Using ρ-cone arcwise connectedness on parametric set-valued optimization problems

Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption.


Introduction
The class of parametric optimization problems (POPs) is a special type of optimization problems (OPs). It has applications in various fields of mathematical science, economics, and operational research. Many authors like Ioffe, Khanh, and Samei studied vector OPs with parameters [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. It has applications in inferring the Pontryagin maximum principle for control problems with state constraints. Khanh et al. [15][16][17][18] studied POPs for set-valued case. They established the Fritz John and Kuhn-Tucker necessary optimality conditions of set-valued parametric optimization problems (S-VPOPs) under relaxed differentiability assumptions on the state variable and convexlikeness assumptions on the parameter. The S-VPOPs arise in such a situation where OPs involve set-valued maps and the equality constraint represents equations, like differential equations and initial conditions. This class of OPs also arises in the case where the differential inclusions replace the differential equations to describe the system under consideration.
The arcwise connectedness is a generalization of convexity by replacing the line segment joining two points by a continuous arc which Avriel introduced in [19] in 1976. Later, in 2003, Fu et al. [20] and Lalitha et al. [21] introduced the concept of cone arcwise connected set-valued maps (CACS-VM) which is an extension of the class of convex set-valued maps. Lalitha et al. [21] established the sufficient optimality condition of S-VOPs using contingent epiderivative and CAC assumptions. In 2013, Yu [22] established the necessary and sufficient optimality conditions for the existence of global proper efficient points of vector OPs involving CACS-VMs. Yihong et al. [23] introduced the notion of α-order nearly CACS-VMs and derived the necessary and sufficient optimality conditions of S-VOPs.
In 2016, Yu [24] established the necessary and sufficient optimality conditions for the existence of global proper efficient elements of vector OPs involving CACS-VMs. In 2018, Peng et al. [25] introduced the notion of cone subarcwise connected set-valued maps (CSCS-VM) and established the second-order necessary optimality conditions for the existence of local global proper efficient elements of S-VOPs. For other different but connected points of view regarding this subject, the reader is directed to Ahmad et al. [26][27][28].
In this paper, we consider S-VPOPs Here, W = ∅ is a subset of normed space U, u is the state variable, and a ∈ A is the parameter, A is an arbitrary set, are set-valued maps, and p : V j (j = 1, 2, 3) are real normed spaces and 2 is a solid pointed convex cone in V 1 , where the objective function and functions attached to constraints are set-valued maps. We establish the sufficient Karush-Kuhan-Tucker (KKT) optimality conditions of problem (1) with the help of contingent epiderivative and ρ-cone arcwise connectedness (ρ-CAC) assumptions. Further, we formulate different types of duality relationships between the primal problem (1) and the corresponding dual problems. This paper is organized as follows. Section 2 deals with some definitions and preliminary concepts of set-valued maps. In Sect. 3, a parametric S-VPOP (1) is considered and the sufficient KKT optimality conditions are established for problem (1). Various types of duality theorems are studied under contingent epiderivative and ρ-CAC assumptions.

Definitions and preliminaries
Let be a nonempty subset of a real normed space V . Then is called a cone if ηv ∈ , ∀v ∈ , η ≥ 0. Furthermore, is called nontrivial if = {θ V }. Here, θ V is the zero element of V , proper if = V , pointed if ∩ (-) = {θ V }, solid whenever int( ) = ∅, closed whenever = , and convex whenever where int( ) and denote the interior and closure of , respectively. We denote the space of all continuous linear functionals on V and being a solid pointed convex cone in V by V * . Then the dual cone + to and quasi-interior +i of + are defined as where ·, · is the canonical bilinear form with respect to the duality between V * and V .
for any W ⊂ V . For any two nonempty subsets W ,Ẃ of V and v * ∈ V * , we also use the following notations: There are two types of cone-orderings in V with respect to a solid pointed convex cone in V . For any two elements v 1 For any two nonempty subsets W ,Ẃ of V , we use the following notations: [29,30] introduced the notion of contingent cone to a nonempty subset of a real normed space. Also, Aubin [29,30] and Cambini et al. [31] introduced the notion of second-order contingent set to a nonempty subset of a real normed space. 29,30]) Let W = ∅ be a subset of a real normed space V andv ∈ W . The contingent cone to W atv is denoted by T (W ,v) and is defined as follows: Let U, V be real normed spaces and F : U → 2 V be a set-valued map such that F(u) ⊆ V for all u ∈ U; here, 2 V is the power set of V . The effective domain, image, graph, and epigraph of F are defined respectively by Jahn and Rauh [32] introduced the notion of contingent epiderivative of set-valued maps which plays a vital role in various aspects of S-VPOPs.

Definition 2.2 ([32]) A single-valued map
is said to be the contingent epiderivative of F at (ú,v).
We now turn our attention to the notion of cone convexity of set-valued maps, introduced by Borwein [33]. Let W be a nonempty convex subset of a real normed space U. [19] introduced the concept of arcwise connectedness as a generalization of convexity by replacing the line segment joining two points by a continuous arc. W is said to be an arcwise connected set if for all u 1 , u 2 ∈ W there exists a continuous arc H u 1 ,u 2 (η) defined on [0, 1] with a value in W such that H u 1 ,u 2 (0) = u 1 and H u 1 ,u 2 (1) = u 2 [19]. Fu and Wang [20] and Lalitha et al. [21] introduced the notion of cone arcwise connected set-valued maps as an extension of the class of convex set-valued maps. Let W be an arcwise connected subset of a real normed space U and F : U → 2 V be a set-valued map with W ⊆ dom(F). Then F is said to be -arcwise connected on W if [20,21]. Peng and Xu [25] introduced the notion of cone subarcwise connected set-valued maps.
Let W be an arcwise connected subset of a real normed space U, a ∈ int( ), and F :
Further, we construct an example of ρ-CACS-VM, which is not cone arcwise connected.

Formulation of the main problem
Let U, V 1 , V 2 , and V 3 be real normed spaces and 1 , 2 , and 3 be solid pointed convex cones in V 1 , V 2 , and V 3 , respectively. Let A be an arbitrary set and W be a nonempty subset of U. Suppose that are set-valued maps and p : We consider a parametric S-VPOPs (1), where u is the state variable and a is the parameter. The minimizer and weak minimizer of problem (1) are defined in the following ways. A point (ú,á,v 1 ) ∈ U × A × V 1 , with (ú,á) ∈Š andv 1 ∈ F(ú,á), is called a minimizer of problem (1) if there exists no point (u, a, v 1 ) ∈ U × A × V 1 , with (u, a) ∈Š and v 1 ∈ F(u, a), such that and is called a weak minimizer of problem (1) if there exists no point with (u, a) ∈Š and v 1 ∈ F(u, a), such that v 1 -v 1 ∈ -int(V 1 ).
Hence, from Eq.
We establish the sufficient KKT optimality conditions of the parametric S-VPOPs (1) under contingent epiderivative and ρ-CAC assumptions.
satisfying all the constraints of (15) is called a feasible point of problem (15). The feasible point (15) is called a weak maximizer of (15) if there exists no feasible point (ú,á,v 1 ,v 2 ,ṽ * 1 ,ṽ * 2 ,ṽ * 3 ) of (15) such that We prove the duality results of Wolfe type of problem (1). The proofs are very similar to Theorems 3.10-3.12, and hence omitted.

Conclusions
In this paper, we establish the sufficient KKT optimality conditions for the parametric S-VPOPs (1) under ρ--AC and contingent epiderivative assumptions. We also construct the duals of Mond-Weir (16) and Wolfe (15) types and derive the duality results for weak minimizers between the primal problem (1) and corresponding dual problems.