A Generalized $\Gamma$-Convergence Concept for a Class of Equilibrium Problems

Abstract

A novel generalization of $\Gamma$-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our $\Gamma$-convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature.

1 Introduction

Equilibrium problems form an umbrella of various problems relevant in applied mathematics (see Kassay and Rădulescu 2018, Chapter 2 for an overview). Among them, especially optimization problems are a powerful tool for various applications. Thus, it is not surprising that equilibrium problems such as Nash equilibrium problems and (quasi-)variational inequalities (abbr.: (Q)VIs) enjoyed growing interest in the recent literature (cf. Facchinei and Kanzow 2007; Pang and Fukushima 2005 for an overview, as well as the recent articles Hintermüller and Surowiec 2013; Hintermüller et al. 2015; Kanzow et al. 2019; Alphonse et al. 2019). Many of these problems contain constraints or suffer from a lack of smoothness and are often addressed via a sequence of more regular problems. Such a sequential approximation aims at approaching the original problem in the limit via a sequence of more tractable approximating problems. A powerful concept in this context utilized in optimization is $\Gamma$-convergence (cf. Braides 2002; Dal Maso 2012, see also epiconvergence, e.g., in Attouch 1984).

Let a reflexive Banach space U be given. Consider a subset $U_{\textrm{ad}}\ \subseteq U$ and functionals $({\mathcal {E}}^n)_{n\in {\mathbb {N}}}, {\mathcal {E}}: U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}$. The sequence $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}$ is called $\Gamma$-convergent to ${\mathcal {E}}$, denoted by ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$, if the following two conditions are fulfilled:

• (i) For all sequences $u^n \rightarrow u$ holds ${\mathcal {E}}(u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n)$.

• (ii) For all $v \in U_{\textrm{ad}}{}$ exists a sequence $v^n \rightarrow v$ with $\limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n) \le {\mathcal {E}}(v)$,

where the sequences are chosen in $U_{\textrm{ad}}$.

The above displayed notion of $\Gamma$-convergence is tailored to optimization problems and in principle not immediately applicable to broader problem classes like general (quasi-)variational inequalities or Nash games. The main difficulty therein is the dependence of the constraint set on the solution, respectively, of the objective on the strategies of the other players. This article provides a generalization of $\Gamma$-convergence to equilibrium problems covering the aforementioned applications.

The rest of this work is organized as follows: In Sect. 2, we introduce and discuss the equilibrium concept addressed in this work and provide applications embedding into our concept. In Sect. 3, we derive an abstract existence result and apply it to Nash games. Returning to the applications discussed in Sect. 2, we derive our generalized $\Gamma$-convergence notion in Sect. 4. The work is finished with the comparison of our approach to previous ones that can be found in the literature in Sect. 5.

2 Equilibrium Problems

First, we introduce the type of equilibrium problem under investigation in this article. For this sake, we denote for a given functional $f:U \rightarrow {\mathbb {R}}\cup \{\infty \}$ the domain by ${\text {dom}}\left( f\right) := \{ u \in U: f(u) < \infty \}$.

Definition 1

Consider a reflexive Banach space U and a subset $U_{\textrm{ad}}\subseteq U$ as well as a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow \overline{{\mathbb {R}}}$ with ${\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right) \ne \emptyset$ for all $u \in U_{\textrm{ad}}$. A point $u \in U_{\textrm{ad}}$ is called an equilibrium (of ${\mathcal {E}}$) if

$$\begin{aligned} {\mathcal {E}}(u,u) \le {\mathcal {E}}(v,u) \text { holds for all } v \in U_{\textrm{ad}}. \end{aligned}$$

(1)

Occasionally, we call the first argument in ${\mathcal {E}}$ the control component and the second one the feedback component. Evidently, optimization problems are a special case of equilibrium problems without feedback component. In Blum and Oettli (1994), the term equilibrium problem has been introduced for problems of the following kind:

Let a set $C \subseteq U$ and a bifunction $\Psi : C \times C \rightarrow {\mathbb {R}}$ be given. Seek $u \in C$ such that

$$\begin{aligned} \Psi (u,v) \le 0 \text { for all } v \in C. \end{aligned}$$

There exists a generalization of this problem called quasi-equilibrium problem (cf. Noor and Oettli 1994), where a dependence of the feasible set on the selected point occurs. Thus, a set-valued operator $C:U \rightrightarrows U$ is considered. The problem reads as follows: Seek $u \in C(u)$ such that

$$\begin{aligned} \Psi (u,v) \le 0 \text { for all } v \in C(u). \end{aligned}$$

Both equilibrium and quasi-equilibrium problems have been extensively discussed in the literature with special emphasis on providing existence results. Here, we refer again to Kassay and Rădulescu (2018) and Aussel et al. (2017). Using the difference of the right and left hand side in (1) we can introduce the Nikaido–Isoda functional (compare to Nikaidô and Isoda 1955) reading for $u,v \in U_{\textrm{ad}}$ with $v{} \in {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ as

$$\begin{aligned} \Psi (u,v) = {\mathcal {E}}(u,u) - {\mathcal {E}}(v,u). \end{aligned}$$

(2)

In comparison with Blum and Oettli (1994) and above, the Nikaido–Isoda functional provides the link between (quasi)-equilibrium problems and the one in Definition 1. Our approach addresses both of them, as the set-valued mapping can be hidden in the (feedback dependent) domain. The Nikaido–Isoda functional allows an optimization reformulation, which is given in the next theorem.

Theorem 2

(compare to Nikaidô and Isoda 1955, Lemma 3.1) Let a functional ${\mathcal {E}}$ as in Definition 1 be given. Then, u is an equilibrium, if and only if $V(u) = 0$, where $V: U_{\textrm{ad}}\rightarrow [0,\infty ]$ denotes the value function defined by

$$\begin{aligned} V(u) = \sup _{v \in {\text {dom}}\left( {\mathcal {E}}(\cdot , u)\right) } \Psi (u,v) \end{aligned}$$

for given $u \in U_{\textrm{ad}}$ with $\Psi$ being the Nikaido–Isoda functional defined in (2).

Proof

Let $u \in U_{\textrm{ad}}$ be an equilibrium of ${\mathcal {E}}$. By Definition 1, there holds ${\mathcal {E}}(u,u) \le {\mathcal {E}}(v,u)$ for all $v \in U_{\textrm{ad}}$. Thus, as ${\text {dom}}\left( {\mathcal {E}}(\cdot ,u)\right) \ne \emptyset$ also ${\mathcal {E}}(u,u) < \infty$, respectively, $u \in {\text {dom}}\left( {\mathcal {E}}(\cdot ,u)\right)$ hold true. This yields

$$\begin{aligned} 0 = {\mathcal {E}}(u,u) - {\mathcal {E}}(u,u) = \Psi (u,u) \le V(u) = \sup _{v \in {\text {dom}}\left( {\mathcal {E}}(\cdot ,u)\right) } \Psi (u,v) \le 0, \end{aligned}$$

which implies $V(u) = 0$.

For the other direction, assume $V(u) = 0$. Choosing an arbitrary $v \in {\text {dom}}\left( {\mathcal {E}}(\cdot ,u)\right)$ yields

$$\begin{aligned} {\mathcal {E}}(u,u) - {\mathcal {E}}(v,u) \le V(u) = 0. \end{aligned}$$

Thus, u is an equilibrium. $\square$

The formulation in Definition 1 emphasizes the optimization-related structure in comparison with Blum and Oettli (1994). For every $u \in U_{\textrm{ad}}$, we require the optimization problem $\min {\mathcal {E}}(\, \cdot \, ,u)$ to have a solution. This motivates the following definition.

Definition 3

(Best response operator, see Kanzow and Schwartz 2018, Definition 1.7) Consider a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}$ as in Definition 1. The best response operator is the set-valued mapping ${\mathcal {B}}: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$ defined by

$$\begin{aligned} {\mathcal {B}}(u):= {{\text {argmin}}}_{v \in U_{\textrm{ad}}} {\mathcal {E}}(v,u). \end{aligned}$$

Within this work, we assume the best response operator to have nonempty values. By Definition 1, a point $u \in U_{\textrm{ad}}$ is an equilibrium of ${\mathcal {E}}$, if and only if

$$\begin{aligned} u \in {\mathcal {B}}(u) \end{aligned}$$

holds true. Thus, the equilibrium problem proposed in Definition 1 can equivalently be interpreted as the minimization of a merit functional as proposed in Theorem 2 or as a fixed point problem of a set-valued operator. Regarding existence, the latter will be of importance for us. But before addressing this issue, we draw our attention to a few examples that show the applicability of the introduced concept.

2.1 Partial Differential Equations

Consider the following partial differential equation (abbr.: PDE) related to a Ginzburg–Landau model for superconductivity in the absence of a magnetic field, see Tinkham (2004, Chapters 1 and 4). For given $\alpha > 0$, seek $u \in H_0^1(\Omega )$ such that

$$\begin{aligned} -\Delta u + u^3 - \alpha u&= 0 \text { in } \Omega ,\\ u&= 0 \text { on } \partial \Omega , \end{aligned}$$

(PDE)

where $\Omega \subseteq {\mathbb {R}}^d$, $d \le 3$ denotes an open, bounded domain with boundary $\partial \Omega$. Besides its application in physics, this example is of mathematical interest, since it contains a nonmonotone operator. One can indeed interpret it as the first-order system of an optimization problem. However, (PDE) does not need to be a sufficient optimality condition. Moreover, one can show that for sufficiently large $\alpha$, this system has besides its trivial solution as well another nontrivial one, see the techniques in Badiale and Serra (2010, Section 2.3.2). As the operator is odd, (PDE) has at least three solutions. The trivial solution might not be local minimizer. Thus, this equation cannot be approached by the sole use of optimization techniques to obtain all solutions. Therefore, we embed this equation into the equilibrium framework of Definition 1:

Consider for an arbitrary $u \in H_0^1(\Omega )$ the following equation. Seek $v \in H_0^1(\Omega )$, such that

$$\begin{aligned} -\Delta v + v^3= & {} \alpha u \text { in } \Omega ,\nonumber \\ v= & {} 0 \text { on } \partial \Omega . \end{aligned}$$

(3)

Here, $H_0^1(\Omega )$ denotes the classical Sobolev space as defined, e.g., in Adams and Fournier (2003, Definition 3.2). Using standard arguments of calculus of variations and convex analysis, it is straightforward to see that (3) can be interpreted as a first-order system of a convex optimization problem in v, with u treated as fixed parameter, using $U = U_{\textrm{ad}}= H_0^1(\Omega )$ and the functional ${\mathcal {E}}: H_0^1(\Omega ) \times H_0^1(\Omega ) \rightarrow {\mathbb {R}}\cup \{\infty \}$ defined by

$$\begin{aligned} {\mathcal {E}}(v,u):= \frac{1}{2}\Vert \nabla v \Vert _{L^2(\Omega ;{\mathbb {R}}^d)}^2 + \frac{1}{4}\Vert v\Vert _{L^4(\Omega )}^4 - \alpha (v,u)_{L^2(\Omega )}. \end{aligned}$$

Returning to (PDE) and using the best response operator ${\mathcal {B}}$ of the above-defined ${\mathcal {E}}$ (cf. Definition 3), we can rewrite (3) as ${\mathcal {B}}(u) = v$. Then, (PDE) is equivalent to $u = {\mathcal {B}}(u)$. Hence, u is an equilibrium of ${\mathcal {E}}$, if and only if u solves (PDE).

Alternatively, one could have associated the equation to the following one. Seek $v \in H_0^1(\Omega )$, such that

$$\begin{aligned} -\Delta v= & {} \alpha u - u^3 \text { in } \Omega ,\\ v= & {} 0 \text { on } \partial \Omega . \end{aligned}$$

Then, one can relate the following functional

$$\begin{aligned} {\mathcal {E}}(v,u):= \frac{1}{2}\Vert \nabla v \Vert _{L^2(\Omega ;{\mathbb {R}}^d)}^2 + (u^3 - \alpha u, v)_{L^2(\Omega )} \end{aligned}$$

instead. Returning to Definition 1, we can associate (PDE) with the equilibrium of the second functional. Hence, a given system can be embedded into the equilibrium framework in different ways.

2.2 Nash Equilibrium Problems

Let a family of Banach spaces $U_i$ for $i = 1, \dots , N$ and $N \ge 1$ be given. Define the space $U:= U_1 \times \cdots \times U_N$ as well as the strategy sets $U_{\textrm{ad}}^i \subseteq U_i$ and the joint strategy set $U_{\textrm{ad}}:= \prod _{i = 1}^N U_{\textrm{ad}}^i$ together with a family of real-valued functionals ${\mathcal {J}}_i: U_{\textrm{ad}} \rightarrow {\mathbb {R}}$ for all $i = 1, \dots , N$. Every index is associated with a player. They seek to minimize their objectives choosing an argument called strategy from their admissible set $U_{\textrm{ad}}^i$. With the index $-i$ we denote $(N-1)$-tuples of strategies, where the i-th component has been omitted. A joint strategy $(u_1, \dots , u_{i - 1}, v_i, u_{i + 1}, \dots , u_N) \in U$ is written as $(v_i,u_{-i})$—with no change of the ordering. In particular, we have $(u_i, u_{-i}) = (u_1, \dots , u_N)$. Similarly, the set $U_{\textrm{ad}}^{-i}$ is defined as $\prod _{j \ne i} U_{\textrm{ad}}^j$.

Consider the Nash equilibrium problem (abbr.: NEP) (cf. Nash 1990, 1950) reading as follows. Seek $u \in U_{\textrm{ad}}$ such that for all indices $i = 1, \dots , N$ the relations

$$\begin{aligned} {\mathcal {J}}_i(u_i,u_{-i}) \le {\mathcal {J}}_i(v_i,u_{-i}) \text { for all } v_i \in U_{\textrm{ad}}^i \end{aligned}$$

(NEP)

hold true. Now, we are dealing with a system of N coupled optimization problems instead of only one. Note, however, that they are formulated in separate components, i.e., only $u_i$ is used to minimize ${\mathcal {J}}_i(\cdot ,u_{-i})$. In the sense of Definition 1, we formulate the following functional

$$\begin{aligned} {\mathcal {E}}_{\textrm{NEP}}(v,u):= \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}). \end{aligned}$$

By the product structure, it is straightforward to show the equilibrium problem induced by ${\mathcal {E}}_\textrm{NEP}$ to be equivalent to (NEP).

In a similar fashion, we can as well treat generalized NEPs (abbr.: GNEPs), see Facchinei and Kanzow (2007). Here, an additional dependency of the feasible set on the other players’ strategies occurs. The described mechanism can be modeled by a set-valued mapping $C_i: U_{\textrm{ad}}^{-i} \rightrightarrows U_{\textrm{ad}}^i$ for each player. This map is called the strategy map of the i-th player. Their combination $C: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$ defined by $C(u):= C_1(u_{-1}) \times \dots \times C_N(u_{-N})$ is called the (joint) strategy mapping.

With this notation, we propose the generalized Nash equilibrium problem: Seek $u \in U_{\textrm{ad}}$ with $u \in C(u)$, such that for all $i = 1, \dots , N$ the relations

$$\begin{aligned} {\mathcal {J}}_i(u_i,u_{-i}) \le {\mathcal {J}}_i(v_i,u_{-i}) \text { for all } v_i \in C_i(u_{-i}) \end{aligned}$$

(GNEP)

hold true. This problem embeds as well into Definition 1 by adding to ${\mathcal {E}}^\textrm{NEP}$ the indicator function realizing the additional constraint. This yields

$$\begin{aligned} {\mathcal {E}}_\textrm{GNEP}(v,u):= \sum _{i = 1}^N \left( {\mathcal {J}}_i(v_i,u_{-i}) + I_{C_i(u_{-i})}(v_i) \right) . \end{aligned}$$

Here, as well as in the remainder of this article, the indicator function of a set $M \subseteq U$ is defined by

$$\begin{aligned} I_M(u) = \left\{ \begin{array}{ll} 0 &{} \text {if } x \in M,\\ \infty &{}\text {else.} \end{array}\right. \end{aligned}$$

A frequently encountered special case is a single condition that needs to be fulfilled by all players simultaneously. We say that a GNEP is jointly constrained or is said to have shared constraints, if there exists ${\mathcal {F}}\subseteq U_{\textrm{ad}}$ such that $v_i \in C_i(u_{-i})$ holds if and only if $(v_i,u_{-i}) \in {\mathcal {F}}$. This condition is sometimes called Rosen’s law, see (2011, p. 484).

This motivates the formulation of a modified solution concept called variational equilibrium (also known as normalized equilibrium) that has been introduced in Rosen (1965). Seek $u \in {\mathcal {F}}$ such that

$$\begin{aligned} \sum _{i = 1}^N {\mathcal {J}}_i(u_i,u_{-i}) \le \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) \text { for all } v \in {\mathcal {F}}\end{aligned}$$

Here, the values of the strategy mapping are replaced by the set ${\mathcal {F}}$. To return to Definition 1, we formulate the functional ${\mathcal {E}}_\textrm{VEP}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}$ defined by

$$\begin{aligned} {\mathcal {E}}_{\textrm{VEP}}(v,u) = \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) + I_{\mathcal {F}}(v). \end{aligned}$$

Again, the equilibrium problem induced by ${\mathcal {E}}_\textrm{VEP}$ is equivalent to the variational equilibrium problem. In conclusion, Nash equilibrium problems are covered by the framework presented in this paper.

2.3 Saddle Point Problems

Let X, Y be Banach spaces and let nonempty, closed, convex subsets $X_{\textrm{ad}}\subseteq X$ and $Y_{\textrm{ad}}\subseteq Y$ be given. For a bifunction $F: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$, we consider the following saddle point problem. Seek $x \in X_{\textrm{ad}}$ and $y \in Y_{\textrm{ad}}$ such that

$$\begin{aligned} F(x,y') \le F(x',y) \text {~for all~} x' \in X_{\textrm{ad}}, y' \in Y_{\textrm{ad}}. \end{aligned}$$

We take $U_{\textrm{ad}}:= X_{\textrm{ad}}\times Y_{\textrm{ad}}$ and define the equilibrium functional ${\mathcal {E}}_{\textrm{SP}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} {\mathcal {E}}_{\textrm{SP}}((x',y'),(x,y)):= F(x',y) - F(x,y'). \end{aligned}$$

A point $(x,y) \in U_{\textrm{ad}}$ is an equilibrium of ${\mathcal {E}}_{\textrm{SP}}$, if and only if

$$\begin{aligned} 0 = F(x,y) - F(x,y)= & {} {\mathcal {E}}_{\textrm{SP}}((x,y),(x,y)) \le {\mathcal {E}}_{\textrm{SP}}((x',y'),(x,y))\\\le & {} F(x',y) - F(x,y') \end{aligned}$$

for all $(x',y') \in U_{\textrm{ad}}$. This is equivalent to

$$\begin{aligned} F(x,y') \le F(x',y) \text {~for all~} x' \in X, y' \in Y. \end{aligned}$$

Hence, saddle point problems fall into the equilibrium framework in Definition 1.

In fact, they are already a special case of a two player Nash equilibrium problem with $u_1 = x, u_2 = y$, objectives ${\mathcal {J}}_1 = F$, ${\mathcal {J}}_2 = -F$ and strategy sets $U_{\textrm{ad}}^1 = X_{\textrm{ad}}$, $U_{\textrm{ad}}^2 = Y_{\textrm{ad}}$. This viewpoint also yields the above.

2.4 Quasi-Variational Inequalities

Let $C: U \rightrightarrows U$ be a set-valued mapping with nonempty, closed, convex values and $f: U \rightarrow U^*$ be a single-valued operator, where U denotes a Banach space and $U^*$ its topological dual. Further, assume $A \in {\mathcal {L}}(U,U^*)$ to be a bounded, linear and coercive operator. Consider the following type of quasi-variational inequality. Seek $u \in C(u)$, such that

$$\begin{aligned} f(u) \in A u + N_{C(u)}(u) \end{aligned}$$

(QVI)

holds. Here, $N_{C(u)}(\cdot )$ denotes the normal cone mapping associated with C(u); see e.g., (1990, Definition 4.4.2).

We rewrite (QVI) as follows. First, we decompose $A = A_{\textrm{sym}}+ A_{\textrm{anti}}$ into its symmetric and antisymmetric (skew-symmetric) part. Then, we formulate the corresponding variational inequality for given $u \in U$ as: Seek $v \in U$ such that

$$\begin{aligned} f(u) - A_{\textrm{anti}}u \in A_{\textrm{sym}}v + N_{C(u)}(v) \text { in } U^*. \end{aligned}$$

This can be interpreted as v being the best response of

$$\begin{aligned} {\mathcal {E}}_\textrm{QVI}(v,u):= \frac{1}{2}\langle A_{\textrm{sym}}v,v \rangle _{U^*,U} + \langle A_{\textrm{anti}}u - f(u), v \rangle _{U^*,U} + I_{C(u)}(v). \end{aligned}$$

In many cases, this procedure can be transferred to nonlinear, cyclically monotone operators A using (cf. Bauschke and Combettes 2017, Theorem 22.18). The above shows that QVIs can formally be brought into the framework of Definition 1.

2.5 Eigenvalue Problems

Consider a (real) Hilbert space H with inner product $(\cdot ,\cdot )_H$ and its dual space $H^*$ identified with H. Let a linear, bounded operator $A \in {\mathcal {L}}(H,H)$ be given. The associated eigenvalue problem (EVP) reads as: Seek $\lambda \in {\mathbb {R}}$ and $u \in H$, $\Vert u\Vert _H = 1$ such that

$$\begin{aligned} Au = \lambda u.{} \end{aligned}$$

(EVP)

First note that any eigenvalue of A fulfills the well-known inequality $|\lambda | \le \Vert A\Vert _{{\mathcal {L}}(H,H)}$. Hence, we can take $\alpha = \Vert A\Vert _{{\mathcal {L}}(H,H)} + 1$ and rewrite (EVP) equivalently as

$$\begin{aligned} (A + \alpha \cdot \textrm{id}) v = (\lambda + \alpha ) v.{} \end{aligned}$$

Thus, we obtain again an eigenvalue problem with a coercive operator with shifted eigenvalues. Therefore, we assume without loss of generality A to be coercive with modulus $\ge 1$. We formulate the following QVI. Seek $u \in C(u)$ such that

$$\begin{aligned} 0 \in A u + N_{C(u)}(u), \end{aligned}$$

(4)

with $C: H \rightrightarrows H$, $C(u):= \{ v \in H: (v,u)_H = 1 \}$. To show the equivalence of (EVP) and (4), we calculate the normal cone of C(u) in $u \in C(u){}$:

Take first $v \in C(u)$ and decompose it as $v = \mu u + u^\perp$ for $\mu \in {\mathbb {R}}$ with $(u^\perp ,u)_{H} = 0$. Clearly, $u \in C(u)$ if and only if $\Vert u\Vert _H = 1$. To determine $\mu$, we calculate

$$\begin{aligned} 1 = (v,u)_H = \mu \Vert u\Vert _H^2 = \mu \end{aligned}$$

as $v,u \in C(u)$. This implies $C(u) \subseteq u + ({\mathbb {R}}u)^\perp$ To obtain the other direction, one can reuse the previous arguments and show that every $v = u + u^\perp$, $(u^\perp , u)_H = 0$ belongs to C(u). Thus, for $u^* \in N_{C(u)}(u)$, it holds that

$$\begin{aligned} 0 \ge (u^*, v - u)_H = (u^*,u^\perp )_{H}, \end{aligned}$$

and thus $(u^*, u^\perp )_{H} = 0$ for all $u^\perp \in H{}$ with $(u^\perp , u)_{H} = 0$. Hence, $u^* \in {\mathbb {R}}u$.

The other direction ${\mathbb {R}}u \subseteq N_{C(u)}(u)$ holds by the same calculation as well and yields $N_{C(u)}(u) = {\mathbb {R}}u$. Thus, the QVI in (4) is equivalent to (EVP). The reformulation into the form in Definition 1 follows the discussion in Sect. 2.4.

3 Existence

For the sake of self-containment, we want to draw the attention shortly to the existence of equilibria. There are several approaches for equilibria based on bifunctions (cf. Kassay and Rădulescu 2018; Yuan 1999). However, we seek to apply fixed point results involving the best response operator. One of the classical results is the Kakutani fixed point theorem (cf. Kakutani 1941). As we aim at the infinite-dimensional situation, we cite here the Glicksberg fixed point theorem serving as the corresponding generalization of Kakutani’s result.

Theorem 4

(cf. Glicksberg 1952) Given a closed point-to-(non-void)-convex-set mapping $\Phi : Q \rightrightarrows Q$ of a convex compact subset Q of a convex Hausdorff linear topological space into itself, then there exists a fixed point $x \in \Phi (x)$.

Next, we apply Theorem 4 to derive an existence result for an equilibrium problem of the type presented in Definition 1. The existence discussion follows in close proximity the arguments used for the existence of Nash equilibria. We just refer to Dutang (2013) for finite-dimensional Nash games. The following result generalizes the aforementioned existence.

Theorem 5

(Existence for (1)) Consider a nonempty, closed, convex and bounded subset $U_{\textrm{ad}}\subseteq U$ of a reflexive Banach space U and a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}$. There exists an equilibrium of ${\mathcal {E}}$, if the following assumptions are fulfilled:

• (i) For all $u \in U_{\textrm{ad}}$, the functional ${\mathcal {E}}(\, \cdot \, ,u)$ is quasiconvex and bounded from below.

• (ii) The functional ${\mathcal {E}}$ is weakly lower semicontinuous and the following recovery condition is fulfilled. For all sequences $u^n \rightharpoonup u$ and $v \in U_{\textrm{ad}}$, there exists $v^n \rightharpoonup v$ such that

  $$\begin{aligned} \limsup _{n \rightarrow \infty } {\mathcal {E}}(v^n,u^n) \le {\mathcal {E}}(v,u). \end{aligned}$$

• (iii) The set-valued operator $u \mapsto {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ has as effective domain the set $U_{\textrm{ad}}$ and has a weakly closed graph.

Proof

To prove the existence of an equilibrium, we demonstrate the existence of a fixed point of the associated best response mapping ${\mathcal {B}}: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$ (cf. Definition 3). For this sake, we use Theorem 4 and check the assumptions therein.

We equip $U_{\textrm{ad}}$ with the weak topology. Due to its closedness and convexity, it is weakly closed. Since $U_{\textrm{ad}}$ is as well-bounded and U is reflexive, it is weakly compact as well.

As for given $u \in U_{\textrm{ad}}$, the functional ${\mathcal {E}}(\, \cdot \, ,u)$ is bounded from below, and we choose an infimizing sequence $(v^n)_{n \in {\mathbb {N}}}$. Since ${\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ is nonempty and ${\mathcal {E}}(\, \cdot \, ,u)$ is quasiconvex, all sublevel sets are convex and thus the domain is convex as well. As the graph of $u \mapsto {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ is closed, the domain is closed as well. Since $U_{\textrm{ad}}$ is weakly compact, we can extract a weakly convergent subsequence with limit $v \in U_{\textrm{ad}}$. Using the convexity of the domain, the Mazur’s lemma yields $v \in {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ and the assumed lower semicontinuity implies

$$\begin{aligned} \inf {\mathcal {E}}(\, \cdot \, ,u) \le {\mathcal {E}}(v,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}(v^n,u) = \inf {\mathcal {E}}(\, \cdot \, ,u). \end{aligned}$$

Thus, ${\mathcal {B}}(u)$ is nonempty. By the quasiconvexity and the lower semicontinuity, we obtain that ${\mathcal {B}}(u)$ is nonempty, closed and convex.

It is left to show the closedness of ${{\text {gph}}({\mathcal {B}})}$. Therefore, take a sequence $(v^n,u^n) \subseteq {{\text {gph}}({\mathcal {B}})}$ with $(v^n,u^n) \rightharpoonup (v,u)$. Moreover, take without loss of generality an arbitrary $w \in {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$. By assumption (ii), we can find for every sequence $u^n \rightharpoonup u$ a recovery sequence $w^n \rightharpoonup w$ with $\limsup _{n \rightarrow \infty } {\mathcal {E}}(w^n,u^n) \le {\mathcal {E}}(w,u)$. Since $v^n$ are minimizers of ${\mathcal {E}}(\, \cdot \, ,u^n)$, we obtain

$$\begin{aligned} {\mathcal {E}}(v,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}(v^n,u^n) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}(w^n,u^n) \le \limsup _{n \rightarrow \infty } {\mathcal {E}}(w^n,u^n){} \le {\mathcal {E}}(w,u). \end{aligned}$$

Thus, $v \in {\mathcal {B}}(u)$ and ${\mathcal {B}}$ has a closed graph. Subsequently, we can use Theorem 4 and obtain the existence of a fixed point of ${\mathcal {B}}$. This is equivalent to the existence of an equilibrium of ${\mathcal {E}}$. $\square$

The assumption on the existence of a recovery sequence in Theorem 5 in (ii) implies the domain to be a lower semicontinuous set-valued operator (see Aubin and Frankowska 1990, Definition 1.4.2), i.e., for all $v \in {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u)\right)$ and all sequences $u^n \rightharpoonup u$ there exists a sequence $v^n \rightharpoonup v$ with $v^n \in {\text {dom}}\left( {\mathcal {E}}(\, \cdot \, ,u^n)\right)$.

Next, we seek to utilize Theorem 5 and the arguments in its proof to derive the existence of an equilibrium for (generalized) Nash equilibrium problems.

Theorem 6

(Existence for GNEPs, cf. Dutang 2013) Let $U_{\textrm{ad}}^i \subseteq U_i$ be a family of nonempty, closed, convex and bounded sets and consider the following generalized Nash equilibrium problem. Seek $u \in U_{\textrm{ad}}$ with $u \in C(u)$ such that

$$\begin{aligned} {\mathcal {J}}_i(u_i,u_{-i}) \le {\mathcal {J}}_i(v_i,u_{-i}) \text { for all } v_i \in C_i(u_{-i}). \end{aligned}$$

There exists a Nash equilibrium, if the following assumptions are fulfilled:

• (i) The objectives $v_i \mapsto {\mathcal {J}}_i(v_i,u_{-i})$ are quasiconvex and bounded from below for all $i = 1, \dots , N$ and $u_{-i} \in U_{\textrm{ad}}^{-i}$.

• (ii) The set-valued operator $C: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$ has nonempty, closed, convex and bounded values, has a weakly closed graph and its effective domain is $U_{\textrm{ad}}$.

• (iii) Let C be a completely lower semi-continuous mapping, i.e., for all sequences $u^n \rightharpoonup u$ and all $v \in C(u)$ there exists a sequence $v^n \in C(u^n)$, such that $v^n \rightarrow v$.

• (iv) The functionals $u \mapsto {\mathcal {J}}_i(u)$ are weakly lower semi-continuous on ${{\text {gph}}(C)}$ and moreover upper semicontinuous on ${{\text {gph}}(C)}$ with respect to the strong topology on $U_i$ and the weak topology on $U_{-i}$, i.e., for all $i = 1, \dots , N$ and all sequences $u_i^n \rightarrow u_i$ in $U_i$ and $u_{-i}^n \rightharpoonup u_{-i}$ in $U_{-i}$ holds ${\mathcal {J}}_i(u) \ge \limsup _{n \rightarrow \infty }{\mathcal {J}}_i(u^n)$ (cf. Aubin and Frankowska 1990, Definition 1.4.2).

Proof

Unfortunately, we cannot use Theorem 5 directly as the quasiconvexity of each $v_i \mapsto {\mathcal {J}}_i(v_i,u_{-i})$ does not imply the quasiconvexity of ${\mathcal {E}}_\textrm{GNEP}(\, \cdot \, ,u)$. However, we can still guarantee the best response operator to have nonempty, closed and convex values.

For that sake, we exploit the product structure of the strategy mapping in the underlying minimization problem and rewrite

$$\begin{aligned} {\mathcal {B}}(u)= & {} {{\text {argmin}}}_{v \in C(u)} \left( \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) \right) = \prod _{i = 1}^N {{\text {argmin}}}_{v_i \in C_i(u_{-i})} {\mathcal {J}}_i(v_i,u_{-i}) \\= & {} \prod _{i = 1}^N {\mathcal {B}}_i(u_{-i}), \end{aligned}$$

where the product is taken in the canonical ordering $1, \dots , N$. Then, the arguments used in Theorem 5 can be used as well to prove nonemptyness, closedness and convexity of ${\mathcal {B}}_i(u_{-i})$ and thus of ${\mathcal {B}}(u)$.

The remaining assumptions translate using the example given in Sect. 2.2. As $({\mathcal {J}}_i)_{i = 1}^N$ are defined on $U_{\textrm{ad}}$, we obtain ${\text {dom}}\left( {\mathcal {E}}_\textrm{GNEP}(\, \cdot \, ,u)\right) = C(u)$. Thus, the remaining requirements on C are translated accordingly. The continuity requirements on ${\mathcal {E}}_\textrm{GNEP}$ are translated via the continuity requirements on the functionals ${\mathcal {J}}_i$. To check the continuity condition on $(v,u) \mapsto I_{C(u)}(v)$, take first a sequence $(v^n,u^n) \rightharpoonup (v,u)$. There are two cases:

(i)::

There are infinitely many indices with $v^n \in C(u^n)$. Then, along this subsequence, we obtain $\liminf _{n \rightarrow \infty } I_{C(u^n)}(v^n) = 0$ and by the assumed weak closedness of the graph $v \in C(u)$. Thus, $I_{C(u)}(v) = 0$, and the desired weak lower semicontinuity is proven.

(ii)::

Otherwise, if for almost all indices $v^n \notin C(u^n)$ holds true, then $\lim _{n \rightarrow \infty } I_{C(u^n)}(v^n) = \infty$ and the desired lower semicontinuity is proven as well.

For the recovery condition, take without loss of generality, $w \in C(u)$ and a sequence $u^n \rightharpoonup u$. Then, there exists a sequence $w^n \rightarrow w$ with $w^n \in C(u^n)$. By the upper semicontinuity condition on ${\mathcal {J}}_i$ for $i = 1, \dots , N$, we obtain

$$\begin{aligned} \limsup _{n \rightarrow \infty } {\mathcal {E}}_\textrm{GNEP}(w^n,u^n) = \limsup _{n \rightarrow \infty } \sum _{i = 1}^N {\mathcal {J}}_i(w_i^n,u_{-i}^n) \le \sum _{i = 1}^N {\mathcal {J}}_i(w_i,u_{-i}) = {\mathcal {E}}_\textrm{GNEP}(w,u). \end{aligned}$$

Thus, the remaining arguments in the proof of Theorem 5 yield the existence of a Nash equilibrium. $\square$

In comparison with Theorem 5, we demanded a strongly convergent recovery sequence, but only combined it with an upper semicontinuity condition, that used strong continuity in the control component. Alternatively, one could have used weak convergence for both. This choice however might depend on the application in mind.

Analogously, we proceed with the existence of variational equilibria.

Theorem 7

(Existence for VEPs, cf. Dutang 2013) Let $U_{\textrm{ad}}^i \subseteq U_i$ for $i = 1, \dots , N$ be a family of nonempty, closed, convex and bounded sets. Consider the following variational equilibrium problem. Seek $u \in {\mathcal {F}}$ such that

$$\begin{aligned} \sum _{i = 1}^N {\mathcal {J}}_i(u_i,u_{-i}) \le \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) \text { for all } v \in {\mathcal {F}}. \end{aligned}$$

There exists a variational equilibrium, if the following assumptions are fulfilled:

• (i) The objective $v \mapsto \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i})$ is quasiconvex and bounded from below for all $u \in U_{\textrm{ad}}$.

• (ii) The set of shared constraints ${\mathcal {F}}\subseteq U_{\textrm{ad}}$ is nonempty, closed and convex.

• (iii) The functional $(v,u) \mapsto \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i})$ is weakly lower semicontinuous on ${\mathcal {F}}$. Moreover, assume that for every weakly convergent sequence $u^n \rightharpoonup u$ in ${\mathcal {F}}$ and $v \in {\mathcal {F}}$ there exists a sequence $(v^n)_{n \in {\mathbb {N}}} \subseteq {\mathcal {F}}$ with $v^n \rightarrow v$ such that

  $$\begin{aligned} \limsup _{n \rightarrow \infty } \sum _{i = 1}^N {\mathcal {J}}_i(v_i^n,u_{-i}^n) \le \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}). \end{aligned}$$

Proof

The proof uses Theorem 5 and the arguments in the proof of Theorem 6 and is omitted for brevity. $\square$

Here, we decided again to use a strongly convergent recovery sequence. With these results at hand, we close our existence discussion. It is worth noting that the aforementioned approach via a bifunction can alternatively be used. Moreover, especially in the context of QVIs, other fixed point results are of interest (cf. Alphonse et al. 2019; Birkhoff 1973).

4 Extended $\Gamma$-Convergence of Equilibrium Problems

The constraint in the optimization procedure, which induces the equilibrium problem, leads to analytical and as a consequence to numerical difficulties. To address them, one group of techniques are penalization and regularization schemes, see e.g., Hintermüller and Kunisch (2006), Hintermüller and Rasch (2015) and Adam et al. (2018). The basic idea is the substitution of the addressed functional with a sequence of more regular objects, which are easier to handle. Then, one hopes to recover the originally formulated problem in the limit. A successful concept in mathematical programming that is capable to provide such a convergence statement for optimization problems is $\Gamma$-convergence.

For our purpose, we generalize this notion to equilibrium problems as in Definition 1.

Definition 8

(Extended $\Gamma$-convergence) Let $U_{\textrm{ad}}$ be a subset of a reflexive Banach space U. A sequence of functionals $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$ is called extended (weakly) $\Gamma$-convergent to a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$, denoted by ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$ (resp. ${\mathcal {E}}^n \overset{\Gamma }{\rightharpoonup }\ {\mathcal {E}}$), if the following two conditions hold:

• (i) For all sequences $u^n \rightarrow u$ ($u^n \rightharpoonup u$) holds

  $$\begin{aligned} {\mathcal {E}}(u,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

  (5)

• (ii) For all $v \in U_{\textrm{ad}}$ and all sequences $u^n \rightarrow u$ ($u^n \rightharpoonup u$) there exists a sequence $v^n \rightarrow v$ ($v^n \rightharpoonup v$), such that

  $$\begin{aligned} {\mathcal {E}}(v,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n). \end{aligned}$$

  (6)

There have been previous extensions of $\Gamma$-, respectively epiconvergence to various type of equilibrium problems. Their relation to our investigation is studied in Sect. 5.

A strengthened concept that addresses the differences between strong and weak convergence in the infinite-dimensional case is Mosco convergence, which we generalize next.

Definition 9

(Extended Mosco convergence) Let $U_{\textrm{ad}}$ be a subset of a reflexive Banach space U. A sequence of functionals $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$ is called extended Mosco convergent to a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$, denoted by ${\mathcal {E}}^n \overset{M}{\longrightarrow }{\mathcal {E}}$, if the following two conditions hold:

• (i) For all sequences $u^n \rightharpoonup u$ it holds that

  $$\begin{aligned} {\mathcal {E}}(u,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

  (7)

• (ii) For all $v \in U_{\textrm{ad}}$ and all sequences $u^n \rightharpoonup u$ there exists a sequence $v^n \rightarrow v$, such that

  $$\begin{aligned} {\mathcal {E}}(v,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n). \end{aligned}$$

  (8)

The direct comparison between extended $\Gamma$-convergence with respect to weak topology and extended Mosco convergence yields a strong convergence for the recovery sequence with a weak convergent sequence in the feedback component. It is worth noting that the above definitions do only require the first condition to hold true on the diagonal.

Alternatively, for a given sequence $u^n \rightharpoonup u$, one can consider the sequence ${\mathcal {E}}(\, \cdot \, ,u^n)$ and use the usual $\Gamma$-/Mosco convergence known from optimization. We refer to this as sequential $\Gamma -$/Mosco convergence.

Definition 10

(Sequential $\Gamma$-and Mosco convergence) Let $U_{\textrm{ad}}$ be a subset of a reflexive Banach space U. A sequence of functionals $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$ is called sequentially (weakly) $\Gamma$-convergent, respectively sequentially Mosco convergent to a functional ${\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}\cup \{\infty \}{}$, if for all $u^n \rightharpoonup u$ the convergence ${\mathcal {E}}^n(\, \cdot \, ,u^n) \overset{\Gamma }{\longrightarrow }{\mathcal {E}}(\, \cdot \, ,u)$ (${\mathcal {E}}^n(\, \cdot \, ,u^n) \overset{\Gamma }{\rightharpoonup }\ {\mathcal {E}}(\, \cdot \, ,u)$), respectively, ${\mathcal {E}}^n(\, \cdot \, ,u^n) \overset{M}{\longrightarrow }{\mathcal {E}}(\, \cdot \, ,u)$ holds true. Then, we may write ${\mathcal {E}}^n \overset{\mathrm {seq.}~\Gamma }{\longrightarrow }{\mathcal {E}}$ (${\mathcal {E}}^n \overset{\mathrm {seq.}~\Gamma }{\rightharpoonup }{\mathcal {E}}$) and ${\mathcal {E}}^n \overset{{\mathrm{seq.~Mosco}}}{\longrightarrow }{\mathcal {E}}$, respectively.

In comparison with Definitions 8 and 9, the first requirement is placed on the whole product set and not only on the diagonal. Hence, sequential $\Gamma$-/Mosco convergence implies $\Gamma$- respectively Mosco convergence.

In fact, considering the sequence ${\mathcal {E}}(v,u) = {\mathcal {E}}^n(v,u):= (v,u)_U$ with $U_{\textrm{ad}}= U$ being a real Hilbert space yields the extended Mosco convergence, but not sequential Mosco convergence. To see this, take two weakly convergent sequences $v^n \rightharpoonup v$ and $u^n \rightharpoonup u$. Then, the relation $(v,u)_U \le \liminf _{n \rightarrow \infty } (v^n,u^n)_U$ does not need to be true in general.

Moreover, Mosco convergence of convex sets in Mosco (1969) can be interpreted as sequential Mosco convergence for the indicator functions $I_{C(u)}$ of a set-valued mapping $C: U \rightrightarrows U$.

However, it is straightforward to show that (quasi)convexity in the control component is preserved by sequential convergence. This is not clear to the authors for the extended convergence introduced in Definitions 8 and 9.

As we solve a sequence of equilibrium problems, we wish them to cluster around an equilibrium of the original one. This is shown in the following result.

Theorem 11

Let $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}$ be an extended (weakly) $\Gamma$-convergent sequence of functionals with limit ${\mathcal {E}}$ as in Definition 8. Then, every (weak) accumulation point of a sequence of corresponding equilibria $(u^n)_{n \in {\mathbb {N}}}$ is an equilibrium of the limit.

Proof

Let u be a (weak) accumulation point of $(u^n)_{n \in {\mathbb {N}}}$ along a (not relabeled) subsequence. Let $v \in U_{\textrm{ad}}$ be arbitrary. Then, there exists a recovery sequence $v^n \rightarrow v$ ($v^n \rightharpoonup v$) by (6). We deduce

$$\begin{aligned} {\mathcal {E}}(u,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n) \le \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n) \le {\mathcal {E}}(v,u). \end{aligned}$$

This proves the assertion. $\square$

In the upcoming two subsections, the announced penalization technique is addressed next. We describe it and derive extended $\Gamma -$, respectively, Mosco convergence results to a selection of equilibrium problems—Nash games as well as quasi-variational inequalities.

4.1 Application to Penalized Nash Equilibrium Problems

To a given set-valued mapping $C: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$, a penalty functional $\pi _C: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow [0, \infty )$ is associated. We assume that $\pi _C(v,u) = 0$ holds if and only if $v \in C(u)$. In the same fashion, we associate with a given set ${\mathcal {F}}\subseteq U_{\textrm{ad}}$ a penalty functional $\pi _{\mathcal {F}}: U_{\textrm{ad}}\rightarrow [0,\infty )$ with $\pi _{\mathcal {F}}(v) = 0$ if and only if $v \in {\mathcal {F}}$. A fairly general example of such penalty functionals is given by

$$\begin{aligned} \pi _C(v,u)&:= {\text {dist}}(v,C(u)) \text { and } \pi _{\mathcal {F}}(v) := {\text {dist}}(v,{\mathcal {F}}) \text { with} \nonumber \\ {\text {dist}}(v,M)&:= \inf \{ \Vert v - v'\Vert _U : v' \in M\} \text { for a subset } M \subseteq U. \end{aligned}$$

(9)

Here, the sets C(u) and ${\mathcal {F}}$ are assumed to be closed. The latter guarantees $u \in M$ if and only if ${\text {dist}}(u,M) = 0$. Next, we establish a $\Gamma$-convergence result for the penalized version of (GNEP).

Theorem 12

(Extended Mosco convergence of penalized GNEPs) Let a sequence of positive penalty parameters $\gamma _n \rightarrow \infty$ be given. Moreover, assume the following conditions to be fulfilled:

• (i) The set-valued operator $C: U_{\textrm{ad}}\rightrightarrows U_{\textrm{ad}}$ has a weakly closed graph and the set $U_{\textrm{ad}}$ as its domain.

• (ii) Moreover, let C be a completely lower semicontinuous mapping, i.e., for all sequences $u^n \rightharpoonup u$ and all $v \in C(u)$, there exists a sequence $v^n \in C(u^n)$ such that $v^n \rightarrow v$.

• (iii) The functionals $U_{\textrm{ad}}\ni u \mapsto {\mathcal {J}}_i(u) \in {\mathbb {R}}$ are bounded from below and weakly lower semi-continuous on $U_{\textrm{ad}}$ as well as upper semicontinuous on $U_{\textrm{ad}}$ with respect to the strong topology on $U_i$ and the weak topology on $U_{-i}$, i.e., for all $i = 1, \dots , N$ and all sequences $u_i^n \rightarrow u_i$ in $U_i$ and $u_{-i}^n \rightharpoonup u_{-i}$ in $U_{-i}$ holds ${\mathcal {J}}_i(u) \ge \limsup _{n \rightarrow \infty }{\mathcal {J}}_i(u^n)$.

• (iv) Let the penalty functional $\pi _C: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow [0,\infty )$ be weakly lower semicontinuous.

Then, the sequence of functionals ${\mathcal {E}}_\textrm{GNEP}^\gamma : U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {E}}_\textrm{GNEP}^\gamma (v,u):= \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) + \gamma \pi _C(v,u) \end{aligned}$$

(10)

is extended Mosco convergent to

$$\begin{aligned} {\mathcal {E}}_\textrm{GNEP}(v,u):= \sum _{i = 1}^N \left( {\mathcal {J}}_i(v_i,u_{-i}) + I_{C_i(u_{-i})}(v_i) \right) . \end{aligned}$$

Proof

Take a sequence $(\gamma ^n)_{n \in {\mathbb {N}}}$ with $\gamma ^n > 0$, $\gamma ^n \rightarrow \infty$. To check the condition in (5), take an arbitrary sequence $(u^n)_{n \in {\mathbb {N}}} \subseteq U_{\textrm{ad}}$ with $u^n \rightharpoonup u$ in U. Since $U_{\textrm{ad}}$ is assumed to be a nonempty, closed, convex set, we have $u \in U_{\textrm{ad}}$. First, consider the case $u \in C(u)$. By the lower semicontinuity of the functionals ${\mathcal {J}}_i$, one obtains

$$\begin{aligned} {\mathcal {E}}_\textrm{GNEP}(u,u)= & {} \sum _{i = 1}^N {\mathcal {J}}_i(u_i, u_{-i}) \le \liminf _{n \rightarrow \infty } \left( \sum _{i = 1}^N {\mathcal {J}}_i(u_i^n, u_{-i}^n) \right) \\\le & {} \liminf _{n \rightarrow \infty } \left( \sum _{i = 1}^N {\mathcal {J}}_i(u_i^n, u_{-i}^n) + \gamma ^n \pi _C(u^n,u^n) \right) = \liminf _{n \rightarrow \infty } {\mathcal {E}}_\textrm{GNEP}^{\gamma ^n}(u^n, u^n). \end{aligned}$$

If $u \notin C(u)$, then holds $\pi _C(u,u) > 0$. The assumed weak lower semicontinuity of $\pi _C$ yields

$$\begin{aligned} 0 < \pi _C(u,u) \le \liminf _{n \rightarrow \infty } \pi _C(u^n,u^n). \end{aligned}$$

Hence, $\pi _C(u^n,u^n) \ge \frac{1}{2}\pi _C(u,u)$ holds for almost all indices n and therefore

$$\begin{aligned} \lim _{n \rightarrow \infty } \gamma ^n \pi _C(u^n,u^n) = \infty . \end{aligned}$$

In combination with the boundedness of ${\mathcal {J}}_i$ from below, we obtain

$$\begin{aligned} \lim _{n \rightarrow \infty } \left( \sum _{i=1}^N {\mathcal {J}}_i(u_i^n,u_{-i}^n) + \gamma ^n \pi _C(u^n,u^n) \right) = \infty = {\mathcal {E}}_\textrm{GNEP}(u,u) \end{aligned}$$

This proves the condition in (5).

To show (6), choose again an arbitrary sequence $u^n \rightharpoonup u$ in $U_{\textrm{ad}}$. Moreover, take an arbitrary $v \in C(u)$. Then, $\pi _C(v,u) = 0$ holds. Taking by assumption, a sequence $(v^n)_{n \in {\mathbb {N}}} \subset U$ with $v^n \in C(u^n)$ and $v^n \rightarrow v$ yields

$$\begin{aligned} {\mathcal {E}}_\textrm{GNEP}(v,u)= & {} \sum _{i = 1}^N {\mathcal {J}}_i(v_i, u_{-i}) \ge \limsup _{n \rightarrow \infty }{} \sum _{i = 1}^N {\mathcal {J}}_i(v^n_i, u_{-i}^n) = \limsup _{n \rightarrow \infty }{} \sum _{i = 1}^N {\mathcal {J}}_i(v_i^n, u_{-i}^n)\\= & {} \limsup _{n \rightarrow \infty }{} \left( \sum _{i = 1}^N {\mathcal {J}}_i(v_i^n,u_{-i}^n) + \gamma ^n \pi _C(v^n,u^n) \right) = \limsup _{n \rightarrow \infty }{} {\mathcal {E}}_\textrm{GNEP}^{\gamma ^n}(v^n, u^n). \end{aligned}$$

$\square$

As in Theorem 6 and Theorem 7, the treatment of the weak convergence in infinite dimensions was of significant importance. Analogously, the corresponding result for variational equilibrium is derived in the following theorem.

Theorem 13

(Extended Mosco convergence of penalized VEPs) Let a sequence of penalty parameters $\gamma _n \rightarrow \infty$ be given. Moreover, assume the following conditions to be fulfilled:

• (i) The joint constraint set ${\mathcal {F}}\subseteq U_{\textrm{ad}}$ is nonempty, bounded, closed and convex.

• (ii) The functional $(v,u) \mapsto \sum _{i = 1}^N{} {\mathcal {J}}_i(v_i,u_{-i})$ is weakly lower semicontinuous on ${\mathcal {F}}$. Moreover, assume that for every weakly convergent sequence $u^n \rightharpoonup u$ and $v \in {\mathcal {F}}$, there exists a sequence $(v^n)_{n \in {\mathbb {N}}} \subseteq {\mathcal {F}}$ with $v^n \rightarrow v$ such that

  $$\begin{aligned} \limsup _{n \rightarrow \infty } \sum _{i = 1}^N {\mathcal {J}}_i(v_i^n,u_{-i}^n) \le \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}). \end{aligned}$$

• (iii) The penalty functional $\pi _{\mathcal {F}}: U_{\textrm{ad}}\rightarrow [0,\infty )$ is weakly lower semicontinuous.

Then, the sequence of functionals ${\mathcal {E}}^\gamma : U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {E}}_\textrm{VEP}^\gamma (v,u):= \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) + \gamma \pi _{\mathcal {F}}(v)\nonumber \\ \end{aligned}$$

(11)

is extended Mosco convergent to

$$\begin{aligned} {\mathcal {E}}_\textrm{VEP}(v,u):= \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) + I_{\mathcal {F}}(v). \end{aligned}$$

Proof

The proof is analogous to the one of Theorem 12 and therefore omitted. $\square$

Returning to the penalty functionals in (9), we would like to discuss the interplay between the conditions listed in Theorems 12 and 13. We consider the latter setting first.

If ${\mathcal {F}}$ is nonempty, closed and convex, also $v \mapsto {\text {dist}}(v,{\mathcal {F}})$ is continuous and convex and thus weakly lower semicontinuous. Thus, the condition on the feasible set ${\mathcal {F}}$ induces the requested properties of the penalty functional.

The reasoning for $\pi _C$ requires more effort. Take weakly convergent sequences $u^n \rightharpoonup u$ and $v^n \rightharpoonup v$. Then, choose the subsequence (not relabeled) realizing $\liminf _{n \rightarrow \infty } {\text {dist}}(v^n, C(u^n))$. For an arbitrary sequence $\varepsilon ^n \searrow 0$, there exists a sequence $(w^n)_{n \in {\mathbb {N}}}, w^n \in C(u^n)$ such that $\Vert v^n - w^n\Vert _U \le {\text {dist}}(v^n, C(u^n)) + \varepsilon ^n$. To show its boundedness, we take an arbitrary $w \in C(u)$. By the complete lower semicontinuity of C, there exists a sequence ${{\tilde{w}}}^n \rightarrow w$ with ${{\tilde{w}}}^n \in C(u^n)$. Then, a direct estimate yields

$$\begin{aligned} \Vert w^n\Vert _U\le & {} \Vert v^n - w^n\Vert _U + \Vert v^n\Vert _U \le {\text {dist}}(v^n, C(u^n)) + \Vert v^n\Vert _U + \varepsilon ^n\\\le & {} \Vert v^n - {{\tilde{w}}}^n\Vert _U + \Vert v^n\Vert _U + \varepsilon ^n \le 2\Vert v^n\Vert _U + \Vert {{\tilde{w}}}^n\Vert _U + \varepsilon ^n. \end{aligned}$$

This implies the boundedness of $(w^n)_{n \in {\mathbb {N}}}$ by the boundedness of $(v^n)_{n \in {\mathbb {N}}}$ and $({{\tilde{w}}}^n)_{n \in {\mathbb {N}}}$. Since U is reflexive, we can extract a weakly convergent subsequence (not relabeled) converging toward $w^* \in U$. By the assumed weak closedness of the graph of C, we deduce using ${{\tilde{w}}}^n \in C(u^n)$ that $w^* \in C(u)$. Utilizing the weak convergence along the previously constructed subsequence, we obtain the estimate

$$\begin{aligned} {\text {dist}}(v,C(u))\le & {} \Vert v - w^*\Vert _U \le \liminf _{n \rightarrow \infty } \Vert v^n - w^n\Vert _U\\\le & {} \lim _{n \rightarrow \infty } \left( {\text {dist}}(v^n, C(u^n)) + \varepsilon ^n \right) = \liminf _{n \rightarrow \infty } {\text {dist}}(v^n, C(u^n)). \end{aligned}$$

This proves the weak lower semicontinuity of $\pi _C$ on $U_{\textrm{ad}}\times U_{\textrm{ad}}$.

4.2 Application to Penalized Quasi-Variational Inequalities

In principle, the arguments in the proofs of Theorems 12 and 13 can be used to derive an analogous result for penalized quasi-variational inequalities. This is addressed in the next theorem.

Theorem 14

(Extended Mosco convergence of penalized QVIs) Let a sequence of positive penalty parameters $\gamma ^n \rightarrow \infty$ be given. Moreover, assume the following assumptions to be fulfilled:

• (i) The set-valued operator $C: U \rightrightarrows U$ has a weakly closed graph and the set U as its domain.

• (ii) Moreover, let C be a completely lower semi-continuous mapping, i.e., for all sequences $u^n \rightharpoonup u$ and all $v \in C(u)$, there exists a sequence $v^n \in C(u^n)$ such that $v^n \rightarrow v$.

• (iii) The operator $f: U \rightarrow U$ is weakly continuous, i.e., $u^n \rightharpoonup u$ implies $f(u^n) \rightarrow f(u)$, and for every $u^n \rightharpoonup u$ holds

  $$\begin{aligned} \limsup _{n \rightarrow \infty } \langle f(u^n),u^n \rangle _{U^*,U} \le \langle f(u), u \rangle _{U^*,U}. \end{aligned}$$

• (iv) Let the penalty functional $\pi _C: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow [0,\infty )$ be weakly lower semicontinuous.

Then, the sequence of functionals ${\mathcal {E}}_\textrm{QVI}^\gamma : U \times U \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {E}}_\textrm{QVI}^\gamma (v,u):= \frac{1}{2}\langle A_\textrm{sym} v,v \rangle _{U^*,U} + \langle A_\textrm{anti}u,v \rangle _{U^*,U} - \langle f(u),v \rangle _{U^*,U} + \gamma \pi _C(v,u)\nonumber \\ \end{aligned}$$

(12)

is extended Mosco convergent to

$$\begin{aligned} {\mathcal {E}}_\textrm{QVI}(v,u):= \frac{1}{2}\langle A_\textrm{sym} v,v \rangle _{U^*,U} + \langle A_\textrm{anti}u,v \rangle _{U^*,U} - \langle f(u),v \rangle _{U^*,U} + I_{C(u)}(v). \end{aligned}$$

Proof

Take sequences $(\gamma ^n)_{n \in {\mathbb {N}}}$ with $\gamma ^n > 0$, $\gamma ^n \rightarrow \infty$ as well as $u^n \rightharpoonup u$ in U and consider the case $u \in C(u)$. The antisymmetry yields $\langle A_\textrm{anti} u^n,u^n \rangle _{U^*,U} = \langle A_\textrm{anti} u,u \rangle _{U^*,U} = 0$. By the coercivity of $A_\textrm{sym}$, the functional $v \mapsto \frac{1}{2}\langle A_\textrm{sym}v,v \rangle _{U^*,U}$ is continuous and convex and thus weakly lower semicontinuous. Then, using assumption (iii), we obtain

$$\begin{aligned} {\mathcal {E}}_\textrm{QVI}(u,u)= & {} \frac{1}{2} \langle A_\textrm{sym} u, u \rangle _{U^*,U} - \langle f(u), u \rangle _{U^*,U}\\\le & {} \liminf _{n \rightarrow \infty } \left( \frac{1}{2} \langle A_\textrm{sym} u^n, u^n \rangle _{U^*,U} - \langle f(u^n), u^n \rangle _{U^*,U} \right) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^{\gamma ^n}(u^n,u^n). \end{aligned}$$

If $u \notin C(u)$ we have $\pi _C(u,u) > 0$. The assumed weak lower semicontinuity of $\pi _C$ yields

$$\begin{aligned} 0 < \pi _C(u,u) \le \liminf _{n \rightarrow \infty } \pi _C(u^n,u^n). \end{aligned}$$

Hence, $\pi _C(u^n,u^n) \ge \frac{1}{2}\pi _C(u,u)$ holds for almost all indices n and therefore

$$\begin{aligned} \lim _{n \rightarrow \infty } \gamma ^n \pi _C(u^n,u^n) = \infty . \end{aligned}$$

The rest of the functional is bounded from below as $\frac{1}{2}\langle A_\textrm{sym} u^n, u^n \rangle _{U^*,U} \ge 0$ by coercivity. By the weak continuity of f we obtain the boundedness of $\Vert f(u^n)\Vert _{U^*}$. This yields

$$\begin{aligned} {\mathcal {E}}_\textrm{QVI}^{\gamma ^n}(u^n,u^n) \ge - \Vert f(u^n)\Vert _{U^*}\cdot \Vert u^n\Vert _{U} + \gamma ^n \pi _C(u^n,u^n) \rightarrow \infty \text { as } n \rightarrow \infty \end{aligned}$$

and thus ${\mathcal {E}}_\textrm{QVI}(u,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}_\textrm{QVI}^{\gamma ^n}(u^n,u^n)$.

Let now again a sequence $u^n \rightharpoonup u$ and $v \in U$ be given. Assume without loss of generality $v \in C(u)$. Choose as recovery sequence the one for the operator C such that $v^n \rightarrow v$ and $v^n \in C(u^n)$. Then, we obtain by the weak continuity of f and the strong convergence of $(v^n)_{n \in {\mathbb {N}}}$ that

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathcal {E}}_\textrm{QVI}^{\gamma ^n}(v^n,u^n)= & {} \lim _{n \rightarrow \infty } \left( \frac{1}{2} \langle A_\textrm{sym} v^n,v^n \rangle _{U^*,U} + \langle A_{\textrm{anti}}v^n,u^n \rangle _{U^*,U} - \langle f(u^n),v^n \rangle _{U^*,U} \right) \\= & {} \frac{1}{2} \langle A_\textrm{sym} v,v \rangle _{U^*,U} + \langle A_{\textrm{anti}}v,u \rangle _{U^*,U} - \langle f(u),v \rangle _{U^*,U} = {\mathcal {E}}_\textrm{QVI}(v,u). \end{aligned}$$

Thus, we obtain the requested extended Mosco convergence. $\square$

5 Relation to Other Convergence Theories

As mentioned earlier, there have been previous attempts to generalize $\Gamma -$, respectively, epiconvergence beyond optimization problems, see e.g., Attouch and Wets (1983), Jofré and Wets (1983), Royset and Wets (2019) and Gürkan and Pang (2009). In the remainder of this work, we compare our results with those and explore their relationship. As most of these concepts have been developed for finite dimensions, they do not distinguish between weak and strong topology. Hence, we perform our comparison for strong topology only. An extension to weak and mixed notions, like in Mosco convergence, should be nevertheless possible with reasonable effort. Moreover, some of these concepts have been developed for a changing sequence of admissible sets, but we stick to fixed ones for the sake of exposition.

5.1 Epi/Hypo-Convergence

In Attouch and Wets (1983), an analog of epiconvergence tailored to saddle point problems called epi/hypo-convergence has been developed. We adapt hereafter the notation in Sect. 2.3.

Definition 15

(epi/hypo-convergence, cf. Attouch and Wets 1983, pp. 8–9) A sequence of bifunctions $(F^n)_{n \in {\mathbb {N}}}: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$ is called epi/hypo-convergent (to $F: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$) if the following two conditions are fulfilled:

• (i) For every sequence $(x^n)_{n \in {\mathbb {N}}} \subseteq X_{\textrm{ad}}$ with $x^n \rightarrow x$ and all $y \in Y_{\textrm{ad}}$, there exists a sequence $(y^n)_{n \in {\mathbb {N}}} \subseteq Y_{\textrm{ad}}$ with $y^n \rightarrow y$ such that

  $$\begin{aligned} F(x,y) \le \liminf _{n \rightarrow \infty } F^n(x^n,y^n). \end{aligned}$$

  (13)

• (ii) For every sequence $(y^n)_{n \in {\mathbb {N}}} \subseteq Y_{\textrm{ad}}$ with $y^n \rightarrow y$ and all $x \in X_{\textrm{ad}}$, there exists a sequence $(x^n)_{n \in {\mathbb {N}}} \subseteq X_{\textrm{ad}}$ with $x^n \rightarrow x$ such that

  $$\begin{aligned} \limsup _{n \rightarrow \infty } F^n(x^n,y^n) \le F(x,y). \end{aligned}$$

  (14)

Then, we may write $F^n \overset{\mathrm {epi/hypo}}{\longrightarrow }F$.

In Attouch and Wets (1983), the above properties have been requested along subsequences. We, however, work with this version. Next, we discuss the relation between Definitions 15 and 8. As epi/hypo-convergence has only been formulated for saddle point problems, it only makes sense to discuss this case.

Theorem 16

In the setting of Sect. 2.3, consider bifunctions $(F^n)_{n \in {\mathbb {N}}}, F: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$ and define $({\mathcal {E}}_{\textrm{SP}}^n)_{n \in {\mathbb {N}}}, {\mathcal {E}}_{\textrm{SP}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} {\mathcal {E}}_{\textrm{SP}}^n((x',y'),(x,y))&:= F^n(x',y) - F^n(x,y') \text {~and~} {\mathcal {E}}_{\textrm{SP}}((x',y'),(x,y)) \\&:= F(x',y) - F(x,y'). \end{aligned}$$

If $F^n \overset{\mathrm {epi/hypo}}{\longrightarrow }F$, then ${\mathcal {E}}_{\textrm{SP}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}_{\textrm{SP}}$.

Proof

Clearly, we have ${\mathcal {E}}(u,u) = F(x,y) - F(x,y) = 0$ and in the same way ${\mathcal {E}}^n(u,u) = 0$ for all $u = (x,y) \in X_{\textrm{ad}}\times Y_{\textrm{ad}}$. Hence, we have trivially (5), and it is left to show (6). We take arbitrary $x' \in X_{\textrm{ad}}, y' \in Y_{\textrm{ad}}$ and sequences $x^n \rightarrow x$ in $X_{\textrm{ad}}$ and $y^n \rightarrow y$ in $Y_{\textrm{ad}}$. By (13) and (14), there exist sequences $x'^n \rightarrow x'$ in $X_{\textrm{ad}}$ and $y'^n \rightarrow y'$ in $Y_{\textrm{ad}}$ such that

$$\begin{aligned} F(x,y') \le \liminf _{n \rightarrow \infty } F^n(x^n, y'^n) \text {~and~} F(x',y) \ge \limsup _{n \rightarrow \infty } F^n(x'^n, y^n) \end{aligned}$$

hold. Then, we get

$$\begin{aligned} {\mathcal {E}}(v,u)= & {} F(x',y) - F(x,y') \ge \limsup _{n \rightarrow \infty } F^n(x'^n,y^n) - \liminf _{n \rightarrow \infty } F^n(x^n,y'^n)\\\ge & {} \limsup _{n \rightarrow \infty } \left( F^n(x'^n,y^n) - F^n(x^n,y'^n) \right) = \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n). \end{aligned}$$

This yields the assertion. $\square$

Hence, extended $\Gamma$-convergence can be interpreted as a weaker notion than epi/hypo-convergence.

5.2 Multi-Epiconvergence

For Nash equilibrium problems, a generalization called multi-epiconvergence has been given in Gürkan and Pang (2009). In the following, we use the notation in Sect. 2.2.

Definition 17

(Multi-epiconvergence) A sequence of families of functionals $(({\mathcal {J}}_i^n)_{i = 1, \dots , N})_{n \in {\mathbb {N}}}$ is called multi-epiconvergent, if the following two conditions hold:

• (i) For all $i = 1, \dots , N$ and sequences $u^n \rightarrow u$ in $U_{\textrm{ad}}$ holds

  $$\begin{aligned} {\mathcal {J}}_i(u) \le \liminf _{n \rightarrow \infty } {\mathcal {J}}_i^n(u^n). \end{aligned}$$

  (15)

• (ii) For $i = 1, \dots , N$ and all sequences $u_{-i}^n \rightarrow u_{-i}$ in $U_{\textrm{ad}}^{-i}$ and $v_i \in U_{\textrm{ad}}^i$ exists a sequence $v_i^n \rightarrow v^i$ in $U_{\textrm{ad}}^i$ such that

  $$\begin{aligned} {\mathcal {J}}_i(v_i, u_{-i}) \ge \limsup _{n \rightarrow \infty } {\mathcal {J}}_i^n(v_i^n, u_{-i}^n). \end{aligned}$$

  (16)

Then, we may write $({\mathcal {J}}_i^n)_{i = 1, \dots , N} \overset{\mathrm {multi-epi}}{\longrightarrow }({\mathcal {J}}_i)_{i = 1, \dots , N}$.

Just as previously, we discuss the relation to our work for Nash games only.

Theorem 18

Consider the families of functionals $(({\mathcal {J}}_i^n)_{i = 1, \dots , N})_{n \in {\mathbb {N}}}, ({\mathcal {J}}_i)_{i = 1, \dots , N}: U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ and associate to them the equilibrium functionals $({\mathcal {E}}_{\textrm{NEP}}^n)_{n \in {\mathbb {N}}}, {\mathcal {E}}_{\textrm{NEP}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {\mathcal {E}}_{\textrm{NEP}}^n(v,u):= \sum _{i = 1}^N {\mathcal {J}}_i^n(v_i,u_{-i}) \text {~and~} {\mathcal {E}}_{\textrm{NEP}}(v,u):= \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}). \end{aligned}$$

The following assertions hold.

• (i) If $({\mathcal {J}}_i^n)_{i = 1, \dots , N} \overset{\mathrm {multi-epi}}{\longrightarrow }({\mathcal {J}}_i)_{i = 1, \dots , N}$, then ${\mathcal {E}}^n_{\textrm{NEP}}\overset{\mathrm {seq.}~\Gamma }{\longrightarrow }{\mathcal {E}}_{\textrm{NEP}}$.

• (ii) Assume ${\mathcal {E}}^n_{\textrm{NEP}}\overset{\mathrm {seq.}~\Gamma }{\longrightarrow }{\mathcal {E}}_{\textrm{NEP}}$. If (15) or (16) holds, then $({\mathcal {J}}_i^n)_{i = 1, \dots , N} \overset{\mathrm {multi-epi}}{\longrightarrow }({\mathcal {J}}_i)_{i = 1}^N$.

Proof

ad (i): For $i = 1, \dots , N$ and all sequences $v_i^n \rightarrow v_i$ in $U_{\textrm{ad}}^i$ and $u_{-i}^n \rightarrow u_{-i}$ in $U_{\textrm{ad}}^{-i}$, we have

$$\begin{aligned} {\mathcal {J}}_i(v_i, u_{-i}) \le \liminf _{n \rightarrow \infty } {\mathcal {J}}^n_i(v_i^n, u_{-i}^n). \end{aligned}$$

This yields

$$\begin{aligned} {\mathcal {E}}_\textrm{NEP}(v,u)= & {} \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) \le \sum _{i = 1}^N \liminf _{n \rightarrow \infty } {\mathcal {J}}^n_i(v_i^n, u_{-i}^n)\\\le & {} \liminf _{n \rightarrow \infty } \sum _{i = 1}^N {\mathcal {J}}^n_i(v_i^n, u_{-i}^n) = \liminf _{n \rightarrow \infty } {\mathcal {E}}_\textrm{NEP}^n(v^n,u^n). \end{aligned}$$

Using (16), we obtain for all $i = 1, \dots , N$ as well as $v_i \in U_{\textrm{ad}}^i$ and all sequences $u_{-i}^n \rightarrow u_{-i}$ in $U_{\textrm{ad}}^{-i}$ the existence of a sequence $v_i^n \rightarrow v_i$ in $U_{\textrm{ad}}^i$ with

$$\begin{aligned} {\mathcal {J}}_i(v_i,u_{-i}) \ge \limsup _{n \rightarrow \infty } {\mathcal {J}}_i(v_i^n, u_{-i}^n). \end{aligned}$$

Then, we deduce

$$\begin{aligned} {\mathcal {E}}_\textrm{NEP}(v,u)= & {} \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i}) \ge \sum _{i = 1}^N \limsup _{n \rightarrow \infty } {\mathcal {J}}^n_i(v_i^n,u_{-i}^n)\\\ge & {} \limsup _{n \rightarrow \infty } \sum _{i = 1}^N {\mathcal {J}}_i^n(v_i^n,u_{-i}^n) = \limsup _{n \rightarrow \infty } {\mathcal {E}}_\textrm{NEP}^n(v^n,u^n). \end{aligned}$$

This shows (i).

ad (ii): We make the following general observation. For sequences of real numbers $(a_n)_{n \in {\mathbb {N}}}$ and $(b_n)_{n \in {\mathbb {N}}}$, we have

$$\begin{aligned} \liminf _{n \rightarrow \infty } a_n= & {} \liminf _{n \rightarrow \infty } \left( a_n + b_n - b_n \right) \ge \liminf _{n \rightarrow \infty } (a_n + b_n) + \liminf _{n \rightarrow \infty } (-b_n)\\= & {} \liminf _{n \rightarrow \infty } (a_n + b_n) - \limsup _{n \rightarrow \infty } b_n. \end{aligned}$$

This yields

$$\begin{aligned} \liminf _{n \rightarrow \infty } (a_n + b_n) \le \liminf _{n \rightarrow \infty } a_n + \limsup _{n \rightarrow \infty } b_n. \end{aligned}$$

(17)

With an analog argument, one obtains

$$\begin{aligned} \limsup _{n \rightarrow \infty } (a_n + b_n) \ge \limsup _{n \rightarrow \infty } a_n + \liminf _{n \rightarrow \infty } b_n. \end{aligned}$$

(18)

Assume first (15) to hold. Take an arbitrary $k \in \{1, \dots , N\}$. From the sequential $\Gamma$-convergence, we get for arbitrary $v \in U_{\textrm{ad}}$ and $u^n \rightarrow u$ in $U_{\textrm{ad}}$ the existence of a recovery sequence $v^n \rightarrow v$ in $U_{\textrm{ad}}$ with

$$\begin{aligned} \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i})= & {} {\mathcal {E}}_{\textrm{NEP}}(v,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}_\textrm{NEP}^n(v^n,u^n). \end{aligned}$$

Using (18), we get

$$\begin{aligned} \limsup _{n \rightarrow \infty } {\mathcal {E}}_\textrm{NEP}^n(v^n,u^n)= & {} \limsup _{n \rightarrow \infty } \left( {\mathcal {J}}_k^n(v_k^n, u_{-k}^n) + \sum _{i \ne k} {\mathcal {J}}_i^n(v_i^n,u_{-i}^n) \right) \\\ge & {} \limsup _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k,u_{-k}^n) + \liminf _{n \rightarrow \infty } \sum _{i \ne k} {\mathcal {J}}_i^n(v_i^n,u_{-i}^n)\\\ge & {} \limsup _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k^n, u_{-k}^n) + \sum _{i \ne k} \liminf _{n \rightarrow \infty } {\mathcal {J}}_i^n(v_i^n, u_{-i}^n)\\\ge & {} \limsup _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k^n, u_{-k}^n) + \sum _{i \ne k} {\mathcal {J}}_i(v_i, u_{-i}). \end{aligned}$$

Subtracting $\sum _{i \ne k}{\mathcal {J}}_i(v_i,u_{-i})$ yields (16) with $(v_k^n)_{n \in {\mathbb {N}}}$ as recovery sequence.

Second, assume now (16) to hold. Let a sequence $u^n \rightarrow u$ in $U_{\textrm{ad}}$ be given. Take $k \in \{1, \dots , N\}$ and an arbitrary sequence $v_k^n \rightarrow v_k$. Moreover, take for each $i \in \{1, \dots , N\}$, $i \ne k$ and arbitrary $v_i \in U_{\textrm{ad}}$ a sequence $v_i^k \rightarrow v_i$ in $U_{\textrm{ad}}$ with

$$\begin{aligned} {\mathcal {J}}_i(v_i,u_{-i}) \ge \limsup _{n \rightarrow \infty } {\mathcal {J}}_i^n(v_i^n,u_{-i}^n). \end{aligned}$$

Then, using (17), we get

$$\begin{aligned} \sum _{i = 1}^N {\mathcal {J}}_i(v_i,u_{-i})= & {} {\mathcal {E}}_{\textrm{NEP}}(v,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}_{\textrm{NEP}}^n(v^n,u^n)\\= & {} \liminf _{n \rightarrow \infty } \left( {\mathcal {J}}_k^n(v_k^n, u_{-k}^n) + \sum _{i \ne k} {\mathcal {J}}_i^n(v_i,u_{-i})\right) \\\le & {} \liminf _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k^n,u_{-k}^n) + \limsup _{n \rightarrow \infty } \sum _{i \ne k} {\mathcal {J}}_i^n(v_i^n,u_{-i}^n)\\\le & {} \liminf _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k^n,u_{-k}^n) + \sum _{i \ne k} \limsup _{n \rightarrow \infty } {\mathcal {J}}_i^n(v_i^n,u_{-i}^n)\\\le & {} \liminf _{n \rightarrow \infty } {\mathcal {J}}_k^n(v_k^n,u_{-k}^n) + \sum _{i \ne k} {\mathcal {J}}_i(v_i,u_{-i}^n). \end{aligned}$$

Subtracting $\sum _{i \ne k} {\mathcal {J}}_i(v_i,u_{-i})$ on both sides yields (15) and finishes the proof. $\square$

5.3 Lopsided Convergence

Lopsided convergence has been introduced in Jofré and Wets (1983) as a generalization of the concept in Sect. 5.1 for so-called min-sup problems reading as follows. Take a bifunction $F: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$.

$$\begin{aligned} \text {Seek}~ {{\bar{x}}} \in {\text {argminsup}} F:= {\text {argmin}}_{x \in X_{\textrm{ad}}}\left( \sup _{y \in Y_{\textrm{ad}}} F(x,y) \right) . \end{aligned}$$

Based on this formulation, it makes sense to apply it to the Nikaido–Isoda formulation for an equilibrium functional as in Definition 1.

Definition 19

(Lopsided convergence, see Jofré and Wets 1983and Royset and Wets 2019, Definition 3.1) A sequence of bifunctions $(F^n)_{n \in {\mathbb {N}}}: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$ lop-converges or converges lopsided (to $F: X_{\textrm{ad}}\times Y_{\textrm{ad}}\rightarrow {\mathbb {R}}$), if the following two conditions hold:

• (i) For all $y \in Y_{\textrm{ad}}$ and all sequences $x^n \rightarrow x \in X_{\textrm{ad}}$ exists a sequence $y^n \rightarrow y$ such that

  $$\begin{aligned} F(x,y) \le \liminf _{n \rightarrow \infty } F^n(x^n,y^n) \end{aligned}$$

  (19)

  holds.

• (ii) For all $x \in X_{\textrm{ad}}$ exists $x^n \rightarrow x$ such that for all $y^n \rightarrow y$ holds

  $$\begin{aligned} F(x,y) \ge \limsup _{n \rightarrow \infty } F^n(x^n,y^n). \end{aligned}$$

  (20)

For Nash equilibrium problems, another concept called system-epiconvergence has been considered, see Gürkan and Pang (2009). Therein, also its relation to multi-epi- and lopsided convergence has been discussed. Here, we propose the following adaptation to Definition 1, which coincides with the aforementioned.

Definition 20

(System-epiconvergence, see Gürkan and Pang 2009, p. 240, 241, (Sa), (Sb)) Consider equilibrium functionals $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}, {\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$. We say that $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}$ is system-epiconvergent (to ${\mathcal {E}}$) if the following two conditions hold:

• (i) For all sequences $u^n \rightarrow u$ in $U_{\textrm{ad}}$ holds

  $$\begin{aligned} {\mathcal {E}}(u,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

  (21)

• (ii) For all $u \in U_{\textrm{ad}}$ exists $u^n \rightarrow u$ in $U_{\textrm{ad}}$ such that

  $$\begin{aligned} {\mathcal {E}}(u,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

  (22)

This differs from Definition 8 only in (22).

Theorem 21

Consider the equilibrium functionals $({\mathcal {E}}^n)_{n \in {\mathbb {N}}}, {\mathcal {E}}: U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ with associated Nikaido–Isoda functionals $(\Psi ^n)_{n \in {\mathbb {N}}}, \Psi : U_{\textrm{ad}}\times U_{\textrm{ad}}\rightarrow {\mathbb {R}}$ as in (2). The following assertions hold:

• (i) If ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$, then $(\Psi ^n)_{n \in {\mathbb {N}}}, \Psi$ satisfy (19).

• (ii) If ${\mathcal {E}}^n \overset{\mathrm {seq.}~\Gamma }{\longrightarrow }{\mathcal {E}}$ and ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$, then $\Psi ^n \overset{\textrm{lop}}{\longrightarrow }\Psi$.

• (iii) If ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$ and $(\Psi ^n)_{n \in {\mathbb {N}}}, \Psi$ satisfy (20), then ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$.

• (iv) If ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$ and $(\Psi ^n)_{n \in {\mathbb {N}}}, \Psi$ satisfy (19), then ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$.

Proof

ad (i): Take a sequence $u^n \rightarrow u$ and $v \in U_{\textrm{ad}}$. Then, there exists $v^n \rightarrow v$ such that

$$\begin{aligned} {\mathcal {E}}(v,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n). \end{aligned}$$

Then, we get

$$\begin{aligned} \Psi (u,v)= & {} {\mathcal {E}}(u,u) - {\mathcal {E}}(v,u) \le \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n) - \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n) \\\le & {} \liminf _{n \in \infty } \Psi ^n(u^n,v^n). \end{aligned}$$

This shows (i).

ad (ii): By (22), there exists $u^n \rightarrow u$ such that

$$\begin{aligned} {\mathcal {E}}(u,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

Using the sequential $\Gamma$-convergence, we obtain for an arbitrary $v_n \rightarrow v$ in $U_{\textrm{ad}}$ that

$$\begin{aligned} \Psi (u,v)= & {} {\mathcal {E}}(u,u) - {\mathcal {E}}(v,u) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n) - \liminf _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n)\\\ge & {} \limsup _{n \rightarrow \infty } \left( {\mathcal {E}}^n(u^n,u^n) - {\mathcal {E}}^n(v^n,u^n) \right) = \limsup _{n \rightarrow \infty } \Psi ^n(u^n,v^n). \end{aligned}$$

In combination with statement (i), we obtain the assertion.

ad (iii): The condition in (21) is guaranteed by the extended $\Gamma$-convergence. To show (22), we take for $u \in U_{\textrm{ad}}$ a sequence $u^n \rightarrow u$ in $U_{\textrm{ad}}$ according to (20) such that

$$\begin{aligned} \Psi (u,v) \ge \limsup _{n \rightarrow \infty } \Psi ^n(u^n,v^n) \end{aligned}$$

for all $v^n \rightarrow v$. Then, we get for an arbitrary $v \in U_{\textrm{ad}}$ with recovery sequence $v^n \rightarrow v$ in (6) that

$$\begin{aligned} {\mathcal {E}}(u,u)= & {} \Psi (u,v) + {\mathcal {E}}(v,u) \ge \limsup _{n \rightarrow \infty } \Psi ^n(u^n,v^n) + \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n)\\\ge & {} \limsup _{n \rightarrow \infty } \left( \Psi ^n(u^n,v^n) + {\mathcal {E}}^n(v^n,u^n) \right) = \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n). \end{aligned}$$

This shows ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$.

ad (iv): It is sufficient to show the condition in (6). For this, take $v \in U_{\textrm{ad}}$ and $u^n \rightarrow u$ in $U_{\textrm{ad}}$. By (19), there exists a sequence $v^n \rightarrow v$ in $U_{\textrm{ad}}$ such that

$$\begin{aligned} \Psi (u,v) \le \liminf _{n \rightarrow \infty } \Psi ^n(u^n,v^n). \end{aligned}$$

Using (18), we get

$$\begin{aligned} {\mathcal {E}}(v,u)= & {} {\mathcal {E}}(u,u) - \Psi (u,v) \ge {\mathcal {E}}(u,u) - \liminf _{n \rightarrow \infty } \Psi ^n(u^n,v^n)\\= & {} {\mathcal {E}}(u,u) - \liminf _{n \rightarrow \infty } \left( {\mathcal {E}}^n(u^n,u^n) - {\mathcal {E}}^n(v^n,u^n) \right) \\= & {} {\mathcal {E}}(u,u) + \limsup _{n \rightarrow \infty } \left( {\mathcal {E}}^n(v^n,u^n) - {\mathcal {E}}^n(u^n,u^n) \right) \\\ge & {} {\mathcal {E}}(u,u) + \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n) + \liminf _{n \rightarrow \infty } (-{\mathcal {E}}^n(u^n,u^n))\\= & {} {\mathcal {E}}(u,u) - \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(u^n,u^n)+ \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n) \ge \limsup _{n \rightarrow \infty } {\mathcal {E}}^n(v^n,u^n), \end{aligned}$$

where we used the assumed system-epiconvergence in the last step. This shows the last assertion. $\square$

In particular, we see that ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$ and $\Psi ^n \overset{\textrm{lop}}{\longrightarrow }\Psi$ imply ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$ as well as that ${\mathcal {E}}^n \overset{\mathrm {sys-epi}}{\longrightarrow }{\mathcal {E}}$ and $\Psi ^n \overset{\textrm{lop}}{\longrightarrow }\Psi$ imply ${\mathcal {E}}^n \overset{\Gamma }{\longrightarrow }{\mathcal {E}}$.

6 Conclusion

Within the scope of this text, we discussed a type of equilibrium problem, formulated equivalent characterizations of equilibria and derived existence results in the abstract case as well as for Nash-type equilibrium problems. The generalized $\Gamma$-convergence concept has been analyzed and applied to a penalization technique for Nash games and QVIs. Moreover, our investigation has been compared to and embedded into the landscape of previous generalizations.

We expect that the presented results are as well-suitable for other approximation techniques in the context of equilibrium problems and serve as a strong theoretical foundation of a convergence analysis and its numerical realization.