Optimality Conditions and Duality for Multiobjective Variational Problems with Generalized ρ − ( η , θ )-B-Type-I Functions

The relationship between mathematical programming and classical calculus of variation was explored and extended by Hanson [1]. Thereafter variational programming problems have some attention in the literature. Mond and Hanson [2] obtained optimality conditions and duality results for scalar valued variational problems under convexity assumptions. Motivated by the approach by Bector and Husain [3], Nahak and Nanda [4] and later Bhatia and Mehra [5] extended the results of Mond et al. [6] to multiobjective variational problems involving invex functions and generalized B-invex functions, respectively. Analogous results were developed by Zalmai [7] for fractional variational programming problem containing arbitrary norms and by Liu [8] for generalized fractional case involving (F, ρ)-convex functions. Type-I functions were first introduced by Hanson and Mond [9], and Rueda and Hanson [10] defined a class of pseudo-type-I and quasi-type-I functions as generalization of type-I functions. Bhatia and Mehra [5] studied the optimality conditions and duality results for multiobjective variational problems involving generalized B-invexity. We use ρ − (η, θ)-B-type-I and generalized ρ − (η, θ)-B-typeI functions to establish sufficient optimality conditions and duality results for multiobjective variational problems. In Section 2 of this paper, we introduce ρ − (η, θ)-Btype-I and generalized ρ − (η, θ)-B-type-I functions for the continuous case. Using these new classes of functions, we establish various sufficient optimality conditions in Section 3 of this paper. Section 4 is devoted to the duality results. Some related problems are discussed in Section 5. The results obtained in this paper are more general than those obtained in [4, 5, 11]. Let I = [a,b] be a real interval, let f : I × Rn × Rn → Rp , and let h : I × Rn × Rn → Rm be continuously differentiable functions with respect to their arguments and x : I → Rn, ẋ denotes the derivative of x with respect t. Denote the partial derivative of scalar valued function g : I × Rn × Rn → R, with respect to t, x, and ẋ, respectively, by gt , gx, and gẋ such that gx = [∂g/∂x1, . . . ,∂g/∂xn], gẋ = [∂g/∂ẋ1, . . . ,∂g/∂ẋn]. Similarly we write the partial derivative of the vector functions f and h using matrices with p and m rows, respectively, instead of one. Let C(I ,Rn) denote the space of piecewise smooth functions x with norm ‖x‖ = ‖x‖∞ + ‖Dx‖∞, where the differential operator D is given by


Introduction
The relationship between mathematical programming and classical calculus of variation was explored and extended by Hanson [1].Thereafter variational programming problems have some attention in the literature.Mond and Hanson [2] obtained optimality conditions and duality results for scalar valued variational problems under convexity assumptions.Motivated by the approach by Bector and Husain [3], Nahak and Nanda [4] and later Bhatia and Mehra [5] extended the results of Mond et al. [6] to multiobjective variational problems involving invex functions and generalized B-invex functions, respectively.Analogous results were developed by Zalmai [7] for fractional variational programming problem containing arbitrary norms and by Liu [8] for generalized fractional case involving (F, ρ)-convex functions.
Type-I functions were first introduced by Hanson and Mond [9], and Rueda and Hanson [10] defined a class of pseudo-type-I and quasi-type-I functions as generalization of type-I functions.Bhatia and Mehra [5] studied the optimality conditions and duality results for multiobjective variational problems involving generalized B-invexity.We use ρ − (η, θ)-B-type-I and generalized ρ − (η, θ)-B-type-I functions to establish sufficient optimality conditions and duality results for multiobjective variational problems.
In Section 2 of this paper, we introduce ρ − (η, θ)-Btype-I and generalized ρ − (η, θ)-B-type-I functions for the continuous case.Using these new classes of functions, we establish various sufficient optimality conditions in Section 3 of this paper.Section 4 is devoted to the duality results.Some related problems are discussed in Section 5.The results obtained in this paper are more general than those obtained in [4,5,11].
Let I = [a, b] be a real interval, let f : and let h : I × R n × R n → R m be continuously differentiable functions with respect to their arguments and x : I → R n , ẋ denotes the derivative of x with respect t.Denote the partial derivative of scalar valued function g : I × R n × R n → R, with respect to t, x, and ẋ, respectively, by g t , g x , and g ẋ such that g x = [∂g/∂x 1 , . . ., ∂g/∂x n ], g ẋ = [∂g/∂ ẋ1 , . . ., ∂g/∂ ẋn ].Similarly we write the partial derivative of the vector functions f and h using matrices with p and m rows, respectively, instead of one.Let C(I, R n ) denote the space of piecewise smooth functions x with norm x = x ∞ + Dx ∞ , where the differential operator D is given by Therefore, D = d/dt except at discontinuities.
Consider the following multiobjective variational problem: Efficiency and proper efficiency are defined in their usual sense as defined in [4].
In relation to (VP), we introduce the following multiobjective problem (P * k ), for each k = 1, . . ., p, each problem with single objective: ( The following lemma can be established on the lines of Chankong and Haimes [12]. Definition 1 (see [4]).A point * x in K is said to be an efficient solution of (VP) if for all Definition 2 (see [13]).A point * x in K is said to be a properly efficient solution of (VP) if for all x ∈ K there exists a scalar for some j such that x is an efficient solution of (VP) if and only if * x is an optimal solution of (P * k ) for each k = 1, 2, . . ., p.
If, in the above definition, inequality ( 14) is satisfied as then we say that a pair ( If, in the above definition, inequality ( 16) is satisfied as η, and θ and ρ 0 , ρ 1 ∈ R.
Remark 10.Let ψ : I × R n × R n → R be a continuously differentiable function with respect to each of the arguments.Let x, u : Proof.It is clear that the statement every (ρ 0 , ρ 1 ) − (η, θ)-Btype-I function is (ρ 0 , ρ 1 ) − (η, θ)-B-strongly pseudo-type-I function but the converse is not true follows from the following counter example.

First we verify that (
From Cases 1 and 2, it is clear that the pair (
Proof.Let * x be not; an efficient solution of (VP), then there exist x ∈ K and an index 1 r P such that the following inequalities x is a properly efficient solution of (VP).
Proof.From (30), we have   51) along with the fact that ( x is a properly efficient solution of (VP).

Proof. If *
x is not an efficient solution of (VP), then there exist x ∈ K and an index r, 1 r p, such that Because * λ> 0 and b 0 (x, * x) > 0, the previous relations give Journal of Control Science and Engineering which along with the fact that ( Using Remark 10, (56) becomes Now using (29) in (58), we have As Again using Remark 10, we obtain which in view of a given hypothesis implies that Since b 1 (x, * x) > 0, we obtain which contradicts (30).Hence * x is a properly efficient solution of (VP).

Duality
The Mond-Weir-type dual problem associated with (VP) is given by (VD) maximize We now establish duality results between (VP) and (VD) under generalized B-type-I conditions.

Some Related Problems
All the optimality conditions and duality results developed for (VP) in the previous sections can easily be modified for several other classes of variational problems.If the problems (VP 0 ) and (VD 0 ) are independent of t, then these problems essentially reduce to the static case (V P) and (V D) of multiobjective nonlinear programs: V P minimize f (x) = f 1 (x), f 2 (x), . . ., f p (x) subject to h(x) ≤ 0, V D maximize f (u) = f 1 (u), f 2 (u), . . ., f p (u) , subject to λ T f x + y(t) T h x = 0, y 0, λ 0, λ T e = 1, e = (1, 1, . . ., 1) ∈ R p . (90)
(t, x, ẋ)dt − b a λ T f (t, u, u)dt 0 (74)which is the same as (41) with *x replaced by u and * λ replaced by λ.The rest of the proof runs on the same line of that of Theorem 15 and hence is omitted.