Efficiency and Duality in Nondifferentiable Multiobjective Programming Involving Directional Derivative

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doi:10.4236/am.2011.24057 Published Online April 2011 (http://www.SciRP.org/journal/am)

Efficiency and Duality in Nondifferentiable Multiobjective

Programming Involving Directional Derivative

Izhar Ahmad

Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Saudi Arabia, India

E-mail: [email protected]; [email protected]

Received December 24, 2010; revised February 24, 2011; accepted February 26, 2011

Abstract

In this paper, we introduce a new class of generalized dI-univexity in which each component of the objective

and constraint functions is directionally differentiable in its own direction di for a nondifferentiable

multiob-jective programming problem. Based upon these generalized functions, sufficient optimality conditions are

established for a feasible point to be efficient and properly efficient under the generalised dI-univexity

re-quirements. Moreover, weak, strong and strict converse duality theorems are also derived for Mond-Weir type dual programs.

Keywords:Multiobjective Programming, Nondifferentiable Programming, Generalized dI-Univexity,

Sufficiency, Duality

1. Introduction

The field of multiobjective programming, also known as vector programming, has grown remarkably in different directions in the setting of optimality conditions and dual-ity theory. It has been enriched by the applications of var-ious types of generalizations of convexity theory, with and without differentiability assumptions, and in the frame-work of continuous time programming, fractional programming, inverse vector optimization, saddle point theory, symmetric duality and vector variational inequalities etc.

Hanson [1] introduced a class of functions by genera-lizing the difference vector xx in the definition of a convex function to any vector function 



x x,



. These functions were named invex by Craven [2] and -convex by Kaul and Kaur [3]. Hanson and Mond [4] defined two new classes of functions called Type I and Type II functions, which were further generalized to pseudo Type I and quasi Type I functions by Rueda and Hanson [5]. Zhao [6] established optimality conditions and dual-ity in nonsmooth scalar programming problems assum-ing Clarke [7] generalized subgradients under Type I functions.

Kaul et al. [8] extended the concept of type I and its generalizations for a multiobjective programming prob-lem. They investigated optimality conditions and derived Wolfe type and Mond-Weir type duality results. Suneja

and Srivastava [9] introduced generalized d-type I func-tions in terms of directional derivative for a multiobjec-tive programming problem and discussed Wolfe type and Mond-Weir type duality results. In [10], Kuk and Tanino derived optimality conditions and duality theorems for non-smooth multiobjective programming problems in-volving generalized Type I vector valued functions. Gu-lati and Agarwal [11] discussed sufficiency and duality results for nonsmooth multiobjective problems under

( , , ,F   d-type I functions. Agarwal et al. [12] estab-lished sufficient conditions and duality theorems for nonsmooth multiobjective problems under V-type I func-tions. Recently, Jayswal et al. [13] obtained some opti-mality conditions and duality results for nonsmooth mul-tiobjective problems involving generalized



F, , ,  



d V

  -univexity.

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corrrected the Karush-Kuhn-Tucker necessary conditions in [14] and discussed the sufficiency and duality under d r type I functions. Recently, Silmani and Radjef [20] introduced generalzed dI-invexity in which each

component of the objective and constraint functions is directionally differentiable in its own direction and es-tablished the necessary and sufficient conditions for effi-cient and properly effieffi-cient solutions. The duality results for a Mond-Weir type dual are also derived in [20]. They also observed that the Karush-Kuhn-Tucker sufficient conditions discussed in [16-18] are not applicable. More recently, Agarwal et al. [21] introduced a new class of generalized d   



,



 type I for a non-smooth multiobjective programming problem and discussed op-timality conditions and duality results.

In this paper, we introduce dI V -univexity and

ge-neralized dIV-univexity in which each component of

the objective and constraint functions of a multiobjective programming problem is semidirectionally differentiable in its own direction di. Various Karush-Kuhn-Tucker sufficient optimality conditions for efficient and properly efficient solutions to the problem are established involving new classes of semidirectionally differentiable generalized type I functions. Moreover, usual duality theorems are discussed for a Mond-Weir type dual involving afo-resaid assumptions. The results in this paper extend many earlier work appeared in the literature [9,10,12,14-16, 19].

2. Preliminaries and Definitions

The following conventions for equalities and inequalities will be used. If =



1, ,



, =



1, ,



n

n n

x  x y  y 

x y , then

= x_i = y i_i, = 1,,n

x y ; x< yx_i< y i_i, = 1,,n;

, = 1, , ;

i i

x y i n

   

x y x  y x yandx y ,

We also note q (resp. q or q) the set of vectors q



y with y0 (resp. y0 or y> 0).

Definition 1 [22]. Let D be a nonempty subset of n,

:D D n

   and let x0 be an arbitrary point of D.The set D is said to be invex at x0 with respect to , if for each xD,





 

0 , 0 , 0,1 .

x  x x D 

D is said to be an invex set with respect to , if D is invex at each x0D with respect to the same .

Definition 2 [23]. Let n

D be an invex set with respect to :_{D D}_{ }_n

.A function f D:  is called pre-invex on D with respect to , if for all

0

,

x x D,

  

1

  

0



0



, 0



,

 

0,1 .

f x f x f x x x

      

Definition 3 [14]. Let Dn be an invex set with respect to :D D n . A m-dimensional vector

valued function :Dm is pre-invex with respect to , if each of its components is pre-invex on D with respect to the same function .

Definition 4 [7]. Let D be a nonempty open set in n. A function f D:  is said to be locally Lipschitz at

0

x D, if there exist a neighborhood 

 

x0 of x0 and a constant K> 0 such that

 

, ,

 

0 , f y f x K yx x y x

where . denotes the Euclidean norm. We say that f is locally Lipschitz on D if its locally Lipschitz at any point of D.

Definition 5 [7]. If : n

f D R is locally Lip-schitz at x0D, the Clarke generalized directional de-rivative of f at x0 in the direction

n

d , denoted by









 

0 0

; = limsup .

y x t

f y td f y f x d

t

 

 

 

 

 

And the usual one-sided directional derivative of f at 0

x in the direction d is defined by







0



 

0

; = lim f x d f x , f x d



 

 

 



whenever this limit exists. Obviously,









0

0; 0; f x d  f x d .

We say that f is directionally differentiable at x0 if its directional derivative f



x d0;



exists finite for all

n

d .

Definition 6 [15]. Let _{f D}: __N

be a function de-fined on a nonempty open set _D__n

and directional-ly differentiable at x0D. f is called d-invex at x0 on D with respect to , if there exists a vector function

:_{D D} n,

   such that for any xD,

 

   

0

 

0



0



0



,

, ; , ,

for all = 1, , ,

i i i

f y f x K y x

x y x f x f x f x x x

i N

 

 



  





where f xi



0;



x x, 0



denotes the directional derivative of fi at x0 in the direction

















 

0 0 0

0

, : ; ,

,

=lim .

i

i i

x x f x x x

f x x x f x



 

 

 



 

If Inequalities (1) are satisfied at any point x0D, then f is said to be d-invex on D with respect to . Definition 7 [20]. Let D be a nonempty set in n and :D D n a function.

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differentiable at x0D,if there exist a nonempty subset Sn such that f



x d0;



exists finite for all dS

 We say that f is a semi-directionally differentiable at x0D in the direction 



x x, 0



, if its direc-tional derivative f



x0;



x x, 0



exists finite for all xD.

Definition 8 [20]. Let f D: N be a function de-fined on a nonempty open set Dn and for all

= 1, , , i

i  N f is semi-directionally differentiable at 0

x D in the direction _i :D D n. f is called dI-invex at x0 on D with respect to

 

i i=1,N, if for any xD,

 

0



0;



, 0



, for all , 2, , ,

i i i i

f x  f x f x  x x i  N

where f xi



₀;i



x x, ₀



denotes the directional deriva-tive of fi at x0 in the direction

















 

0 0 0

0

, : ; ,

,

. lim

i i i

x x f x x x

f x x x f x



 

 

 



 



If Inequalities (2) are satisfied at any point x0D, then f is said to be dI-invex on D with respect to

 

_{i i}_=1,_N_,

Consider the following multiobjective programming problem



MP



Minimizef x

 

=



f1

   

x ,f2 x ,,fN

 

x



 

Subject to g x 0, ,

xD

where f D: N, g D: k, D is a nonempty open subset of n. Let X =



xD g x:

 

0



be the set of feasible solutions of (MP). For x0D, we denote by

 

0

J x the set



j



1, 2,,k



:gj

 

x0 = 0 ,



J = J x

 

0 and by J x

 

0



resp.J x

 

0



the set





 



j 1, 2,,k :gj x0 0 (resp. gj

 

x0 > 0





. we have J x

 

0 J x

 

0 J x

  

0  1, 2,,k



and if

 

0 , =0

x X J x .

We recall some optimality concepts, the most often studied in the literature, for the problem (MP).

Definition 9. A point x0X is said to be a local weakly efficient solution of the problem (MP), if there exists a neighborhood N x

 

₀ around x0 such that

 

0 for all

 

0

f x f x xN x X

Definition 10. A Point x0X is said to be a weakly efficient (an efficient) solution of the problem (MP), if there exists no xX such that

 

<

   

0



 

0



.

f x f x f x  f x

Definition 11. An efficient solution x0X of (MP) is said to be properly efficient, if there exists a positive real number M such that inequality

 

0

 

0

i i j j

f x f x M_f x f x _

is verified for all i



1,,N



and xX such that

 

<

 

0 i i

f x f x , and for a certain j



1,,N



such that fj

 

x > fj

 

x0 .

Following Jeyakumar and Mond [24], Kaul et al. [8] and Slimani and Radjef [20], we give the following defi-nitions.

Definition 12.



f g,



is dIV -univex type I at

0

x D if there exist positive real valued functions and

i j

  defined on XD, nonnegative functions 0 and 1

b b , also defined on XD,0:RR,1:R

; : n, and : n

i j

R X D R  X D R

such that





 









0 , 0 0 i i 0 i , 0 i 0; i , 0 b x x  _f x f x _ x x f x  x x

(3) and





 









1 1, 0 1 j 0 j , 0 j 0; j , 0

b x x  g x   x x g x  x x

 _ _ (4)

for every xX and for all i= 1, 2,,N and = 1, 2, ,

j  k.

If the inequality in (3) is strict (whenever xx0), we say that (MP) is of semistrictly dI V -univex type I

at x0 with respect to

 

i i=1,N and

 

j j=1,k.

Definition 13.



f g,



is quasi-dIV-univex type I

at x0D if there exist positive real valued functions

 and j, defined on XD, nonnegative functions

0 and 1

b b , also defined on XD, :0 RR, 1:R R

  and



Nk



dimensional vector functions

: n, = 1,

i X D R i N

   and j:X D Rn, = 1,j k such that for some vectors R_N and R_k:







  



 



0 0 0 0 0

=1

0 0

=1

, ,

0 ( ; ( , )) 0

N

i i i i i

N

i i i i

b x x x x f x f x

f x x x x X

  

 

 _ 

 

 



  



 

(5)

and







  









1 0 1 0 0

=1

0 0 =1

, , 0

; , 0 .

k

j j j

j k

j j j

j

b x x x x g x

g x x x x X

  

 

 

 

 



  





(6)

If the second inequality in (5) is strict



xx0



, we say that (MP) is of semi-strictly quasi dI-V-univex type

I at X with respect to

 

_=1,

 

=1,

and

i i N j j k

(4)

Definition 14.



f g,



is pseudo-dIV-univex type

I at x0D if there exist positive real valued functions

i

 and j, defined on XD, nonnegative functions

0

b and b1, also defined on XD , 0:RR , 1:R R

  and



Nk



dimensions vector functions

: n, = 1,

i X D R i N

   and _j:X D Rn, = 1,j k

such that for some vectors R_N and R_k:















  

 

0 0

=1

0 0 0 0 0

=1

; , 0

, , ( ) 0

N

i i i i

N

i i i i i

f x x x

b x x x x f x f x x X

 

  

 

 _  _{ }

 

 







(7) and















  

0 0 =1

1 0 1 0 0

=1

; , 0

, , 0 .

k

j j j

j

k

j j j

j

g x x x

b x x x x g x x X

 

  

 

 

 

 

 







(8)

Definition 15.



f g,



is quasi pseudo-dI V -univex

type I at x0D if there exist positive real valued func-tions i and j, defined on XD, nonnegative fun-

ctions b0 and b1, also defined on XD,0:RR, 1:R R

  and (Nk) dimensions vector functions

: n,

i X D R

   i= 1,N and : n,

j X D R

   j= 1,k

such that the relation (5) and (8) are satisfied. If the second inequality in (8) is strict (xx0, we say that

(VP) is of quasi strictly-pseudo d -I Vtype I at x0 with respect to

 

=1, and =1, .

i _i _N j _j _k

 

Definition 16.



f g,



is pseudoquasi -dI-V-univex type I at x0D if there exist positive real valued func-tions i and j , defined on XD , nonnegative

functions b0 and b1 , also defined on XD , 0:R R, 1:R R

    and (Nk) dimensions vector functions : n, = 1,

i X D R i N

   and

: n, = 1,

j X D R j k

   , such that R_k the rela-tions (7) and (6) are satisfied. If the second inequality in (7) is strict



xx0



, we say that



MP



is of strictly- pseudo quasi dI Vtype I at x0 with respect to

 

i i=1,N and

 

j j=1,k.

3. Optimality Conditions

In this section, we discuss some sufficient conditions for a point to be an efficient or properly efficient for (MP) under generalized dI V univex type I assumptions.

Theorem 3.1. Let x0 be a feasible solution for (MP) and suppose that there exist



NJ



vector functions

: n, = 1, ,

i X X R i N

   _j:XX Rn,jJ x

 

₀

and scalars i 0, = 1, , N₌₁ _i = 1;

i

i N

 



 0, _j

 

0

jJ x such that







0 0





0



0



=1 (₀)

; , ; , 0,

,

N

i _i _i _j _j _j

i j J x

f x x x g x x x

x X

   



  

 





(9) Further, assume that one of the following conditions is satisfied:

a) i)



f g,



is quasi strictly-pseudo dI V -univex

type I at x0 with respect to

 

=1,

 

  0

, , ,

i i N j j J x

  _  

and for some positive functions _i, = 1,i N, j,

 

0

jJ x ,

ii) for any uR, u00

 

u 0; 1

 

u < 0 < 0;

u

 b₀



x x, ₀



> 0, b x x₁



, ₀



> 0;

b) i)



f g,



is strictly-pseudo dI V -univex type I at

0

x with respect to

 

_{i i}_=1,_N,

 

_{ } 0

, ,

j _{j J x}

 _   and for some positive functions _i, = 1,i N, ,_j

 

0 ,

jJ x

ii) for any uR, 0( ) > 0u u> 0; 1

0 ( ) 0,

u  u  b x x₀( , ₀) > 0, b x x₁( , ₀)0.

Then x0 is an efficient solution for



MP



.

Proof: Condition a). Suppose that x0 is not an effi-cient solution of



MP



. Then there exists an xX

such that

 

0 ,

f x  f x

which implies that



0

  

 

0

=1

, 0.

N

i _i _i _i i

x x f x f x

  _  _



 (10)

Since b0



x x, 0



> 0; u00

 

u 0, the above inequality gives







  

 

0 0 0 0 0

=1

, , 0.

N

i _i _i _i

i

b x x  _   x x _f x  f x _







From the above inequality and Hypothesis i) of a), we have







0 0



=1

; , 0.

N

i _i _i i

f x x x

  





By using the Inequality (9) we deduce that







0 0



(₀)

; , 0,

j j j j J x

g x x x

 









which implies from the condition part ii) of a) that





 0



  

1 , 0 1 j j , 0 j 0 < 0. j J x

b x x    x x g x



 

 







(5)

 0



0

  

0 , < 0.

j j j j J x

x x g x

 





(11)

As 0 and gj

 

x0 = 0;  j J x

 

0 , it follows that

 

0 = 0, ,

 

0

jg_j x j J x

   which implies that

 0



0

  

0

, = 0.

j j j j J x

x x g x

 





The above equation contradicts Inequality (11) and hence the conclusion of the theorem follows:

Condition b): Since gj

 

x0 = 0, 0, j  j J x

 

0 , and j



x x, 0



> 0, jJ x

 

0 , we obtain

 0



0

  

0

, = 0, .

j j j j J x

x x g x x X

 



 



By Hypothesis ii) of b), we get





 





1 0 1 0

0

, j j , 0.

j J x

b x x    x x



 

 









From the above inequality and the Hypothesis i) of b)( in view of reverse implication in (8), if follows that

 0



0



0



 

0

; , < 0, \ .

j j j

j J x

g x x x x X x

 



  



By using Inequality (9), we deduce that







0 0



 

0

=1

; , > 0, \ ,

N

i i i i

f x x x x X x

    



(12)

which by virtue of relation (7) implies that







  



 



 

0 0 0 0 0

=1

0

, , > 0,

\ .

N

i i i i i

b x x x x f x f x

x X x

 _    _

 

 



The above inequality along with Hypothesis ii) of b) gives



0

  



 

0



 

0

=1

, > 0, \ .

N

i i i i i

x x f x f x x X x

    



(13)

Since (10) and (13) contradicts each other, and hence the conclusion follows:

Theorem3.2. Let x0 be a feasible solution for (MP) and suppose that there exist



NJ



vector functions

 

0

: n, = 1, , : n,

i X X R i N j X X R j J x

      

and scalars 0, = 1, , =1 = 1, 0,

 

0 N

i _i _j

i

i N j J x

 



   

such that Inequality (9) of Theorem 3.1 is satisfied. Moreover, assume that one of the following conditions is satisfied.

a) i)



f g,



is pseudo quasi dI  V univex type I

at x0 with respect to

 

_=1,

 

_{ } 0

, , ,

i i N j j J x

   

 and for

some positive functions

 

0

, = 1, and , ,

i i N j j J x

  

ii) for any uR,

 

1 0

0 0, 0 0,

u  u   u  u









0 , 0 > 0, 1 , 0 0;

b x x b x x 

b) i)



f g,



is strictly pseudo dI  V univex type

I at x0 with respect to

 

_=1,

 

_{ } 0

, , ,

i i N j j J x

  _   and

for positive functions _i = 1,N and j,jJ x

 

0 , ii) for any uR

 









0 1

0 0 1 0

0 0; 0 0;

, > 0, , 0.

u u u u

b x x b x x

 

 

   



Then x0 is an efficient solution for



MP



. Further Suppose that these exist positive real numbers n_i, m_i

such that ni<i



x x, 0



<mi, i= 1,N for all feasible

x

.Then

x

0 is a properly efficient solution for



MP



Proof: Condition a). Suppose that x0 is not an efficient solution of



MP



. Then there exists an xX

such that f x

 

 f x

 

0 which implies that



0

  



 

0



=1

, < 0.

N

i i i i i

x x f x f x

  



(14)

Since gj

 

x0 = 0, 0j  and



, 0



> 0,

 

0 j x x j J x

  

we obtain

 0



0

  

0

, = 0.

j j j j J x

x x g x

 





From the above inequality and Hypothesis ii) of a), we have







  

1 0 1 0 0

(₀)

, _j _j , _j 0.

j J x

b x x    x x g x



 

 







Using Hypothesis i) of a), we deduce that



0





0



0



(₀)

, ; , 0.

j j j j j J x

x x g x x x

  







 (15)

The Inequalities (9) and (14) yield that







0 0



=1 ; , 0,

N

i i i i  f x  x x





which by Hypothesis i) of a), we obtain







  



 



0 0 0 0 0

=1

, , 0,

N

i i i i

i

b x x  _   x x f x  f x _





 (16)

The Inequality (16) and Hypothesis ii) of a) give



0

  



 

0



=1

, 0.

N

i i i i i

x x f x f x

  



 (17)

(6)

that x0 is not an efficient solution of



MP



. The properly efficient solution follows as in Hanson et al. [25]. For the proof of part b), we proceed as in part b) of Theorem 3.1, we get Inequality (17). Thus complete the proof.

4 .

Mond-Weir Type Duality

Consider the following multiobjective dual to problem



MP





MD



Maximize f y

 

=



f1

   

y ,f2 y ,,fN

 

y



subject to

 







 



=1 =1

; , ; , 0,

N k

i i i j j j

i j

f y x y g y x y x X

        





 

0, = 1, 2, , , , N, k

jgj y j k y D R R

     _  _

: , = 1, 2, , ,

: , = 1, 2, , .

n i

n j

X D R i N X D R j k

 

  

 

 

Let Y be the set of feasible solutions of problem



MD



; that is,

 







 



2





 



=1 =1

= , , , , :

; , ; , 0,

i i j j

N k

i i j j j

i j

Y y

f y x y g y x y

   

      





 

0, ; , , ;

: = 1, 2, , ;

N k j j

n i

g y x X y D R R

X D R i N

  



    

   

 





: n, = 1, 2, , .

j X D R j k

    

We denote by P YrD , the projection of set Y on D.

Theorem 4.1. (Weak Duality). Let x and

 



y, , ,  i i=1,N, j j=1,k



be feasible solution for (MP) and (MD) respectively. Moreover, assume that one of the following conditions is satisfied:

a) i)



f g,



is pseudo quasi dI-V-univex type I at

y with respect to > 0, ,

 

_{i i}_=1, ,

N

  

 

=1,

j _j _k

 and

for some positive functions i,j for i= 1, 2,,N

and j= 1, 2,,k, ii) for any uR

 

0 1

0 0; 0 0;

, > 0, , 0

u u u u

b x y b x y

      



b) i)



f g,



is strictly-pseudo quasi dI-V-univex

type I at y with respect to   , ,

 

_{i i}_=1,_N,

 

=1,

j _j _k  and for some positive function i, j for i= 1, 2,,N

and j= 1, 2,,k, ii) for any uR,

 

0 1

1 0

0 > 0; 0 0;

, 0, , > 0;

u u u u

b x y b x y

     



c) i)



f g,



is quasi strictly-pseudo dI V -univex

type I at y with respect to

 

_=1,

 

=1,

, , _i , _j

i N j k

   

and for some positive functions  i, j for =i

1, 2,,N and j= 1, 2,,k, ii) for any uR,

 

0 1

> 0 > 0; > 0 > 0;

, > 0, , > 0.

u u u u

b x y b x y

  

Then f x

 

f y

 

. Proof: Since

 

1 1

0, = 1, 2, , ,

0 0, , > 0

and , > 0, = 1, 2, ,

j j

j

g y j k

u u b x y x y j k



 







  ,

we have

 

   

1 , 1 =1 , 0.

k

j j j j

b x y  _



  x y g y _ By Condition a) (in view of definition 16), it follows that

 



 



=1

, ; , 0.

k

j j j j

j

x y g y x y

  



 (18)

Since



 

_=1,

 



=1, , , , _i , _j

i N j k

y     is a feasible solution for (MD), the first dual constraint with (18) implies that

 





=1

; , 0.

N

i i i i

f y x y

  



 (19)

From (19) and Hypothesis i) of a), we obtain

 

   



 



0 0 =1

, , 0.

N

i i i i

i

b x y  _   x y f x  f y _





 (20)

Condition ii) of a) and Inequality (20) give

   



 



=1

, 0.

N

i i i i i

x y f x f y

  



 (21)

Assume that f x

 

 f y

 

. Since

> 0, = 1, 2, , and > 0

i i N

   , we obtain

   



 



=1

, < 0,

N

i i i i i

x y f x f y

  



(22)

which contradicts (21), Therefore, the conclusion follows: The proof of part b) and c) are very similar to proof of part a), except that: for part b), the Inequality (21) becomes strict

 

> and Inequality (22) becomes non strict

(7)

it follows that the Inequalities (20) and (21) become strict

 

> . Since 0, then the Inequality (22) becomes non strict

 

 . In this cases, the Inequalities (21) and (22) contradicts each other always.

Remark 1: If we omit the assumption > 0 in the condition i) of a) or the word “strictly” in the condition b),we obtain, for this part of theorem, f x

 

f y

 

.

Theorem 4.2. (Weak Duality). Let x and

 



y, , ,  i i=1,N, j j=1,K



be feasible solutions for



MP



and



MD



respectively, Assume that

1)



f g,



is semi-strictly dIV-univex type I at y

with respect to > 0,  ,

 

_{i i}_=1,_N,

 

=1,

j _j _k

 and for some positive functions

 

=1, , =1,

i _i _N j _j _k

 

, 2) for any uR,

 

0 1

> 0 > 0, 0 0,

, > 0, , 0.

u a u u

b x y b x y

    



Then f x

 

 f y

 

.

Proof: Since _jg_j

 

y 0, = 1, 2,j , ,k which implies that

   

=1

, 0.

k

j j j

j

x y g y

 



 

(23) By (23) and Hypothesis i) (with b x y1

 

, j

 

x y, ) in Definition 12 replaced by

 

,

j x y 

it follows that

 





=1

, , 0.

k

j j j

j

g y x y

  



 (24)

The first dual constraint and (24) give

 





=1

, , 0.

N

i i i i

f y x y

  



 (25)

Dividing both sides of

(3)

by _i

 

x y, and taking x y, by Hypothesis i), we get

   

 



 



0 0

1

, > , , ,

,

= 1, 2, , .

i i i i

i

b x y f x f y f y x y x y

i N

 

    



On Multiplying by i and taking

 

1 =

,

i

i x y 





, we get

 



 



0 , 0 > , , ,

= 1, 2, ,

i i i i i i i

b x y f x f y f y x y

i N

   _  _   





Adding with respect to i, and applying (25) and Hy- pothesis ii), we have

   



 



=1

, > 0.

N

i i i i i

x y f x f y

  



(26)

Assume that f x

 

 f y

 

. Since _i > 0 and> 0, we have

   



 



=1

, < 0,

N

i i i i i

x y f x f y

  



which contradicts (26).

Theorem 4.3. (Strong Duality ).Let x0 be a weakly efficient solution for



MP



. Assume that the function g

satisfies the dI-constraint qualification at x0 with respect to

 

=1,

j _j _k

 . Then there exist > and >

N K

R R

  such that



x0, , ,  i



_i_=1,_N,

 

j _j_=1,_kY and objective

functions of



MP



and



MD



have the same values at x0 and



x0, , ,  

 

i _i_=1,_N,

 

j _j_=1,_k



, respectively. If,

further, the weak duality between



MP



and



MD



in theorem holds with the condition a) without > 0 (resp. with the condition b) or c)), then

 



x0, , ,  i i=1,N, j j=1,k



Y is a weakly efficient (resp. an efficient) solutions of



MD



.

Proof. By the Theorem 31 [20], there exists k and  J x 0



 such that







0 0





0



0



=1 ; , =1 ; , 0,

.

N k

i i i j j j

i f x x x j g x x x

x X

      

 





It follows that



x0, , ,  

 

i i=1,N,

 

j j=1,k



Y. Tri-

vially, the objective function values of (MP) and (MD) are equal.

Suppose that



₀

 



=1, =1, , , , i i N, j j k

x     Y is not a weakly efficient solution of



MD



. Then there exists

 



y ,  , , i i=1,N, j j=1,k



Y

    

such that

 

0 <

 

f x f y which violates the weak duality theorem. Hence



x0, , ,  

 

i i=1,N,

 

j j=1,K



Y is in- deed a weakly efficient solution of (MD).

Theorem 4.4. (Strict Converse Duality). Let x0 and

 



y0, , ,  i i=1,N, j j=1,k



be feasible solutions for (MP)

and (MD) respectively, such that

 

0

 

0

=1 =1

= .

N N

i i i i

i i

f x f y

 



(27)

Moreover, assume that



f g,



is strictly pseudo quasi

I

d  V type I at yo with respect to

 

i _i_=1,_N,

 

j _j_=1,_k

and for  and . Then x0=y0.

(8)







  

1 0 0 1 0 0 0 =1

, 0.

k

j j j

j

b x y  _   x y g y _







Using the second part of the hypothesis, we get







0 0 0



=1

; , 0.

k

j j j

j

g y x y

  



 (28)

The Inequality (28) and feasibility of

 



y0, , ,  i i=1,N, j j=1,k



for (MD) give







0 0 0



=1

; , 0,

N

i i i i

f y x y

  





which by the first part of Hypothesis ii), we obtain







  



 



0 0 0 0 0 0 0 0 =1

, , > 0,

.

N

i i i i

i

b x y x y f x f y

x X

 _    _

 

 



The above inequality along with Hypothesis iii) gives



0 0

  



0

 

0



=1

, > 0.

N

i i i i

i

x y f x f y

  



(29)

By Hypothesis i), iii) and i



x y0, 0



> 0, = 1, 2, ,

i  N we have



0 0

  



0

 

0



=1

, = 0.

N

i i i i

i

x y f x f y

  



(30)

Now (29) and (30) contradict each other. Hence the conclusion follows.

5. Conclusion and Future Developments

In this paper, generalized dIV -univex functions have

been introduced. The sufficient optimality conditions are discussed for a point to be an efficient or properly efficient for (MP) under the introduced functions. App- ropriate Mond-Weir type duality relations are established under these assumptions. Sufficiency and duality with generalized dIV-univex functions will be studied for

nonsmooth variational and nonsmooth control problems, which will orient the future research of the author.

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