OPTIMALITY, DUALITY AND SADDLE POINT CRITERIA FOR A ROBUST FRACTIONAL INTERVAL-VALUED OPTIMIZATION PROBLEM WITH UNCERTAIN INEQUALITY CONSTRAINTS VIA CONVEXIFICATORS

. This article focuses on optimality conditions for a robust fractional interval-valued optimization problem with uncertain inequality constraints (RNFIVP) based on convexificators. Using the tools of convexity, an example of sufficient optimality conditions is demonstrated. Robust parametric duality for (RNFIVP) is formulated and utilizing the concept of convexity, usual duality results be-tween the primal and dual problems are investigated. Further, the equivalence between the saddle point criteria of a Lagrangian type function and a robust ℒ𝒰 -optimal solution for (RNFIVP) with convexity is also examined


Introduction
Robust optimization has developed as a powerful conceptual structure for analyzing interval-valued optimization problems involving data uncertainty.This is an emerging area of study that motivates researchers to solve a spectrum of optimization problems concerning real-life situations such as industrial settings, where the data input for an interval-valued program is usually uncertain or noisy due to inaccuracies which occur in measurement or prediction.In the function space, the objective and constraint functions belong to "uncertainty sets".A few examples of real-world applications of robust optimization include topology design problem [25], dynamic power generation [29] and shelter location-allocation problem [14].For a comprehensive analysis of robust optimization, the readers may refer to [5-7, 17, 18].
Demyanov [11] first proposed the concept of convexificators in 1994 as a generalization of upper convex and lower concave approximations.Convexificators are considered as weaker forms of the concept of subdifferentials as they are often closed sets, unlike the well-known subdifferentials which are compact and convex sets.In the case of extended real-valued functions, the concept of noncompact convexificator and characterization of quasiconvexity was proposed by Jeyakumar and Luc [20].Furthermore, they presented numerous calculus laws which included extremality and mean value properties.Subsequently, Dutta and Chandra [12] introduced a new concept of nonsmooth pseudoconvex function, investigated its properties and also analyzed optimality criteria for vector minimization using convexificators.Moreover, as an application of chain rule for a mathematical programming problem involving inequality constraints, Dutta and Chandra [13] derived necessary optimality conditions.By employing the tools of convexificator, Li and Zhang [22] derived Kuhn-Tucker type necessary optimality criteria for nonsmooth optimization problems using locally Lipschitz functions.Recently, Ahmad et al. [3] examined the optimality and duality conditions for nonsmooth minimax programming problems with locally Lipschitz functions using the approach of convexificator.On the other hand, Jayswal et al. [19] analyzed optimality criteria for nonsmooth multi-objective programming problems and also discussed the duality results of two types of dual models based on the notion of convexificator.
The objective function in fractional programming problems is the ratio of two functions.When many rates need to be optimized at the same time, such as health care and hospital planning, production planning, and financial and corporate planning, these models naturally occur.There are many applications of fractional programming, namely, engineering designs [27], sustainable management of electric power systems [30] and production planning [4] etc.For more information on fractional programming, one can refer to Stancu-Minasian [26].
In the last few years, there has been a lot of research on fractional interval-valued programming problems.An interval-valued linear fractional programming problem was introduced by Effati and Pakdaman [15], which is reduced to an interval-valued objective function with the bounds being fractional functions.In addition, Ahmad et al. [1] studied the Karush-Kuhn-Tucker optimality conditions for a multi-objective programming problem with interval-valued objective functions utilizing generalized convexity and generalized differentiability.Furthermore, Ahmad et al. [2] examined the Fritz-John and Kuhn-Tucker type optimality conditions for a non-differentiable interval-valued multi-objective model using the concept of ℒ-convexity.The Karush-Kuhn-Tucker optimality conditions for multiple objective fractional interval-valued optimization problems were examined by Debnath and Gupta [10], presuming that the functions involved are gH-differentiable.Very recently, Dar et al. [8] derived optimality conditions of the Fritz-John and Kuhn-Tucker type for an interval multiobjective fractional model (IVMFP) using the concept of ℒ-convexity and ℒ-concavity.Also, they examined parametric duality results.On the other hand, Rani and Kummari [23] discussed optimality and saddle point criteria for a fractional interval-valued optimization problem using convexificator.
We perceive that, in the literature, there is no work addressing robust fractional interval-valued optimization problems with uncertain inequality constraints.Therefore, the focus of this paper is to analyze optimality, duality and saddle point criteria for (RNFIVP).This paper is organized in the following way: Some preliminary and fundamental notions are recalled in Section .Robust optimality criteria for (RNFIVP) are analyzed based on the concept of convexificators in Section 3, while Section 4 deals with the formulation of robust parametric duality for (RNFIVP) and the usual duality results between the primal and dual problems are also investigated.Using the notion of convexificators, Section 5 illustrates how the saddle point conditions of a Lagrange type function and a robust ℒ-optimal solution for (RNFIVP) are equivalent.Section 6 examines special cases.Finally, Section 7 provides a conclusion to this article.

Preliminaries
All spaces in this paper, unless otherwise mentioned, are real Banach spaces with norm denoted by ‖.‖.Let  * be the topological dual of a given real Banach space  with the canonical dual pairing ⟨., .⟩;which stands for the norm on  and  * .Let R  + be the non-negative orthant of an -dimensional Euclidean space R  .Let  :  → R ∪ {+∞} be an extended real-valued function.Then the lower and upper Dini directional derivatives of  at  ∈  in the direction of  is defined as follows as: . The fundamental concepts of interval mathematics are as follows: An order relation ⪯ ℒ between two intervals E H and J G is stated as given below: Consider the subsequent robust non-differentiable fractional interval-valued optimization problem with data uncertainty in constraints: where  is an arbitrary index set (possibly infinite) and  ℒ (),   () ≥ 0,  ℒ (),   () > 0, and   :  ×R  → R are continuous functions on  and   ∈ R  is an uncertain parameter which pertains to the convex compact set   ⊂ R  ,  ∈ .The uncertainty set-valued function  :  ⇒ R  , is given by () :=   , ∀  ∈ , so, and Let M denote the robust feasible set for the problem (RNFIVP).That is, Definition 2.7 (Wu [28]).A robust feasible solution α is termed as a robust ℒ-optimal solution for (RNFIVP) if and only if there exists no robust feasible solution  such that ]︃ .

Robust optimality conditions
Consider the subsequent two fractional problems for the given robust feasible solution α: On the lines of Debnath and Gupta [9], we give the below mentioned Lemmas 3.1 and 3.2.
Lemma 3.1.Let α be a robust ℒ-optimal solution of the problem (RNFIVP) if and only if α is a robust optimal solution for the problems (RFP1) and (RFP2).
Proof.Let α be a robust ℒ-optimal solution of the problem (RNFIVP).Then there is no robust feasible solution  such that ]︃ .
This implies that there is no robust feasible solution  such that Suppose that α is not a robust optimal solution for the problem (RFP1), then there exists an  satisfying the constraints of the problem (RFP1), that is, the presumption that α is a robust ℒ-optimal solution of the problem (RNFIVP).
Conversely, let α be a robust optimal solution for the problems (RFP1) and (RFP2).Suppose that α is not a robust ℒ-optimal solution of the problem (RNFIVP), then there exists a robust feasible solution , that is, Thus, by the definition of the relation ≺ ℒ , we get or or By   (,   ) ≤ 0, ∀  ∈ , ∀   ∈   and second inequality of (3.1), which implies that  is a robust feasible solution of the problem (RFP1).Thus, the first inequality of (3.1) contradicts the presumption that α is a robust optimal solution of the problem (RFP1).Further, by   (,   ) ≤ 0, ∀  ∈ , ∀   ∈   and first inequality of (3.2), which implies that  is a robust feasible solution of the problem (RFP2).Thus, the second inequality of (3.2) contradicts the presumption that α is a robust optimal solution of the problem (RFP2).
Lemma 3.2.Let α be a robust ℒ-optimal of the problem (RNFIVP) if and only if α minimizes ℎ 0 ℒ  0 ℒ () on the subsequent uncertainty constraint set Suppose that α is not a robust ℒ-optimal of the problem (RNFIVP), then there exists a robust feasible solution Thus, by the definition of the relation ≺ ℒ , we get , or , or Conversely, let α be a robust ℒ-optimal of the problem (RNFIVP).Suppose that α does not minimizes ℎ 0 ℒ  0 ℒ () on ℱ ℒ , then there exists a robust feasible solution  0 such that which contradicts that α is a robust ℒ-optimal of the problem (RNFIVP).
Let α be a robust ℒ-optimal solution of the problem (RNFIVP).
The following theorem for the problem () is stated, based on Theorem 6 of Gadhi [16] and Lee and Lee [21].
Theorem 3.5 (Robust sufficient ℒ-optimality conditions).Suppose that α is a robust feasible solution of the problem (RNFIVP), then there exists + such that the conditions (3.7)-(3.9)hold at α. Also, presume that then α is a robust ℒ-optimal solution of the problem (RNFIVP).
The sufficient ℒ-optimality conditions specified in Theorem 3.5 is demonstrated by the subsequent example.

Robust parametric duality
The aforementioned robust parametric duality of the primal problem (RNFIVP) is discussed in the present section.
Corollary 4.2.If in addition to the presumptions of Theorem 4.1, we consider α ∈ M and robust ℒ-optimal solution of the problems (RNFIVP) and (RNFIVD), respectively.
Theorem 4.3 (Strong duality). .Suppose that α is a robust ℒ-optimal solution of the problem (RNFIVP) and both the Slater's constraint qualification for the problems (RFP1) and (RFP2) are satisfied at α. Then there Also, if the presumptions of Theorem 4.1 hold for all robust feasible solution of the problems (RNFIVP) and (RNFIVD), respectively then (︁ α, κ , τ ℒ , τ  , θ)︁ is a robust ℒ-optimal solution of the problem (RNFIVD).

Lagrangian type function and saddle-point analysis
In the present section, for the robust feasible solution α ∈ M, we define the Lagrangian type function for the primal problem (RNFIVP) as stated below: where  ∈  ,  ℒ ≥ 0,   ≥ 0 and (  ) ∈ ∈ R .Then α is a robust ℒ-optimal solution of the problem (RNFIVP).

Special Cases
(P2) min (iii) From Dutta and Chandra [12,13] and Jeykumar and Luc [20], it is evident that the convexificators are not necessarily compact or convex.These exemptions permit applications to an extensive category of nonsmooth continuous functions.Thus, the results of this paper are sharper when compared to the other papers like robust interval-valued optimization problems with uncertain inequality constraints [17,18].

Conclusion
In this article, the notion of convexificator is employed to discuss the optimality conditions for a robust fractional interval-valued optimization problem with uncertainty constraints (RNFIVP) and an example is provided to illustrate sufficient optimality criteria.Furthermore, the robust parametric duality of (RNFIVP) is described and the duality results between the primal and dual problems are investigated by utilizing the concept of convexity.Finally, the equivalence between the saddle point criteria of a Lagrangian type function and a robust ℒ-optimal solution of (RNFIVP) with convexity is also examined.The approaches used in this study, from our viewpoint, can be used to demonstrate equivalent outcomes for other types of fractional programming problems involving convexificators.This could be the focus of research in the future.

Figure 1 .
Figure 1.Graphical view of the objective function of the problem (RFP3).