Optimality conditions for a class of variational inequalities with cone constraints

Abstract

In this paper, through the use of image space analysis, optimality conditions for a class of variational inequalities with cone constraints are proposed. By virtue of the nonlinear scalarization function, known as the Gerstewitz function, three nonlinear weak separation functions, two nonlinear regular weak separation functions and a nonlinear strong separation function are introduced. According to nonlinear separation functions, some optimality conditions of the weak and strong alternative for variational inequalities with cone constraints are derived.

Keywords

Variational inequalities with constraints / image space analysis / nonlinear separation function / optimality condition

1 Introduction and preliminary knowledge

As is well-known, variational inequalities have been widely applied in control theory, equilibrium problems, complementarity problems, fixed-point problems, and game theory, among others. In recent years, the theory and methods related to variational inequalities have gained increasing attention from scholars [5, 6, 11]. Castellani and Giannessi [2] first introduced mimetic space analysis, which serves as a very powerful tool in studying constrained extrema, classical variational inequalities, and Ky Fan inequalities, and has been further explored in [8]. Mimetic space analysis primarily studies optimization problems that can be equivalently expressed as the infeasibility of a parametric system and the separation of two sets in the mimetic space on constrained optimization problems. With the help of mimetic space analysis, Li and Huang [12] studied the optimality conditions for variational inequality problems and applied them to traffic equilibrium problems. Mastroeni et al. [14] studied saddle points and gap functions for vector quasi-equilibrium problems with cone constraints. Gu and Li [10] explored vector quasi-equilibrium problems and used linear separation functions to investigate the optimality conditions and error bounds of vector quasi-equilibrium problems. Based on mimetic space analysis, Chen et al. [4] studied vector-like variational inequalities with cone constraints, thereby deriving the optimality conditions and continuity of gap functions. Xu [15] employed mimetic space analysis to study nonlinear separation problems of inverse variational inequalities with cone constraints. However, it fails to successfully transform the variational inequality problem into the infeasibility form of a parametric system and separate the corresponding sets in its mimetic space in this paper, which is the core to study the problem with the help of mimetic space analysis. To resolve this issue, [15] will be improved in this paper.

Some symbols and definitions are reviewed first. The closure, relative interior, interior, and boundary of a set M\subseteq {\mathbb{R}}^n are denoted by clM, riM, intM and \mathrm{\partial }M, respectively. Let K\subset {\mathbb{R}}^n be an interior non-empty closed convex cone.

Given a function f:{\mathbb{R}}^n\to {\mathbb{R}}^n, the following considers variational inequality with cone constraints. Find x^{\ast }\in {\mathbb{R}}^n such that

(1.1)f\left(x^{\ast }\right)\in \mathrm{\Omega },(y-f(x^{\ast }){)}^{\mathrm{T}}{x}^{\ast }\ge 0,\mathrm{\forall }y\in \mathrm{\Omega },

where \mathrm{\Omega }:=\left\{y\in {\mathbb{R}}^n:g\left(y\right)\in D\right\},\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}g:{\mathbb{R}}^n\to {\mathbb{R}}^m being a vector-valued mapping, and D\subseteq {\mathbb{R}}^m an interior non-empty closed convex cone. In [15], the variational inequality (1.1) is also referred to as the inverse variational inequality.

The following shows the main characteristics of the image in the variational inequality (1.1). Given x^{\ast }\in {\mathbb{R}}^n, define the map

A_{x^{\ast }}:{\mathbb{R}}^n\to \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m,A_{x^{\ast }}\left(y\right):=\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right),y\in {\mathbb{R}}^n.

Consider the set

\mathcal{K}\left(x^{\ast }\right):=\left\{\right(u,v,\tau )\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m:(u,v,\tau )=A_{x^{\ast }}(y),y\in {\mathbb{R}}^n\},

\mathcal{H}:=\left\{\right(u,v,\tau )\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m:u>0,(v,\tau )\in D\times D\}.

\mathcal{K}\left(x^{\ast }\right) is called the image of the variational inequality (1.1), and \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m is referred to as the image space. Clearly, x^{\ast }\in {\mathbb{R}}^n is a solution to the variational inequality (1.1) if and only if f\left(x^{\ast }\right)\in \mathrm{\Omega } and the generalized system

(1.2)A_{x^{\ast }}\left(y\right)\in \mathcal{H},y\in {\mathbb{R}}^n

is infeasible, or equivalently, f\left(x^{\ast }\right)\in \mathrm{\Omega } and \mathcal{K}\left(x^{\ast }\right)\cap \mathcal{H}=\varnothing.

For a function h defined on a set X{\subseteq \mathbb{R}}^n and a\in \mathbb{R}, the sets

{\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge \alpha }h:=\left\{x\in X:h\left(x\right)\ge \alpha \right\},

{\mathrm{l}\mathrm{e}\mathrm{v}}_{>\alpha }h:=\{x\in X:h(x)>\alpha \}

are called the non-negative level set and the positive level set of the function h, respectively.

Definition 1.1 [9]　Given a function \omega :\mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathrm{\Pi }\to \mathbb{R}, the function \omega is called a weak separation function class, denoted as W\left(\mathrm{\Pi }\right), if it satisfies the following conditions:

(i) {\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}\omega (\cdot ;\pi )\supseteq \mathcal{H},\mathrm{\forall }\pi \in \mathrm{\Pi },

(ii) \bigcap _{\pi \in \mathrm{\Pi }} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\omega (\cdot ;\pi )\subseteq \mathcal{H},

where \mathrm{\Pi } is a certain parameter set.

Definition 1.2 [9]　Given a function \omega :\mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathrm{\Pi }\to \mathbb{R}, the function ω is called a regular weak separation function class, denoted as W_R\left(\mathrm{\Pi }\right), if it satisfies the condition: \bigcap _{\pi \in \mathrm{\Pi }} {lev}_{>0}\omega (\cdot ;\pi )=\mathcal{H}, where П is a certain parameter set.

Definition 1.3 [9]　Given a function s:\mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m\times \mathrm{\Pi }\to \mathbb{R}, the function s is called a strong separation function class, denoted as S\left(\mathrm{\Pi }\right), if it satisfies the following conditions:

(i) {\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}\omega (\cdot ;\pi )\supseteq \mathcal{H},\mathrm{\forall }\pi \in \mathrm{\Pi },

(ii) \bigcup _{\pi \in \mathrm{\Pi }} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)=\mathrm{r}\mathrm{i}\mathcal{H},

where П is a certain parameter set.

Note 1.1　When \text{int}\mathcal{H}\ne \varnothing, \text{ri}\mathcal{H}=\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}. Thus, when \text{int}\mathcal{H}\ne \varnothing, there is

\bigcup _{\pi \in \mathrm{\Pi }} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)=\mathrm{r}\mathrm{i}\mathcal{H}\iff \bigcup _{\pi \in \mathrm{\Pi }} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)=\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}.

Lemma 1.1 [1]　 Let Y be a normed space, and let A,E\subseteq Y and E is an interior non-empty convex cone. Then, \mathrm{i}\mathrm{n}\mathrm{t}\left(A+E\right)=A+\mathrm{i}\mathrm{n}\mathrm{t}E.

Given e\in -\mathrm{i}\mathrm{n}\mathrm{t}K, define the Gerstewitz function {\xi }_{e,K}:{\mathbb{R}}^n\to \mathbb{R} as {\xi }_{e,K}\left(y\right)=\mathrm{m}\mathrm{i}\mathrm{n}\{r\in \mathbb{R}:y\in re+K\},y\in {\mathbb{R}}^n.

Proposition 1.1 [3, 7, 13]　 For any given e\in -\mathrm{i}\mathrm{n}\mathrm{t}K, y\in {\mathbb{R}}^n, and r\in \mathbb{R}:

(i)

(ii) {\xi }_{e,K}\left(y\right)\le r\iff y\in re+K;

(iii) The function {\xi }_{e,K} is decreasing on {\mathbb{R}}^n, that is, y^2-y^1\in K\Rightarrow {\xi }_{e,K}\left(y^2\right)\le {\xi }_{e,K}\left(y^1\right).

2 Separation functions

This section introduces four nonlinear functions and discusses their properties.

(2.1){\omega }_1\left(u,v,\tau ;\theta ,\lambda ,\beta \right):=\theta u-\lambda {\xi }_{e,D}\left(v\right)-\beta {\xi }_{e,D}\left(\tau \right),\left(u,v,\tau \right)\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m,(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1,

where {\mathrm{\Pi }}_1:={\mathbb{R}}_{+}^3\setminus \left\{0_{{\mathbb{R}}^3}\right\},e\in -\mathrm{i}\mathrm{n}\mathrm{t}D.

(2.2){\omega }_2\left(z;\pi \right):=-{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left(z+\pi \right),z=\left(u,v,\tau \right)\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m,\pi \in {\mathrm{\Pi }}_2,

(2.3)s\left(z;\pi \right):=-{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left(z-\pi \right),z=\left(u,v,\tau \right)\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m,\pi \in {\mathrm{\Pi }}_2,

where {\mathrm{\Pi }}_2:=\mathrm{c}\mathrm{l}\mathcal{H},e^0\in -\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}.

(2.4)\begin{array}{rl}{\omega }_3(u,v,\tau ;\theta ,\lambda ,\beta ):=& \theta u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right),(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1,\end{array}

where e\in -\mathrm{i}\mathrm{n}\mathrm{t}D,r is a positive real number, and the expansion function \sigma :{\mathbb{R}}^m\to \mathbb{R} is upper semicontinuous with

\underset{z\in {\mathbb{R}}^m}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\sigma \left(z\right)=\left\{0_{{\mathbb{R}}^m}\right\},\sigma \left(0_{{\mathbb{R}}^m}\right)=0.

Proposition 2.1　(i) The nonlinear function {\omega }_1 is a weak separation function.

(ii) When (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_3={\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}, the nonlinear function {\omega }_1 is a regular weak separation function.

(iii) The nonlinear function {\omega }_2 is a weak separation function.

(iv) If \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}\ne \varnothing, then the nonlinear function s is a strong separation function.

(v) The nonlinear function {\omega }_3 is a weak separation function.

(vi) If (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_3, then the nonlinear function {\omega }_3 is a regular weak separation function.

Proof　(i) For any (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1,(u,v,\tau )\in \mathcal{H}, there is {\theta }_u\ge 0. From Proposition 1.1(ii), we know

{\omega }_1(u,v,\tau ;\theta ,\lambda ,\beta ):=\theta u-\lambda {\xi }_{e,D}\left(v\right)-\beta {\xi }_{e,D}\left(\tau \right)\ge 0.

Thus,

(2.5)\begin{array}{cc}{\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}{\omega }_1(\cdot ;\theta ,\lambda ,\beta )\supseteq \mathcal{H},& \mathrm{\forall }(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1.\end{array}

Now, it is required to prove

(2.6)\bigcap _{\left(\theta ,\lambda ,\beta \right)\in {\mathrm{\Pi }}_1} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_1\left(\cdot ;\theta ,\lambda ,\beta \right)\subseteq \mathcal{H}.

To prove this, it is sufficient to show that if (u,v,\tau )\notin \mathcal{H}, then (u,v,\tau )\notin \bigcap _{(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_1(\cdot ;\theta ,\lambda ,\beta ), i.e., when (u,v,\tau )\notin \mathcal{H}, there exists (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1 such that {\omega }_1(u,v,\tau ,\theta ,\lambda ,\beta )\le 0. Let (u,v,\tau )\notin \mathcal{H}. The following discusses the three cases.

Case 1: If u\le 0, and (v,\tau )\in D\times D, then there exists (\theta ,\lambda ,\beta )=(1,0,0)\in {\mathrm{\Pi }}_1 such that {\omega }_1(u,v,\tau ;\theta ,\lambda ,\beta )={\theta }_u=u\le 0.

Case 2: If u>0, and v\notin D,\tau \in D, then there exists (\theta ,\lambda ,\beta )=(0,1,0)\in {\mathrm{\Pi }}_1. According to Proposition 1.1(ii),

Case 3: if u>0, and v\in D,\tau \notin D, then there exists (\theta ,\lambda ,\beta )=(0,0,1)\in {\mathrm{\Pi }}_1. According to Proposition 1.1(ii),

According to Equations (2.5) and (2.6), {\omega }_1\in W\left({\mathrm{\Pi }}_1\right).

(ii) From (i), it is proved

(2.7)\bigcap _{\pi \in {\mathrm{\Pi }}_3} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_1\left(\cdot ;\pi \right)\supseteq \mathcal{H}.

Now, it is required to prove

(2.8)\bigcap _{\pi \in {\mathrm{\Pi }}_3} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_1\left(\cdot ;\pi \right)\subseteq \mathcal{H}.

If (u,v,\tau )\notin \mathcal{H}, there exists (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_3 such that {\omega }_1\left(u,v,\tau ;\theta ,\lambda ,\beta \right)\le 0. Let (u,v,\tau )\notin \mathcal{H}. The following discusses the three cases:

Case 1: If u\le 0, and (v,\tau )\in D\times D, assuming \pi =(1,0,0)\in {\prod }_3, {\omega }_1\left(u,v,\tau ;\pi \right)=u\le 0.

Case 2: If u>0,v\notin D, and \tau \in D, assuming \pi =(0,-\frac{{\xi }_{e,D}\left(\tau \right)}{{\xi }_{e,D}\left(v\right)},1). According to Proposition 1.1(ii), \pi \in {\mathrm{\Pi }}_3, and

{\omega }_1(u,v,\tau ;\pi )=\frac{{\xi }_{e,D}\left(\tau \right)}{{\xi }_{e,D}\left(v\right)}\cdot {\xi }_{e,D}\left(v\right)-{\xi }_{e,D}\left(\tau \right)=0.

Case 3: If u>0,v\in D, and \tau \notin D, we can assume \pi =(0,1,-\frac{{\xi }_{e,D}\left(v\right)}{{\xi }_{e,D}\left(\tau \right)}). According to Proposition 1.1(ii), \pi \in {\mathrm{\Pi }}_3, and

{\omega }_1(u,v,\tau ;\pi )=-{\xi }_{e,D}\left(v\right)+\frac{{\xi }_{e,D}\left(v\right)}{{\xi }_{e,D}\left(\tau \right)}\cdot {\xi }_{e,D}\left(\tau \right)=0.

From Equations (2.7) and (2.8), it is concluded that {\omega }_1\in W_R\left({\mathrm{\Pi }}_3\right).

(iii) For any \pi \in {\mathrm{\Pi }}_2 and (u,v,\tau )\in \mathcal{H}, there is (u,v,\tau )+\pi \in \mathcal{H}+\mathrm{c}\mathrm{l}\mathcal{H}=\mathrm{c}\mathrm{l}\mathcal{H}. Moreover, according to Proposition 1.1(ii), it is known that {\omega }_2(u,v,\tau ;\pi )\ge 0, which implies

(2.9)\bigcap _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}{\omega }_2\left(\cdot ;\pi \right)\supseteq \mathcal{H}.

Now, it is required to prove

(2.10)\bigcap _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_2\left(\cdot ;\pi \right)\subseteq \mathcal{H}.

In fact, let {\pi }_0=0_{{\mathbb{R}}^{m+1}}\in \mathrm{c}\mathrm{l}\mathcal{H}. According to Proposition 1.1(i), it is known

(2.11)\begin{array}{c}{\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_2\left(\cdot ;{\pi }_0\right)=\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}\subseteq \mathcal{H}.\end{array}

Furthermore, from \bigcap _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_2(\cdot ;\pi )\subseteq {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_2(\cdot ;{\pi }_0) and Equation (2.11), it is concluded that Equation (2.10) holds. Therefore, according to Equations (2.9) and (2.10), {\omega }_2\in W\left({\mathrm{\Pi }}_2\right).

(iv) From (iii), it is proved

(2.12){\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)\subseteq \mathcal{H},\mathrm{\forall }\pi \in {\mathrm{\Pi }}_2.

Now, it is required to prove

(2.13)\bigcup _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)=\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}.

First, it is required to prove

(2.14)\bigcup _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)\subseteq \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}.

In fact, take any (u,v,\tau )\in \bigcup _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s(\cdot ;\pi ). Then there exists {\pi }_0\in {\mathrm{\Pi }}_2 such that (u,v,\tau )\in {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s(\cdot ;{\pi }_0). According to Proposition 1.1(i), (u,v,\tau )-{\pi }_0\in \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}. Then, by Lemma 1.1, there is (u,v,\tau )\in \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}. Therefore, Equation (2.14) holds.

Next, it is required to prove

(2.15)\bigcup _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s\left(\cdot ;\pi \right)\supseteq \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}.

In fact, let {\pi }_0=0_{{\mathbb{R}}^{m+1}}\in \mathrm{c}\mathrm{l}\mathcal{H}. From Proposition 1.1(i), it is known that {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s(\cdot ;{\pi }_0)=\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}. Thus,

\mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}={\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s(\cdot ;{\pi }_0)\subseteq \bigcup _{\pi \in {\mathrm{\Pi }}_2} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}s(\cdot ;\pi ).

From Equations (2.14) and (2.15), it is concluded that Equation (2.13) holds. Therefore, based on Equations (2.12) and (2.13), s\in S\left({\mathrm{\Pi }}_2\right).

(v) For any (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1, and (u,v,\tau )\in \mathcal{H}, there is \theta u\ge 0. From Proposition 1.1(ii), it is known that 0_{{\mathbb{R}}^m}\in \left(\right\{v\}-D)\cap \left(\right\{\tau \}-D) and \sigma \left(0_{{\mathbb{R}}^m}\right)=0, and it is deduced that

\begin{array}{rl}{\omega }_3(u,v,\tau ;\theta ,\lambda ,\beta )=& \theta u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)\ge 0\text{,}\end{array}

which implies

(2.16)\bigcap _{\left(\theta ,\lambda ,\beta \right)\in {\mathrm{\Pi }}_1} {\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}{\omega }_3\left(\cdot ;\theta ,\lambda ,\beta \right)\supseteq \mathcal{H}.

Now, it is required to prove

(2.17)\bigcap _{\left(\theta ,\lambda ,\beta \right)\in {\mathrm{\Pi }}_1} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_3\left(\cdot ;\theta ,\lambda ,\beta \right)\subseteq \mathcal{H}.

To do this, if (u,v,\tau )\notin \mathcal{H}, there exists (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1 such that {\omega }_3\left(u,v,\tau ;\theta ,\lambda ,\beta \right)\le 0. Let (u,v,\tau )\notin \mathcal{H}. The following discusses the three cases:

Case 1: If u\le 0, and (v,\tau )\in D\times D, let \left(\theta ,\lambda ,\beta \right)=\left(1,0,0\right)\in {\mathrm{\Pi }}_1. Since \mathrm{a}\mathrm{r}\mathrm{g}\underset{}{{\mathrm{m}\mathrm{i}\mathrm{n}}_{z\in {\mathbb{R}}^m}} \sigma \left(z\right)=\left\{0_{{\mathbb{R}}^m}\right\},\sigma \left(0_{{\mathbb{R}}^m}\right)=0, it follows that \sigma \left(z\right)\ge 0 for all z\in {\mathbb{R}}^m. Also, since 0_{{\mathbb{R}}^m}\in \left(\right\{v\}-D)\cap \left(\right\{\tau \}-D) and \sigma \left(0_{{\mathbb{R}}^m}\right)=0, there is

\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-r\sigma \left(z\right)\right)=0,\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-r\sigma (t\left)\right)=0.

Therefore, it is obtained that {\omega }_3\left(u,v,\tau ;\theta ,\lambda ,\beta \right)=u\le 0.

Case 2: If u>0,v\notin D, and \tau \in D, let (\theta ,\lambda ,\beta )=(0,-\frac{{\xi }_{e,D}\left(\tau \right)}{{\xi }_{e,D}\left(v\right)},1). From Proposition 1.1(ii), it is known that (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1. Furthermore, by Proposition 1.1(iii) and since \sigma \left(z\right)\ge 0 for all z\in {\mathbb{R}}^m, it is deduced that

\begin{array}{rl}{\omega }_3(u,v,\tau ;\theta ,\lambda ,\beta )& \le \underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z\left)\right)+\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t\left)\right)\\ & \le -\lambda {\xi }_{e,D}\left(v\right)-\beta {\xi }_{e,D}\left(\tau \right)=0.\end{array}

Case 3: If u>0,v\in D, and \tau \notin D, let \left(\theta ,\lambda ,\beta \right)=\left(0,1,-\frac{{\xi }_{e,D\left(v\right)}}{{\xi }_{e,D\left(\tau \right)}}\right). Based on Proposition 1.1(ii), it is known that (\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1. Additionally, by Proposition 1.1(iii) and since \sigma \left(z\right)\ge 0 for all z\in {\mathbb{R}}^m, there is

{\omega }_3(u,v,\tau ;\theta ,\lambda ,\beta )\le -\lambda {\xi }_{e,D}\left(v\right)-\beta {\xi }_{e,D}\left(\tau \right)=0.

From Equations (2.16) and (2.17), it is concluded that {\omega }_3\in W\left({\mathrm{\Pi }}_1\right).

(vi) The proof for (vi) is similar to (v). □

Note 2.1　(i) From the definitions of {\omega }_1 and {\omega }_3, it is known that {\omega }_3\le {\omega }_1. In fact, by Proposition 1.1(iii) and since \sigma \left(z\right)\ge 0 for all z\in {\mathbb{R}}^m, there is

\begin{array}{rl}{\omega }_3(u,v,\tau ;\theta ,\lambda ,\beta )& =\theta u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)+\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)\\ & \le \theta u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z\left)\right)+\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t\left)\right)\\ & \le \theta u-\lambda {\xi }_{e,D}\left(v\right)-\beta {\xi }_{e,D}\left(\tau \right)\\ & ={\omega }_1(u,v,\tau ;\theta ,\lambda ,\beta ),\mathrm{\forall }(u,v,\tau )\in R\times {\mathbb{R}}^m\times {\mathbb{R}}^m,\mathrm{\forall }(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1.\end{array}

(ii) Let \overline{\theta }\in {\mathbb{R}}_{++}, and define

\overline{\omega }(u,v,\tau ;\overline{\theta },\lambda ,\beta )=\overline{\theta }u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)+\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right),

where \lambda ,\beta \in {\mathbb{R}}_{+}, and (u,v,\tau )\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m. Then the function \overline{\omega } is a regular weak separation function.

3 Optimality conditions

This section establishes some weak and strong selection theorems, and discusses the optimality conditions for the variational inequality (1.1).

Theorem 3.1　(i) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.1)

(ii) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.2)\mathrm{\exists }{\pi }_0\in {\mathrm{\Pi }}_2,s.t.{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left[\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)+{\pi }_0\right]>0,\mathrm{\forall }y\in {\mathbb{R}}^n.

(iii) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.3)

(iv) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.4)\begin{array}{c}\mathrm{\exists }{\pi }_0\in {\mathrm{\Pi }}_2,s.t.{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left[\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)-{\pi }_0\right]>0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

Proof　(i) Assume that System (1.2) is feasible. Then, there exists y\in {\mathbb{R}}^n such that \left(\right(f\left(x^{\ast }\right)-y{)}^{\mathrm{T}}{x}^{\ast },g\left(y\right),g\left(f\right(x^{\ast }\left)\right))\in \mathcal{H}. From Proposition 2.1(i), it is known that

\theta \left[\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right]-\lambda {\xi }_{e,D}\left(g\left(y\right)\right)-\beta {\xi }_{e,D}\left(g\left(f\left(x^{\ast }\right)\right)\right)\ge 0,\mathrm{\forall }(\theta ,\lambda ,\beta )\in {\mathrm{\Pi }}_1.

Thus, System (3.1) is infeasible.

Conversely, assume that the system (3.1) is feasible. Then,

That is,

\mathcal{K}\left(x^{\ast }\right)\cap \left\{\left(u,v,\tau \right)\in \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m:{\omega }_1\left(u,v,\tau ;\overline{\theta },\overline{\lambda },\overline{\beta }\right)\ge 0\right\}=\varnothing \text{,}

which implies \mathcal{K}\left(x^{\ast }\right)\cap {\mathrm{l}\mathrm{e}\mathrm{v}}_{\ge 0}{\omega }_1(\cdot ;\overline{\theta },\overline{\lambda },\overline{\beta })=\varnothing . From Proposition 2.1(i), it is known that \mathcal{K}\left(x^{\ast }\right)\cap \mathcal{H}=\varnothing, which means that for any y\in {\mathbb{R}}^n, A_{x^{\ast }}\left(y\right)\notin \mathcal{H}. Hence, the system (1.2) with respect to the variable y is infeasible.

(ii) By combining Proposition 2.1(iii) and the proof of part (i), the conclusion follows easily.

(iii) By combining Proposition 2.1(v) and the proof of part (i), the conclusion follows easily.

(iv) Assume that System (1.2) is infeasible. Then A_{x\ast }\left(y\right)\notin \mathcal{H},\mathrm{\forall }y\in {\mathbb{R}}^n. Therefore, from Proposition 2.1(iv), for any y\in {\mathbb{R}}^n, there exists {\pi }_0\in {\mathrm{\Pi }}_2 such that s\left(A_{x\ast }\left(y\right);{\pi }_0\right)\le 0, i.e.,

{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left[\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)-{\pi }_0\right]\ge 0,\mathrm{\forall }y\in {\mathbb{R}}^n.

Thus, System (3.4) is feasible.

Conversely, assume that System (3.4) is infeasible. Then, for any \pi \in {\mathrm{\Pi }}_2, there exists \overline{y}\in {\mathbb{R}}^n such that

i.e., s\left(A_{x\ast }\left(\overline{y}\right);\pi \right)>0. From Proposition 2.1(iv), it is known that A_{x\ast }\left(y\right)\in \mathcal{H}. Therefore, System (1.2) with respect to the variable y is feasible. □

Theorem 3.2　(i) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.5)\begin{array}{rl}\mathrm{\exists }(\overline{\theta },& \overline{\lambda },\overline{\beta })\in {\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},\\ & s.t.\overline{\theta }\left[\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right]-\overline{\lambda }{\xi }_{e,D}\left(g\left(y\right)\right)-\overline{\beta }{\xi }_{e,D}\left(g\left(f\left(x^{\ast }\right)\right)\right)\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

(ii) System (1.2) with respect to the variable y is not simultaneously feasible with the system

(3.6)\begin{array}{rl}\mathrm{\exists }(\overline{\theta },\overline{\lambda },\overline{\beta })\in & {\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},s.t.\overline{\theta }\left[\right(f\left(x^{\ast }\right)-y{)}^{\mathrm{T}}{x}^{\ast }]+\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}\left(z\right)\\ & -|r\sigma (z))+\underset{t\in g\left(f\left(x^{\ast }\right)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

Proof　The proof of this theorem is similar to the proof of Theorem 3.1(i). □

According to Theorems 3.1 and 3.2, it is derived the necessary and sufficient optimality conditions for the variational inequality (1.1).

Theorem 3.3　(i) Let f\left(x^{\ast }\right)\in \mathrm{\Omega } and there exists \left(\overline{\theta },\overline{\lambda },\overline{\beta }\right)\in {\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+} such that

(3.7)\overline{\theta }\left[\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right]-\overline{\lambda }{\xi }_{e,D}\left(g\left(y\right)\right)-\overline{\beta }{\xi }_{e,D}\left(g\left(f\left(x^{\ast }\right)\right)\right)\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.

Then x* is a solution to the variational inequality (1.1).

(ii) Let f\left(x^{\ast }\right)\in \mathrm{\Omega } and there exists \left(\overline{\theta },\overline{\lambda },\overline{\beta }\right)\in {\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+} such that

(3.8)\begin{array}{rl}\overline{\theta }\left[\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right]& +\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\overline{\lambda }{\xi }_{e,D}\left(z\right)-r\sigma \left(z\right)\right)\\ & +\underset{t\in g\left(f\left(x^{\ast }\right)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\overline{\beta }{\xi }_{e,D}\left(t\right)-r\sigma \left(t\right)\right)\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

Then x* is a solution to the variational inequality (1.1).

(iii) If x* is a solution to the variational inequality (1.1) and \mathrm{i}\mathrm{n}\mathrm{t}\mathcal{H}\ne \varnothing, then there exists \pi \in \mathrm{c}\mathrm{l}\mathcal{H} such that

(3.9){\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left[\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)-\pi \right]\ge 0,\mathrm{\forall }y\in {\mathbb{R}}^n.

Proof　By combining Theorem 3.1 and Theorem 3.2, the conclusion can be drawn. □

The following example illustrates the feasibility of Theorem 3.3.

Example 3.1　Let D={\mathbb{R}}_{+}^3, and e={(-1,-1,-1)}^{\mathrm{T}}. Define

g\left(y\right)=(y,y-3,f(y){)}^{\mathrm{T}}, so that \mathrm{\Omega }=[3,+\mathrm{\infty }).

Let x^{\ast }=\text{5}, then f\left(x^{\ast }\right)=3\in \mathrm{\Omega } and \left(f\left(x^{\ast }\right)-y\right)x^{\ast }=-5(y-3). Let \left(\overline{\theta },\overline{\lambda },\overline{\beta }\right)=(1,5,0). By the definition of {\xi }_{e,D}\left(y\right) and Note 2.1(i), it is known that

\begin{array}{rl}{\omega }_3(A_{x^{\ast }}(y);\overline{\theta },\overline{\lambda },\overline{\beta })& \le {\omega }_1(A_{x^{\ast }}(y);\overline{\theta },\overline{\lambda },\overline{\beta })\\ & =-5(y-3)-5{\xi }_{e,D}\left((y,y-3,f(y){)}^{\mathrm{T}}\right)\\ & =-5(y-3)+5(y-3)\\ & =0,y\in R.\end{array}

Thus, Equations (3.7) and (3.8) hold. By calculation, x^{\ast }=5 is a solution to the variational inequality (1.1).

Let \pi =(0,(0,0,0{)}^{\mathrm{T}},{(1,1,1)}^{\mathrm{T}})\in \mathrm{c}\mathrm{l}\mathcal{H}, then

\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)-\pi =\left(-5(y-3),g\left(y\right),(2,-1,0{)}^{\mathrm{T}}\right)\notin \mathcal{H}.

Based on Proposition 1.1(ii), it is known

{\xi }_{e^0,\mathrm{c}\mathrm{l}\mathcal{H}}\left[\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast },g(y),g(f\left(x^{\ast }\right))\right)-\pi \right]\ge 0,\mathrm{\forall }y\in \mathbb{R},

which implies that Equation (3.9) holds.

Using the nonlinear regular weak separation functions (3.1) and (3.4), the necessary and sufficient optimality conditions for the variational inequality (1.1) can be obtained.

Theorem 3.4　 Let \overline{\theta }\in {\mathbb{R}}_{++}. Then x^{\ast }\in {\mathbb{R}}^n is a solution to the variational inequality (1.1) if and only if f\left(x^{\ast }\right)\in \Omega and

(3.10)\begin{array}{rl}\underset{y\in {\mathbb{R}}^n}{\mathrm{s}\mathrm{u}\mathrm{p}}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& \left[\overline{\theta }\right({\left(f\right(x^{\ast })-y)}^{\mathrm{T}}{x}^{\ast })+\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}}(-\lambda {\xi }_{e,D}\left(z\right)-r\sigma \left(z\right))\\ & +\underset{t\in g\left(f\left(x^{\ast }\right)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}}(-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]=0,\end{array}

or

(3.11)\underset{y\in {\mathbb{R}}^n}{\mathrm{s}\mathrm{u}\mathrm{p}} \underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} \left[\overline{\theta }\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right)-\lambda {\xi }_{e,D}\left(g\right(y\left)\right)-\beta {\xi }_{e,D}\left(g\right(f\left(x^{\ast }\right)\left)\right)\right]=0.

Proof 　We only need to prove Equation (3.10); the proof for Equation (3.11) is similar. Let

\begin{array}{rl}{\omega }_4\left(u,v,\tau ;\overline{\theta },\lambda ,\beta \right):=& \overline{\theta }u+\underset{z\in \left\{v\right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\lambda {\xi }_{e,D}\left(z\right)-r\sigma \left(z\right)\right)\\ & +\underset{t\in \left\{\tau \right\}-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\beta {\xi }_{e,D}\left(t\right)-r\sigma \left(t\right)\right),\lambda ,\beta \in {\mathbb{R}}_{+}.\end{array}

First, it is required to prove

(3.12)\mathcal{H}=\bigcap _{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}} {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}{\omega }_4\left(\cdot ;\overline{\theta },\lambda ,\beta \right)\supseteq {\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4\left(\cdot ;\overline{\theta },\lambda ,\beta \right)\supseteq \mathcal{H}.

In fact, by Note 2.1(ii), it is known that {\omega }_4 is a regular weak separation function, so the first equality in Equation (3.12) holds. The second inclusion is obviously valid. Now it is required to prove

{\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(\cdot ;\overline{\theta },\lambda ,\beta )\supseteq \mathcal{H}.

For any (u,v,\tau )\in \mathcal{H}, by Proposition 1.1(ii) and 0_{{\mathbb{R}}^m}\in (\left\{v\right\}-D)\cap (\left\{\tau \right\}-D), there is

\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(u,v,\tau ;\overline{\theta },\lambda ,\beta )\ge & \underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} [\overline{\theta }u-\lambda {\xi }_{e,D}\left(0_{{\mathbb{R}}^m}\right)\\ & -\beta {\xi }_{e,D}\left(0_{{\mathbb{R}}^m}\right)-2r\sigma \left(0_{{\mathbb{R}}^m}\right)]=\overline{\theta }u>0.\end{array}

This implies that

{\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(\cdot ;\overline{\theta },\lambda ,\beta )\supseteq \mathcal{H}.

From Equation (3.12), it is known that

(3.13)\mathcal{H}={\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(\cdot ;\overline{\theta },\lambda ,\beta ).

If x^{\ast }\in {\mathbb{R}}^n is a solution to the variational inequality (1.1), then \mathcal{H}\cap \mathcal{K}\left(x^{\ast }\right)=\varnothing. According to Equation (3.13), there is

{\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(\cdot ;\overline{\theta },\lambda ,\beta )\cap \mathcal{K}\left(x^{\ast }\right)=\varnothing ,

which means

(3.14)\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& [\overline{\theta }\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right)+\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

In particular, when y=f\left(x^{\ast }\right), there is

\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& [\overline{\theta }\times 0+\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]\le 0.\end{array}

On the other hand, since 0_{{\mathbb{R}}^m}\in g\left(f\left(x^{\ast }\right)\right)-D, there is

\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& [\overline{\theta }\times 0+\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]\\ \ge & \underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} [-\lambda {\xi }_{e,D}(0_{{\mathbb{R}}^m})-\beta {\xi }_{e,D}(0_{{\mathbb{R}}^m})-2r\sigma (0_{{\mathbb{R}}^m}\left)\right]=0.\end{array}

Thus, there is

\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& [\overline{\theta }\times 0+\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]=0.\end{array}

By combining this with Equation (3.14), it is concluded that Equation (3.10) holds.

Conversely, assume that Equation (3.10) holds. Then, there is

\begin{array}{rl}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}}& [\overline{\theta }\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right)+\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)]\le 0,\mathrm{\forall }y\in {\mathbb{R}}^n.\end{array}

At the same time, from Equation (3.12), it is known that

{\mathrm{l}\mathrm{e}\mathrm{v}}_{>0}\underset{\lambda \in {\mathbb{R}}_{+},\beta \in {\mathbb{R}}_{+}}{\mathrm{i}\mathrm{n}\mathrm{f}} {\omega }_4(\cdot ;\overline{\theta },\lambda ,\beta )\supseteq \mathcal{H}.

Therefore, \mathcal{H}\cap \mathcal{K}\left(x^{\ast }\right)=\varnothing, and since f\left(x^{\ast }\right)\in \mathrm{\Omega }, it is concluded that x^{\ast }\in {\mathbb{R}}^n is a solution to the variational inequality (1.1). □

Next, the definition of nonlinear separation is provided to proceed to investigate the optimality conditions for the variational inequality (1.1).

Definition 3.1　The sets \mathcal{K}\left(x^{\ast }\right) and \mathcal{H} satisfy nonlinear separation if and only if there exist (\overline{\theta },\overline{\lambda },\overline{\beta })\in {\mathbb{R}}_{+}^3\setminus \left\{0_{{\mathbb{R}}^3}\right\} such that

(3.15)\begin{array}{rl}{\omega }_4\left(A_{x^{\ast }}\right(y);\overline{\theta },\overline{\lambda },\overline{\beta })=& \overline{\theta }\left(\left(f\right(x^{\ast })-y{)}^{\mathrm{T}}{x}^{\ast }\right)+\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}}(-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)\\ & +\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\le 0,y\in {\mathbb{R}}^n.\end{array}

Additionally, if \overline{\theta }>0, the separation is called regular.

Proposition 3.1　 If \mathcal{K}\left(x^{\ast }\right) and \mathcal{H}satisfy regular nonlinear separation and f\left(x^{\ast }\right)\in \mathrm{\Omega }, then x^{\ast }\in {\mathbb{R}}^n is a solution to the variational inequality (1.1).

Proof　This proof is similar to Proposition 4.1 in literature [15]. □

Assume x^{\ast }\in {\mathbb{R}}^n. Define the generalized Lagrange function \mathcal{L}:{\mathbb{R}}^n\times {\mathbb{R}}_{+}^3\setminus \left\{0_{{\mathbb{R}}^3}\right\}\to \mathbb{R} as

(3.16)\begin{array}{rl}L(x^{\ast },y;\theta ,\lambda ,\beta ):=& \theta (y-f(x^{\ast }){)}^{\mathrm{T}}{x}^{\ast }-\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}\left(z\right)-r\sigma \left(z\right))\\ & -\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right),y\in {\mathbb{R}}^n,(\theta ,\lambda ,\beta )\in {\mathbb{R}}_{+}^3\setminus \left\{0_{{\mathbb{R}}^3}\right\}.\end{array}

Theorem 3.5　 Let f\left(x^{\ast }\right)\in \mathrm{\Omega }. The following two statements are equivalent:

(i) \mathcal{K}\left(x^{\ast }\right) and \mathcal{H} satisfy nonlinear separation.

(ii) There exist \left(\overline{\theta },\overline{\lambda },\overline{\beta }\right)\in {\mathbb{R}}_{+}^3\mathrm{\setminus }\left\{0_{{\mathbb{R}}^3}\right\} such that (f\left(x^{\ast }\right),\overline{\lambda },\overline{\beta }) is a saddle point of \mathcal{L}(x^{\ast },y;\overline{\theta },\lambda ,\beta ) on {\mathbb{R}}^n\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}, i.e.,

(3.17)\mathcal{L}\left(x^{\ast },f\left(x^{\ast }\right);\overline{\theta },\lambda ,\beta \right)\le \mathcal{L}\left(x^{\ast },f\left(x^{\ast }\right);\overline{\theta },\overline{\lambda },\overline{\beta }\right)\le \mathcal{L}\left(x^{\ast },y;\overline{\theta },\overline{\lambda },\overline{\beta }\right),\mathrm{\forall }y\in {\mathbb{R}}^n,\lambda ,\beta \in {\mathbb{R}}_{+}.

Proof　(i)\Rightarrow(ii). Assume that \mathcal{K}\left(x^{\ast }\right) and \mathcal{H} satisfy nonlinear separation. Then there exist (\overline{\theta },\overline{\lambda },\overline{\beta })\in {\mathbb{R}}_{+}^3\setminus \left\{0_{{\mathbb{R}}^3}\right\} such that Equation (3.15) holds. In Equation (3.15), let y=f\left(x^{\ast }\right). Then

\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)+\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\le 0.

Since f\left(x^{\ast }\right)\in \mathrm{\Omega }, there is 0_{{\mathbb{R}}^m}\in g\left(f\left(x^{\ast }\right)\right)-D, and thus

\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\lambda {\xi }_{e,D}\left(z\right)-r\sigma \left(z\right)\right)+\underset{t\in g\left(f\left(x^{\ast }\right)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\beta {\xi }_{e,D}\left(t\right)-r\sigma \left(t\right)\right)\ge 0,\mathrm{\forall }\lambda ,\beta \in {\mathbb{R}}_{+}.

Therefore,

\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)+\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)=0.

Since

\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\lambda {\xi }_{e,D}\left(z\right)-r\sigma \left(z\right)\right)+\underset{t\in g\left(f\left(x^{\ast }\right)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} \left(-\beta {\xi }_{e,D}\left(t\right)-r\sigma \left(t\right)\right)\ge 0,\mathrm{\forall }\lambda ,\beta \in {\mathbb{R}}_{+}.

There is

\begin{array}{rl}L(x^{\ast },& f(x^{\ast });\overline{\theta },\lambda ,\beta )\\ & =\overline{\theta }\times 0-\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)-\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)\\ & \le \overline{\theta }\times 0-\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)-\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\\ & =L(x^{\ast },f(x^{\ast });\overline{\theta },\overline{\lambda },\overline{\beta }),\mathrm{\forall }\lambda ,\beta \in {\mathbb{R}}_{+}.\end{array}

Next, it is required to prove the second inequality of Equation (3.17). In fact, since Equation (3.15) holds, there is

\mathcal{L}\left(x^{\ast },y;\overline{\theta },\overline{\lambda },\overline{\beta }\right)\ge 0=\mathcal{L}\left(x^{\ast },f\left(x^{\ast }\right);\overline{\theta },\overline{\lambda },\overline{\beta }\right),\mathrm{\forall }y\in {\mathbb{R}}^n.

(ii)\Rightarrow(i). Assume that f\left(x^{\ast }\right);\overline{\lambda },\overline{\beta }) is a saddle point of \mathcal{L}(x^{\ast },y;\overline{\theta },\lambda ,\beta ) on {\mathbb{R}}^n\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}, i.e., for all {y\in \mathbb{R}}^n, and \lambda ,\beta \in {\mathbb{R}}_{+},

(3.18)\begin{array}{rl}& \overline{\theta }\times 0-\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\lambda {\xi }_{e,D}(z)-r\sigma (z\left)\right)-\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\beta {\xi }_{e,D}(t)-r\sigma (t\left)\right)\\ & \le \overline{\theta }\times 0-\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)-\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\\ & \le \overline{\theta }(y-f(x^{\ast }){)}^{\mathrm{T}}{x}^{\ast }-\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}\left(z\right)-r\sigma \left(z\right))\\ & -\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}\left(t\right)-r\sigma \left(t\right)).\end{array}

In the first inequality of Equation (3.18), let \lambda ,\beta =0. Then

\underset{z\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}(z)-r\sigma (z\left)\right)+\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}(t)-r\sigma (t\left)\right)\le 0.

From the second inequality of Equation (3.18), it is obtained

\begin{array}{rl}\overline{\theta }(y-f(x^{\ast }){)}^{\mathrm{T}}{x}^{\ast }& -\underset{z\in g\left(y\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\lambda }{\xi }_{e,D}\left(z\right)-r\sigma \left(z\right))\\ & -\underset{t\in g\left(f\right(x^{\ast }\left)\right)-D}{\mathrm{s}\mathrm{u}\mathrm{p}} (-\overline{\beta }{\xi }_{e,D}\left(t\right)-r\sigma \left(t\right))\ge 0.\end{array}

Therefore, Equation (3.15) holds, and \mathcal{K}\left(x^{\ast }\right) and \mathcal{H} satisfy nonlinear separation. □

Corollary 3.1　 Let f\left(x^{\ast }\right)\in \mathrm{\Omega } and there exist (\overline{\theta },\overline{\lambda },\overline{\beta })\in {\mathbb{R}}_{++}\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+} such that (f\left({\mathrm{x}}^{\ast }\right),\overline{\lambda },\overline{\beta }) is a saddle point of \mathcal{L}(x^{\ast },y;\overline{\theta },\lambda ,\beta ) on {\mathbb{R}}^n\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}, then x^{\ast } is a solution to the variational inequality (1.1).

Proof　By combining Theorem 3.5 and Proposition 3.1, the conclusion follows. □

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