Duality for a Class of Multiobjective Semi-inﬁnite Programming

Problems

Jie Zhao

and Xiaofeng Yan

1,2

College of Foreign Trade and Business, Chongqing Normal University, Chongqing,China

{Jie Zhao,Xiaofeng Yan}zhaojie42@126.com,da68da68@163.com

Keywords: Semi-inﬁnite programming, multiobjective optimization, suﬃcient optimality condition, duality,

generalized convexity.

Abstract: In this paper, a class of multiobjective semi-inﬁnite programming problems is considered. Suﬃcient

optimality condition is established for an eﬃcient solution ﬁrstly. Furthermore, we formulate Mond-Weir

type dual for multiobjective semi-inﬁnite programming problems and establish weak, strong and converse

duality theorems relating the problem and the dual problems under G-invex assumptions.

1 INTRODUCTION AND

PRELIMINARIES

Generalized convexity has been playing a vital role

in mathematical programming and optimization

theory. A class of generalized convex functions

called G-invex functions was defined by Antczak

(2007) for scalar differentiable functions. Then, the

definition of a real-valued G-invex function

introduced by Antczak was generalized to the

vectorial case in (2009). They used vector G-

invexity to develop optimality and duality for

differentiable multiobjective programming problems

with both inequality and equality constraints.

A semi-infinite programming problem is an

optimization problem on a feasible set described by

infinite number of inequality constraints. Recently,

semi-infinite optimization became an active field of

research. Many scholars have been interested in

semi-infinite programming problem, especially their

optimality conditions and duality results(see

(Heettich R., 1993; Jeyakumar V., 2008; Lopez

M.2007; Shapiro A., 2009; Kanzi N., 2010) and the

references therein). S.K.Mishra et al. studied the

duality results of this nonsmooth semi-infinite

programming problem.

Motivated by the works of (T. Antczak., 2009),

(T. Antczak., 2009), and (Mishra S.K.), in this

paper, we study a class of multiobjective semi-

infinite optimization problems. We formulate Mond-

Weir type dual for multiobjective semi-infinite

programming problems. Furthermore, by using G-

invex assumption, related duality theorems are

established.

Next, we first introduce some basic concepts

and results which will be used in the sequel. The

following convention for equalities and inequalities

will be used throughout the paper.

We define:

 

12 12

,,, , ,,,

,1,2,,;

,,1.

xx x y yy y

xy x yi n

xy xyxyn

 

  

  

  

  





Throughout the paper, we will use the same

notation for row and column vectors when the inter-

pretation is obvious.

We say that a vector



is negative if



and strictly negative if

0z 

Definition 1.1

A function

R

is said to

be strictly increasing if and only if

 

,, .xy xR

fx fy  

Let





,,, :

fff f RX

be a vector-

valued differentiable function defined on a

nonempty open set



, and





,1,2,,

IXi k 

be the range of

, that is,

the image of

under

314

Zhao, J. and Yan, X.

Duality for a Class of Multiobjective Semi-inﬁnite Programming Problems.

In 3rd International Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 314-317

ISBN: 978-989-758-312-4

Definition 1.2

[2]

Let

fX R

be a vector-

valued differentiable function defined on a

nonempty open set



and

uX

. If there

exists a differentiable vector-valued function

(,, ):

ff f

GG GRR

such that any its

component

:()

GIX

 is a strictly

increasing function on its domain. And we assume

that there exists a vector-valued function

XRX





such that, for any

1, 2, ,ik 

and all



Xx u



(()) (())

fi fi

Gfx Gfu







(()) () , 0()

fi i

Gfu fu xu





 



Then

is said to be a (strictly) vector

G -invex

function at u on

with respect to



. If (1) is

satisfied for each

uX

, then

is vector -

invex function on

with respect to



Remark 1.1 In order to define an analogous

class of (strictly) vector

G -incave functions with

respect to



, the direction of the inequality in the

definition of these functions should be changed to

the opposite one.

Remark 1.2 In the case, When



1, 2, ,

Ga ai k，

, for any

()

aIX

, we

obtain a definition of a vector-valued invex function.

Definition 1.3 A point

X

is said to be an

efficient solution of the problem if there is no

X

such that

 

xfx

In this paper, we consider the following

multiobjective semi-infinite programming problem

(SIMP).

  







min , , ,

.. 0, .

xfxfx fx

st g x j J







where

is an (possibly infinite) index set,





:,1,2,,

XRiI k 

are vector

differentiable functions,

gX RjJ

,are

strictly vector differentiable functions. Let







:0,

DxXgx jJ   be the set of all

feasible solutions for problem (SIMP). Further, We

assume that, if

D is an efficient point, then

there exits





,, , 0, ,

 









for finite



, such that

() ()

ifi i

Gfx fx





















jg j j

Ggx gx













(1)









jg j

Ggx jJ





(2)

0, 0, 0



 

for finitely many

J

(3)

2 OPTIMALITY CONDITION

In this section, we give Karush-Kuhn-Tucker

suffcient optimality condition of efficient solution

for the problem (SIMP).

Theorem 2.1 Suppose that

D

is a feasible

solution of the problem (SIMP), and



are

G -invex with respect to



gX RjJ are strictly

invex with

respect to



, Then

is the efficient solution of the

problem (SIMP).

Proof Contrary to the result of theorem. Suppose

that there exist



such that









xfx

Since



are strictly increasing functions

and (2)-(3), we get



 

() () 0.

gj gj

Ggx Ggx





By assumption

gjJ



， are strictly

-invex,

then



 

() ()

gj gj

Ggx Ggx







() () , 0, .

gj j

Ggx gx xx jJ









From (1) and (3) it follows that







() () , 0, .

gj j

Ggx gx xx jJ

















() () , 0.

fi i

Gfx fx xx









Since



are

G -invex functions, we have



 

() ()

fi fi

Gfx Gfx



 

() () , 0, .

fi i

Gfx fx xx iI









Moreover, for

GiI



， are strictly increasing

functions, then

Duality for a Class of Multiobjective Semi-inﬁnite Programming Problems

315

() ()

xfx

which is a contradiction to the assumption. The

proof is completed.□

3 VECTOR DUALITY

Now, we consider the following Mond-Weir type

dual (SMWD) for the problem (SIMP).

  



max , ,

yfy fy 

.. () ()

ifi i

st G f y f y

















jg j j

Ggy gy











(4)









y0.

jg j









(5)



,0, 1, 1,1,,1 .

kT k

ee R





  

(6)

0, 0

j J and



 

for finitely many

.jJ

(7)

Theorem 3.1 (weak duality). Let

be feasible

for (SIMP) and



,,y





where









be feasible for (SMWD). Let



invex functions with respect to



gX RjJ be strictly

invex

functions with respect to





00,

GjJ

Then

() ()

xfy

Proof We proceed by contradiction. Suppose that

() ()

xfy



For

iI

invex functions with respect



G are strictly increasing functions, we can

obtain that

 

() () , 0, .

fi i

Gfy fy xy iI







From the (4), (6-7), it follows that



() () , 0, .

gj j

Ggy gy xy jJ







By strict

invexity of ,

gjJ , we have



() () 0.

gj gj

Ggy Ggx

Again from (7), we get









jg j

Ggy









which is a contradiction to (5).

□

Theorem 3.2 (strong duality). Let



invex functions with respect to



, ,

gjJ



be strictly

invex functions with respect to



is efficient solution for (SIMP), then



0, 1, 1, 1, ,1

kT k

ee R





    ，，





0, , 0.

 



 

for finitely many





Jx

such that





,,x





is an efficient

solution of (SMWD), and the respective objective

values are equal.

Proof As

is efficient solution for (SIMP) and

the suitable constraint qualification is satisfied, that

is,





,, , 0, , 0

kj j

  





for finite



, such that (1-2) are satisfied.

Since





1, 1, 1, ,1

ee R







,then





,,x





is a feasible solution of (SMWD).

On the other hand by weak theorem, we have

() ()

xfy

for any efficient solution





,,y





. Hence we get

that





,,x





is a feasible solution of (SMWD)

and the respective objective values are equal.

□

Theorem 3.3 (converse duality). Let





,,y





be efficient solution of (SMWD) and



. Assume



G -invex functions

with respect to



, ,

gjJ



be strictly

invex

functions with respect to





00,

GjJ

then

be efficient solution of (SIMP).

Proof Contrary to the result of theorem. Assume





such that

() ()

xfy







,,y





be efficient solution of (SMWD),

then

() ()

ifi i

Gfy fy





















jg j j

Ggy gy













(8)

ICECTT 2018 - 3rd International Conference on Electromechanical Control Technology and Transportation

316









jg j

Ggy









(9)

From

yD

, it follows that





0, .

gy jJ

(10)

Combining (9)-(10) and



, we obtain



jg j

Ggy









(11)

From strict

invexity of ,

gjJ ,

G invexity of

iI

and

0, , 0,



 

J , we have

 

() ()

if i if i

iI iI

Gfx Gfy













 

() , , .

if i i

Gfy fy xyiI











(12)









() ()

jg j jg j

jJ jJ

Ggx Ggy











() () , , .

jg j j

Ggy gy xyjJ











(13)

Adding both side of (12-13), using (8), we get

 

() ()

if i if i

iI iI

Gfx Gfy









 

() () 0.

jg j jg j

jJ jJ

Ggx Ggy





 



(14)

From

D





00,

GjJ

are strictly

increasing functions, we have





() 0.

Ggx (15)

Combining (11), (14-15), and

0, , 0,

iI jJ



  

, it is obvious that

 

() ().

fi fi

Gfx Gfy

Furthermore, from

GiI



are strictly increasing

functions, it follows that

() ()

xfy

which is a contradiction to the suppose.

4 CONCLUSIONS

Suﬃcient optimality condition is established for an

eﬃcient solution of a multiobjective semi-infinite

programming problem called (SIMP). Mond-Weir

type dual for (SIMP) is formulated. And we

establish weak, strong and converse duality

theorems under G-invex assumptions.

REFERENCES

T. Antczak. New optimality conditions and duality results

of G-type in differentiable mathematical

programming[J]. Nonlinear Analysis, 2007, 66: 1617-

1632.

T. Antczak. On G-invex multiobjective programming. Part

I. Optimality[J]. Journal of Global Optimization, 2009,

43:97-109.

T. Antczak. On G-invex multiobjective programming. Part

II. Duality[J]. Journal of Global Optimization, 2009,

43:111-140.

Heettich R., Kortanek K.O. Smi-infinite programming [J].

SIMA, 1993, 35:380-429.

Jeyakumar V. A note on strong duality in convex semi-

infinite optimization[J]. Optimization Letter, 2008,

2(1):15-25.

Lopez M. Semi-infinite programming[J]. European

Journal of Operational Research, 2007, 180:491-518.

Shapiro A. Semi-infinite programming, duality,

discretization and optimality condition[J].

Optimization,2009, 58(2):133-161.

Kanzi N., Nobakhtian S. Optimality conditions for non-

smooth semi-infinnite programming[J]. Optimization,

2010, 59(5):717-727.

Mishra S.K., Jaiswal M., Le Thi H.A. Duality for

nonsmooth semi-infinite programming problems[J].

Optimization Letter, DOI 10.1007/s11590-010-0240-

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317