Sufficiency and Wolfe Type Duality for Nonsmooth Multiobjective Programming Problems ()
1. Introduction
The field of multiobjective programming, also called vector programming, has grown remarkably in different directions in the settings of optimality conditions and duality theory since the 1980s. It has been enriched by the applications of various types of generalizations of convexity theory, with and without differentiability assumptions. The Clarke subdifferential [1] (also called the Clarke generalized gradient) is an important tool to derive optimality conditions for nonsmooth optimization problems. Together with the Clarke’s subdifferential, many generalized convexity or invexity functions were generalized to locally Lipschitz functions. Based upon the generalized functions, several sufficient optimality conditions and duality results were established for the optimization problems. We can see for examples [2] - [8] . In [9] Upadhyay introduced some new generalizations of the concept of
-invexity and established the necessary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems. In [10] the new concepts of
type I were introduced. Sufficient optimality conditions and Mond-Weir duality results were obtained for nonsmooth multiobjective programming problems. Recently, many researchers have been interested in other types of solution concepts. One of them is higher order strict minimizer. In [11] and [12] some sufficient conditions and duality results were obtained for the new concept of strict minimizer of higher order for a multiobjective optimization problem.
In this paper, we consider the nonsmooth multiobjective programming including the locally Lipschitz functions. The new concepts of invex of order
type II functions are introduced. Then, a sufficient optimality condition is obtained for the nondifferentiable multiobjective programming problem under the new functions and the Wolfe type duality results are obtained.
2. Preliminaries and Definitions
Let
be the n-dimensional Euclidean space and let X be a nonempty open subset of
. For
, we denote:
Definition 2.1. [1] The function
is said to be locally Lipschitz at
, if there exist scalars
and
, such that
. (1)
where
is the open ball of radius
about x.
Definition 2.2. [1] The generalized directional derivative of a locally Lipschitz function f at x in the direction d, denoted by
, is as follows:
. (2)
Definition 2.3. [1] The generalized gradient of
at
, denoted by
, is defined as follows:
. (3)
where
is the inner product in
.
Consider the following nonsmooth multiobjective programming problem:
(MP)
where
and
are locally Lipschitz functions and X is a convex set in
.
Let
be the set of feasible solutions of (MP), and
.
Definition 2.4. A point
is a strict minimizer of order
for (MP) with respect to a nonlinear function
, if for a constant
, such that
. (4)
Throughout the paper, we suppose that
;
;
.
Definition 2.5.
is said to be invex of order
type II at
, if there exist
and some vectors
and
such that for all
the following inequalities hold:
(5)
(6)
Definition 2.6.
is said to be (pseudo, quasi) invex of order
type II at
, if there exist
and some vectors
and
such that for all
the following inequalities hold:
(7)
(8)
3. Optimality Condition
In this section, we establish sufficient optimality conditions for a strict minimizer of (MP).
Theorem 3.1. Let
. Suppose that
1) There exist
,
, such that
2)
is invex of order
type II at
,
3)
.
Then
is a strict minimizer of order
for (MP).
Proof: Since
, there exists
,
, such that
. (9)
whence
. (10)
Suppose that
is not a strict minimizer of order
for (MP). Then there exists
and
, such that
. (11)
By
and hypothesis 3), we have
. (12)
Since
and
, and hypothesis 3), we get
. (13)
In view of the hypothesis 1), one finds from (12) and (13) that
. (14)
(15)
From
and
, we obtain
. (16)
. (17)
Also
(18)
which contradicts (10). Hence the result is true.
4. Wolfe Type Duality
In this section, we consider the Wolfe type dual for the primal problem (MP) and establish various duality theorems. Let e be the vector of
whose components are all ones.
Let
be the set of all feasible solutions in problem (MD).
Theorem 4.1. (weak duality) Let
and
be feasible solutions for (MP) and (MD), respectively. Moreover, assume that
1)
is invex of order
type II at u,
2)
.
Then the following can hold:
. (19)
Proof: Suppose contrary to the result that
holds, then we have
(20)
which implies
(21)
Using
, we have
(22)
By hypothesis 2), we have
(23)
with hypothesis 1) and 2), the above inequality yields
(24)
That is
(25)
From
, which implies
(26)
On the other hand, by using the constraint conditions of (MD), there exist
and
,
such that
(27)
Also,
(28)
which contradicts (26). Then the result is true.
Theorem 4.2. (weak duality) Let
and
be feasible solutions for (MP) and (MD), respectively. Moreover, assume that
1)
is (pseudo,quasi) invex of order
type II at u,
2)
Then the following can hold:
. (29)
Proof: Suppose contrary to the result that
holds, then we have
(30)
Also
(31)
Since
, which yields
(32)
It follows from hypothesis 2) that
(33)
In the view of hypothesis 1), one finds from (33) that
(34)
For
, we have
(35)
Since
is a feasible solution for (MD), there exist
and
such that
(36)
whence
(37)
It follows from (35) that
(38)
For
and
, which yields
(39)
From hypothesis 1), it follows that
(40)
whence
(41)
which contradicts (33). Then the result is true.
The following definition is needed in the proof of the strong duality theorem.
Definition 4.1. A point
is called a strict maximizer of order
for (MD) with respect to a nonlinear function
, if there exists a constant
such that
. (42)
Theorem 4.3. (strong duality) Assume that
is a strict minimizer of order
with respect to
for (MP), also there exist
and
, such that
and
. Furthermore, if all the hypothesis of Theorem 4.1 are satisfied for all feasible solutions of (MP) and (MD), then
is a strict maximizer of order
for (MD) with respect to
.
Proof: The hypothesis implies that
is a feasible solution of (MD). By Theorem 4.1, for any feasible
of (MD), we have
(43)
That is
(44)
Using
, which yields
(45)
whence
(46)
Thus
is a strict maximizer of order
for (MD) with respect to
.
5. Conclusion
In this paper, we have defined a class of new generalized functions. By using the new functions, we have presented a sufficient optimality condition and Wolfe type duality results for a nondifferentiable multiobjective problem. The present results can be further generalized for other programming problems.
Acknowledgements
This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 15JK1456); Natural Science Foundation of Shaanxi Province of China (Program No. 2017JM1041).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.