Duality Relations for a Class of a Multiobjective Fractional Programming Problem Involving Support Functions ()
1. Introduction
Consider the following nonlinear programming problem (P) Minimize
subject to
, where
and
are twice differen- tiable functions. The Mangasarian [1] second-order dualof (P) is (DP) Maximize
such that
By introducing two differentiable functions
and
, Mangasarian [1] formulated the following higher-order dual of (P): (DP)1 Maximize
such that
where
denotes the
gradient of
with respect to p and
denotes the
, gradient of
with respect to p.
Further, Egudo [2] studied the following multiobjective fractional program- ming problem: (MFPP) Minimize
subject to
where
and
are differentiable on X. Also, he discussed duality results for Mond-Weir and Schaible type dual programs under generalized convexity.
For the nondifferentiable multiobjective programming problem: (MPP) Mini- mize
subject to
where
and
are differentiable func- tions.
and
are compact convex sets in
and
and
denote the support func- tions of compact convex sets, various researchers have worked. Gulati and Agarwal [3] introduced the higher-order Wolfe-type dual model of (MPP) and proved duality theorems under higher-order
-type I-assump- tions.
In last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems. In Chen [4] , multiobjective fractional problem and its duality theorems have been considered under higher- order
-convexity. Later on, Suneja et al. [5] discussed higher-order Mond-Weir and Schaible type nondifferentiable dual programs and their duality theorems under higher-order
-type I-assumptions. Several researchers have also worked in this directions such as ( [6] [7] ).
In this paper, we first introduce the definition of higher-order
- invex with respect to differentiable function
. We also construct a nontrivial numerical example which illustrates the existence of such a function. We then formulate three higher-order dual problems corresponding to the multiobjective nondifferentiable fractional programming problem. Further, we establish usual duality relations for these primal-dual pairs under aforesaid assumptions.
2. Preliminaries
Let
be an open set and
be differentiable functions.
,
,
and
.
Definition 2.1.
is said to be (strictly) higher-order
-invex at u with respect to
, if there exist
and
such that, for any
and
,
Example 2.1. Let
be such that
.
Let
.
Also, suppose
.
Now,
(at
).
.
Hence,
is higher-order
-invex at
with respect to
.
Remark 2.1.
1) If
, then the Definition 2.1 reduces to
-invex function introduced by Kuk et al. [8] .
2) If
and
, then the Definition 2.1 becomes that of V-invexity introduced by Jeyakumar and Mond [9] .
3) If
and
, then above definition yields in η-bonvexity given by Pandey [10] .
4) If
, then the Definition 2.1 reduced in
-invex given by Gulati and Geeta [11] .
A differentiable function
is
-invex if for all
,
is
-invex.
Definition 2.2. [12] . Let C be a compact convex set in
. The support function of C is defined by
3. Problem Formulation
Consider the multiobjective programming problem with support function given as: (MFP) Minimize
subject to
where
,
and
are differentiable on X,
and
. Let
be differentiable functions,
and
are compact convex sets in
, for all
.
Definition 3.1. [3] . A point
is said to be an efficient solution (or Pareto optimal) of (MFP), if there exists no
such that for every
,
and for some
,
.
We now state theorems 3.1-3.2, whose proof follows on the lines [13] .
Theorem 3.1. For some t, if
and
are higher- order
-invex at u with respect to
for same
. Then, the fractional function
is higher-order
-invex at u with respect to
, where
,
,
,
and
.
Theorem 3.2. In Theorem 3.1,if either
is strictly higher- order
-invex at u with respect to
and
or
is strictly higher-order
- invex at u with respect to
, then
is strictly higher- order
-invex at
with respect to
.
Theorem 3.3 (Necessary Condition) [14] . Assume that
is an efficient solution of (MFP) and the Slater’s constraint qualification is satisfied on X. Then there exist
and
, such that
(1)
(2)
(3)
(4)
(5)
(6)
Theorem 3.4. (Sufficient Condition). Let u be a feasible solution of (MFP). Then, there exist
and
, such that
(7)
(8)
(9)
(10)
(11)
(12)
Let, for
,
1)
and
be higher-order
- invex at u with respect to
,
2)
be higher-order
-invex at u with res- pect to
,
3)
4)
,
and
,
5)
where
,
,
and
.
Then, u is an efficient solution of (MFP).
Proof. Suppose u is not an efficient solution of (MFP). Then there exists
such that
and
which implies
(13)
and
(14)
Since
, inequalities (13) and (14) gives
(15)
From Theorem 3.1, for each
,
is higher-order
-invex at
with respect to
, we have
(16)
where
,
,
,
and
.
By hypothesis 2), we get
(17)
Adding the two inequalities after multiplying (16) by
and (17) by
, we obtain
(18)
Using hypothesis 3)-4), we get
(19)
Further, using (7)-(8), therefore
(20)
Since x is feasible solution for (MFP), it follows that
This contradicts (15). Therefore, u is an efficient solution of (MFP).
4. Duality Model-I
Consider the following dual (MFD)1 of (MFP): (MFD)1 Maximize
subject to
(21)
Let
be feasible solution for (MFD)1.
Theorem 4.1. (Weak duality theorem). Let
and
. Suppose that
1) for any
,
and
are higher- order
-invex at u with respect to
,
2) for any
,
is higher-order
-invex at u with respect to
,
3)
4)
where
,
,
,
and
.
Then, the following cannot hold
(22)
and
(23)
Proof. Suppose that (22) and (23) hold, then using
,
,
,
,
, we have
(24)
From hypothesis 1) and Theorem 3.1, for
,
is higher-order
-invex at u with respect to
, we get
(25)
For any
,
is higher-order
-invex at u with respect to
, we have
(26)
Adding the two inequalities after multiplying (25) by
and (26) by
, we obtain
(27)
Using hypothesis 3) and (21), we get
(28)
Finally, using hypothesis 4) and x is feasible solution for (MFP), it follows that
This contradicts Equation (24). Hence, the result.
Theorem 4.2. (Strong duality theorem). If
is an efficient solution of (MFP) and the Slater’s constraint qualification holds. Also, if for any
,
(29)
then there exist
and
, such that
is a feasible solution of (MFD)1 and the objective function values of (MFP) and (MFD)1 are equal. Furthermore, if the hypotheses of Theorem 4.1 hold for all feasible solutions of (MFP) and (MFD)1 then,
is an efficient solution of (MFD)1.
Proof. Since
is an efficient solution of (MFP) and the Slater’s constraint qualification holds, then by Theorem 3.3, there exist
and
, such that
(30)
(31)
(32)
(33)
(34)
Thus,
is feasible for (MFD)1 and the objective func- tion values of (MFP) and (MFD)1 are equal.
We now show that
is an efficient solution of (MFD)1. If not, then there exists
of (MFD)1 such that
and
By equation (31), we obtain
and
This contradicts the Theorem 4.1. This complete the result.
Theorem 4.3. (Strict converse duality theorem). Let
and
. Let
1)
2) for any
,
be strictly higher-order
-invex at
with respect to
and
be higher-order
-invex at
with respect to
,
3) for any
,
be higher-order
-invex at
with respect to
,
4)
5)
Then,
.
Proof. Using hypothesis 2) and Theorem 3.2, we have
(35)
For any
,
is higher-order
-invex at u with respect to
, we have
(36)
Adding the two inequalities after multiplying (35) by
and (36) by
, we obtain
(37)
Using hypothesis 3) and (21), we get
(38)
Finally, using hypothesis 4) and
is feasible solution for (MFP), it follows that
This contradicts the hypothesis 1). Hence, the result.
5. Duality Model-II
Consider the following dual (MFD)2 of (MFP): (MFD)2 Maximize
subject to
(39)
(40)
(41)
(42)
Let
be the feasible solution for (MFD)2.
Theorem 5.1. (Weak duality theorem). Let
and
. Let for
,
1)
be higher-order
-invex at u with res- pect to
,
2)
be higher-order
-invex at u with res- pect to
,
3)
4)
Then the following cannot hold
(43)
and
(44)
Proof. The proof follows on the lines of Theorem 4.1.
Theorem 5.2 (Strong duality theorem). If
is an efficient solution of (MFP) and the Slater’s constraint qualification hold. Also, if for any
,
(45)
then there exist
and
, such that
is a feasible solution of (MFD)2 and the objective function values of (MFP) and (MFD)2 are equal. Furthermore, if the conditions of Theorem 5.1 hold for all feasible solu- tions of (MFP) and (MFD)2 then,
is an efficient solution of (MFD)2.
Proof. The proof follows on the lines of Theorem 4.2.
Theorem 5.3. (Strict converse duality theorem). Let
and
. Let
,
1)
2)
be strictly higher-order
-invex at
with respect to
,
3)
be higher-order
-invex at
with respect to
,
4)
5)
Then,
.
Proof. The proof follows on the lines of Theorem 4.3.
6. Duality Model-III
Consider the following dual (MFD)3 of (MFP): (MFD)3 Maximize
subject to
(46)
(47)
(48)
(49)
Let
be feasible solution of (MFD)3.
Theorem 6.1. (Weak duality theorem). Let
and
. Let
,
1)
and
be higher-order
-invex at u with respect to
,
2)
be higher-order
-invex at u with res- pect to
,
3)
4)
where
,
,
,
and
.
Then, the following cannot hold
(50)
and
(51)
Proof. The proof follows on the lines of Theorem 4.1.
Theorem 6.2. (Strong duality theorem). If
is an efficient solution of (MFP) and let the Slater’s constraint qualification be satisfied. Also, if for any
,
(52)
then there exist
and
, such that
is a feasible solution of (MFD)3 and the objective function values of (MFP) and (MFD)3 are equal. Furthermore, if the conditions of Theorem 6.1 hold for all feasible solutions of (MFP) and (MFD)3 then,
is an efficient solution of (MFD)3.
Proof. The proof follows on the lines of Theorem 4.2.
Theorem 6.3. (Strict converse duality theorem). Let
and
be feasible for (MFD)3. Suppose that:
1)
2) for any
,
be strictly higher-order
-invex at
with respect to
and
be higher-order
-invex at
with respect to
,
3) for any
,
is higher-order
-invex at
with respect to
,
4)
5)
Then,
.
Proof. The proof follows on the lines of Theorem 4.3.
7. Conclusion
In this paper, we consider a class of non differentiable multiobjective fractional programming (MFP) with higher-order terms in which each numerator and denominator of the objective function contains the support function of a compact convex set. Furthermore, various duality models for higher-order have been formulated for (MFP) and appropriate duality relations have been obtained under higher-order
-invexity assumptions.
Acknowledgements
The second author is grateful to the Ministry of Human Resource and Development, India for financial support, to carry this work.
NOTES
*Corresponding author.