Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems

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ISSN Online: 2160-0384 ISSN Print: 2160-0368

DOI: 10.4236/apm.2018.88045 Aug. 20, 2018 755 Advances in Pure Mathematics

Sufficiency and Wolfe Type Duality for

Nonsmooth Multiobjective Programming

Problems

Gang An

1

, Xiaoyan Gao

2

1_{College of Science, Xi’an Shiyou University, Xi’an, China}

2_{College of Science, Xi’an University of Science and Technology, Xi’an, China}

Abstract

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concept of invex of order σ ,

(

Bϕ

)

− −V

type II for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. The new functions are used to derive the sufficient op-timality 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.

Keywords

Multiobjective Programming, Optimality Condition, Locally Lipschitz Function, Wolfe Type Dual Problem

1. Introduction

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 differen-tiability assumptions. The Clarke subdifferential [1] (also called the Clarke ge-neralized gradient) is an important tool to derive optimality conditions for nonsmooth optimization problems. Together with the Clarke’s subdifferential, many generalized convexity or invexity functions were generalized to locally Lipschitz functions. Based upon the generalized functions, several sufficient op-timality conditions and duality results were established for the optimization problems. We can see for examples [2]-[8]. In [9] Upadhyay introduced some

How to cite 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

Received: July 30, 2018 Accepted: August 17, 2018 Published: August 20, 2018

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/apm.2018.88045 756 Advances in Pure Mathematics

new generalizations of the concept of

(

φ ρ,

)

-invexity and established the neces-sary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems. In [10] the new concepts of

(

φ ρ,

)

− −V type I were introduced. Sufficient optimality conditions and Mond-Weir duality results were obtained for nonsmooth multiobjective programming problems. Recently, many researchers have been interested in other types of solution concepts. One of them is higher order strict minimizer. In [11] and [12] some sufficient condi-tions and duality results were obtained for the new concept of strict minimizer of higher order for a multiobjective optimization problem.

In this paper, we consider the nonsmooth multiobjective programming in-cluding the locally Lipschitz functions. The new concepts of invex of order

(

)

,B V

σ ϕ − −_{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 ob-tained.

2. Preliminaries and Definitions

Let _Rn_{be the}_{n-dimensional Euclidean space and let X be a nonempty open}

subset of _Rn_{. For}

(

)

T

(

)

T

1, , ,2 n , , , ,1 2 n n

x= x x  x y= y y  y ∈R , we denote:

, 1,2, , ; , 1,2, , ;

, 1,2, , and ; , 1,2, , ;

0.

i i

n

x y x y i n x y x y i n

x y x y i n x y x y x y i n

x R+ x

⇔ = =

⇔ ≤ =

≤ ⇔ = ≠

< ⇔ < =

∈ ⇔

    =





Definition 2.1. [1] The function f X: →R_{is said to be locally Lipschitz at}

x X∈ , if there exist scalars k>0_and ε>0, such that

( )

, for all ,

(

,

)

f y − f z k y z− y z B x∈ ε . (1) where B x

( )

,ε is the open ball of radius ε about x.

Definition 2.2. [1] The generalized directional derivative of a locally Lipschitz function f at x in the direction d, denoted by _{f x d}0

(

_;

)

_{, is as follows:}

(

)

(

)

( )

0

0 ; lim sup

y x

f y d f y

f x d

λ

λ λ

+

→ _→

+ −

= _{. (2)}

Definition 2.3. [1] The generalized gradient of f X: →R_at x X∈ , de-noted by ∂f x

( )

_{, is defined as follows:}

( )

{

n_: 0

(

_;

)

_{, ,} n

}

f x ξ R f x d ξ d d R

∂ = ∈ ≥ ∀ ∈ . (3)

where ⋅ ⋅, is the inner product in _Rn_.

Consider the following nonsmooth multiobjective programming problem:

(MP)

( )

(

( ) ( )

( )

)

( )

1 2

Minimize , , , ,

. . 0, 1,2, , , .

k

j

f x f x f x f x s t g x j m

x X

= = ∈

 

(3)

where f Xi: →R i K

(

∈ =

{

1,2, , k

}

)

and g Xj: →R j M

(

∈ =

{

1,2, , m

}

)

are locally Lipschitz functions and X is a convex set in _Rn_.

Let X0=

{

x g xj

( )

0,j M∈

}

be the set of feasible solutions of (MP), and

( )

{

j( ) 0, 0,

}

J x = j g x = x X 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}_× _→_Rn_{, if for a constant}_ρ _int_Rk

+

∈ ,

such that

( )

(

,

)

, for all 0

f x _ f x +ρ ψ x x σ x X∈ _{. (4)} Throughout the paper, we suppose that η:X X× →R b b X Xn; , :0 1 × →R+;

0, :1 R R

ϕ ϕ → _; α β, :X X× →R+\ 0 ; ,

{ }

ρ τi ∈R i K+, ∈ .

Definition 2.5.

(

f g,

)

is said to be invex of order σ ,

(

Bϕ

)

− −V type II at

x X∈ , if there exist η, , , , , , ,b b0 1ϕ ϕ α β ρ0 1 i

(

i K∈

)

,τ and some vectors λ∈R+k

and _µ _Rm +

∈ such that for all x X∈ the following inequalities hold:

(

)

(

( )

(

)

(

)

(

)

( )

0 0 1 1 , , , , , , , k

i i i i

i k

i i i i

i

b x x f x f x x x

x x x x f x i K

σ

ϕ λ ρ ψ

α λ ξ η ξ

= =  ₋ ₊      ∀ ∈ ∂ ∈

∑

 (5)

( )

1 1 1 1 , , , , , , , . m m

j j j j

j j

b x x g x x x x x x x

g x j M

σ

ϕ µ β µ ζ η τ ψ

ζ = =   − _ _ +   ∀ ∈ ∂ ∈

∑



∑

(6)

Definition 2.6.

(

f g,

)

is said to be (pseudo, quasi) invex of order

(

)

,B V

σ ϕ − −_{type II at} x X∈ , if there exist η, , , , , , ,b b0 1ϕ ϕ α β ρ0 1 i

(

i K∈

)

,τ and some vectors _λ _Rk

+

∈ _and _µ _Rm +

∈ _{such that for all} x X∈ _{the following} inequalities hold:

( )

(

( )

)

1 0 0 1

, , , 0, ,

, , 0,

k

i i i i

i k

i i i i

i

x x x x f x i K

b x x f x f x x x σ

α λ ξ η ξ

ϕ λ ρ ψ

= = ∀ ∈ ∂ ∈   ⇒ _ − + _  

∑

  (7)

( )

1 1 1 1 , 0

, , , , 0, , .

m j j j

m

j j j j

j

b x x g x

x x x x x x σ g x j M

ϕ

µ

β

µ ζ η

τ ψ

ζ

= =   − _ _   ⇒ + ∀ ∈ ∂ ∈

∑

  (8)

3. Optimality Condition

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

Theorem 3.1. Let x X∈ 0. Suppose that

1) There exist λi0,i K∈ ,

( )

1

1, 0,

k

i j

i j J x

λ µ

=

= ∈

(4)

( )

1

0 k i i j j ,

i f x j J x g x

λ µ

= ∈

∈

∑

∂ +

∑

∂

2)

(

f g, _J

)

is invex of order σ ,

(

Bϕ

)

− −V _{type II at}x,

3) b x x0

(

,

)

>0, ,b x x1

(

)

0; 0a< ⇒ϕ0

( )

a <0, 0a= ⇒ϕ1

( )

a =0. Then x is a strict minimizer of order σ for (MP).

Proof: Since

( )

1

0 k i i j j

i f x j J x g x

λ µ

= ∈

∈

∑

∂ +

∑

∂ _{, there exists}ξ_i∈∂f x i K_i

( )

, ∈ _,

( )

,

( )

j g x j J xj

ζ ∈∂ ∈ _{, such that}

( )

1 0

k

i i j j

i j J x

λ ξ µ ζ

= ∈

+ =

∑

. (9)

whence

( )

(

)

1

, , 0

k

i i j j

i j J x

x x

λ ξ µ ζ η

= ∈

+ =

∑

. (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

1

0, , k 1

i i

i

i K

λ λ

=

∈

∑

=

 and hypothesis 3), we have

(

)

(

( )

(

)

0 0

1

, k i i i i , 0

i

b x x ϕ λ f x f x ρ ψ x x σ

=

 ₋ ₊ _<

 



∑

 . (12)

Since g x_j

( )

=0, j J x∈

( )

_and µ_j_0, j J x∈

( )

_{, and hypothesis 3), we get}

(

)

( )

1 , 1 j j 0

j J x

b x x ϕ µ g x

∈

 

−  =

 



∑

 . (13)

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

(

)

(

)

( )

1

, k i i, , 0, , i i

i

x x x x f x i K

α λ ξ η ξ

=

< ∀ ∈∂ ∈

∑

. (14)

(

)

( )

(

)

(

)

( )

, _{j j}, , , 0, _j _j , .

j J x

x x x x x x σ g x j J x

β µ ζ η τ ψ ζ

∈

+ ∀ ∈ ∂ ∈

∑

 (15)

From α

(

x x,

)

>0,β

(

x x,

)

>0_and τ0, we obtain

(

)

( )

1 , , 0, ,

k

i i i i

i x x f x i K

λ ξ η ξ

=

< ∀ ∈∂ ∈

∑

. (16)

( ) j j,

(

,

)

0, j j

( )

,

( )

j J x

x x g x j J x

µ ζ η ζ

∈

∀ ∈ ∂ ∈

∑

 . (17)

Also

( )

(

)

( )

1

, , 0, , , ,

k

i i j j i i j j

i j J x

x x f x i K g x j J x

λ ξ µ ζ η ξ ζ

= ∈

+ < ∀ ∈∂ ∈ ∀ ∈∂ ∈

(5)

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

4. Wolfe Type Duality

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 _Rk_whose com-ponents are all ones.

(

)

( )

T

1

1 1

MD Maximize

, ,

subject to 0 ,

m m

j j k j j

j j

k m

i i j j

i j

F u f u g u e

f u g u f u g u

f u g u

µ

µ µ

λ µ

= =

= +

 

=_ + + _

 

∈ ∂ + ∂

∑



1

0, 1, , 0, .

k

i i

i

j

i K

j M

λ λ

µ

=

= ∈

∈

∑

 

Let

(

)

( )

1 1 1

, , k m 0 k m , 0, k 1, 0

i i j j i

i j i

U u λ µ X R₊ R₊ λ f u µ g u λ λ µ

= = =

 

 

=_ ∈ × × ∈ ∂ + ∂ ≥ = _

 



∑

 

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

Theorem 4.1. (weak duality) Let ∀ ∈x X0 and ∀

(

u, ,λ µ

)

∈W be feasible solutions for (MP) and (MD), respectively. Moreover, assume that

1)

(

f g,

)

is invex of order σ ,

(

Bϕ

)

− −V _{type II at u,}

2) b x u1

( )

, >b x u0

( )

, >0; ;a b< ⇒ϕ0

( )

a <ϕ1

( ) ( )

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

( )

1 , , .

m

i i j j i

j

f x f u µ g u ρ ψ x u σ i K

=

< +

∑

− ∈ ₍₂₀₎

which implies

( )

1

, m , .

i i i j j

j

f x f u ρ ψ x u σ µ g u i K

=

− + <

∑

∈ ₍₂₁₎

Using

1

0, , k 1

i i

i

i K

λ λ

=

∈

∑

=

 , we have

( )

(

)

(

)

( )

1 , 1 .

k m

i i i i j j

i f x f u x x j g u

σ

λ ρ ψ µ

= =

− + <

∑

(22)

By hypothesis 2), we have

( )

(

( )

)

( )

0 0 1 1

1 1

, k i i i i , , m j j .

i j

b x u ϕ λ f x f u ρ ψ x u σ b x u ϕ µ g u

= =

 

 ₋ ₊ _<

 



∑

 

∑

 (23)

(6)

( )

1 1

, , , , , , , ,

, , , .

k m

i i j j

i j

i i j j

x u x u x u x u x u

f u i K g u j M

σ

α λ ξ η α µ ζ η τ ψ

ξ ζ

= =

< − −

∀ ∈∂ ∈ ∀ ∈∂ ∈

∑

(24)

That is

( )

1 1 , , ( , ) , ,

, , , .

k m

i i j j

i j

i i j j

x u x u

x u

f u i K g u j M

σ

τ

λ ξ µ ζ η ψ

α

ξ ζ

= =

+ < −

∀ ∈∂ ∈ ∀ ∈∂ ∈

∑

₍₂₅₎

From τ0, which implies

( )

1 1

, , 0, , , , .

k m

i i j j i i j j

i j x u f u i K g u j M

λ ξ µ ζ η ξ ζ

= =

+ < ∀ ∈∂ ∈ ∀ ∈∂ ∈

∑

(26)

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

( )

,

i f u i Ki

ξ ∈∂ ∈ _and ζ_j∈∂g u j M_j

( )

, ∈ _, such that

1 1 0.

k m

i i j j

i j

λ ξ µ ζ

= =

+ =

∑

(27)

Also,

( )

1 1

, , 0.

k m

i i j j

i j x u

λ ξ µ ζ η

= =

+ =

∑

(28)

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

Theorem 4.2. (weak duality) Let ∀ ∈x X0 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 ,

(

Bϕ

)

− −V _{type II at u,}

2) b x u1

( )

, >b x u0

( )

, >0; 0.a b< ⇒ϕ0

( )

a <ϕ1

( )

b =

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

( )

1 , , .

m

i i j j i

j

f x f u µ g u ρ ψ x u σ i K

=

< +

∑

− ∈ ₍₃₀₎

Also

( )

1

, m , .

i i i j i

j

f x f u ρ ψ x u σ µ g u i K

=

− + <

∑

∈ ₍₃₁₎

Since

1

0, , k 1

i i

i

i K

λ λ

=

∈

∑

=

 , which yields

1 1

( ( ) ( ) ( , ) ) ( ).

k m

i i i i j j

i j

f x f u x x σ g u

λ ρ ψ µ

= =

− + <

∑

(32)

(7)

( )

(

( )

)

( )

0 0 1 1

1 1

, k i i i i , , m j j 0.

i j

b x u ϕ λ f x f u ρ ψ x u σ b x u ϕ µ g u

= =

 

 ₋ ₊ _< ₌

 



∑

 

∑

 (33)

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

( )

1

, k i i, , 0, i i , .

i

x u x u f u i K

α λ ξ η ξ

=

< ∀ ∈∂ ∈

∑

(34)

For α

( )

x u, >0_{, we have}

( )

1 , , 0, , .

k

i i i i

i x u f u i K

λ ξ η ξ

=

< ∀ ∈∂ ∈

∑

(35)

Since

(

u, ,λ µ

)

∈M is a feasible solution for (MD), there exist

( )

,

i f u i Ki

ξ ∈∂ ∈ _and ζ_j∈∂g u j M_j

( )

, ∈ _{such that}

1 1 0.

k m

i i j j

i j

λ ξ µ ζ

= =

+ =

∑

(36)

whence

( )

1 1

, , , , 0.

k m

i i j j

i x u j x u

λ ξ η µ ζ η

= =

+ =

∑

(37)

It follows from (35) that

( )

1

, , 0.

m j j

j x u

µ ζ η

=

>

∑

(38)

For β

( )

x u, >0 and τ0, which yields

( )

1

, m _{j j}, , , 0.

j

x u x u x u σ

β µ ζ η τ ψ

=

+ >

∑

(39)

From hypothesis 1), it follows that

( )

1 1

1

, m j j 0.

j

b x u ϕ µ g u

=

 

− _ _>



∑

 (40)

whence

( )

1 1

1

, m _{j j} 0.

j

b x u ϕ µ g u

=

 

<

 



∑

 (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}_× _→_Rn_{, if there exists a} constant _ρ int_Rk

+

∈ _{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

1

0, k _i 1

i

λ λ

=

≥

∑

= and

0

µ_ _{, such that}

( )

1 1

0 k i i m j j

i f x j g x

λ µ

= =

∈

∑

∂ +

∑

∂ _and

( )

1 0

m j j j g x

µ

=

(8)

Fur-DOI: 10.4236/apm.2018.88045 762 Advances in Pure Mathematics

thermore, if all the hypothesis of Theorem 4.1 are satisfied for all feasible solu-tions 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

( )

(

)

1 , , .

m

i i j j i

j

f x f y µ g y ρ ψ x y σ i K

=

+

∑

− ∈

 (44)

Using

( )

1 0

m j j j g x

µ

=

∑

, which yields

( )

(

)

( )

1 1

+m , m , .

i j j i i j j

j j

f x µ g x ρ ψ x y σ f y µ g 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 .

5. Conclusion

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.

Acknowledgements

This work is supported by Scientific Research Program Funded by Shaanxi Pro-vincial Education Department (Program No. 15JK1456); Natural Science Foun-dation of Shaanxi Province of China (Program No. 2017JM1041).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

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