Optimality Conditions and Second-Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems

Fritz John and Karush-Kuhn-Tucker type optimality conditions for a nondifferentiable multiobjective variational problem are derived. As an application of Karush-Kuhn-Tucker type optimality conditions, Mond-weir type second-order nondifferentiable multiobjective dual variational problems is constructed. Various duality results for the pair of Mond-Weir type second-order dual variational problems are proved under second-order pseudoinvexity and second-order quasi-invexity. A pair of Mond-Weir type dual variational problems with natural boundary values is formulated to derive various duality results. Finally, it is pointed out that our results can be considered as dynamic generalizations of their static counterparts existing in the literature.


Introduction
Second-order duality in mathematical programming has been extensively investigated in the literature.In [1], Chen formulated second order dual for a constrained variational problem and established various duality results under an involved invexity-like assumptions.Subsequently, Husain et al. [2], have presented Mond-Weir type second order duality for the problem of [1], by introducing continuous-time version of second-order invexity and generalized second-order invexity.Husain and Masoodi [3] formulated a Wolfe type dual for a nondifferentiable variational problem and proved usual duality theorems under second-order pseudoinvexity condition while Husain and Srivastav [4] presented a Mond-Weir type dual to the problem of [2] to study duality under second-order pseudo-invexity and second-order quasiinvexity.
The purpose of this research is to present multiobjective version of the nondifferentiable variational problems considered in [2,4] and study various duality in terms of efficient solutions.The relationship between these multiobjective variational problems and their static counterparts is established through problems with natural boundary values.  the first order derivatives of  with respect to    

Definitions and Related
x t and x t  , respectively, that is, where satisfies : , , , , is second-order invex with respect to    and the space of n-dimensional continuous vector functions.

 
, , d then is said to be second-order pseudoinvex with respect to  .

 
, , d  is said to be second-order pseudoinvex with respect to  .

Definition 4. (Second-order
 is said to be second-order quasi-invex with respect to  . does not depend explicitly on t, then the above definitions reduce to those for static cases.

Remark 1. If
The following inequality will also be required in the forthcoming analysis of the research: Lemma: 1 (Schwartz inequality): It states that Throughout the analysis of this research, the following conventions for the inequalities will be used: , , , n If with , .

Statement of the Problem and Necessary Optimality Conditions
Consider the following nondifferentiable Multiobjective variational problem: , where differentiation operator D already defined.

2)
: , 1, 2, , , : are assumed to be continuously differentiable functions, and 3) for each continuous on I.In this section we will derive Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions for (VCP).
Definition: A point x X  is said to be efficient solution of (VCP) if there exist   , , The following result which is a recast of a result of Chankong and Haimes [5] giving a linkage between an efficient solution of (VCP) and an optimal solution of p-single objective variational problem: r K Proposition 1. (Chankong and Haimes [5] , , for obtaining the optimal conditions for (VCP) we will use the optimal conditions obtained by Chandra et al. [6] for a single-objective variational problem which does not contain integral inequality constraints of r P .The validity of the following proposition is quite essential in obtaining the optimality conditions for (VCP), Proposition 2. If ( ) x t X  is an efficient solution of (VCP), then ( )  x t is an optimal solution of the following problem for each t be an efficient solution of (VCP).Suppose that  x ( ) t is not optimal solution of The inequalities (3) along with (4) contradicts the fact that x ( ) t is an efficient solution of (VCP).Hence is an optimal solution of , for some  Theorem 1. (Fritz John Type necessary optimality condition): Let x ( ) t be an efficient solution of (VCP).Then there exist , and piecewise smooth functions : Proof: Since x ( ) t is an efficient solution of (VCP), by Proposition 2, t is an efficient solution of for each and hence in particular 1 .So by the results of [6] there exist , , , , k y t


The above conditions yield the relations ( 5) to (9).Theorem 2 (Kuhn-Tucker type necessary optimality condition): Let ( )  x t r k   P be an efficient solution of (VCP) and let for each , the conditions of  r satisfy Slaters or Robinson condition [6] at ( )  x t .Then there exist p R  and piecewise smooth functions : Proof: Since ( ) x t is an efficient solution of (VCP) by Proposition 2, ( )   x t is an optimal solution of  

Mond-Weir Type Second Order Duality
In this section, we present the following Mond-Weir type second-order dual to (VCP) and validate duality results: , where , . and We denote by C P and C D the sets of feasible solutions to (VCP) and (M-WD) respectively.
(A 3 ) is second-order quasi-invex. Then for some 2 and Proof.Suppose to the contrary, that (20) and (21) hold.

 
Theorem 4 (Strong duality): Let x t be normal and is an efficient solution of (VP).Then there exist is feasible for (M-WD) and the two objective functions are equal.Furthermore, if the hypotheses of Theorem 3 hold for all feasible solutions of (VCP) and (M-WD) ,then   1 ( ), ( ), , ( ), , ( ), ( )

t y t z t z t t  
 is an efficient solution of (M-WD).

 
Proof: Since x t is normal and an efficient solution of (VP), by Proposition 2, there exist


From (24) along with  0, satisfies the constraints of (M-WD) and That is, the two objective functionals have the same value.
Suppose that , , , , p is not the efficient solution of (M-WD).Then there exists This contradicts the of Theorem 3. Hence
Theorem 5 (Converse duality): is an efficient solution of (M-WD) are linearly independent and (A 4 ) for either a) ( ) 0 Then x is feasible for (VCP) and the two objective functionals have the same value.Also, if Theorem 3 holds for all feasible solutions of (CP) and (M-WD), the x is an efficient solution of (VCP).   is an efficient solution of (M-WD), there exist such that the following Fritz John optimality conditions (Theorem 1) , , From (31), we have This, by the hypothesis (A 2 ) gives and Using ( 40), ( 41) and ( 17), we have This, because of (A 3 ) yields and from (43) Hence    Multiplying (30) by j y t


The relation (32) with and gives 2 , , This by Schwartz inequality gives ˆT is not an efficient of (VCP).Then, there exists x t X  such that for some and This contradicts Theorem 3. Hence   x t is an efficient solution for (VCP).

Problems with Natural Boundary Values
In   We shall not repeat the proofs of Theorems 3-6 for the above problems, as these follow on the lines of the analysis of the preceding section with slight modifications.   

Non-Linear Multiobjective Programming Problem
derivative x  , denoted by x  and x

(
By integrating by parts)This, by using η = 0, at t = a and t = b, implies these problems reduce to the following nondifferentiable second-order nonlinear problems already studied in the literature: this section, we formulate a pair of nondifferentiable Mond-Weir type dual variational problems with natural boundary values rather than fixed end points given bellow 1,