Vector optimization with cone semilocally preinvex functions

In this paper we introduce cone semilocally preinvex, cone semilocally quasi preinvex and cone semilocally pseudo preinvex functions and study their properties. These functions are further used to establish necessary and sufficient optimality conditions for a vector minimization problem over cones. A Mond-Weir type dual is formulated for the vector optimization problem and various duality theorems are proved.


Introduction
The concept of semilocally convex functions was introduced by Ewing [1] who applied the notion to provide sufficient optimality conditions in variational and control problems.These functions have some important properties such as local minimum of a semilocally convex function defined on a locally star shaped set is a global minimum and non-negative linear combination of semilocally convex functions is also semilocally convex.Kaul and Kaur [3] defined semilocally quasi convex and semilocally pseudo convex functions.Suneja and Gupta [14] defined the (strict) semilocally pseudo convexity at a point with respect to a set.
By using these concepts Kaul and Kaur [4,5] Kaur [7] and Suneja and Gupta [14] obtained optimality conditions and duality results for a class of non-linear programming problems.
Gupta and Vartak [8] defined -semilocally convex and related functions and studied sufficient optimality conditions for a non-linear program involving these functions.Mukherjee and Mishra [9] and Preda [10] discussed optimality results for a multiobjective programming problem using semilocally convex functions.Weir [16] introduced conesemilocally convex functions and studied optimality conditions and duality theorems for vector optimization problems over cones.
Preda and Stancu-Minasian [12] discussed the Fritz-John and Karush-Kuhn-Tucker type optimality conditions for weak vector minima using semilocally preinvex functions.Stancu-Minasian [13] established optimality and duality results for a non-linear fractional programming problem where the functions involved were semilocally preinvex, semilocally quasi preinvex and semilocally pseudo preinvex.Preda [11] studied optimality and duality for a multiobjective fractional programming problem involving semilocally preinvex functions.Suneja et al. [15] introduced -semilocally preinvex, semilocally quasi preinvex and -semilocally pseudo preinvex functions and proved optimality conditions and duality results for a multiobjective non-linear programming problem using the above defined functions.
In this paper we introduce K-semilocally preinvex, K-semilocally naturally quasi preinvex, K-semilocally quasi preinvex and K-semilocally pseudo preinvex functions where K is a closed convex cone with nonempty interior.Their properties and interrelations are established.Necessary and sufficient optimality conditions are obtained for a vector optimization problem over cones by using the above defined functions.A Mond-Weir type dual is associated with the optimization problem and duality results are studied.

Preliminaries and Definitions Let
The following theorem gives a characterization of cone semilocally preinvex functions. ) ) Epi( ) for 0 ( , ) then K-semilocally preinvex functions reduce to semilocally preinvex functions defined by Preda [11].We now give an example of a function which is K-slpi but fails to be slpi.
The function f fails to be slpi at 1 x  because for x = 1, there does not exist any positive number ( , ) then Ksemilocally preinvex functions reduce to Ksemilocally convex functions defined by Weir [16].We now give an example of a K-slpi function which fails to be K-semilocally convex.Example 2. 2 The function f considered in Example 2.1 is K-slpi at 1 x  but it fails to be K-semilocally convex at 1 x  because for x = 1, there does not exist any positive real number Since S is an  -locally star shaped set and , x x S  , therefore there exists a maximum positive number a  ( , xx )  1 such that ( , ) ).As f is K-slpi on S, there exists a positive number ( , ) for 0 ( , ) On using (2.1) we have, Since f is K-slpi with respect to  therefore there exists a positive number ( , ) for 0 ( , ) for 0 ( , ) Hence S f (y) is -locally star shaped.


We now give the definition of -semi differentiable function.
(1) If (2) If m = 1 and ( , ) In the following result we give another property of K-slpi functions.
Proof.Since the function f is K-slpi at x with respect to , therefore corresponding to each xS  there exists a positive number Since K is a closed cone, therefore taking limit as


We now introduce semilocally naturally quasi preinvex functions over cones.
Definition 2.4.The function f is said to be K-semilocally naturally quasi preinvex (Kslnqpi) at , . Proof.Let S f (y) be -locally star shaped for Since S f (y) is -locally star shaped, therefore there exists a maximum positive number Thus, ( ) ( ( , )) , fo r 0 ( , ) Since K is a closed cone, therefore taking limit as x with respect to same  .
Proof.Let f be K-slpi at x , then there exists a positive number ( , ) (2.4) Adding (2.3) and (2.4) we have for 0 ( , ) Since K is a closed cone, therefore taking limit as  .Hence f is K-slnqpi at x with respect to same .


The converse of the above theorem may not hold as can be seen from the following example.
The function f is K-slnqpi at 2 x  because  .The function f fails to be K-slpi at 2 x  by Theorem 2.4, because for ( , )) .
Theorem 2.7.If K is a pointed cone and f is Kslqpi at x then f is K-slnqpi at x with respect to same .
Proof.Let K be a pointed cone and f be K-slqpi In view of (2.5) we get ( ) Thus by (2.6) we have ( ) ( , ( , ))  The converse of the above theorem may not hold, as can be seen by the following example.x  because for whereas ( ) ( , ( , )) (12, 0) .

K-slpi K-slnqpi K-slqpi
It can be seen from Example 2.1 that f fails to be K-slqpi because for We now give an example to show that a K-slqpi function need not be K-slpi.

Figure 5
The next definition introduces cone semilocally pseudo preinvex functions.

Optimality Conditions
We consider the vector optimization problem be the set of all feasible solutions of (VOP).Let 0 : ( , , ) : where is a nonempty set and to be a closed set, then, 00 () becomes convex and the following alternative theorem follows on the lines of Illés and Kassay [2].Theorem 3.1 (Theorem of Alternative).Let 0 F be 0 K -slpi on S such that 00 () is closed with nonempty interior then exactly one of the following holds (i) there exists We shall be using the following constraint qualification to prove the necessary optimality conditions for (VOP). .We now establish the necessary optimality conditions for (VOP).

(Necessary Optimality Conditions
be closed with nonempty interior.Let 0 xX  be a weak minimum of (VOP), f be K-slpi, g be Q-slpi and h be O-slpi with respect to same .Suppose that the pair ( , ) gh satisfies generalized Slater type constraint qualification and ( , ) 0 xx   , then there exist 0 Proof.Since x is a weak minimum of (VOP), therefore there does not exist By Theorem 3.1, there exist 3) and using ( ) 0 hx  , we get (3.4) From (3.2), (3.3) and ( ) 0 hx  , we have which can be rewritten as, Dividing by 0 t  and taking limit as (3.8) Adding (3.7), (3.8) and using (3.4) and h   (3.12)If 0   , then from (3.12), ( ) Since g is Q-slqpi at x , we get Since x  X 0 is arbitrarily chosen, therefore x is a weak minimum of (VOP).

Duality
The following Mond-Weir type dual is associated with the primal problem (VOP).
and   R k , u  S Theorem 4.1 (Weak Duality).Let x be feasible for (VOP) and ( , , , )  u    be feasible for (VOD).Let f be K-slppi, g be Q-slqpi and h be O-slnqpi at u, with respect to same .Then Proof.Since x is feasible for (VOP) and x    is a weak maximum of (VOD).
Proof.Since x is a weak minimum of (VOP), x    be not a weak maximum of (VOD).Then there exists ( , , , ) u    feasible for (VOD) such that f (u)  f ( x ) int K which contradicts Weak Duality Theorem (Theorem 4.1) as x is feasible for (VOP).

2 . 4 .
If m = n and cone K = , n R  Kslnqpi functions reduce to slqpi functions defined by Preda [11].star shaped for each y  R m , then f is K-semilocally naturally quasi preinvex on S with respect to same .

Example 2 . 4 .
The function f considered in Example 2.3 is K-slnqpi at 2 x  .But f fails to be K-slqpi at 2

Definition 3 . 1 .
The constraint pair( , )  gh is said to satisfy generalized Slater type constraint qualification at x if there exists * x  S.(3.6)Since (g, h) is () QO  -slpi at x , therefore we have for every x  S,  int Q and h(x * )  O


f be K-slppi, g be Q-slqpi and h be O-slnqpi at x with respect to same  .If there exist 0 such that (3.1) and (3.2) hold  x  X 0 then x is a weak minimum of (VOP).

Theorem 4 . 2 (
Again using feasibility of x and u, we have h is O-slnqpi at u, Because f is K-slppi at u, we get Strong Duality).Let f be K-slpi, g be Q-slpi and h be O-slpi with respect to same.Let F 1 (S) + (K  Q  O) beclosed with nonempty interior.Suppose that the pair (g, h) satisfies generalized Slater type constraint qualification.If x is a weak minimum of (solution of (VOD).Moreover if the conditions of Weak Duality Theorem 4.1 are satisfied for all feasible solutions of (VOP) and (VOD) then * * * ( , , , ) The set S is said to be -locally star shaped at xS  if for each xS  , [1]hen -locally starshaped set reduces to locally star shaped set[1].Let n SR  be an -locally star shaped at xS  ,