Nondifferentiable Multiobjective Programming with Equality and Inequality Constraints

In this paper, we derive optimality conditions 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. As an application of Karush-Kuhn-Tucker type optimality conditions, a Mond-Weir type dual to this problem is formulated and various duality results are established under generalized invexity assumptions. Finally, a special case is deduced from our result.


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 objectives, and/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.

Related Pre-Requisites and Expression of the Problem
In [1], the following problem is considered: and are continuously differentiable.
2) B is an symmetric positive semi definite matrix.

 n n
The following generalized Schwartz inequality [7] will be needed in the present analysis: The equality in the above holds if, for 0, The function     , being convex and everywhere finite, has a subdifferential in the sense of convex analysis.The subdifferential of   , where , and 1 We also require the Mangasarian-Fromovitz constraint qualification which is described as the following: Let x   be the set of feasible solution of the problem (EP), that is, and by  , A x the set of inequality active constraint indices, that is, where x   .We say the Mangasarian-Fromovitz constraint qualification holds at x   when the equality constraint gradients 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 x is an optimal solution of (EP), then there exist Lagrange multipliers If Mangasarian-Fromovitz constrain qualification (MFCQ) holds at x , then the above theorem reduces to th : e following theorem giving Karush-Kuhn-Tucker optimality conditions Theorem 2.2 (Karush-Kuhn-Tucker optimality conditions): If x is an optimal solution of (EP) and MFCQ holds at x , then there exist , The following conventions for inequalities will be used in the subsequent analysis: If , then ,  (VEP): Minimize where f, g and h are the same as Let in (EP).
The following results relate an efficient solution of Copyright © 2013 SciRes.OJMSi (EP) of k-scalar objective programming problems.

Lemma 2.2 (Chankong and Haimes [9]): A point x   is an efficient solution of (EP) if and only if x is an
We recall the following definitions of generalized invexity which will be used to derive various duality resu asi-invex with respect function lts.Definitions 2.2: 3)  is said to be th udoi e strictly pse nvex with re-

Optimality Conditions
In this section, the optimality conditions for the problem (EP) are obtained.Theorem 3.1 (Fritz John Type Optimality Conditions): If x be an efficient solution of (EP), then there exist , for , , and and hence n particular of   1 EP .Therefore by 2.1 there exist , for , , The theorem follows.Theorem 3.2 (Kuhn-Tucker type necessary optimality conditions): If x be an optimal solution of (VEP) and let for r K  , the constraints f o   r EP satisfy MECQ.Then there exist , , Proof: Since x is an optimal solution of (VEP), by Lemma 3.1, x is an optimal solution of for each r.As for some r, the constraint of MFCQ at From the above relation it is obvious that the theorem follows.
In Theorem 3.2, we assume MFCQ fo some r x is an optim solution of , by Kuhn-Tucker type necessary optimality conditions for eac , there exist Dividing throughout the above relation and setting, by

Mond-Weir Type Duality
We formulate the following differentiable multiobjective dual nonlinear problem for (VEP): In the following, we shall use for the set of feasible solutions of (M-WED) ,for some ca t hold.f: Su ry that (9) and (10) hold.the above inequalities (9) and (10) give Combining these, we give Using the equality constraint of (M-WED), this yields, Hence the result follows.


is an efficient solution of the (M-WED).
Proof: Since x is an efficient solution (VEP) satisfy ), there exist and , i.e. the two objective functions have the same Now we claim that , , , , , k  x y z w w   cient sol tion of (M-WED).

is an effiu
If not, then there exist This contradicts Theorem 4.1 Hence e same If with respect to th , e have By hypothesis (A 2 ) and (A 3 ) w  Combining (12), ( 13) and ( 14), we have which contradict the equality con Hence positive or negative definite and 2) the vectors is quasithe same  , then x is an efficient soluti eorem 3.3, there exist on (EP).Proof: By Th , , , , , , , 0 (25)  and summing over i, we have From ( 18) and ( 19), we have Using these in (15), we have which because of the hypothesis 1) gives 0 θ . Using 0 θ  and the hypothesis 2), we have Thus by (33), we have special case is also results can be revisited in the multiobjective setting of a nondifferentiable control problem.
implying the two objective functions hav e value.
w, ass e the sam No ume that x is not an efficient solution of (VEP), then there exists such that x is an efficient solution of (VEP).

A Special Case
Consider the following multiobjective programming problem containing square root of a certain quadratic form in each component of the objective.
Theorem 3.3 (Kuhn-Tucker type optimality conditions): If x be an efficient of (VEP) and let for each r K  , the constraints of   r EP satisfy MECQ at x .Then there exi jective functions are equal.Furthe (M-WED) and the two obrmore, if the weak ufor all feasible solution of (VEP) and d ality holds (M-WED), then  , , , ,  x y z w WED), suc be an efficient solution of (VEP) and h that V. Chankong and Y. Y. Haimes, "Multiobjective Decision Making Theory and Methodology," North-Holland, New York, 1983.