Bilevel programming problems are a type of problem where one participant, called the leader, tries to make decisions that consider how another participant, called the follower, will react. The leader’s decisions impact the follower’s choices. The upper-level part of the problem deals with what the leader wants to achieve and the conditions they need to satisfy, while the lower-level deals with the follower’s goals and the constraints associated with them. The origin of the concept of BLPP can be attributed to the influential contributions of Von Stackelberg [41]. The first mathematical bilevel model was developed in 1973 by Bracken and McGill [7]. Subsequently, numerous researchers have created fascinating theories and applications, including optimistic and pessimistic methods, single-level reformulation, optimality criteria, duality results, algorithms, etc. BLPPs have important applications across multiple fields of modern research, including engineering, medicine, and economics as well as BLPPs offer numerous benefits from both theoretical and practical standpoints see, for instance, [4,5,21] and the references cited therein.

Bilevel programming problems are hierarchical problems consisting of two decision parameters. These variables are not independent of each other but act according to a certain hierarchy. In this type of problem, the variables of the first problem act as leaders, and the variables of the second problem act as followers. Obtaining the optimal solution for the second problem is crucial for determining the objective function value of the first problem. Numerous researchers, such as Bard [1,2,3] and Outrata [38], have explored bilevel programming problems due to their fascinating attributes and significant relevance. Dempe [13] derived the necessary and sufficient optimality criteria for BLPP. Yezza [48] established the first-order necessary optimality criteria for general BLPPs. Moreover, Yezza [48] formulated the general multilevel programming problem and deduced the necessary conditions of optimality in the general case. Dempe [14] rectified deficiencies in [48], specifically addressing Proposition 2.1 and Theorem 5.1. The optimistic version of BLPPs and its necessary optimality conditions are studied by Dempe [16]. For further insights into bilevel programming problems, we refer to [2,15] and the references cited therein.

The concept of variational inequality (abbreviated as, VI) was introduced by Hartman and Stampacchia [22]. Variational inequalities appear in the forms of Minty VI [30] and Stampacchia VI [42]. Variational inequalities have several applications in the fields of economics, game theory, and traffic analysis, see, [11,17,23]. Giannessi [17] introduced the notion of vector VI for finite-dimensional Euclidean spaces. VI problems have been studied by several scholars as tools for solving optimization problems, for more exposition, see, [17,18,26,44,45,46,47], and the references cited therein. Komlósi [25] derived the equivalence among the solutions of Minty and Stampacchia VI and the optimal solution of the minimization problem. Kinderlehrer and Stampacchia [23] studied the relations between the solutions of VI and minimization problems. Crespi et al. [10] investigated the relations between the solutions of Minty VI and scalar optimization problems. Kohli [24] studied the relations between variational inequalities and BLPP involving generalized convex functions in terms of convexificators.

The main motivation and objective for investigating the relationships between the solutions of AVI, namely, AMTVI, ASTVI, and the local $ϵ$-quasi solutions of BLPP in the notion of limiting subdifferential are threefold. Firstly, in several real-world problems, nonsmooth phenomena occur frequently. To deal with such problems, Clarke introduced the notion of subdifferential for a certain class of locally Lipschitz functions [9]. However, the convexity of the Clarke subdifferential has led to various limitations. To overcome these shortcomings stemming from its convexity, Mordukhovich proposed the concept of the limiting subdifferential [35]. Limiting subdifferential offers an improved Lagrange multiplier rule compared to the Clarke subdifferential and is recognized as the smallest subdifferential among all known robust subdifferentials. Secondly, Convexity and generalized convexity are pivotal in the domains of operations research, economics, and engineering. Moreover, within optimization theory, convexity plays a crucial role as it ensures that a stationary point serves as a global minimizer, and the first-order optimality criteria are transformed into sufficient conditions for identifying a point as a global minimizer. Mangasarian [29] generalized the notion of convex function by introducing the class of pseudoconvex functions. For a detailed study of generalized convex functions, we refer to [8,31,34]. Ngai et al. [36] introduced the notion of approximate convex function. Recently, several generalizations of approximate convex functions have been introduced, for example, Bhatia et al. [6] and Gupta et al. [20]. Thirdly, to the best of our knowledge, there is only one research paper (see, [25]) available in the literature that investigates the relationships between the solutions of BLPP and variational inequalities. However, the investigation of the relationships between the solutions of AVI and the local $ϵ$-quasi solutions of BLPP in the notion of limiting subdifferentials has not been explored yet. Consequently, this paper aims to address this specific research gap by establishing the results that make relationships between the solutions of AMTVI, ASTVI, and the local $ϵ$-quasi solutions of BLPP in the notion of limiting subdifferentials.

Motivated by the works of [6,10,24,32,43], in this paper, we consider BLPP and establish the relationships between the solutions of approximate convex functions, namely, AMTVI and ASTVI, and the local $ϵ$-quasi solutions of BLPP. Moreover, we deduce some existence results for the solutions of AMTVI and ASTVI, by employing the assumption of generalized KKM-Fan’s lemma.

The novelty and contributions of this paper are threefold. Firstly, the results of this paper extend the analogous results in [10,25] from single-level optimization problems to more general optimization problems, namely, bilevel optimization problems. Secondly, since the limiting subdifferential is the least among all the known robust subdifferentials and offers an enhanced Lagrange multiplier rule compared to the Clarke subdifferential, therefore our findings naturally sharpen the analogous results of [6,24,32,43]. Thirdly, the established results of this paper extend the analogous results in [6,24,27,43] for a broader class of approximate convex functions.

The organization of this article is as follows. In Section 2, we recall some basic definitions and preliminaries. In Section 3, employing the potent tool of limiting subdifferential, we investigate the equivalence among the solutions of AMTVI and ASTVI, and the local $ϵ$-quasi solution of nonsmooth BLPP. In Section 4, a generalized Fan lemma has been employed to establish the existence results for the solutions of AVI.

Throughout this paper, we use the notation $\langle ·,·\rangle$ to denote the Euclidean inner product in the n-dimensional Euclidean space ${\mathbb{R}}^n$. For a nonempty subset $\Omega$ of ${\mathbb{R}}^n\times {\mathbb{R}}^m$ equipped with the Euclidean norm $\parallel ·\parallel$, we signify the closure and interior of $\Omega$ by $\mathrm{cl}\Omega$ and $\mathrm{int}\Omega$, respectively.

The definition of a convex set, provided below is from [34].

Now, we recall the following definitions related to nonsmooth analysis from [35].

Now, we consider the following bilevel programming problem:

BLPP: $\underset{{\wp }_1,{\wp }_2}{min}\Phi \left({\wp }_1,{\wp }_2\right)$ where $\Phi :{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R}$, $\varphi :{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R},H_j:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R},j\in J$, and $h_i:{\mathbb{R}}^n\times {\mathbb{R}}^m\to \mathbb{R},i\in I$ are real valued functions and

So, the basic idea is that based on the choice of the leader, the follower minimizes his objective function and the leader then uses the obtained solution ${\wp }_2={\wp }_2\left({\wp }_1\right)$ to minimize his objective function. BLPP is said to be well defined if we can uniquely determine the optimal solution of the lower level problem for every ${\wp }_1\in {\mathbb{R}}^n$. In literature, two types of solution concepts have been studied for the problems having more than one optimal solutions for lower level problem, such as optimistic solution and pessimistic solution.

In the optimistic approach, the follower considers an optimal solution which is the best from the leader’s perspective. Therefore, one has the following optimistic bilevel programming problem:

OBLPP: $\underset{{\wp }_1}{min}{\psi }_{\circ }\left({\wp }_1\right),{\wp }_1\in {\mathbb{R}}^n$ and $\psi \left({\wp }_1\right)$ is the set of optimal solutions for the following lower level problem

Let, S be the set of all feasible solutions to the problem BLPP, that is,

Now, we introduce two following notations which will be used in this sequel.

Therefore, it is evident that $S=\{({\wp }_1,{\wp }_2)\in \Omega :{\wp }_2=argmin{\mathcal{F}}_{{\wp }_1}\left({\wp }_2\right)\}.$

In the rest of the paper, we will assume that $\Omega$ is a non-empty convex subset of ${\mathbb{R}}^n\times {\mathbb{R}}^m.$

The following definition from Ngai and Penot [37] represents the notion of approximate convexity of a real-valued function.

We have the following characterization for lower semicontinuous approximate convex functions from Ngai and Penot [37].

On the lines of Bhatia et al. [6] and Golestani et al. [19], in the following definitions, we define the notion of generalized approximate convex functions in terms of limiting subdifferentials.

The subsequent mean value theorem for locally Lipschitz functions from [34], will be used in the sequel.

The following notions of $ϵ$-quasi solution and local $ϵ$-quasi solution for BLPP are adaptations of the notions of $ϵ$-quasi solution and local $ϵ$-quasi solution for scalar optimization problems from Loridan [28].

Now, we consider the following AMTVI and ASTVI in terms of limiting subdifferentials:

AMTVI: Find $\left({\overline{\wp }}_1,{\overline{\wp }}_2\right)\in \Omega$ such that for an $ϵ>0,\exists d>0$, such that for each $\left({\wp }_1,{\wp }_2\right)\in B\left(\left({\overline{\wp }}_1,{\overline{\wp }}_2\right),d\right)\cap \Omega$ and all ${\xi }_1\in {\partial }_M\Phi \left({\wp }_1,{\wp }_2\right)$ and ${\xi }_2\in {\partial }_M{\mathcal{F}}_{{\wp }_1}\left({\wp }_2\right)$, the following inequalities hold:

ASTVI: Find $\left({\overline{\wp }}_1,{\overline{\wp }}_2\right)\in \Omega$ such that for an $ϵ>0,\exists d>0$, such that for each $\left({\wp }_1,{\wp }_2\right)\in B\left(\left({\overline{\wp }}_1,{\overline{\wp }}_2\right),d\right)\cap \Omega$, there exists ${\zeta }_1\in {\partial }_M\Phi \left({\overline{\wp }}_1,{\overline{\wp }}_2\right)$ and ${\zeta }_2\in {\partial }_M{\mathcal{F}}_{{\wp }_1}\left({\overline{\wp }}_2\right)$ such that the following inequalities hold:

This section is devoted to studying the equivalence relationships between the solutions of AVI, namely, AMTVI, ASTVI, and the local $ϵ$-quasi solutions of the BLPP within the framework of limiting subdifferential.

From now onwards, let $ϵ>0$ be given and $\Phi$ be a lower semicontinuous function unless otherwise specified.