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The purpose of this paper is to define the concept of mixed saddle point for a vector-valued Lagrangian of the non-smooth multiobjective vector-valued constrained optimization problem and establish the equivalence of the mixed saddle point and an efficient solution under generalized (V,
*p*)-invexity assumptions.

Jeyakumar and Mond [

Kuk, Lee and Kim [

In this paper, we define the concept of mixed saddle point for a vector-valued constrained optimization problem and establish the equivalence of the mixed saddle point and an efficient solution under generalized (V, r)- invexity assumptions. Further mixed saddle point theorems are obtained.

In this section we require some definitions and results.

Let

a)

b)

c)

The following non-smooth multiobjective programming problem is studied in this paper:

where

1)

2) Let

clearly

Problem (MOP) can be associated to problem

Now, we introduce the following definitions:

Definition 1. A vector function

for every

for every

Definition 2. A vector function

for every

for every

Definition 3. A vector function

for every

If

with

From the definitions it is clear that every strictly (V, r)-pseudoinvex on

Definition 4. A feasible point

and

for all

Definition 5. The vector valued mixed Lagrangian function

where

Definition 6. A vector

Definition 7. A function

1)

2)

For

Now, we have established our main results, to prove equivalence between mixed saddle point and an efficient solution.

Theorem 1. Let

Further, let

with

Proof. Since

As

Hence, there exist

such that

Now for any

As

From (2) and (10) it follows that

Using the (V, r)-quasiinvexity of

(12) along with the fact

From (8) and (13) and using the sublinearity of

Now using (V, r)-pseudoinvex of

Since

Again for any

(18) along with (2) implies

Therefore, from (19)

Hence

From (17) and (21) and the fact that

Theorem 2. Let

quasiinvex at

Proof. Since

where

Now, for any

Using strict (V, r)-pseudoinvexity of

The fact of

From the sublinearty of V

(25) along with (26) gives

From (V, r)-quasiinvexity of

From (28), proceeding in the same manner as in Theorem (1) we obtain that

Theorem 3. Let _{r}) is calm at

and

Proof. Since

Now as

(30), (31) and (32) imply that conditions (1) to (4) are satisfied. As

Now, using strict (V, r)-pseudoinvexity of

Since,

Again, proceeding in the same manner as in Theorem (1), it is proved that

In the next theorem no invexity or generalized invexity is used.

Theorem 4. If

Proof: Since

From (34), we get

Taking

Moreover,

Thus, we have

Hence,

But as

Now contrary to the result, let

and

(39) and (40) along with (38) give

and

that is

(43) and (44) are contradiction to the fact that

The research work presented in this paper is supported by grants to the first author from “University Grants Commission, New Delhi, India”, Sch. No./JRF/AA/283/2011-12.

ArvindKumar,Pankaj KumarGarg, (2015) Mixed Saddle Point and Its Equivalence with an Efficient Solution under Generalized (V, p)-Invexity. Applied Mathematics,06,1630-1637. doi: 10.4236/am.2015.69145