Nondifferentiable Multiobjective Programming with Equality and Inequality Constraints ()
1. Introduction
A number of researchers have discussed optimality and duality for a class of nondifferentiable problem containing the square root of a positive semi-definite quadratic form. Mond [1] presented Wolfe type duality while Chandra et al. [2] investigated Mond-Weir type duality for this class of problems. Later, Zhang and Mond [3] validated various duality results for the problem under generalized invexity conditions, it is observed that the popularity of this kind of problems seems to originate from the fact that, even through the objective functions, and/or constraint function are non-smooth, a simple and elegant representation for the dual to this type of problems may be obtained. Obviously non-smooth mathematical programming with more general type functions by means of generalized sub differentials. However, the square root of positive semi-definite quadratic form is one of some of a nondifferentiable function for which sub differentials can be explicitly be written.
Multiobjective optimization problems have been applied in various field of science, where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Researchers study multiobjective optimization problems from different viewpoints and, then there exist different goals when setting and solving them. The goal may be finding a representation set of Pareto optimal solutions, and/or qualifying the trade-offs in satisfying the different objectivesand/or finding a single solution that satisfies the preferences of a human decisions making. Motivated with these observations, there has been an increasing interest in studying optimality and duality for nondifferentiable multiobjective programming problems. Duality results for nondifferentiable multiobjective programming problems with square root term appearing in each component of the vector objective derived by Lal et al. [4]. In nondifferentiable multiobjective programming problems, having a support function in each component of the vector objective, further developments for duality results are found in Kim et al. [5] and Yang et al. [6].
In this paper, we obtain optimality conditions for a class of nondifferentiable multiobjective programming problems with equality and inequality involving a square root terms in each component of the objective. For this class of problems, Mond-Weir type dual is formulated and usual duality results are obtained. In the end a special case is generated.
2. Related Pre-Requisites and Expression of the Problem
In [1], the following problem is considered:
Problem (EP): Minimize
subject to 
where 1)
, 
and 
are continuously differentiable.
2) B is an 
symmetric positive semi definite matrix.
The following generalized Schwartz inequality [7] will be needed in the present analysis:
The equality in the above holds if, for 
.
The function
, being convex and everywhere finite, has a subdifferential in the sense of convex analysis. The subdifferential of 
is given by
We also require the Mangasarian-Fromovitz constraint qualification which is described as the following:
Let 
be the set of feasible solution of the problem (EP), that is,
and by 
the set of inequality active constraint indices, that is,
where
. We say the Mangasarian-Fromovitz constraint qualification holds at 
when the equality constraint gradients 
are linearly independent and there exist a vector
such that
The following theorems (Theorem 2.1 and Theorem 2.2) give Fritz John and Karush-Kuhn-Tucker type optimality conditions using the concept of sub differential obtained by Husain and Srivastav [8] using the concept of subdifferential:
Theorem 2.1 (Fritz John Optimality Conditions): If 
is an optimal solution of (EP), then there exist Lagrange multipliers 
such that
If Mangasarian-Fromovitz constrain qualification (MFCQ) holds at
, then the above theorem reduces to the following theorem giving Karush-Kuhn-Tucker optimality conditions:
Theorem 2.2 (Karush-Kuhn-Tucker optimality conditions): If 
is an optimal solution of (EP) and MFCQ holds at
, then there exist 
such that
The following conventions for inequalities will be used in the subsequent analysis: If
, then
Consider the following multiobjective programming problem containing square root of a certain quadratic form in each component of the objective.
(VEP): Minimize
Subject to
(1)
(2)
where f, g and h are the same as in (EP).
Let 
Definition 2.1 A point 
is said to be an efficient solution of (EP) if there exists no 
such that
for some 
and
for
.
The following results relate an efficient solution of (EP) of k-scalar objective programming problems.
Lemma 2.2 (Chankong and Haimes [9]): A point 
is an efficient solution of (EP) if and only if 
is an optimal solution of 
for each
.
: Minimize 
subject to 
We recall the following definitions of generalized invexity which will be used to derive various duality results.
Definitions 2.2: 1) A function 
is said to be quasi-invex with respect to a vector function 
if
2) A function 
is said to be pseudo-invex with respect to a vector function
if
.
3) 
is said to be the strictly pseudoinvex with respect to 
if 
.
Equivalently, if
3. Optimality Conditions
In this section, the optimality conditions for the problem (EP) are obtained.
Theorem 3.1 (Fritz John Type Optimality Conditions): If 
be an efficient solution of (EP), then there exist 
such that
Proof: Since 
is an efficient solution of (EP), by Lemma 2.1
is an optimal solution of 
for each 
and hence in particular of
. Therefore by Theorem 2.1 there exist 
such that
The theorem follows.
Theorem 3.2 (Kuhn-Tucker type necessary optimality conditions): If 
be an optimal solution of (VEP) and let for
, the constraints of 
satisfy MECQ. Then there exist 
and 
such that
Proof: Since 
is an optimal solution of (VEP), by Lemma 3.1, 
is an optimal solution of
for each r. As for some r, the constraint of 
satisfy MFCQ at
, by Theorem 2.2 of their exist 
such that
yielding
From the above relation it is obvious that the theorem follows.
In Theorem 3.2, we assume MFCQ for some
, which implies
. In the following theorem, we assume MFCQ for every 
and obtain
.
Theorem 3.3 (Kuhn-Tucker type optimality conditions): If 
be an efficient of (VEP) and let for each
, the constraints of 
satisfy MECQ at
. Then there exist 
and
such that
Proof: Since 
is an efficient of (VEP) by Lemma 3.1, 
is an optimal solution of
, by Kuhn-Tucker type necessary optimality conditions, for each
, there exist 
such that
Summing over
, we get
where 
where 
Dividing throughout the above relation and setting, by
We obtain,
or
4. Mond-Weir Type Duality
We formulate the following differentiable multiobjective dual nonlinear problem for (VEP):
(M-WED): Maximize 
subject to
(3)
(4)
(5)
(6)
(7)
(8)
In the following, we shall use 
for the set of feasible solutions of (M-WED)
Theorem 4.1 (Weak Duality): Let 
and 
such that with respect to the same 
1) 
is pseudoinvex 2) 
is quasi-invex, and 3) 
is quasi-invex.
Then
(9)
(10)
cannot hold.
Proof: Suppose the contrary that (9) and (10) hold. Since 
the above inequalities (9) and (10) give
These inequalities because of quasi-invexity of 
and 
imply
Combining these, we give
Using the equality constraint of (M-WED), this yields,
This, because of 1), implies
Using 
this yields,
Hence the result follows.
Theorem 4.2 (Strong Duality): Let
satisfy MFCQ and be an efficient solution (VEP). Then there exist 
such that 
is feasible for (M-WED) and the two objective functions are equal. Furthermore, if the weak duality holds for all feasible solution of (VEP) and (M-WED), then 
is an efficient solution of the (M-WED).
Proof: Since
is an efficient solution (VEP) satisfy MFCQ, therefore by Theorem (3.3), there exist 
satisfy
Hence 
satisfies the constraints of (M-WED) and
i.e. the two objective functions have the same value.
Now we claim that 
is an efficient solution of (M-WED). If not, then there exist 
As
, we have
This contradicts Theorem 4.1 Hence 
is an efficient solution.
Theorem 4.3 (Strict-converse duality): Let 
and 
be an efficient solution of (VEP) and (M-WED), such that
(11)
If with respect to the same 
(A1) 
is strictly pseudoinvex(A2) 
is quasi-invex, and
(A3) 
is quasi-invex then 
Proof: Let 
By hypothesis (A1), we have from (11)
(12)
By hypothesis (A2) and (A3) we have
(13)
(14)
Combining (12), (13) and (14), we have
which contradict the equality constraint of (M-WED). Hence 
Theorem 4.4 (Converse duality):
Let 
be an efficient solution of
(M-WED) at which 1) the matrix 
is positive or negative definite and 2) the vectors 
are linearly independent .
If, for all feasible
,
is pseudoinvex, 
is qausi-invex and 
is quasi-invex with respect to the same
, then 
is an efficient solution (EP).
Proof: By Theorem 3.3, there exist 
and 
such that
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
Multiplying (15) by 
and summing over i, we have
(27)
Using the equality constraint, we have
(28)
(29)
From (18) and (19), we have
Using these in (15), we have
which because of the hypothesis 1) gives 
Using 
and the hypothesis 2), we have
(30)
Let
, i = 1, 2, ∙∙∙, k. Then (30) implies 
The relations (17) and (18) implies 
and
. Using 
and 
along with (24) in (16), we get
.
Thus 
a contradiction to (26). Hence 
Consequently 
Using
, 
, in (18) and (19), we have
This implies that
.
If 
in (16), we have
(31)
Hence by Schwartz inequality
(32)
If 
then (24) implies
Consequently (32) yields
If 
then (31) implies 
So we still get
(33)
Thus by (33), we have
implying the two objective functions have the same value.
Now, assume that 
is not an efficient solution of (VEP), then there exists 
such that
for some 
and
Using
, we have
for some 
This contradicts Theorem 4.1. Hence
is an efficient solution of (VEP).
5. A Special Case
If 
and
our problems reduce to the following problems recently studied by Husain and Srivastav [8]:
(EP): Minimize 
subject to
(M-WED): Maximize 
subject to
6. Conclusion
In this research optimality conditions are derived for a nondifferentiable multiobjective programming problem containing a certain square root of a quadratic form in each component of the objective function in the presence of equality and inequality constraints embodying many realistic problems. A Mond-Weir type dual to this problem is formulated and usual duality theorems are proved under appropriate generalized invexity conditions. A special case is also obtained from our duality results. Our results can be revisited in the multiobjective setting of a nondifferentiable control problem.
7. Acknowledgements
The authors acknowledge anonymous referees for their valuable comments which have improved the presentation of this research paper.