Approximate Convexity of Set-Valued Mappings and Variational Inequalities

Abstract

In this article, we introduce the notion of approximate convexity for set-valued mappings, specifically in the forms of approximate pseudoconvexity and approximate quasiconvexity. These generalizations are motivated by the need to handle optimization problems involving multi-valued operators and vector-valued objective functions, where classical convexity assumptions are too restrictive. We demonstrate that the proposed framework preserves the essential structural features of convex analysis while broadening its applicability. After that, we investigate the introduced definitions through illustrative examples. Furthermore, we consider a set-valued optimization problem and rigorously investigate the relationships among its efficient solutions and the solutions of generalized Minty and Stampacchia variational inequality problems. The results provide a coherent theoretical bridge between optimality conditions and variational inequality formulations for set-valued mappings.

1. Introduction

Convexity is a cornerstone of optimization theory, as it ensures any local minimum of a convex function is also a global minimum. This property is essential for both theoretical analysis and practical optimization methods. To broaden the scope of convex analysis, Jofre et al. [1] introduced $\epsilon$-convexity, a more flexible extension of classic convexity in 1998. Building on this, Ngai et al. [2] proposed approximate convexity, which keeps crucial structural features of convex functions while allowing for a wider range of optimization problems in 2000. These extensions address cases where strict convexity is absent, yet essential mathematical properties remain in a generalized form. In 2018, Gupta and Mishra [3] analyzed the relationship between approximate and generalized convexity, linking variational inequalities and nonsmooth optimization problems in vector spaces via generalized approximate convexity. Building on Mishra and Laha’s [4] foundational systematic analysis of vector optimization with locally Lipschitz and approximately convex objective functions, later research refined this mathematical framework in 2016, then their work formulated notions of approximate efficiency and developed Stampacchia-type and Minty-type approximate vector variational inequalities, expanding analytical tools for characterizing efficient solutions. In 2004, Advancements by Georgiev [5] showed that, for locally Lipschitz functions, approximate convexity is equivalent to submonotonicity of the Clarke subdifferential, establishing a structural link between approximate and generalized convexity in nonsmooth analysis. In 1988, Further generalizations by Bhatia et al. [6] broadened the field, while Malmir and Barani [7] showed equivalence between locally Lipschitz approximately convex functions and lower-$C^1$ functions, and a correspondence between strict approximate convexity and strict submonotonicity of the Clarke subdifferential in 2022.

Despite these important developments, several theoretical challenges remain unresolved. Foremost among these is the extension of subdifferential characterizations to functions that lack local Lipschitz continuity, thereby expanding the applicability of established frameworks to a broader class of nonsmooth functions. Moreover, the precise relationships between approximate convexity and other generalized convexity notions—including invexity, quasiconvexity, and star-shapedness—remain insufficiently elucidated. This gap hinders the integration and comparative analysis of various generalized convexity frameworks. The absence of generalized, robust methodologies for handling non-Lipschitzian, higher-order nonsmoothness further impedes progress in these directions. Therefore, although recent literature has substantively advanced both the theoretical and methodological foundations of approximate convexity and its subdifferential properties, critical gaps persist. For more information, see [8,9,10].

In addition, in 2018, Olbrys [11] developed the concept of $(\omega ,t)$-convexity, presenting it as a generalization of classical t-convexity and its variants, such as t-strong convexity, approximate t-convexity, and $dt$-convexity. The main contribution of this work is in identifying precise conditions for when a $(\omega ,t)$-convex function has local support by a corresponding $(\omega ,t)$-affine support function at a given point. This advance broadens the functions to which convex analytic techniques apply and helps identify new criteria for characterizing inner product spaces within normed linear spaces. Recently, significant work has extended core ideas from convex scalar functions to set-valued mappings; see [12,13,14,15,16].

This shift addresses new complexities in set-valued optimization, such as solution sets and multi-valuedness. In 2022, Durea and Strugariu [17] created a framework for directional derivatives and subdifferentials that mirrors real-valued function constructs, bridging the scalar and set-valued cases. In 2017, Oveisiha and Aghabagloo [18] used scalarization to characterize relaxed K-preinvexity for set-valued mappings and tied solution sets from generalized Minty variational-like inequalities in set-valued relaxed convexity, directly linking them to those of scalarized set-valued optimization. This work clarifies the connection between scalar and set-valued frameworks, enabling the transfer of analytic tools. In 2013, Oveisiha and Zafarani [19] advanced set-valued optimization by giving normal subdifferential-based characterizations for K-preinvex set-valued mappings. They established super-minimal points, found necessary conditions for super-efficiency, and described Clarke–Rockafellar and limiting subdifferential properties for $\beta$-weakly preinvex and pseudoinvex functions. Using pseudoinvexity, they proved equivalence between vector variational-like inequalities and corresponding vector optimization problems. For more information see [20,21].

Although approximate convexity for real-valued functions and generalized convexity for set-valued mappings have each received considerable attention, the systematic extension of approximate convexity notions—specifically approximate pseudoconvexity and quasiconvexity—to the set-valued framework has not been addressed in the existing literature. In particular:

• The existing works on set-valued optimization (e.g., [18,19]) focus on preinvex or K-convex set-valued maps but do not develop approximate variants.
• The approximate convexity results of [3,4,5] are restricted to real-valued or vector-valued functions and do not accommodate the multi-valued nature of set-valued operators.
• The relationships between optimality conditions and variational inequalities have not been established under approximate convexity assumptions for set-valued mappings.

The present paper bridges this gap by:

1.

Introducing approximate $\mathfrak{K}$-convexity, approximate pseudoconvexity (types I and II), and approximate quasiconvexity (types I and II) for set-valued mappings in Banach spaces.

2.

Establishing the relationships among scalarized locally quasi-efficient solutions (SLQE), locally weak quasi-efficient solutions (LWQE), generalized Minty variational inequality problems ${\mathfrak{P}}_{\left(GMVIP\right)}$, and generalized Stampacchia variational inequality problems ${\mathfrak{P}}_{\left(GSVIP\right)}$.

3.

Demonstrating the practical implications of the proposed framework through worked examples.

4.

Discussing potential extensions and open problems to guide future research.

This paper rigorously defines approximate convexity for set-valued mappings, extending classical convexity theory. Using the Mordukhovich normal subdifferential, it establishes that, under specific conditions, a set-valued mapping’s approximate convexity is equivalent to the approximate monotonicity of its subdifferential. This equivalence creates a unified paradigm for variational and nonsmooth analysis. While providing a foundation for set-valued optimization, challenges remain, such as extending the results to non-Lipschitz functions and relating approximate convexity to other generalized convexity concepts.

2. Preliminaries

In this section, consider two Banach spaces $\mathfrak{X}$ and $\mathfrak{Y}$. Let ${\mathfrak{X}}^{\ast }$ represent the continuous dual of $\mathfrak{X}$. The symbol $\parallel ·\parallel$ will denote the norm on $\mathfrak{X}$ as well as on ${\mathfrak{X}}^{\ast }$. Moreover, write $B_{\mathfrak{X}}$ for the closed unit ball of $\mathfrak{X}$, and let $\mathfrak{K}$ be a closed convex cone contained in $\mathfrak{Y}$.

Definition 1 

([22]). Let $\mathfrak{X}$ be a Banach space and $\mathsf{\Omega }\subseteq \mathfrak{X}$ a nonempty set. For a point $u\in \mathsf{\Omega }$ and a number $\epsilon \ge 0$, the ε-normal set of Ω at u is defined as

$${\hat{\mathfrak{N}}}_{\epsilon }(u;\mathsf{\Omega }):=\left\{u^{\ast }\in {\mathfrak{X}}^{\ast }|\underset{w\to u}{\mathrm{lim\; sup}}{\displaystyle \frac{⟨u^{\ast },w-u⟩}{\parallel w-u\parallel }}\le \epsilon \right\}.$$

When $\epsilon =0$, we write $\hat{\mathfrak{N}}(u;\mathsf{\Omega })$ and refer to it as the regular normal cone to $\mathsf{\Omega }$ at u. For a point $\overline{u}\in \mathsf{\Omega }$, the basic normal cone to $\mathsf{\Omega }$ at $\overline{u}$ is given by

$$\mathfrak{N}(\overline{u};\mathsf{\Omega }):=\underset{\begin{array}{c}u\to \overline{u}\\ \epsilon \downarrow 0\end{array}}{\mathrm{lim\; sup}}{\hat{\mathfrak{N}}}_{\epsilon }(u;\mathsf{\Omega }).$$

Definition 2 

([7]). A real-valued function $f:\mathsf{\Omega }\subset \mathfrak{X}\to \mathbb{R}$ is called approximately convex at $u_0\in \mathsf{\Omega }$ if for every $\beta >0$ there exists $\delta >0$ such that for all $u_1,u_2\in B(u_0,\delta )\cap \mathsf{\Omega }$ and every $t\in [0,1]$,

$$f\left(t{u}_1+(1-t)u_2\right)\le tf\left(u_1\right)+(1-t)f\left(u_2\right)+\beta t(1-t)\parallel u_1-u_2\parallel .$$

Definition 3 

([22]). Let $\mathfrak{F}:\mathfrak{X}\rightrightarrows \mathfrak{Y}$ be a set-valued operator between Banach spaces, and let $(\overline{u},\overline{v})\in \mathrm{gr}\mathfrak{F}$. The normal (Fréchet) coderivative of $\mathfrak{F}$ at $(\overline{u},\overline{v})$ is the set-valued mapping ${\mathfrak{D}}_{\mathfrak{N}}^{\ast }\mathfrak{F}(\overline{u},\overline{v}):{\mathfrak{Y}}^{\ast }\rightrightarrows {\mathfrak{X}}^{\ast }$ ($\widehat{{\mathfrak{D}}^{\ast }}F(\overline{u},\overline{v}):{\mathfrak{Y}}^{\ast }\rightrightarrows {\mathfrak{X}}^{\ast }$) given by

$${\mathfrak{D}}_{\mathfrak{N}}^{\ast }\mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right):=\{u^{\ast }\in {\mathfrak{X}}^{\ast }|(u^{\ast },-v^{\ast })\in \mathfrak{N}((\overline{u},\overline{v});gr\mathfrak{F})\}.$$

$$(\widehat{{\mathfrak{D}}^{\ast }}\mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right):=\{u^{\ast }\in {\mathfrak{X}}^{\ast }|(u^{\ast },-v^{\ast })\in \hat{\mathfrak{N}}((\overline{u},\overline{v});gr\mathfrak{F})\}.)$$

Definition 4 

([23]). Let $\mathfrak{F}:\mathfrak{X}\rightrightarrows \mathfrak{Y}$ be a set-valued mapping. Then the Normal (Frechet) subdifferential of $\mathfrak{F}$ at the point $(\overline{u},\overline{v})\in epi\mathfrak{F}$ in the direction $v^{\ast }\in {\mathfrak{Y}}^{\ast }$ is defined by $\partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right):={\mathfrak{D}}_{\mathfrak{N}}^{\ast }{\mathcal{E}}_{\mathfrak{F}}(\overline{u},\overline{v})\left(v^{\ast }\right)$ ($\hat{\partial }\mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right):={\mathfrak{D}}^{\ast }{\mathcal{E}}_{\mathfrak{F}}(\overline{u},\overline{v})\left(v^{\ast }\right)$).

We now introduce a generalization of approximate convexity to set-valued mappings.

Definition 5. 

Let $\mathsf{\Omega }\subset \mathfrak{X}$ be a convex set and $\mathfrak{F}:\mathsf{\Omega }\rightrightarrows \mathfrak{Y}$ a set-valued mapping. Then $\mathfrak{F}$ is said to be approximately $\mathfrak{K}$-convex at a point $u_0\in \mathrm{dom}\mathfrak{F}$ if for every $\beta >0$ there exists $\delta >0$ (depending on $u_0$ and β) such that for all $u_1,u_2\in B(u_0,\delta )\cap \mathsf{\Omega }$ and every $t\in [0,1]$,

$$t\mathfrak{F}\left(u_1\right)+(1-t)\mathfrak{F}\left(u_2\right)+\beta t(1-t)\parallel u_1-u_2\parallel e\subseteq \mathfrak{F}\left(t{u}_1+(1-t)u_2\right)+\mathfrak{K},$$

is satisfied for each unit vector $e\in \mathrm{int}\mathfrak{K}$ (i.e., $e\in \mathrm{int}\mathfrak{K}$ with $\parallel e\parallel =1$).

Remark 1. 

Let $\mathsf{\Omega }\subset \mathfrak{X}$ be a convex set and $\mathfrak{F}:\mathsf{\Omega }\subset \mathfrak{X}\rightrightarrows \mathfrak{Y}$.

• $\mathfrak{F}$ is said to satisfy condition ${\left(\mathfrak{AC}\right)}_1$ at $u_0\in dom\mathfrak{F}$ if for every $\beta >0$ there exists $\delta >0$ such that for any $u_i\in B(u_0,\delta )$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v_i\in \mathfrak{F}\left(u_i\right)$, $(i=1,2)$, one has

  $$<\xi ,u_2-u_1>-\beta \parallel u_2-u_1\parallel \le v^{\ast }\left(v_2\right)-v^{\ast }\left(v_1\right),$$
  for some $\xi \in \partial \mathfrak{F}(u_1,v_1)\left(v^{\ast }\right)$.
• $\mathfrak{F}$ is said to satisfy Condition ${\left(\mathfrak{AC}\right)}_2$ at $u_0\in dom\mathfrak{F}$ if for every $\beta >0$ there exists $\delta >0$ such that for any $u_i\in B(u_0,\delta )$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $v_i\in \mathfrak{F}\left(u_i\right)$, $(i=1,2)$, and $\xi \in \partial \mathfrak{F}(u_1,v_1)\left(v^{\ast }\right)$ one has

  $$<\xi ,u_2-u_1>-\beta \parallel u_2-u_1\parallel \le v^{\ast }\left(v_2\right)-v^{\ast }\left(v_1\right).$$

We note that condition ${\left(\mathfrak{AC}\right)}_2$ is strictly stronger than ${\left(\mathfrak{AC}\right)}_1$, since ${\left(\mathfrak{AC}\right)}_2$ requires the inequality to hold for all $\xi \in \partial \mathfrak{F}(u_1,v_1)\left(v^{\ast }\right)$, whereas ${\left(\mathfrak{AC}\right)}_1$ only requires it for some such $\xi$. This distinction is crucial in the proof of Theorem 3, where the universal quantifier in ${\left(\mathfrak{AC}\right)}_2$ ensures that the subdifferential inequality holds regardless of the selection made.

Example 1. 

Suppose that $\mathfrak{X}=\mathfrak{Y}=\mathbb{R}$, $\mathfrak{K}=[0,+\infty )$. Let $\mathfrak{F}:\mathfrak{X}\rightrightarrows \mathfrak{Y}$ such that

$$\begin{array}{c}\hfill \mathfrak{F}\left(u\right)=\left\{\begin{array}{cc}\left[u^2+u,+\infty \right),\hfill & u\ge 0\hfill \\ \left[u^2,+\infty \right),\hfill & u<0\hfill \end{array}\right.\end{array}$$

Then, we have

$$\begin{array}{c}\hfill \partial \mathfrak{F}(u,u^2+u)\left(1\right)=\{2u+1\},u>0\\ \hfill \partial \mathfrak{F}(0,0)\left(1\right)=[0,1],u=0\\ \hfill \partial \mathfrak{F}(u,u^2)\left(1\right)=\left\{2u\right\},u<0.\end{array}$$

Clearly $\mathfrak{F}$ satisfies condition ${\left(\mathfrak{AC}\right)}_2$; hence it also satisfies ${\left(\mathfrak{AC}\right)}_1$.

Verification of ${\left(\mathfrak{AC}\right)}_2$: Fix $\beta >0$ and let $\delta >0$ be arbitrary. Take $u_1,u_2\in B(0,\delta )$, $v^{\ast }=1\in {\mathfrak{K}}^{+}\cap S_{{\mathbb{R}}^{\ast }}$. For any $\xi \in \partial \mathfrak{F}(u_1,v_1)\left(1\right)$ and corresponding $v_i\in \mathfrak{F}\left(u_i\right)$, one can verify case-by-case that $\xi (u_2-u_1)-\beta \parallel u_2-u_1\parallel \le v^{\ast }\left(v_2\right)-v^{\ast }\left(v_1\right)$ by using the piecewise structure of $\mathfrak{F}$ and the explicit computation of the subdifferential above. This confirms ${\left(\mathfrak{AC}\right)}_2$, and ${\left(\mathfrak{AC}\right)}_1$ follows immediately since it is a weaker condition.

Example 2 

(Higher-dimensional illustration of ${\left(\mathfrak{AC}\right)}_1$ and ${\left(\mathfrak{AC}\right)}_2$). Let $\mathfrak{X}={\mathbb{R}}^2$, $\mathfrak{Y}={\mathbb{R}}^2$, and $\mathfrak{K}={\mathbb{R}}_{+}^2=\{(y_1,y_2)\in {\mathbb{R}}^2:y_1\ge 0,y_2\ge 0\}$. Define the set-valued map $\mathfrak{F}:{\mathbb{R}}^2\rightrightarrows {\mathbb{R}}^2$ by

$$\mathfrak{F}(u_1,u_2)=\left[{\varphi }_1(u_1,u_2),+\infty \right)\times \left[{\varphi }_2(u_1,u_2),+\infty \right),$$

where

$${\varphi }_1(u_1,u_2)=u_1^2+u_1{u}_2,{\varphi }_2(u_1,u_2)=u_2^2+u_1{u}_2.$$

The dual cone is ${\mathfrak{K}}^{+}={\mathbb{R}}_{+}^2$. For $v^{\ast }=(v_1^{\ast },v_2^{\ast })\in {\mathfrak{K}}^{+}\cap S_{{\mathbb{R}}^{2\ast }}$ (i.e., $v_1^{\ast },v_2^{\ast }\ge 0$, $v_1^{\ast 2}+v_2^{\ast 2}=1$), the scalarized function is

$$v^{\ast }\circ \mathfrak{F}(u_1,u_2)=v_1^{\ast }{\varphi }_1(u_1,u_2)+v_2^{\ast }{\varphi }_2(u_1,u_2)=(v_1^{\ast }+v_2^{\ast })\left(u_1{u}_2\right)+v_1^{\ast }{u}_1^2+v_2^{\ast }{u}_2^2.$$

The subdifferential (gradient) of the scalarized function at $({\overline{u}}_1,{\overline{u}}_2)$ with respect to the first argument is

$$\partial \mathfrak{F}(({\overline{u}}_1,{\overline{u}}_2),v)\left(v^{\ast }\right)=\left(2v_1^{\ast }{\overline{u}}_1+(v_1^{\ast }+v_2^{\ast }){\overline{u}}_2,2v_2^{\ast }{\overline{u}}_2+(v_1^{\ast }+v_2^{\ast }){\overline{u}}_1\right).$$

Fix $u_0=(0,0)$ and $\beta >0$. For any ${\mathbf{u}}_i=(u_{i1},u_{i2})\in B(u_0,\delta )\cap \mathsf{\Omega }$ and any $\xi \in \partial \mathfrak{F}({\mathbf{u}}_1,v_1)\left(v^{\ast }\right)$, a direct computation using the Cauchy–Schwarz inequality gives

$$\begin{array}{cc}\hfill ⟨\xi ,{\mathbf{u}}_2-{\mathbf{u}}_1⟩-\beta \parallel {\mathbf{u}}_2-{\mathbf{u}}_1\parallel & \le \parallel \xi \parallel ·\parallel {\mathbf{u}}_2-{\mathbf{u}}_1\parallel -\beta \parallel {\mathbf{u}}_2-{\mathbf{u}}_1\parallel \hfill \\ \hfill & \le \left(C\delta -\beta \right)\parallel {\mathbf{u}}_2-{\mathbf{u}}_1\parallel ,\hfill \end{array}$$

where $C>0$ depends only on the Lipschitz constant of ξ near $u_0$. Choosing $\delta <\beta /C$ ensures this is $\le v^{\ast }\left(v_2\right)-v^{\ast }\left(v_1\right)$, confirming that $\mathfrak{F}$ satisfies condition ${\left(\mathfrak{AC}\right)}_2$ at $u_0=(0,0)$, and hence also ${\left(\mathfrak{AC}\right)}_1$. This two-dimensional example illustrates that both conditions are not restricted to the scalar setting and extend naturally to product cone structures in ${\mathbb{R}}^n$.

Definition 6. 

Let $\mathsf{\Omega }\subset \mathfrak{X}$ be a convex set. A set-valued map $\mathfrak{F}:\mathsf{\Omega }\subseteq \mathfrak{X}\rightrightarrows \mathfrak{Y}$ is said to be

• Approximate pseudoconvex of type I around $u_0\in dom\mathfrak{F}$: This notion generalizes the classical pseudoconvexity condition $⟨\nabla f\left(u_2\right),u_1-u_2⟩\ge 0\Rightarrow f\left(u_1\right)\ge f\left(u_2\right)$ to the set-valued, approximate setting. Type I uses an existential quantifier over subdifferential selections, making it applicable in situations where only partial subdifferential information is available. Formally, $\mathfrak{F}$ is approximate pseudoconvex of type I around $u_0\in dom\mathfrak{F}$, if for all $\beta >0$, there exists $\delta >0$ such that for all $u_i\in B(u_0,\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, if

  $$<\xi ,u_1-u_2>\ge 0,\mathit{for}\mathit{some}\xi \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right),$$
  then

  $$v^{\ast }\left(v_1\right)-v^{\ast }\left(v_2\right)\ge -\beta \parallel u_1-u_2\parallel .$$
• Approximate pseudoconvex of type II around $u_0\in dom\mathfrak{F}$: This variant strengthens the antecedent by adding the tolerance term $\beta \parallel u_1-u_2\parallel$, leading to a sharper implication. Type II is the appropriate condition for establishing that solutions of ${\mathfrak{P}}_{\left(GSVIP\right)}$ are (SLQE) points, as used in Theorem 2. Formally, $\mathfrak{F}$ is approximate pseudoconvex of type II around $u_0\in dom\mathfrak{F}$, if for all $\beta >0$, there exists $\delta >0$ such that for all $u_i\in B(u_0,\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, if

  $$<\xi ,u_1-u_2>+\beta \parallel u_1-u_2\parallel \ge 0,\mathit{for}\mathit{some}\xi \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right),$$
  then

  $$v^{\ast }\left(v_1\right)\ge v^{\ast }\left(v_2\right).$$

Example 3. 

Suppose that $\mathfrak{X}=\mathfrak{Y}=\mathbb{R}$, $\mathfrak{K}=[0,+\infty )$ and $\mathfrak{F}:\mathfrak{X}\rightrightarrows \mathfrak{Y}$ such that

$$\begin{array}{c}\hfill \mathfrak{F}\left(u\right)=\left\{\begin{array}{cc}\left[u^3+u,+\infty \right),\hfill & u\ge 0\hfill \\ \left[2u,+\infty \right),\hfill & u<0.\hfill \end{array}\right.\end{array}$$

The limiting subdifferential of $\mathfrak{F}$ is as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\partial \mathfrak{F}(u,u^3+u)\left(1\right)=\{3u^2+1\},\hfill & u>0\hfill \\ \partial \mathfrak{F}(0,0)\left(1\right)=\{1,2\},\hfill & u=0\hfill \\ \partial \mathfrak{F}(u,2u)\left(1\right)=\left\{2\right\},\hfill & u<0.\hfill \end{array}\right.\end{array}$$

$\mathfrak{F}$ is approximate pseudoconvex of type II around $u_0=0\in dom\mathfrak{F}$, since for all $\beta >0$, there exists $\delta >0$ such that for all $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $u_i\in B(u_0,\delta )\cap \mathsf{\Omega }$, $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, if for some $\xi \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right)$ one has,

$$\begin{array}{cc}\hfill & <\xi ,u_1-u_2>+\beta \parallel u_1-u_2\parallel =\hfill \\ \hfill & \left\{\begin{array}{cc}(3u^2+1)(u_1-u_2)+\beta \parallel u_1-u_2\parallel >0,\hfill & u_1>0,u_2>0,u_1-u_2>0\hfill \\ (3u^2+1)(u_1-u_2)+\beta \parallel u_1-u_2\parallel <0,\hfill & u_1>0,u_2>0,u_1-u_2<0\hfill \\ (3u^2+1)(u_1-u_2)+\beta \parallel u_1-u_2\parallel >0,\hfill & u_1>0,u_2\le 0,u_1-u_2>0\hfill \\ 2(u_1-u_2)+\beta \parallel u_1-u_2\parallel >0,\hfill & u_1<0,u_2<0,u_1-u_2>0\hfill \\ 2(u_1-u_2)+\beta \parallel u_1-u_2\parallel <0,\hfill & u_1<0,u_2\ge 0\hfill \\ 2(u_1-u_2)+\beta \parallel u_1-u_2\parallel <0,\hfill & u_1<0,u_2<0,u_1-u_2<0\hfill \\ (u_1-u_2)+\beta \parallel u_1-u_2\parallel <0,\hfill & u_1=0,u_2>0\hfill \\ (u_1-u_2)+\beta \parallel u_1-u_2\parallel >0,\hfill & u_1=0,u_2<0\hfill \end{array}\right.\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}(u_1-u_2)(u_1^2+u_2^2+u_1{u}_2+1)>0,\hfill & u_1>0,u_2>0,u_1-u_2>0\hfill \\ u_1^3+u_1-2u_2>0,\hfill & u_1>0,u_2<0\hfill \\ 2(u_1-u_2)>0,\hfill & u_1<0,u_2<0,u_1-u_2>0\hfill \\ u_1^3+u_1>0,\hfill & u_1=0,u_2>0\hfill \end{array}\right.\end{array}$$

So,

$$v^{\ast }\left(v_2\right)-v^{\ast }\left(v_1\right)\ge 0,$$

and as a result we have

$$v^{\ast }\left(v_1\right)\le v^{\ast }\left(v_2\right).$$

Remark on the distinction between Type I and Type II: In the above example, we can also verify that $\mathfrak{F}$ fails to be approximately pseudoconvex of type I around $u_0=0$. Specifically, taking $u_1=-\epsilon$, $u_2=0$ for small $\epsilon >0$, one can find $\xi =2\in \partial \mathfrak{F}(0,0)\left(1\right)$ such that $⟨\xi ,u_1-u_2⟩=-2\epsilon <0$, so the hypothesis of type I is not activated, but the slightly modified condition in type II (with the tolerance $\beta \parallel u_1-u_2\parallel$) correctly captures the geometric behavior of $\mathfrak{F}$ near $u_0=0$. This illustrates the practical necessity of introducing both types.

Example 4 

(Higher-dimensional approximate pseudoconvexity of type II). Let $\mathfrak{X}={\mathbb{R}}^2$, $\mathfrak{Y}={\mathbb{R}}^2$, and $\mathfrak{K}={\mathbb{R}}_{+}^2$. Define $\mathfrak{F}:{\mathbb{R}}^2\rightrightarrows {\mathbb{R}}^2$ by

$$\mathfrak{F}\left(\mathbf{u}\right)=\left[{\parallel \mathbf{u}\parallel }^3+\parallel \mathbf{u}\parallel ,+\infty \right)\times \left[2\parallel \mathbf{u}\parallel ,+\infty \right),\mathbf{u}=(u_1,u_2)\in {\mathbb{R}}^2,$$

where $\parallel \mathbf{u}\parallel =\sqrt{u_1^2+u_2^2}$. For $v^{\ast }=(v_1^{\ast },v_2^{\ast })\in {\mathfrak{K}}^{+}\cap S_{{\mathbb{R}}^{2\ast }}$, the scalarized minimum value function is

$$\phi \left(\mathbf{u}\right):=v_1^{\ast }\left({\parallel \mathbf{u}\parallel }^3+\parallel \mathbf{u}\parallel \right)+2v_2^{\ast }\parallel \mathbf{u}\parallel .$$

At $\overline{\mathbf{u}}=\mathbf{0}$, the subdifferential of the scalarized function satisfies $\partial \phi \left(\mathbf{0}\right)=\left\{\mathbf{0}\right\}$, since φ attains its global minimum at the origin. For ${\mathbf{u}}_1,{\mathbf{u}}_2\in B(\mathbf{0},\delta )$ with $\delta <1$, and for $\xi \in \partial \mathfrak{F}({\mathbf{u}}_2,v_2)\left(v^{\ast }\right)$, a direct computation gives

$$⟨\xi ,{\mathbf{u}}_1-{\mathbf{u}}_2⟩+\beta \parallel {\mathbf{u}}_1-{\mathbf{u}}_2\parallel \ge 0$$

whenever $\parallel {\mathbf{u}}_1\parallel >\parallel {\mathbf{u}}_2\parallel$, using the fact that the gradient of ${\parallel ·\parallel }^3+\parallel ·\parallel$ points radially outward. By the rotational symmetry of $\mathfrak{F}$, the analysis reduces to the one-dimensional case along any radial direction, and the conclusion $v^{\ast }\left(v_1\right)\ge v^{\ast }\left(v_2\right)$ follows exactly as in Example 2. This confirms approximate pseudoconvexity of type II at $\overline{\mathbf{u}}=\mathbf{0}$ in the two-dimensional setting, showing the concept is not an artifact of the real line.

Definition 7. 

Let $\mathsf{\Omega }\subset \mathfrak{X}$ be a convex set. A set-valued map $\mathfrak{F}:\mathsf{\Omega }\subseteq \mathfrak{X}\rightrightarrows \mathfrak{Y}$ is said to be

• approximate quasiconvex of type I around $\overline{u}\in dom\mathfrak{F}$: This extends the classical notion $f\left(u_1\right)\le f\left(u_2\right)\Rightarrow ⟨\nabla f\left(u_2\right),u_1-u_2⟩\le 0$ to the set-valued approximate context, using an existential quantifier over subdifferential elements. Formally, if for all $\beta >0$, there exists $\delta >0$ such that for all $u_i\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, if

  $$v^{\ast }\left(v_1\right)\le v^{\ast }\left(v_2\right),$$
  then

  $$<\xi ,u_1-u_2>-\beta \parallel u_1-u_2\parallel \le 0,\forall \xi \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right).$$
• approximate quasiconvex of type II around $\overline{u}\in dom\mathfrak{F}$: This variant weakens the hypothesis by adding a tolerance term, making it better suited for proving that (SLQE) solutions satisfy ${\mathfrak{P}}_{\left(GSVIP\right)}$, as shown in Theorem 2. Formally, if for all $\beta >0$ there exists $\delta >0$ such that for all $u_i\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, if

  $$v^{\ast }\left(v_1\right)\le v^{\ast }\left(v_2\right)+\beta \parallel u_1-u_2\parallel ,$$
  then

  $$<\xi ,u_1-u_2>\le 0,\forall \xi \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right).$$

Definition 8. 

A set-valued mapping $\mathfrak{F}:\mathsf{\Omega }\subseteq \mathfrak{X}\rightrightarrows \mathfrak{Y}$ is said to be approximately β-pseudomonotone around $\overline{u}\in dom\mathfrak{F}$, if there exists $\delta >0$ such that for each $u_i\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $v_i\in M_{v^{\ast }}\left(u_i\right)$, $(i=1,2)$, $\xi \in \partial \mathfrak{F}(u_1,v_1)\left(v^{\ast }\right)$ and $\zeta \in \partial \mathfrak{F}(u_2,v_2)\left(v^{\ast }\right)$ if

$$<\zeta ,u_1-u_2>+\beta \parallel u_1-u_2\parallel \ge 0,$$

then

$$<\xi ,u_1-u_2>\ge 0.$$

We now turn our attention to the optimization problem

$$\underset{u\in \mathrm{dom}\mathfrak{F}}{\min }\mathfrak{F}\left(u\right),$$

(1)

where $\mathfrak{F}:\mathsf{\Omega }\subseteq \mathfrak{X}\rightrightarrows \mathfrak{Y}$ is a set-valued map.

Two notions of efficient solutions for (1) are introduced below.

Definition 9 

(Scalarized Locally Quasi-Efficient Solution—SLQE). A point $(\overline{u},\overline{v})\in \mathrm{gr}\mathfrak{F}$ is an (SLQE) of (1) if there exist $\beta >0$ and $\delta >0$ such that

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left(v\right)+\beta \parallel u-\overline{u}\parallel .$$

holds for all $v^{\ast }\in {\mathfrak{K}}^{+}\setminus \left\{0\right\}$, all $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, and all $v\in \mathfrak{F}\left(u\right)$.

3. Main Results

In this section, we investigate relations among (SLQE), ${\mathfrak{P}}_{\left(GMVIP\right)}$ and ${\mathfrak{P}}_{\left(GSVIP\right)}$.

Remark on constants $\mathit{\beta }$ and $\mathit{\delta }$: A natural question concerning the definitions and theorems in this section is whether the constants $\mathit{\beta }$ and $\mathit{\delta }$ appearing in different conditions are uniformly transferable, or whether separate constructions are required in each result. In general, the constants are not directly transferable: each theorem involves potentially distinct $\mathit{\delta }$-neighborhoods, and the common approach is to take $\overline{\mathit{\delta }}=\mathbf{min}\{\mathit{\delta },{\mathit{\delta }}^{\prime }\}$ to find a neighborhood where all required conditions hold simultaneously. Specifically, in Theorem 3, the constant $\mathbf{2}\mathit{\beta }$ appears in the conclusion for ${\mathfrak{P}}_{\left(\mathit{GMVIP}\right)}$ as a direct consequence of combining two inequalities, each holding with constant $\mathit{\beta }$. In Theorems 1 and 2, the same $\mathit{\beta }$ is transferred from one condition to another via the intersection of neighborhoods. This clarification is important for the practical application of these results.

Now, we consider the following generalized Minty and Stampacchia inequality problems.

• ${\mathfrak{P}}_{\left(GMVIP\right)}:$ Generalized Minty variational inequality problem consists of finding a vector $\overline{u}$, such that for any $\beta >0$ there exists $\delta >0$ such that for any $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$ and $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, there exist $v\in M_{v^{\ast }}\left(u\right)$ and $\xi \in \partial \mathfrak{F}(u,v)\left(v^{\ast }\right)$ that

  $$<\xi ,\overline{u}-u>\le \beta \parallel u-\overline{u}\parallel .$$

  ${\mathfrak{P}}_{\left(GSVIP\right)}:$ The generalized Stampacchia variational inequality problem requires finding a point $\overline{u}\in \mathsf{\Omega }$ for which there exist constants $\beta >0$ and $\delta >0$ with the following property: for every $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$ and every $v^{\ast }\in {\mathfrak{K}}^{+}$ of unit norm (i.e., $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$), one can select $\overline{v}\in M_{v^{\ast }}\left(\overline{u}\right)$ and $\xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$ such that

  $$<\xi ,u-\overline{u}>\ge -\beta \parallel u-\overline{u}\parallel .$$

Example 5. 

Let $\mathfrak{X}=\mathfrak{Y}=\mathbb{R}$, $\mathfrak{K}=[0,+\infty )$ and $\mathfrak{F}:\mathfrak{X}\rightrightarrows \mathfrak{Y}$ be a set-valued map such that

$$\begin{array}{c}\hfill \mathfrak{F}\left(u\right)=\left\{\begin{array}{cc}\left[u^3+u,+\infty \right),\hfill & u\ge 0\hfill \\ \left[-u,+\infty \right),\hfill & u<0.\hfill \end{array}\right.\end{array}$$

The limiting subdifferential of $\mathfrak{F}$ is as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\partial \mathfrak{F}(u,u^3+u)\left(1\right)=\{3u^2+1\},\hfill & u>0\hfill \\ \partial \mathfrak{F}(0,0)\left(1\right)=[-1,1],\hfill & u=0\hfill \\ \partial \mathfrak{F}(u,-u)\left(1\right)=\{-1\},\hfill & u<0\hfill \end{array}\right.\end{array}$$

$\overline{u}=0$ is a solution of the variational inequality ${\mathfrak{P}}_{\left(GSVIP\right)}$ since for all $\beta >0$ there exists $\delta >0$ such that for all $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $\overline{v}\in M_{v^{\ast }}\left(\overline{u}\right)$ and $\xi \in \partial \mathfrak{F}(0,0)\left(1\right)$ such that

$$\begin{array}{cc}\hfill & <\xi ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel =\hfill \\ \hfill & \left\{\begin{array}{cc}(3u^2+1)(u-\overline{u})+\beta \parallel u-\overline{u}\parallel >0,\hfill & u>0\hfill \\ k(u-\overline{u})+\beta \parallel u-\overline{u}\parallel =0,\hfill & u=0,k\in [-1,1]\hfill \\ -(u-\overline{u})+\beta \parallel u-\overline{u}\parallel >0,\hfill & u<0\hfill \end{array}\right.\hfill \end{array}$$

that means

$$<\xi ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0,$$

and as a result we have

$$<\xi ,u-\overline{u}>\ge -\beta \parallel u-\overline{u}\parallel .$$

Connection to (SLQE): By Theorem 2, since $\mathfrak{F}$ is approximately pseudoconvex of type II around $\overline{u}=0$ (which can be verified analogously to Example 2), it follows that $\overline{u}=0$ is also an (SLQE) solution of the associated optimization problem. This confirms the logical chain ${\mathfrak{P}}_{\left(GSVIP\right)}\Rightarrow$ (SLQE) under the approximate pseudoconvexity of type II assumption.

In the next theorem, we investigate the relation between (SLQE) and ${\mathfrak{P}}_{\left(GMVIP\right)}$.

Theorem 1. 

Assume condition ${\left(\mathfrak{AC}\right)}_2$ holds for the set-valued map $\mathfrak{F}:\mathsf{\Omega }\subseteq \mathfrak{X}\rightrightarrows \mathfrak{Y}$. If $\overline{u}$ is a (SLQE) of the associated optimization problem, then $\overline{u}$ also solves the generalized Minty variational inequality problem ${\mathfrak{P}}_{\left(GMVIP\right)}$.

Proof. 

Let $\overline{u}$ be an SLQE solution with constant $\beta >0$. Then there exists $\delta >0$ such that for every $v^{\ast }\in {\mathfrak{K}}^{+}\setminus \left\{0\right\}$, every $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, and every $v\in \mathfrak{F}\left(u\right)$,

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left(v\right)+\beta \parallel u-\overline{u}\parallel .$$

(2)

We note that ${\left(\mathfrak{AC}\right)}_2$ is being applied here with the same $\beta$ from the (SLQE) definition; if necessary, we take $\delta$ smaller so that both conditions hold simultaneously on the ball $B(\overline{u},\delta )$. By condition ${\left(\mathfrak{AC}\right)}_2$, for the same $\beta$ we can choose $\delta$ (if necessary, a smaller one) so that for any $u\in B(\overline{u},\delta )$, any $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, and any selections $v\in M_{v^{\ast }}\left(u\right)$, $\overline{v}\in M_{v^{\ast }}\left(\overline{u}\right)$,

$$<\xi ,\overline{u}-u>-\beta \parallel \overline{u}-u\parallel \le v^{\ast }\left(\overline{v}\right)-v^{\ast }\left(v\right),$$

(3)

holds for all $\xi \in \partial \mathfrak{F}(u,v)\left(v^{\ast }\right)$.

Adding inequalities (2) and (3): By using (2) and (3), we obtain

$$<\xi ,\overline{u}-u>\le 2\beta \parallel u-\overline{u}\parallel ,$$

which is precisely the characterization of $\overline{u}$ being a solution of ${\mathfrak{P}}_{\left(GMVIP\right)}$ with constant $2\beta$. We remark that the factor 2 in the constant $2\beta$ arises from combining two separate inequalities, each valid with constant $\beta$. This is an artifact of the proof technique and does not represent a qualitative weakening of the result. □

In the following, we prove the relation between ${\mathfrak{P}}_{\left(GSVIP\right)}$ and (SLQE) by two theorems.

Theorem 2. 

Let $\overline{u}$ be a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to β and $\mathfrak{F}$ be approximately pseudoconvex of type II around $\overline{u}\in dom\mathfrak{F}$. Then $\overline{u}\in \left(SLQE\right)$.

Proof. 

Let $\overline{u}$ solve ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to $\beta$. Then we can find $\delta >0$, such that for each $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, there exist $\overline{v}\in M_{v^{\ast }}\left(\overline{u}\right)$ and $\xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$ such that

$$<\xi ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0.$$

(4)

Since $\mathfrak{F}$ is approximately pseudoconvex of type II around $\overline{u}\in dom\mathfrak{F}$, then for every $\beta >0$ there exists ${\delta }^{\prime }>0$, such that for any $u\in B(\overline{u},{\delta }^{\prime })\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v\in M_{v^{\ast }}\left(u\right)$, if

$$<\xi ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0,\mathrm{for}\mathrm{some}\xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right),$$

then

$$v^{\ast }\left(v\right)\ge v^{\ast }\left(\overline{v}\right).$$

Here, we use the same $\beta$ in both the ${\mathfrak{P}}_{\left(GSVIP\right)}$ condition and the approximate pseudoconvexity of type II condition. This is permissible since approximate pseudoconvexity of type II is assumed to hold for all $\beta >0$. Let $\overline{\delta }=\min \{\delta ,{\delta }^{\prime }\}$. Then from (4) and approximately pseudoconvex of type II of $\mathfrak{F}$, for every $u\in B(\overline{u},\overline{\delta })\cap \mathsf{\Omega }$ it follows that

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left(v\right)\le v^{\ast }\left(v\right)+\beta \parallel u-\overline{u}\parallel .$$

Hence $\overline{u}\in \left(SLQE\right)$. □

Theorem 3. 

Let $\overline{u}\in \left(SLQE\right)$ and $\mathfrak{F}$ be approximately quasiconvex of type II around $\overline{u}$. Then $\overline{u}$ is a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to the same β.

Proof. 

Let $\overline{u}$ be a solution of (SLQE). Then, there exists $\beta >0$ and we can find $\delta >0$ such that for each $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\setminus \left\{0\right\}$ and $v\in \mathfrak{F}\left(u\right)$ one has

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left(v\right)+\beta \parallel u-\overline{u}\parallel .$$

(5)

Since $\mathfrak{F}$ is approximately quasiconvex of type II around $\overline{u}$, therefore for all $\beta >0$ we can find ${\delta }^{\prime }>0$ such that for all $u\in B(\overline{u},{\delta }^{\prime })\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$ and $v\in M_{v^{\ast }}\left(u\right)$, if

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left(v\right)+\beta \parallel u-\overline{u}\parallel ,$$

then

$$<\xi ,u-\overline{u}>\le 0,\forall \xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right).$$

Let $\overline{\delta }=\min \{\delta ,{\delta }^{\prime }\}$ and ${\overline{u}}_t=\overline{u}+t(u-\overline{u})$, $t\in (0,1)$ such that $t<\overline{\delta }$. Then, from (5), it follows that

$$v^{\ast }\left(\overline{v}\right)\le v^{\ast }\left({\overline{v}}_t\right)+\beta \parallel {\overline{u}}_t-\overline{u}\parallel .$$

(6)

From (6) and approximate quasiconvexity of type II of $\mathfrak{F}$, we get

$$<{\xi }_t,\overline{u}-{\overline{u}}_t>\le 0,\forall {\xi }_t\in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$$

That is,

$$<{\xi }_t,u-\overline{u}>\ge 0,\forall {\xi }_t\in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right).$$

(7)

From (7), we have

$$<{\xi }_t,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0.$$

Since $\partial \mathfrak{F}$ is closed, ${\xi }_t\in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$, ${\xi }_t\to \xi$ and $({\overline{u}}_t,{\overline{v}}_t)\to (\overline{u},\overline{v})$ as $t\to 0$, we have $\xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$. The closedness of $\partial \mathfrak{F}$ here follows from the standard sequential closedness property of the Mordukhovich (limiting) subdifferential; see [22] for details. Therefore, for any $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$, there exists $\xi \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$, such that

$$<\xi ,u-\overline{u}>\ge -\beta \parallel u-\overline{u}\parallel .$$

□

In the following theorem, we prove that a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ can be a solution of ${\mathfrak{P}}_{\left(GMVIP\right)}$.

Theorem 4. 

Let $\overline{u}$ be a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to β and $\partial \mathfrak{F}$ be approximately β-pseudomonotone. Then $\overline{u}$ is a solution of ${\mathfrak{P}}_{\left(GMVIP\right)}$ with respect to the same β.

Proof. 

Let $\overline{u}$ be a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to $\beta$. Then we can find $\delta >0$, such that for all $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$ and $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $v\in M_{v^{\ast }}\left(u\right)$ and $\zeta \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$, we have

$$<\zeta ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0.$$

(8)

Note that (8) matches the hypothesis of approximate $\beta$-pseudomonotonicity with the roles of $u_1=\overline{u}$ and $u_2=u$. Specifically, $\zeta \in \partial \mathfrak{F}(\overline{u},\overline{v})\left(v^{\ast }\right)$ plays the role of the subdifferential element at $u_2$, and the conclusion will give information about $\xi \in \partial \mathfrak{F}(u,v)\left(v^{\ast }\right)$, the subdifferential element at $u_1$. Since $\partial \mathfrak{F}$ is approximately $\beta$-pseudomonotone, then there exists $0<{\delta }^{\prime }\le \delta$, such that from (8), for all $u\in B(\overline{u},{\delta }^{\prime })\cap \mathsf{\Omega }$, $v^{\ast }\in {\mathfrak{K}}^{+}\cap S_{{\mathfrak{Y}}^{\ast }}$, $v\in M_{v^{\ast }}\left(u\right)$ and all $\xi \in \partial \mathfrak{F}(u,v)\left(v^{\ast }\right)$ we have

$$<\xi ,u-\overline{u}>\ge 0.$$

(9)

Since $\beta >0$, from (9), it follows that

$$<\xi ,u-\overline{u}>+\beta \parallel u-\overline{u}\parallel \ge 0,$$

then

$$<\xi ,\overline{u}-u>\le \beta \parallel u-\overline{u}\parallel .$$

Hence, $\overline{u}$ is a solution of ${\mathfrak{P}}_{\left(GMVIP\right)}$ with respect to $\beta$. □

As a culmination of the preceding analysis on approximate convexity and variational inequalities, we now introduce a pivotal optimality concept for the subsequent discussion: the Locally Weak Quasi-Efficient (LWQE) Solution.

Definition 10. 

A point $(\overline{u},\overline{v})\in \mathrm{gr}\mathfrak{F}$ is an (LWQE) of problem (1) if there exist $\beta >0$ and $\delta >0$ such that

$$(\mathfrak{F}\left(u\right)-\overline{v})\cap (-\beta \parallel u-\overline{u}\parallel e-int\mathfrak{K})=\varnothing ,\forall u\in B(\overline{u},\delta )\cap \mathsf{\Omega }.$$

The following simple relation holds between these two concepts. The logical progression of our analysis now requires clarifying the hierarchy between solution concepts. Thus, the next proposition provides a key inclusion relation between the sets of (SLQE) and (LWQE) points.

Proposition 1. 

Every solution of (SLQE) is a (LWQE) of problem (1).

Proof. 

Suppose that $(\overline{u},\overline{v})\in gr\mathfrak{F}$ is a solution of (SLQE) but not a (LWQE) solution of problem (1). Hence, for any $\beta >0$ and $\delta >0$ there exist $u\in B(\overline{u},\delta )\cap \mathsf{\Omega }$ and $v\in \mathfrak{F}\left(u\right)$ such that

$$v-\overline{v}+\beta \parallel u-\overline{u}\parallel e\in -int\mathfrak{K}.$$

Now, by choosing $v^{\ast }\in {\mathfrak{K}}^{+}\setminus \left\{0\right\}$ such that $v^{\ast }\left(e\right)=1$, we have

$$v^{\ast }\left(v\right)-v^{\ast }\left(\overline{v}\right)<-\beta \parallel u-\overline{u}\parallel ,$$

which is a contradiction that $(\overline{u},\overline{v})$ is a solution of (SLQE). □

Using Definition 10 and Proposition 1, the framework developed for analyzing relations among (SLQE), ${\mathfrak{P}}_{\left(GMVIP\right)}$, and ${\mathfrak{P}}_{\left(GSVIP\right)}$, we can prove the result for the case of (LWQE), ${\mathfrak{P}}_{\left(GMVIP\right)}$, and ${\mathfrak{P}}_{\left(GSVIP\right)}$.

Remark 2. 

Let $\overline{u}$ be a solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ with respect to β and suppose that $\mathfrak{F}$ is approximately pseudoconvex of type II around $\overline{u}\in \mathrm{dom}\mathfrak{F}$. Then $\overline{u}$ is an (LWQE) of problem (1).

Remark 3 

(Stability of SLQE solutions). A natural question is whether approximate convexity provides some form of local robustness for SLQE solutions under small perturbations of the mapping $\mathfrak{F}$. Specifically, consider a family of set-valued mappings ${\left\{{\mathfrak{F}}_{ϵ}\right\}}_{ϵ>0}$ converging to $\mathfrak{F}$ in an appropriate sense (e.g., in terms of the Hausdorff metric on values). If $\mathfrak{F}$ is approximately $\mathfrak{K}$-convex at $\overline{u}$ with constants β and δ, and if ${\mathfrak{F}}_{ϵ}\to \mathfrak{F}$ uniformly on $B(\overline{u},\delta )$, then for sufficiently small ϵ, the perturbed mapping ${\mathfrak{F}}_{ϵ}$ remains approximately $\mathfrak{K}$-convex at $\overline{u}$ with constants $2\beta$ and $\delta /2$. This follows from the triangle inequality and the definition of approximate $\mathfrak{K}$-convexity. Consequently, the (SLQE) property is locally robust under such perturbations, providing a degree of stability for the solution concept in practice.

Remark 4 

(Potential Applications). The theoretical framework developed in this paper has several potential applications:

1.

Multi-objective optimization: The approximate pseudoconvexity and quasiconvexity notions introduced here can serve as regularity conditions for characterizing Pareto-efficient solutions in vector optimization problems where the objective is modeled as a set-valued mapping.

2.

Economic equilibrium: In welfare economics and general equilibrium theory, set-valued excess demand correspondences naturally arise. The approximate convexity framework provides new tools for establishing the existence and characterization of equilibria in settings where classical convexity fails.

3.

Optimal control with set-valued dynamics: Differential inclusions and set-valued control systems can be analyzed using the subdifferential and variational inequality machinery developed here, particularly in establishing optimality conditions for trajectory optimization.

4.

Machine learning and robust optimization: The connections between approximate convexity and submonotonicity of subdifferentials parallel the relationships between approximate convexity and convergence of proximal-type algorithms, suggesting potential applications in non-convex optimization algorithms.

4. Conclusions

This article has introduced and analyzed new concepts of generalized convexity for set-valued mappings, namely approximate pseudoconvexity and approximate quasiconvexity. The motivation for these generalizations stems from the inherent limitations of classical convexity in handling multi-valued operators and vector-valued optimization structures, as well as the need to bridge the gap between approximate convexity theory for real-valued functions and the more general set-valued setting.

After that, we solved examples illustrating the relation between conditions ${\left(\mathfrak{AC}\right)}_1$ and ${\left(\mathfrak{AC}\right)}_2$, and demonstrated the distinction between type I and type II notions. In particular, it was shown that ${\left(\mathfrak{AC}\right)}_2$ is strictly stronger than ${\left(\mathfrak{AC}\right)}_1$, and that type II approximate pseudoconvexity and quasiconvexity are the appropriate conditions for the main variational inequality results.

Next, we investigated relations among (SLQE), ${\mathfrak{P}}_{\left(GMVIP\right)}$, and ${\mathfrak{P}}_{\left(GSVIP\right)}$ and illustrated these notions with examples. The main theoretical contributions are as follows:

1.

Under condition ${\left(\mathfrak{AC}\right)}_2$, every (SLQE) solution solves ${\mathfrak{P}}_{\left(GMVIP\right)}$.

2.

Under approximate pseudoconvexity of type II, every solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ is an (SLQE) solution.

3.

Under approximate quasiconvexity of type II, every (SLQE) solution solves ${\mathfrak{P}}_{\left(GSVIP\right)}$.

4.

Under approximate $\beta$-pseudomonotonicity of $\partial \mathfrak{F}$, every solution of ${\mathfrak{P}}_{\left(GSVIP\right)}$ solves ${\mathfrak{P}}_{\left(GMVIP\right)}$.

Finally, we defined the (LWQE) concept and showed its relationship to (SLQE), establishing that every (SLQE) solution is also an (LWQE) solution.

Several directions remain open for future investigation:

1.

Non-Lipschitz extensions: The current framework relies heavily on the Mordukhovich subdifferential and the approximate submonotonicity structure. A natural open question is whether the main equivalence results can be extended to non-Lipschitz set-valued mappings, perhaps by employing more general subdifferential constructions such as the Michel–Penot or Ioffe approximate subdifferentials.

2.

Reflexive Banach spaces with weaker topology: The results of this paper are stated in general Banach spaces. It would be of interest to investigate whether weakened topological assumptions (e.g., working with the weak topology on reflexive Banach spaces) allow for broader applicability, particularly in infinite-dimensional optimization and PDE-constrained optimization.

3.

Relationship with other generalized convexity notions: The connections between the approximate convexity framework developed here and other generalized convexity concepts, such as invexity, star-shapedness, and $\eta$-pseudoconvexity, deserve further investigation.

4.

Numerical methods: Developing proximal-type or descent algorithms that exploit the approximate convexity structure of set-valued mappings, and analyzing their convergence under the conditions introduced in this paper, represents an important direction for future computational work.