the following inequalities hold:

b 0 ( x , x ¯ ) φ 0 [ ∑ i = 1 k λ i ( f i ( x ) − f i ( x ¯ ) + ρ i ‖ ψ ( x , x ¯ ) ‖ σ ) ] ≧ α ( x , x ¯ ) 〈 ∑ i = 1 k λ i ξ i , η ( x , x ¯ ) 〉 , ∀ ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K (5)

− b 1 ( x , x ¯ ) φ 1 [ ∑ j = 1 m μ j g j ( x ¯ ) ] ≧ β ( x , x ¯ ) 〈 ∑ j = 1 m μ j ζ j , η ( x , x ¯ ) 〉 + τ ‖ ψ ( x , x ¯ ) ‖ σ , ∀ ζ j ∈ ∂ g j ( x ¯ ) , j ∈ M . (6)

Definition 2.6. ( f , g ) is said to be (pseudo, quasi) invex of order σ ( B , φ ) − V − type II at x ¯ ∈ X , if there exist η , b 0 , b 1 , φ 0 , φ 1 , α , β , ρ i ( i ∈ K ) , τ and some vectors λ ∈ R + k and μ ∈ R + m such that for all x ∈ X the following inequalities hold:

α ( x , x ¯ ) 〈 ∑ i = 1 k λ ¯ i ξ i , η ( x , x ¯ ) 〉 ≧ 0 , ∀ ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K ⇒ b 0 ( x , x ¯ ) φ 0 [ ∑ i = 1 k λ i ( f i ( x ) − f i ( x ¯ ) + ρ i ‖ ψ ( x , x ¯ ) ‖ σ ) ] ≧ 0 , (7)

− b 1 ( x , x ¯ ) φ 1 [ ∑ j = 1 m μ j g j ( x ¯ ) ] ≦ 0 ⇒ β ( x , x ¯ ) 〈 ∑ j = 1 m μ j ζ j , η ( x , x ¯ ) 〉 + τ ‖ ψ ( x , x ¯ ) ‖ σ ≦ 0 , ∀ ζ j ∈ ∂ g j ( x ¯ ) , j ∈ M . (8)

In this section, we establish sufficient optimality conditions for a strict minimizer of (MP).

Theorem 3.1. Let x ¯ ∈ X 0 . Suppose that

1) There exist λ i ≧ 0 , i ∈ K , ∑ i = 1 k λ i = 1 , μ j ≧ 0 , j ∈ J ( x ¯ ) , such that

0 ∈ ∑ i = 1 k λ i ∂ f i ( x ¯ ) + ∑ j ∈ J ( x ¯ ) μ j ∂ g j ( x ¯ ) ,

2) ( f , g J ) is invex of order σ ( B , φ ) − V − type II at x ¯ ,

3) b 0 ( x , x ¯ ) > 0 , b 1 ( x , x ¯ ) ≧ 0 ; a < 0 ⇒ φ 0 ( a ) < 0 , a = 0 ⇒ φ 1 ( a ) = 0 .

Then x ¯ is a strict minimizer of order σ for (MP).

Proof: Since 0 ∈ ∑ i = 1 k λ i ∂ f i ( x ¯ ) + ∑ j ∈ J ( x ¯ ) μ j ∂ g j ( x ¯ ) , there exists ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K , ζ j ∈ ∂ g j ( x ¯ ) , j ∈ J ( x ¯ ) , such that

∑ i = 1 k λ i ξ i + ∑ j ∈ J ( x ¯ ) μ j ζ j = 0 . (9)

whence

〈 ∑ i = 1 k λ i ξ i + ∑ j ∈ J ( x ¯ ) μ j ζ j , η ( x , x ¯ ) 〉 = 0 . (10)

Suppose that x ¯ is not a strict minimizer of order σ for (MP). Then there exists x ∈ X 0 and ρ ∈ R + k , such that

f ( x ¯ ) > f ( x ) + ρ ‖ ψ ( x , x ¯ ) ‖ σ . (11)

By λ i ≧ 0 , i ∈ K , ∑ i = 1 k λ i = 1 and hypothesis 3), we have

b 0 ( x , x ¯ ) φ 0 [ ∑ i = 1 k λ i ( f i ( x ) − f i ( x ¯ ) + ρ i ‖ ψ ( x , x ¯ ) ‖ σ ) ] < 0 . (12)

Since g j ( x ¯ ) = 0 , j ∈ J ( x ¯ ) and μ j ≧ 0, j ∈ J ( x ¯ ) , and hypothesis 3), we get

− b 1 ( x , x ¯ ) φ 1 [ ∑ j ∈ J ( x ¯ ) μ j g j ( x ¯ ) ] = 0 . (13)

In view of the hypothesis 1), one finds from (12) and (13) that

α ( x , x ¯ ) 〈 ∑ i = 1 k λ i ξ i , η ( x , x ¯ ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K . (14)

β ( x , x ¯ ) 〈 ∑ j ∈ J ( x ¯ ) μ j ζ j , η ( x , x ¯ ) 〉 + τ ‖ ψ ( x , x ¯ ) ‖ σ ≦ 0 , ∀ ζ j ∈ ∂ g j ( x ¯ ) , j ∈ J ( x ¯ ) . (15)

From α ( x , x ¯ ) > 0 , β ( x , x ¯ ) > 0 and τ ≧ 0 , we obtain

〈 ∑ i = 1 k λ i ξ i , η ( x , x ¯ ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K . (16)

〈 ∑ j ∈ J ( x ¯ ) μ j ζ j , η ( x , x ¯ ) 〉 ≦ 0 , ∀ ζ j ∈ ∂ g j ( x ¯ ) , j ∈ J ( x ¯ ) . (17)

Also

〈 ∑ i = 1 k λ i ξ i + ∑ j ∈ J ( x ¯ ) μ j ζ j , η ( x , x ¯ ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( x ¯ ) , i ∈ K , ∀ ζ j ∈ ∂ g j ( x ¯ ) , j ∈ J ( x ¯ ) (18)

which contradicts (10). Hence the result is true.

In this section, we consider the Wolfe type dual for the primal problem (MP) and establish various duality theorems. Let e be the vector of R k whose components are all ones.

( MD ) Maximize F ( u ) = f ( u ) + μ ¯ T g ( u ) e = ( f 1 ( u ) + ∑ j = 1 m μ ¯ j g j ( u ) , ⋯ , f k ( u ) + ∑ j = 1 m μ ¯ j g j ( u ) ) subjectto0 ∈ ∑ i = 1 k λ ¯ i ∂ f i ( u ) + ∑ j = 1 m μ ¯ j ∂ g j ( u ) , λ ¯ i ≧ 0, ∑ i = 1 k λ ¯ i = 1, i ∈ K , μ ¯ j ≧ 0, j ∈ M .

Let

U = { ( u , λ ¯ , μ ¯ ) ∈ X × R + k × R + m | 0 ∈ ∑ i = 1 k λ ¯ i ∂ f i ( u ) + ∑ j = 1 m μ ¯ j ∂ g j ( u ) , λ ¯ ≥ 0 , ∑ i = 1 k λ ¯ i = 1 , μ ¯ ≧ 0 }

be the set of all feasible solutions in problem (MD).

Theorem 4.1. (weak duality) Let ∀ x ∈ X 0 and ∀ ( u , λ ¯ , μ ¯ ) ∈ W be feasible solutions for (MP) and (MD), respectively. Moreover, assume that

1) ( f , g ) is invex of order σ ( B , φ ) − V − type II at u,

2) b 1 ( x , u ) > b 0 ( x , u ) > 0 ; a < b ⇒ φ 0 ( a ) < φ 1 ( b ) ; α ( x , u ) = β ( x , u ) .

Then the following can hold:

f ( x ) ≧ F ( u ) − ρ ‖ ψ ( x , u ) ‖ σ . (19)

Proof: Suppose contrary to the result that f ( x ) < F ( u ) − ρ ‖ ψ ( x , u ) ‖ σ holds, then we have

f i ( x ) < f i ( u ) + ∑ j = 1 m μ ¯ j g j ( u ) − ρ i ‖ ψ ( x , u ) ‖ σ , i ∈ K . (20)

which implies

f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , u ) ‖ σ < ∑ j = 1 m μ ¯ j g ( u ) j , i ∈ K . (21)

Using λ ¯ i ≧ 0 , i ∈ K , ∑ i = 1 k λ ¯ i = 1 , we have

∑ i = 1 k λ ¯ i ( f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , x ¯ ) ‖ σ ) < ∑ j = 1 m μ ¯ j g j ( u ) . (22)

By hypothesis 2), we have

b 0 ( x , u ) φ 0 [ ∑ i = 1 k λ ¯ i ( f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , u ) ‖ σ ) ] < b 1 ( x , u ) φ 1 [ ∑ j = 1 m μ ¯ j g ( u ) j ] . (23)

with hypothesis 1) and 2), the above inequality yields

α ( x , u ) 〈 ∑ i = 1 k λ ¯ i ξ i , η ( x , u ) 〉 < − α ( x , u ) 〈 ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 − τ ‖ ψ ( x , u ) ‖ σ , ∀ ξ i ∈ ∂ f i ( u ) , i ∈ K , ∀ ζ j ∈ ∂ g j ( u ) , j ∈ M . (24)

That is

〈 ∑ i = 1 k λ ¯ i ξ i + ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 < − τ α ( x , u ) ‖ ψ ( x , u ) ‖ σ , ∀ ξ i ∈ ∂ f i ( u ) , i ∈ K , ∀ ζ j ∈ ∂ g j ( u ) , j ∈ M . (25)

From τ ≧ 0 , which implies

〈 ∑ i = 1 k λ ¯ i ξ i + ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( u ) , i ∈ K , ∀ ζ j ∈ ∂ g j ( u ) , j ∈ M . (26)

On the other hand, by using the constraint conditions of (MD), there exist

ξ i ∈ ∂ f i ( u ) , i ∈ K and ζ j ∈ ∂ g j ( u ) , j ∈ M ,

such that

∑ i = 1 k λ ¯ i ξ i + ∑ j = 1 m μ ¯ j ζ j = 0. (27)

Also,

〈 ∑ i = 1 k λ ¯ i ξ i + ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 = 0. (28)

which contradicts (26). Then the result is true.

Theorem 4.2. (weak duality) Let ∀ x ∈ X 0 and ∀ ( u , λ ¯ , μ ¯ ) ∈ W be feasible solutions for (MP) and (MD), respectively. Moreover, assume that

1) ( f , g ) is (pseudo,quasi) invex of order σ ( B , φ ) − V − type II at u,

2) b 1 ( x , u ) > b 0 ( x , u ) > 0 ; a < b ⇒ φ 0 ( a ) < φ 1 ( b ) = 0.

Then the following can hold:

f ( x ) ≧ F ( u ) − ρ ‖ ψ ( x , u ) ‖ σ . (29)

Proof: Suppose contrary to the result that f ( x ) < F ( u ) − ρ ‖ ψ ( x , u ) ‖ σ holds, then we have

f i ( x ) < f i ( u ) + ∑ j = 1 m μ ¯ j g j ( u ) − ρ i ‖ ψ ( x , u ) ‖ σ , i ∈ K . (30)

Also

f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , u ) ‖ σ < ∑ j = 1 m μ ¯ j g i ( u ) , i ∈ K . (31)

Since λ ¯ i ≧ 0 , i ∈ K , ∑ i = 1 k λ ¯ i = 1 , which yields

∑ i = 1 k λ ¯ i ( f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , x ¯ ) ‖ σ ) < ∑ j = 1 m μ ¯ j g ( u ) j . (32)

It follows from hypothesis 2) that

b 0 ( x , u ) φ 0 [ ∑ i = 1 k λ ¯ i ( f i ( x ) − f i ( u ) + ρ i ‖ ψ ( x , u ) ‖ σ ) ] < b 1 ( x , u ) φ 1 [ ∑ j = 1 m μ ¯ j g j ( u ) ] = 0. (33)

In the view of hypothesis 1), one finds from (33) that

α ( x , u ) 〈 ∑ i = 1 k λ ¯ i ξ i , η ( x , u ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( u ) , i ∈ K . (34)

For α ( x , u ) > 0 , we have

〈 ∑ i = 1 k λ ¯ i ξ i , η ( x , u ) 〉 < 0 , ∀ ξ i ∈ ∂ f i ( u ) , i ∈ K . (35)

Since ( u , λ ¯ , μ ¯ ) ∈ M is a feasible solution for (MD), there exist ξ i ∈ ∂ f i ( u ) , i ∈ K and ζ j ∈ ∂ g j ( u ) , j ∈ M such that

∑ i = 1 k λ ¯ i ξ i + ∑ j = 1 m μ ¯ j ζ j = 0. (36)

whence

〈 ∑ i = 1 k λ ¯ i ξ i , η ( x , u ) 〉 + 〈 ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 = 0. (37)

It follows from (35) that

〈 ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 > 0. (38)

For β ( x , u ) > 0 and τ ≧ 0 , which yields

β ( x , u ) 〈 ∑ j = 1 m μ ¯ j ζ j , η ( x , u ) 〉 + τ ‖ ψ ( x , u ) ‖ σ > 0. (39)

From hypothesis 1), it follows that

− b 1 ( x , u ) φ 1 [ ∑ j = 1 m μ ¯ j g j ( u ) ] > 0. (40)

whence

b 1 ( x , u ) φ 1 [ ∑ j = 1 m μ ¯ j g j ( u ) ] < 0. (41)

which contradicts (33). Then the result is true.

The following definition is needed in the proof of the strong duality theorem.

Definition 4.1. A point u ∈ X is called a strict maximizer of order σ for (MD) with respect to a nonlinear function ψ : X × X → R n , if there exists a constant ρ ∈ int R + k such that

F ( u ) + ρ ‖ ψ ( x , u ) ‖ σ ≧ F ( x ) , ∀ x ∈ X . (42)

Theorem 4.3. (strong duality) Assume that x ¯ ∈ X 0 is a strict minimizer of order σ with respect to ψ for (MP), also there exist λ ¯ ≥ 0 , ∑ i = 1 k λ ¯ i = 1 and μ ¯ ≧ 0 , such that 0 ∈ ∑ i = 1 k λ ¯ i ∂ f i ( x ¯ ) + ∑ j = 1 m μ ¯ j ∂ g j ( x ¯ ) and ∑ j = 1 m μ ¯ j g j ( x ¯ ) = 0 . Furthermore, if all the hypothesis of Theorem 4.1 are satisfied for all feasible solutions of (MP) and (MD), then ( x ¯ , λ ¯ , μ ¯ ) is a strict maximizer of order σ for (MD) with respect to ψ .

Proof: The hypothesis implies that ( x ¯ , λ ¯ , μ ¯ ) is a feasible solution of (MD). By Theorem 4.1, for any feasible ( y , λ , μ ) of (MD), we have

f ( x ¯ ) ≧ F ( y ) − ρ ‖ ψ ( x ¯ , y ) ‖ σ . (43)

That is

f i ( x ¯ ) ≧ f i ( y ) + ∑ j = 1 m μ j g j ( y ) − ρ i ‖ ψ ( x ¯ , y ) ‖ σ , i ∈ K . (44)

Using ∑ j = 1 m μ ¯ j g j ( x ¯ ) = 0 , which yields

f i ( x ¯ ) + ∑ j = 1 m μ ¯ j g j ( x ¯ ) + ρ i ‖ ψ ( x ¯ , y ) ‖ σ ≧ f i ( y ) + ∑ j = 1 m μ j g j ( y ) , i ∈ K . (45)

whence

F ( x ¯ ) + ρ ‖ ψ ( x ¯ , y ) ‖ σ ≧ F ( y ) . (46)

Thus ( x ¯ , λ ¯ , μ ¯ ) is a strict maximizer of order σ for (MD) with respect to ψ .

In this paper, we have defined a class of new generalized functions. By using the new functions, we have presented a sufficient optimality condition and Wolfe type duality results for a nondifferentiable multiobjective problem. The present results can be further generalized for other programming problems.

This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 15JK1456); Natural Science Foundation of Shaanxi Province of China (Program No. 2017JM1041).

The authors declare no conflicts of interest regarding the publication of this paper.

An, G. and Ga, X.Y. (2018) Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems. Advances in Pure Mathematics, 8, 755-763. https://doi.org/10.4236/apm.2018.88045