Duality Theorems for ( ρ , ψ , d ) -Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

: The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of ( ρ , ψ , d ) -quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

As is well known, the concept of quasiinvexity is a generalization of the notion of quasiconvexity. It (and some modified versions of it) played a fundamental role in formulating and demonstrating sufficient efficiency conditions for certain classes of variational problems (see, for instance, Mititelu [22], Mititelu and Treanţȃ [23], and Treanţȃ and Mititelu [24]). In this paper, by considering the new notion of (ρ, ψ, d)-quasiinvexity associated with an interval-valued multiple-integral functional, we establish Mond-Weir weak, strong, and converse duality results for a new class of multiobjective optimization problems with interval-valued components of the ratio vector. The duality model considered in this paper includes a partition associated with a set of indices used for the inequality type constraints. This element is new in multidimensional multiobjective interval-valued optimization problems. In addition, another novel element of this work is represented by the necessary LU-efficiency conditions derived by using a recent research paper of the author (see Treanţȃ [25]). More precisely, compared with previous research works (see [7,[10][11][12]21,24,25]), the present paper deals with the duality study associated with a new class of multiobjective optimization problems including interval-valued components of the ratio vector. These three highlighted elements, considered together at the same time, are totally new in the related literature. In addition, to illustrate the effectiveness of the results derived in the paper, an example is provided.
The paper is organized as follows: Section 2 includes the notations, preliminary mathematical tools, and formulation of the problem that we are going to study; Section 3 contains the main results of this paper: Mond-Weir weak, strong, and converse duality results are

Preliminaries and Problem Formulation
In the following, we consider the compact domain Ω in the Euclidean real space R m , and denote by t = (t α ), α = 1, m, u = (u j ), j ∈ 1, k, and a = (a i ), i = 1, n, the points in Ω, R k and R n , respectively. Now, we define the following continuously differentiable functions: and we accept that the following Lagrange densities satisfy the closeness conditions (complete integrability conditions) where D ζ is the total derivative operator. We denote by A the space of all piecewise smooth state functions a : Ω → R n , and by U the space of all piecewise continuous control functions u : Ω → R k , endowed with the induced norm. By dt := dt 1 dt 2 · · · dt m , we denote the volume element on R m ⊃ Ω. In addition, in this paper, for any two p-tuples l = l 1 , . . . , l p , c = c 1 , . . . , c p in R p , we use the following partial ordering: Consider that K is the set of all closed and bounded real intervals. We denote by I = [i L , i U ] a closed and bounded real interval, where i L and i U indicate the lower and upper bounds of I, respectively. Throughout this paper, the interval operations are performed as follows: Definition 1. Let I, J ∈ K be two closed and bounded real intervals. We write I ≤ J if and only if i L ≤ j L and i U ≤ j U . Definition 2. Let I, J ∈ K be two closed and bounded real intervals. We write I < J if and only if i L < j L and i U < j U .
In the following (in accordance with Mititelu and Treanţȃ [23], Treanţȃ [25]), in order to formulate and prove the main results included in this paper, we introduce the concept of (ρ, ψ, d)-quasiinvexity associated with an interval-valued multiple-integral functional. Consider an interval-valued continuously differentiable function where a α (t) := ∂a ∂t α (t), and for a ∈ A and u ∈ U , we introduce the following intervalvalued multiple-integral functional: In addition, let ρ be a real number, ψ : then H is said to be (ρ, ψ, d)-quasiinvex at a 0 , u 0 ∈ A × U with respect to ζ and κ; then H is said to be strictly (ρ, ψ, d)-quasiinvex at a 0 , u 0 ∈ A × U with respect to ζ and κ.
Consider the following vector continuously differentiable functions: Now, we are in a position to formulate the following new class of multiobjective fractional variational control problems with interval-valued components, which are called a Primal Problem (in short, PP): where, for r = 1, p, we have denoted Ω f r (t, a(t), u(t))dt Ω g r (t, a(t), u(t))dt The set of all feasible solutions in (PP) is defined by D := {(a, u)|a ∈ A, u ∈ U satisfying (1), (2), (3)}.

Mond-Weir Duality Associated with (PP)
Consider that {Q 1 , Q 2 , · · · , Q s } is a partition of the set Q = {1, 2, · · · , q}, where s < q. For (b, w) ∈ A × U , with the same notations as in Section 2, we associate to (PP) the next multiobjective fractional variational control problem with interval-valued components of the vector, which is called the Dual Problem (in short, DP): Remark 1. In the previous dual problem, the expression µ Q ϑ (t)Y Q ϑ (t, b(t), w(t)) has the following meaning: In this section, we establish that the multiobjective optimization problems with interval-valued components of the ratio vector, (PP) and (DP), are a Mond-Weir (see [26]) dual pair under (ρ, ψ, d)-quasiinvexity hypotheses. Further, consider that is the set of all feasible solutions associated with (DP). Now, we formulate and prove the first duality result.
Theorem 1 (Weak Duality). Let (a, u) ∈ D be a feasible solution of the multiobjective variational control problem with interval-valued components (PP), and let (b, w, θ, λ, µ) ∈ be a feasible solution of the multiobjective variational control problem with interval-valued components (DP).
In addition, consider that the following conditions are fulfilled: (a) Each functional with respect to ζ and κ; or, equivalently, each interval-valued multiple-integral functional . Then, the infimum of (PP) is greater than or equal to the supremum of (DP).
The next result establishes a strong duality between the two considered multiobjective optimization problems with interval-valued components.

Conclusions
In this paper, based on the totally new concept of (ρ, ψ, d)-quasiinvexity associated with an interval-valued multiple-integral functional, we have formulated and proved Mond-Weir weak, strong, and converse duality results for a new class of multiobjective optimization problems with interval-valued components of the ratio vector. Taking into account the applicability of interval analysis and duality theory in optimization and control, the present paper represents an important outcome for researchers and engineers in applied sciences.