Applied Mathematics
Vol.06 No.09(2015), Article ID:59023,7 pages
10.4236/am.2015.69145
Mixed Saddle Point and Its Equivalence with an Efficient Solution under Generalized (V, r)-Invexity
Arvind Kumar1, Pankaj Kumar Garg2
1Department of Mathematics, University of Delhi, Delhi, India
2Department of Mathematics, Rajdhani College, University of Delhi, Delhi, India
Email: arvind.ch83@gmail.com, akumar1@maths.du.ac.in, gargpk08@gmail.com
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 16 July 2015; accepted 21 August 2015; published 24 August 2015
ABSTRACT
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, r)-invexity assumptions.
Keywords:
Nonsmooth Multiobjective Programs, (V, r)-Invexity, Mixed Saddle Point, Vector-Valued Mixed Lagrangian Function
1. Introduction
Jeyakumar and Mond [1] have introduced the notion of V-invexity for vector function and discussed its application to a class of multiobjective problems. Mishra and Mukherjee [2] and Liu [3] extended the concept of V-invexity of multiobjective programming to the case of nonsmooth multiobjective programming problems and duality results are also obtained. Jeyakumar [4] introduced r-invexity for differentiable scalar-valued functions. Also, Jeyakumar [5] defined r-invexity for nonsmooth scalar-valued functions, studied duality theorems for nonsmooth optimization problems, and gave relationship between saddle points and optima. In [6] (Bector), a sufficient optimality theorem is proved for a certain minmax programming problem under the assumptions (B, h)-invexity conditions.
Kuk, Lee and Kim [7] discussed that weak vector saddle-point theorems are obtained under V-r-invexity for vector-valued functions. Bhatia and Garg [8] defined (V, r)-invexity, (V, r)-quasiinvexity and (V, r)-pseudo- invexity for nonsmooth vector-valued Lipschitz functions using Clarke’s generalized subgradients and established duality results for multiobjective programming problems. Bhatia [9] introduced higher order strong convexity for Lipschitz functions. The notion of vector-valued partial Lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided. In [10] -[13] , saddle point theory in terms of Lagrangian functions was introduced. In [14] (Reddy and Mukherjee), some problems consisting of nonsmooth composite multiobjective programs have been treated with (V, r)-invexity type conditions and also vector saddle point theorems were obtained for composite programs. Yuan, Liu and Lai [15] defined new vector generalized convexity.
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.
2. Preliminaries
In this section we require some definitions and results.
Let
be the n-dimensional Euclidean space and
be its nonnegative orthant. Throughout this paper, the following conventions for vectors in
will be used:
a)
if and only if
,
b)
if and only if
,
c)
is the negation of
.
The following non-smooth multiobjective programming problem is studied in this paper:
where
1),
and
,
are locally Lipschitz functions on
.
2) Let
be the set of feasible solution of problem (MOP). Now let
and
,
denotes the cardinality of the index set
and
clearly.
Problem (MOP) can be associated to problem:
Now, we introduce the following definitions:
Definition 1. A vector function, locally Lipschitz at
, is said to be (V, r)-invex at
if there exist functions
, a real number ρ and
such that for all
for
for every
and for
for every
then
is called strictly (V, r)-invex at
.
Definition 2. A vector function, locally Lipschitz at
, is said to be (V, r)-pseudoinvex at
if there exist functions
, a real number ρ and
such that for all
for every
for every
then the function is strictly (V, r)-pseudoinvex at
.
Definition 3. A vector function, locally Lipschitz at
, is said to be (V, r)-quasiinvex at
if there exist functions
, a real number ρ and
such that for all
for every
If
is (V, r)-invex at each
then the function is (V, r)-invex on
. Similar is the definition of other functions. It is evident that every (V, r)-invex function is both (V, r)-pseudoinvex and (V, r)-quasiinvex
with
and
From the definitions it is clear that every strictly (V, r)-pseudoinvex on
is (V, r)-quasiinvex on
.
Definition 4. A feasible point
is said to be efficient solution for MOP if there is no other feasible solution
such that for some
and
for all.
Definition 5. The vector valued mixed Lagrangian function
corresponding to problem (MOP) is defined as
where
Definition 6. A vector
is said to be mixed saddle point of mixed Lagrangian
if
Definition 7. A function
is sublinear if for any
,
1)
2)
for any
and
.
For
Now, we have established our main results, to prove equivalence between mixed saddle point and an efficient solution.
3. Main Results
Theorem 1. Let
satisfy the following conditions
(1)
(2)
(3)
(4)
Further, let
be (V, r)-pseudoinvex at
and
is (V, r)-quasiinvex at
with
Then
is a mixed saddle point of
.
Proof. Since
satisfies (1), we have
(5)
(6)
As, from (6), we obtain
(7)
Hence, there exist
such that
(8)
Now for any
(9)
As, (9) gives
(10)
From (2) and (10) it follows that
(11)
Using the (V, r)-quasiinvexity of
at
, we get
(12)
(12) along with the fact
gives
(13)
From (8) and (13) and using the sublinearity of, we have
(14)
(15)
Now using (V, r)-pseudoinvex of
at
in (15)
(16)
Since, we obtain from (16)
(17)
Again for any
and
we have
(18)
(18) along with (2) implies
(19)
Therefore, from (19)
(20)
Hence
(21)
From (17) and (21) and the fact that, it follows that
is a mixed saddle point of
.
Theorem 2. Let
satisfy the conditions from (1) to (4). If
is (V, r)-
quasiinvex at
and
is strictly (V, r)-pseudoinvex at
with
then
is mixed saddle point.
Proof. Since
satisfies (1), proceeding in the same manner as in the Theorem (1), we have
(22)
where
and
.
Now, for any,
,
which along with (2) gives
(23)
Using strict (V, r)-pseudoinvexity of
at
in (23) we get
(24)
The fact of
and (24) gives
(25)
From the sublinearty of V
(26)
(25) along with (26) gives
(27)
From (V, r)-quasiinvexity of
at
and (27) it follows that
(28)
From (28), proceeding in the same manner as in Theorem (1) we obtain that
is the mixed saddle point of
.
Theorem 3. Let
be an efficient solution for the problem (MOP) and let the functions
be regular at
. Assume that for at least one r, (MOPr) is calm at
. Then there exit
and
such that
satisfies conditions from (1) to (4). Further let
be strictly (V, r)-pseudoinvex at
and
be (V, r)-quasiinvex at
with
then
is a mixed saddle point of
.
Proof. Since
is an efficient solution of (1) and Clarke’s calmness constraint qualification holds. It follows from Fritz John type necessary optimality conditions that
,
such that
(29)
(30)
(31)
Now as
are regular at
, (29) gives
(32)
(30), (31) and (32) imply that conditions (1) to (4) are satisfied. As
satisfies (1) to (4), proceeding in the same manner as in Theorem (1), we obtain (15).
Now, using strict (V, r)-pseudoinvexity of
at
, we get
(33)
Since,
, we obtain from (33)
Again, proceeding in the same manner as in Theorem (1), it is proved that
is a mixed saddle point of
.
In the next theorem no invexity or generalized invexity is used.
Theorem 4. If
is a mixed saddle point of mixed Lagrangian then
is an efficient solution of the problem (MOP).
Proof: Since
is a mixed saddle point of
, we have
and
(34)
From (34), we get
(35)
Taking
in (35), where
is a vector having unity at the
position and zero elsewhere, we get
Moreover,
hence
(36)
Thus, we have
Hence,
is feasible for the problem (MOP). Further, taking
in (35), we get
(37)
But as
and
, from (37), we obtain
(38)
Now contrary to the result, let
be not an efficient solution of the problem (MOP). Then there exist
and an index
, such that
(39)
and
(40)
(39) and (40) along with (38) give
(41)
and
(42)
that is
(43)
(44)
(43) and (44) are contradiction to the fact that
Acknowledgements
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.
Cite this paper
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
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