Introduction
In the first chapter of the thesis on stability, we explore variational convergence, focusing on the key concepts of epi-convergence for univariate functions and the principal variational convergence We also develop a comprehensive theory of epi/hypo convergence for finite-valued bivariate functions The foundational work on these topics was introduced in several studies, with further details available in notable literature Additionally, we discuss lopsided convergence, a modified form of epi/hypo convergence, which has been rigorously analyzed in prior research These studies primarily address extended real-valued functions and bifunctions, contributing to a deeper understanding of optimization problem approximations through variational convergence.
Lop-convergence has not been applied to extended-real-valued bifunctions, focusing instead on finite-valued bifunctions defined on subsets like C × D ⊂ R n × R m, referred to as fv-biv(R n × R m ) This focus is crucial as many practical bifunctions, including Lagrangians in constraint optimization and payoff functions in zero-sum games, are finite-valued Lop-convergence, being stronger than epi/hypo convergence, possesses beneficial variational properties but is inherently one-sided, focusing on either minsup or maxinf points rather than both A more comprehensive approach to variational convergence should address saddle points, which encompass both minsup and maxinf points, suggesting that distinctions between these problems may be unnecessary, particularly in duality contexts This chapter revisits epi/hypo convergence, previously discussed for extended real-valued bifunctions, but emphasizes the significance of finite-valued bifunctions Section 2.2 outlines basic concepts of epi-convergence and explores Legendre-Fenchel conjugates of finite-valued unifunctions Subsequent sections provide detailed characterizations of epi/hypo convergence in fv-biv(R n × R m ), comparing them with lop-convergence These characterizations can be complex due to the non-uniqueness of limits, forming equivalence classes that impact properties like closedness and convexity Ignoring this non-uniqueness may result in errors in analyzing convergence, as highlighted in existing literature.
Understanding the full characterizations of epi/hypo convergence is crucial for both grasping its essence and its practical applications These characterizations serve as the strongest necessary conditions and the weakest sufficient conditions to determine whether a sequence of bifunctions exhibits epi/hypo convergence Additionally, the variational properties associated with epi/hypo convergence are frequently required, which will be further explored in Chapter 3.
Epi-convergence
Epi-convergence
In [42], the epi-convergence defined in [84-86] for the class fcn(R n ) was adjusted for finite-valued unifunctions as follows.
Definition 2.2.1 (epi-convergence, [42]) A sequence {f ν : C ν → R}ν∈ N is called to epi-converge tof :C →R, denoted by f ν → e f or f = e-lim ν f ν , if
(a) for all x ν ∈C ν →x, liminfνf ν (x ν ) ≥f(x) when x∈ C and f ν (x ν )→ ∞ when x /∈C;
(b) for all x∈C, there exists x ν ∈C ν →xsuch that limsupνf ν (x ν )≤f(x).
Regardless of whether we consider fv-fcn(R n) or fcn(R n), the convergence of fν to ef occurs if and only if the epigraph of fν converges to the epigraph of ef in the Painlevé-Kuratowski sense This principle also extends to the definitions of upper and lower epi-limits based on epigraphs.
Definition 2.2.2 (lower and upper epi-limits) Let f ν , f ∈ fv-fcn(R n ).
(i)f is the lower epi-limit of the functionsf ν , denoted byf = e-li ν f ν , if epif is the outer set-limit of epif ν
(ii) f is the upper epi-limit of the functions f ν , denoted by f = e-ls ν f ν , if epif is the inner set-limit of epif ν
Of course, e-li ν f ν ≤e-ls ν f ν and the epi-limit exists if and only if e-li ν f ν = e-ls ν f ν e-lim ν f ν
Proposition 2.2.1 states that the lower and upper epi-limits, as well as the epi-limit of a sequence {f ν }ν∈ N in fv-fcn(R n) or fcn(R n), are lower semicontinuous (lsc) Additionally, if the functions f ν are convex, then the upper epi-limit and the epi-limit (if it exists) are also convex This indicates that the set of lsc convex functions is closed under epi-convergence.
We can identify fv-fcn(R n ) with a subclass of the so-called proper functions of fcn(R n ), denoted by pfcn(R n ), as follows A bijection η between fv-fcn(R n ) and pfcn(R n ) is defined by
In the context of hypo-convergence, we define it symmetrically: a function sequence \( f_\nu \) hypo-converges to \( f \) if and only if the negative sequence \( -f_\nu \) converges to \( -f \) Consequently, the principles governing epi-convergence can be adapted for hypo-convergence by substituting terms such as epi with hypo, and adjusting limits and extrema accordingly, replacing liminf with limsup, inf with sup, argmin with argmax, and recognizing the changes in continuity conditions from lower semicontinuous (lsc) to upper semicontinuous (usc), along with the appropriate inequalities.
Legendre-Fenchel transform and its continuity
In this section, we explore the concepts of Legendre-Fenchel conjugation and skew-conjugation within the framework of fv-fcn(R n) The Legendre-Fenchel conjugate, also referred to as the Fenchel or Young-Fenchel conjugate, is defined for a function f in fcn(R n) as follows: for any u in (R n) *, the conjugate f*(u) is given by the supremum of the expression hu, xi - f(x) over all x in R n, or more specifically, over the domain of f.
In the special case where f has a one-to-one gradient5f, the Legendre-Fenchel trans- form comes down to the classical Legendre transform of the calculus of variations: f ∗ (u) =hu,(5f) −1 ui −f((5f) −1 u).
The skew-conjugate of a convex functionf ∈fcn(R n ) is defined in [69] as g(u) := infx∈ R n {f(x)− hx, ui}=−f ∗ (u) (2.1) Then, g is a (upper) closed (i.e., usc) concave function and
Starting from a concave function g ∈ fcn(R n ), its skew-conjugate is a (lower) closed (i.e., lsc) convex functionf ∈fcn(R n ) defined in [69] by f(x) := sup u∈ R n{g(u) +hu, xi}= (−g) ∗ (x) (2.2)
The skew-conjugate offis establishes a one-to-one correspondence between closed convex functions and closed concave functions within the space of functions defined on R^n.
Consider now a closed convex function f ∈ fv-fcn(R n ) We compute fˆ ∗ (u) := sup x∈ R n{hu, xi −(ηf)(x)}= (ηf) ∗
We define the Legendre-Fenchel conjugate of f as f ∗ := ˆf ∗ | dom ˆ f ∗= (ηf) ∗ | dom(ηf ) ∗ with domf ∗ := dom ˆf ∗ = dom(ηf) ∗ For the closed convex function ηf ∈fcn(R n ) it is known that
The relationship between a function \( f \) and its conjugate \( f^* \) is established through the equation \( \eta_f | \text{dom}(\eta_f) = f \), highlighting a one-to-one correspondence When performing a conjugation, it is essential to restrict the resulting function to its domain, ensuring it remains a valid function within \( fv-\text{fcn}(\mathbb{R}^n) \) For closed convex elements in \( fv-\text{fcn}(\mathbb{R}^n) \), the Legendre-Fenchel transform exhibits bicontinuity concerning epi-convergence This is affirmed by the fact that the epigraph of \( f \) is equivalent to the epigraph of \( \eta_f \) for any \( f \) in \( fv-\text{fcn}(\mathbb{R}^n) \), allowing the application of relevant results from [68, 69] to the proper functions involved.
Now we move on to defining the skew-conjugate of a closed convex function f ∈ fv-fcn(R n ) We see that ˆ g(u) := inf x∈C f {f(x)− hx, ui}
We define the skew-conjugate of f as g := ˆg | domˆ g (then domg = domˆg and g ∈ fv- fcn(R n ) is closed and concave) For the closed convex function ηf ∈ fcn(R n ), one knows from [68] that
(ηf)(x) = sup u∈ R n{ˆg(u) +hu, xi}= sup u∈domg {g(u) +hu, xi}.
Asηf | dom(ηf) =f, the double skew-conjugate of a closed convex functionf ∈fv-fcn(R n ) is againf.
Exactly as for fcn(R n ), for a closed convex function f ∈ fv-fcn(R n ) and its skew- conjugate g, we have f = (−g) ∗ , g = (−f) ∗ and hence f ∗ =−g, g ∗ =−f Therefore, we have
Thus, in fv-fcn(R n ) epi-convergence of closed convex functions is equivalent to hypo- convergence of their skew-conjugates.
Epi/hypo convergence and lopsided convergence, geometric characteri-
The first study of finite-valued bifunctions was [42], where lopsided convergence for a bifunctionK ∈ fv-biv(R n ×R m ) was proposed as follows.
Minsup lopsided convergence refers to the behavior of bifunctions \( K_\nu \) within the space of finite-valued bifunctions \( fv-biv(\mathbb{R}^n \times \mathbb{R}^m) \) Specifically, \( K_\nu \) is said to minsup-lop converge to \( K \) if, for every \( y \) in the domain \( D \) and for every sequence \( x_\nu \) converging to \( x \) in \( C \), there exists a sequence \( y_\nu \) converging to \( y \) such that the limit inferior of \( K_\nu(x_\nu, y_\nu) \) is greater than or equal to \( K(x, y) \) when \( x \) is in \( C \) Conversely, if \( x \) is not in \( C \), the value \( K_\nu(x_\nu, y_\nu) \) approaches infinity.
(b) for all x ∈ C, there exists x ν ∈ C ν → x such that, for all y ν ∈ D ν → y, limsup ν K ν (x ν , y ν )≤K(x, y) ify ∈D orK ν (x ν , y ν )→ −∞ if y /∈D.
A more restrictive definition of lop-convergence includes the condition that \( C \nu \times D \nu \) converges to \( C \times D \) Additionally, an alternative definition of this convergence is weaker than Definition 2.3.1 as it excludes the "infinity condition" for points \( x \notin C \) and \( y \notin D \) It's important to note that Conditions (a) and (b) of Definition 2.3.1 are not symmetric Moreover, since we focus on minimizing with respect to \( x \) and maximizing with respect to \( y \), we introduce the concept of maxinf-lop convergence, which differs from the maxinf-lop convergence defined in [42].
(a) for all x ∈ C and y ν ∈ D ν → y, there exists x ν ∈ C ν such that x ν → x and lim sup ν K ν (x ν , y ν )≤K(x, y) ify ∈Dor K ν (x ν , y ν )→ −∞ if y /∈D;
(b) for all y∈D, there existsy ν ∈D ν →ysuch that, for all x ν ∈C ν withx ν →x, lim inf ν K ν (x ν , y ν )≥K(x, y) if x∈C orK ν (x ν , y ν )→ ∞if x /∈C.
The nonsymmetric notions discussed in [43] arise from various practical applications that often require one-sided convergence, focusing on either minsup points or maxinf points In contrast, the variational convergence of bifunctions primarily targets the convergence of saddle points, which encompass both minsup and maxinf points A saddle point consists of a minsup point and a maxinf point, leading us to propose a new definition for epi/hypo convergence of finite-valued bifunctions This definition serves as a counterpart to the previously established epi/hypo convergence for extended real-valued bifunctions outlined in [11].
Definition 2.3.2 (epi/hypo convergence) Bifunctions K ν , ν ∈N, in fv-biv(R n ×R m ) are called epi/hypo convergent (e/h-convergent) to a bifunction K ∈ fv-biv(R n ×R m ) if
(a) for all y ∈ D and all x ν ∈ C ν → x, there exists y ν ∈ D ν → y such that liminf ν K ν (x ν , y ν )≥K(x, y) if x∈C orK ν (x ν , y ν )→ ∞ if x /∈C;
(b) for all x ∈ C and all y ν ∈ D ν → y, there exists x ν ∈ C ν → x such that limsup ν K ν (x ν , y ν )≤K(x, y) ify ∈D orK ν (x ν , y ν )→ −∞ if y /∈D.
The convergence is denoted as K ν e/h → K or K = e/h-lim ν K ν It is important to note that if the functions K ν are independent of y, then epi/hypo convergence simplifies to epi-convergence, while if they are independent of x, it reduces to hypo-convergence This also applies to lop-convergence However, it is crucial to understand that epi/hypo convergence, similar to lop-convergence, does not equate to the epi-convergence of K ν (ã, y) to K(ã, y) for all y, nor does it imply hypo-convergence.
K ν (x,ã) to K(x,ã) for allx This is a sufficient condition for e/h-convergence, but not necessary.
The conditions for epi/hypo convergence, as outlined in Definition 2.3.2, are symmetric When hypo/epi convergence of K ν to K is defined as the epi/hypo convergence of -K ν to -K, it becomes essential to focus on maximizing with respect to x and minimizing with respect to y By merely altering the order of these operations, we must apply the same Definition 2.3.2 for hypo/epi convergence, confirming that epi/hypo convergence exhibits complete symmetry.
Both minsup-lop convergence and maxinf-lop convergence inherently imply e/h-convergence, as they share a common condition while the other condition of lop-convergence is stronger than that of epi/hypo convergence Specifically, in the case of minsup-lop convergence, if a common sequence {x ν ∈ C ν } can be found such that limsup ν K ν (x ν , y ν ) ≤ K(x, y) or K ν (x ν , y ν ) → −∞, then condition (b) for epi/hypo convergence is satisfied, allowing for the sequence x ν → x to depend on y ν → y However, the reverse implication does not hold true, as demonstrated by subsequent examples.
Clearly K ν e/h → K We show that Condition (b) of Definition 2.3.1 of minsup-lop convergence is violated For x = 0 and any x ν ∈ C ν → x, we take y = 0 and y ν ∈D ν →0 such that y ν 6=x ν for all ν Then, limsupνK ν (x ν , y ν ) = 1>0 =K(x, y).
In the definitions provided, it is important to note that we do not require the condition C ν × D ν to converge to C × D Additionally, when analyzing limits, we focus not only on the points (x, y) within C × D but also on points (x, y) that are limits of sequences (x ν, y ν) belonging to C ν × D ν.
This article explores the relationship between fv-biv(R n × R m) and biv(R n × R m) We introduce two sets of proper bifunctions within biv(R n × R m) that are associated with a specific K in fv-biv(R n × R m), akin to the definition of ηf for functions in fv-fcn(R n).
In the context of fv-biv(R n ×R m ), we establish two bijections, η e/h and η h/e, that map into distinct collections of proper bifunctions in biv(R n ×R m ) According to Definition 2.3.2, the definition includes points (x, y) that belong to C × D, as well as those where x is in C and y is not in D, or vice versa These points are referred to as (e/h-convergence) involved points Notably, for all involved points, the relationship η e/h (x, y) equals η h/e (x, y) To streamline our analysis of e/h- and lop-convergence within fv-biv(R n ×R m ), we will henceforth refer to both bijections simply as η, unless a specific formula is required.
To refine the concept of e/h-convergence in the context of biv(R n × R m), we need to adjust the definition provided in [11], as outlined in Definition 2.3.3 For a given K in biv(R n × R m), the effective domain is defined as domK = dom x K × dom y K Here, dom x K consists of all x in R n such that K(x, y) is finite for every y in R m, while dom y K includes all y in R m for which K(x, y) is greater than or equal to negative infinity for every x in R n.
Thus,K is finite on domK, butK might be finite also at points outside domK.
Definition 2.3.3 (epi/hypo convergence, biv) Let K and K ν , ν ∈N, be in biv(R n ×
R m ) We say that K ν e/h-converge toK if
The definition of e/h-convergence in [11] stipulates that conditions (a xt) and (b xt) must apply to all y ∈ R^m and x ∈ R^n, rather than being limited to y ∈ dom y K and x ∈ dom x K An illustrative example is provided by the sequence {K ν}ν∈N, which demonstrates e/h-convergence according to Definition 2.3.3, yet fails to meet the criteria outlined in [11].
In the context of the convergence of families, we observe that for the domains dom x K = (0,1) and dom y K = [0,1], the sequence K ν e/h-converges to K as per Definition 2.3.3 However, they fail to satisfy the condition (a xt) when replacing ∀y∈dom y K with ∀y∈R m Specifically, for y = −1 and x ν approaching ν −1 → 0, there is no corresponding y ν converging to −1 that meets the condition liminf ν K ν (ν −1 , y ν )≥K(0,−1) =∞ This example illustrates that, according to the definition in [11], e/h-convergent families must be limited to those that converge appropriately.
To characterize e/h- and lop-convergence we introduce the following four limits connected to a given sequence{K ν } ν∈ N of bifunctions in fv-biv(R n ×R m ).
Definition 2.3.4 Let K ν ∈fv-biv(R n ×R m ) forν ∈N We define:
{x ν ∈Cinf ν →x}liminf ν K ν (x ν , y ν ),for all involved points (x, y).
The effective domains of bifunctions in the space biv(R n × R m) are defined as follows: for the bifunction \( l \in K_\nu \), the domain with respect to \( x \) is given by \( \text{dom}_x(l \in K_\nu) = \{ x \in R^n | (l \in K_\nu)(x, y) < \infty \text{ for all related } y \} \), and the domain with respect to \( y \) is \( \text{dom}_y(l \in K_\nu) = \{ y \in R^m | (l \in K_\nu)(x, y) > -\infty \text{ for all related } x \} \) This definition extends similarly to the other three limit bifunctions Consequently, when restricted to their respective domains, these four limit bifunctions are classified within the framework of fv-biv(R n × R m).
In this article, we denote \( l_h K_\nu \leq l_h K_\nu \) to indicate that \( (l_h K_\nu)(x, y) \leq (l_h K_\nu)(x, y) \) for all relevant pairs \( (x, y) \) According to Definition 2.3.4, it follows that \( leK_\nu \leq leK_\nu \), \( lhK_\nu \leq lhK_\nu \), \( lhK_\nu \leq leK_\nu \), \( l_h K_\nu \leq leK_\nu \), and \( l_h K_\nu \leq leK_\nu \) We will now utilize the four limit bifunctions along with \( \eta \) to define and analyze e/h- and lop-convergence.
Theorem 2.3.1 (characterizations of e/h- and lop-convergence) Let K and K ν , ν ∈
(i) K ν e/h-converge to K if and only if, at each involved (x, y),
(ii) K ν minsup-lop-converge to K if and only if, at each involved (x, y),
(l e K ν )(x, y)≤(ηK)(x, y)≤(l h K ν )(x, y), and hence we have in fact the equalities by Remark 2.3.2;
(iii) K ν maxinf-lop-converge to K if and only if, at each involved (x, y),
(leK ν )(x, y)≤(ηK)(x, y)≤(lhK ν )(x, y),and hence we have in fact the equalities by Remark 2.3.2.
Proof (i) (l h K ν )(x, y)≥(η e/h K)(x, y) means that, for all involved (x, y),∀x ν ∈C ν → x, ∃y ν ∈D ν →y, liminf ν K ν (x ν , y ν )≥(η e/h K)(x, y) (2.3)
Therefore, if (x, y)∈C×D, the right side of (2.3) isK(x, y) While ifx /∈C, y ∈D, it is∞ Thus, (i) is sufficient for (a) of the definition of e/h-convergence Conversely, if
K ν e/h → K, then ∀x ν ∈C ν → x, ∀y ∈D, ∃y ν ∈ D ν → y such that liminf ν K ν (x ν , y ν )≥ K(x, y) if x ∈ C or K ν (x ν , y ν ) → ∞ if x /∈ C Then, as (ηe/hK)(x, y) = ∞ in the latter case, we obtain (2.3) For the case where x /∈ C and y /∈ D, (2.3) is evidently fulfilled since (ηe/hK)(x, y) = −∞.
Now we take care of (b) (l e K ν )(x, y) ≤(η h/e K)(x, y) means that, for all involved points (x, y), ∀y ν ∈D ν →y, ∃x ν ∈C ν →x, limsup ν K ν (x ν , y ν )≤(η h/e K)(x, y) (2.4)
If (x, y) ∈ C × D, the right side of (2.4) is K(x, y) If y /∈ D, x ∈ C, it is −∞. Thus, (i) is sufficient for (b) of the mentioned definition Conversely, if K ν e/h → K, then
∀y ν ∈D ν →y,∀x∈C,∃x ν ∈C ν →xsuch that limsup ν K ν (x ν , y ν )≤K(x, y) ify∈D orK ν (x ν , y ν )→ −∞ if y /∈D Then, one has (2.4) for x∈C In the case x /∈ C and y /∈D, (η h/e K)(x, y) = ∞ and (2.4) is obvious.
(ii) Only (b) is different from the case (i) and needs to be checked (l e K ν )(x, y)≤ (η h/e K)(x, y) means that, for all involved (x, y),∃x ν ∈C ν →x, ∀y ν ∈D ν →y, limsup ν K ν (x ν , y ν )≤(η h/e K)(x, y) (2.5)
If (x, y) ∈ C × D, the right side of (2.5) is K(x, y) When x ∈ C and y /∈ D, (η h/e K)(x, y) = −∞ and (2.5) implies that K ν (x ν , y ν ) → −∞ as required in (b) for minsup-lop convergence Conversely, let K ν minsup-lop converge to K For (x, y) ∈
C ×D, (2.5) is fulfilled by definition If x ∈ C and y /∈ D, then ∃x ν ∈ C ν → x,
∀y ν ∈ D ν → y, K ν (x ν , y ν ) → −∞ and (2.5) is also fulfilled If x /∈ C and y /∈ D, (η h/e K)(x, y) =∞ and (2.5) is evident.
(iii) This is similar to (ii)
In Remark 2.3.3 (i), we establish that for a sequence \( K_\nu \in fv-biv(R^n \times R^m) \), Theorem 2.3.1 enables us to determine whether it e/h- or lop-converges and to identify limit bifunctions if it does Specifically, \( K_\nu \) e/h-converges if and only if \( (l_e K_\nu)(x, y) \leq (l_h K_\nu)(x, y) \) for all relevant \( (x, y) \) In this scenario, any \( K \in fv-biv(R^n \times R^m) \) satisfying \( (\eta_{e/h} K)(x, y) \) within the interval \([l_e K_\nu(x, y), l_h K_\nu(x, y)]\) for all pertinent \( (x, y) \) serves as an e/h-limit of \( K_\nu \), thus forming an e/h-equivalence class for the e/h-convergent sequence Regarding lop-convergence, Remark 2.3.2 confirms the uniqueness of both the minsup-lop limit and maxinf-lop limit, while Theorem 2.3.1 further indicates that the existence of either limit guarantees that \( \{K_\nu\} \) also e/h-converges, a conclusion supported by the definitions.
A characterization by e/h-convergence of proper bifunctions
Theorem 2.4.1 Let K, K ν , ν ∈ N, be in fv-biv(R n ×R m ) Then, K ν e/h-converges to K if and only if ηK ν e/h-converges to ηK.
Proof We use the formula of η e/h in this proof, but employ the notation η.
(a xt ) ⇒ (a) of Definition 2.3.2 Suppose y ∈ D = dom y (ηK) and x ν ∈ C ν → x. Then, (a xt ) of Definition 2.3.3 yieldsy ν ∈R m →y such that liminf ν (ηK ν )(x ν , y ν )≥(ηK)(x, y) (2.7)
If we consider a point \( x \) in set \( C \) and assume there is a subsequence \( y_{\nu_k} \) that does not belong to \( D_{\nu_k} \), it leads to a contradiction since \( (\eta K_{\nu_k})(x_{\nu_k}, y_{\nu_k}) \) would equal \(-\infty\) Therefore, for sufficiently large \( \nu \), \( y_{\nu} \) must belong to \( D_{\nu} \), and the condition (2.7) implies that the limit inferior of \( \nu K_{\nu}(x_{\nu}, y_{\nu}) \) is greater than or equal to \( K(x, y) \), as required by condition (a) Conversely, if \( x \) is not in \( C \), then \( (\eta K)(x, y) \) is infinite, and (2.7) indicates that \( (\eta K_{\nu})(x_{\nu}, y_{\nu}) \) approaches infinity A subsequence \( y_{\nu_k} \) not in \( D_{\nu_k} \) would again lead to a contradiction since it would imply \( (\eta K_{\nu_k})(x_{\nu_k}, y_{\nu_k}) \equiv -\infty \) Thus, it follows that \( K_{\nu}(x_{\nu}, y_{\nu}) \equiv (\eta K_{\nu})(x_{\nu}, y_{\nu}) \) approaches infinity, as required by condition (a).
(a) ⇒ (axt) Let y ∈ domy(ηK) = D and x ν → x ∈ R n If x ∈ C and there is a subsequence x ν k ∈C ν k , then (a) gives y ν k ∈D ν k →y such that liminfk(ηK ν k )(x ν k , y ν k ) = liminfkK ν k (x ν k , y ν k )≥(ηK)(x, y) (2.8)
To construct a convergent sequence \( y_\nu \) where \( x_\nu \notin C_\nu \) and \( y_\nu \in D_\nu \), we ensure that for a given \( y \in D \), there exists a corresponding \( y_\nu \) such that \( y_\nu \to y \) By substituting the elements of the sequence \( \{y_\nu\}_{\nu \in \mathbb{N}} \) with \( y_{\nu_k} \) at specific indices \( \nu_k \), we maintain the convergence For instances where \( x_\nu \notin C_\nu \), the expression \( (\eta K_\nu)(x_\nu, y_\nu) \equiv \infty \) does not influence the limit inferior in (2.8) Consequently, if \( x_\nu \notin C_\nu \) for sufficiently large \( \nu \), we can select any \( y_\nu \in D_\nu \to y \), leading to \( (\eta K_\nu)(x_\nu, y_\nu) \equiv \infty \) and confirming (2.7).
If \( x \notin C \) and \( y \in D \), then \( (\eta K)(x, y) = \infty \) For a sequence \( \{x_\nu\}_{\nu \in \mathbb{N}} \) where \( x_\nu \notin C_\nu \) for large \( \nu \), we can choose \( y_\nu \in D_\nu \) such that \( (\eta K_\nu)(x_\nu, y_\nu) \equiv \infty \), leading to equation (2.7) Alternatively, if there exists a subsequence \( x_{\nu_k} \in C_{\nu_k} \), then we can find \( y_{\nu_k} \in D_{\nu_k} \) that converges to \( y \) while ensuring \( K_{\nu_k}(x_{\nu_k}, y_{\nu_k}) \to \infty \) For additional \( \nu \) where \( x_\nu \notin C_\nu \), we can include points \( y_\nu \in D_\nu \) to ensure the entire sequence converges to \( y \), since for those \( \nu \), \( (\eta K_\nu)(x_\nu, y_\nu) \equiv \infty \) guarantees that the full convergent sequence \( K_\nu(x_\nu, y_\nu) \to \infty \), thus confirming equation (2.7).
(b xt ) ⇒ (b) Let x ∈ C and y ν ∈ D ν → y Condition (b xt ) supplies a sequence x ν →x such that limsup ν (ηK ν )(x ν , y ν )≤(ηK)(x, y) (2.9)
If \( x_\nu \in D \) and there exists a subsequence \( x_{\nu_k} \notin C_{\nu_k} \), it leads to a contradiction with equation (2.9), as \( (\eta K_{\nu_k})(x_{\nu_k}, y_{\nu_k}) \equiv \infty \) Thus, for sufficiently large \( \nu \), it follows that \( x_\nu \in C_\nu \) as required by condition (b) If \( x_\nu \) belongs to \( C_\nu \) for all large \( \nu \), then (2.9) ensures that \( \limsup_{\nu} K_\nu(x_\nu, y_\nu) \leq K(x, y) \), satisfying condition (b) Additionally, if \( y \notin D \), then \( (\eta K)(x, y) = -\infty \), reinforcing the contradiction if a subsequence \( x_{\nu_k} \) exists outside \( C_\nu \) Consequently, for all large \( \nu \), we confirm that \( x_\nu \in C_\nu \), thereby validating condition (b).
Let \( x \) belong to the domain of \( x \) where \( \eta K = C \) and \( y \) approaches \( y \) in \( \mathbb{R}^m \) If \( y \) is in \( D/\nu \) for sufficiently large \( \nu \), then \( (\eta K_\nu)(., y_\nu) \) is equivalent to \(-\infty\), allowing us to select any sequence \( x_\nu \) converging to \( x \) to satisfy (2.9) Conversely, if there exists a subsequence \( y_{\nu_k} \) in \( D_{\nu_k} \), then \( (b) \) implies that \( x_{\nu_k} \) belongs to \( C_{\nu_k} \) and converges to \( x \), ensuring that the limit superior of \( (\eta K_{\nu_k})(x_{\nu_k}, y_{\nu_k}) \) is less than or equal to \( (\eta K)(x, y) \) when \( y \) is in \( D \) For indices \( \nu \) where \( y_\nu \) is not in \( D_{\nu} \), since \( (\eta K_\nu)(., y_\nu) \) is \(-\infty\), we can choose any sequence \( x_\nu \) that converges to \( x \) to achieve (2.9).
Now assume that y /∈D and hence (ηK)(x, y) =−∞ Ify ν ∈/ D ν for large ν, then
(ηK ν )(., y ν ) ≡ −∞ and (2.9) is clearly satisfied Otherwise, there is a subsequence y ν k ∈D ν k , then (b) yields x ν k ∈C ν k →x such that
Forν with y ν ∈/ D ν , one has (ηK ν )(., y ν )≡ −∞ and hence one can take x ν to ensure that the entire sequence x ν →x Now (2.9) is fulfilled as limsup ν (ηK ν )(x ν , y ν ) = −∞= (ηK)(x, y).
A characterization by continuity of partial Legendre-Fenchel transform 25 2.6 Conclusions
In this section we consider convex-concave bifunctions in fv-biv(R n × R m ) In [68,
The partial Legendre-Fenchel conjugate and skew-conjugate have been introduced and effectively developed for extended real-valued convex-concave bifunctions It is important to note that, similar to univariate functions, the term "conjugate" is frequently substituted with "transform."
The bicontinuity of these operations for extended real-valued bifunctions was studied in [6, 7, 11] In this section, we establish this bicontinuity for finite-valued bifunctions.
We first summarize basic facts about minimax equivalence and closedness for finite- valued convex-concave bifunctions These facts are derived from the counterparts in
[68-70] for extended real-valued bifunctions.
We define partial Legendre-Fenchel skew-conjugates for a convex-concave bifunction
K in fv-biv(R n ×R m ) as follows For x ∈ C and v ∈R m , take the skew-conjugate of concave functionK(x, ):
Then, ˆF is a closed convex function The effective domain dom ˆF := {(x, v)| x ∈
The function F, defined as the convex parent of K, is represented as F := ˆF| dom ˆF, and is classified as a unifunction in the space fv-fcn(R n+m), with its domain being domF = dom ˆF, distinguishing it from elements in fv-biv(R n × R m) Additionally, for each y in the domain D, we define the skew-conjugate of the convex function K(., y) within the space fv-fcn(R n) for any u in R n.
In the context of bifunctions, G(u, y) is defined as the infimum of K(x, y) minus the inner product of hx, ui, leading to the concave parent of K, denoted as G, with the domain domG Bifunctions K and K' in the space fv-biv(R n × R m) are considered minimax equivalent if they share the same convex and concave parents, forming a minimax equivalence class represented as an interval [K, K] In this class, K and K' are referred to as the lower and upper members, respectively The equivalence class is uniquely determined by the kernel of K, which is defined as the restriction of K to the relative interiors of sets C and D The functions cl1 K and cl2 K are derived by closing K as a convex and concave function, respectively A bifunction K is classified as closed if it is minimax equivalent to both cl1 K and cl2 K, highlighting the distinction between minimax equivalence and e/h-equivalence, as well as the closedness of unifunctions versus bifunctions.
Minimax equivalence is an important concept, and we will outline key characterizations in Proposition 2.5.1 To support this discussion, we will refer to the natural definitions of complete conjugate and skew-conjugate as presented in sources [68-70] It is essential to recognize that our focus extends beyond single bifunctions to encompass minimax equivalence classes Specifically, for a convex-concave bifunction K ∈ fv-biv(R n ×R m ), the minimax equivalence class of its conjugates includes closed convex-concave bifunctions, forming an interval defined by specific lower and upper members.
L(u, v) := sup x∈C y∈Dinf{hx, ui+hy, vi − K(x, y)},
The minimax equivalence class of skew-conjugates includes closed concave-convex bifunctions, represented by the interval of bifunctions defined by specific lower and upper bounds.
J(u, v) := sup x∈C y∈Dinf{K(x, y) − hx, ui+hy, vi},
Proposition 2.5.1 Let E be a collection of closed convex-concave bifunctions in fv- biv(R n ×R m ) Then, the following assertions are equivalent
(ii) all K ∈E have the same (minimax equivalence class of ) conjugates;
(iii) all K ∈E have the same (minimax equivalence class of ) skew-conjugates; (iv) all K ∈E have the same cl 1 K and the same cl 2 K;
(v) all K ∈ E have the same domain C×D and have the same values, except at the “corner” points, i.e., points (x, y)∈C×D with x /∈riC and y /∈riD;
(vi) all K ∈ E have the same values of subdifferential ∂K(x, y) on the (common) domain (it is also known that ∂K(x, y)6=∅ if x∈riC and y ∈riD).
All minimax equivalent convex-concave bifunctions share identical saddle points and saddle values A convex-concave bifunction is considered closed if its convex and concave components are skew-conjugate Additionally, for any K within the interval [K, K], it holds that cl 1 K = K and cl 2 K = K.
By utilizing Theorem 1 from reference [69] on ηK, where K is part of fv-biv(R n × R m) and η is defined as either η e/h or η h/e in Section 2.3, we can establish a significant correspondence relation Given that all bifunctions that are minimax equivalent to ηK share the same effective domain C × D, deriving this relation becomes straightforward.
Proposition 2.5.2 There is an one-to-one correspondence between the finite-valued closed convex functions F in fv-fcn(R n+m ), the finite-valued closed concave functions
The article discusses the minimax equivalence classes of closed convex-concave bifunctions K and closed concave-convex bifunctions H within the framework of fv-biv(R n × R m) It highlights the skew-conjugate relationships between functions F and G, as well as between K and H, indicating that F and G serve as the foundational elements for K and H, respectively Additionally, it addresses the lower and upper members of the minimax equivalence class of K, which correspond to functions F and G.
K(x, y) = inf v∈domF (x,.){F(x, v) − hy, vi}, (2.11) and we restrict K and K to their domains to have bifunctions in fv-biv(R n ×R m ).
The following notion of extended epi/hypo convergence is a counterpart for fv- biv(R n ×R m ) of the notion introduced in [11] for extended-real-valued bifunctions.
Definition 2.5.1 Let K ν , K ∈ fv-biv(R n ×R m ) for ν ∈ N K ν is said to epi/hypo converge in the extended sense to K if cl 1 (l e K ν )≤ηK ≤cl 2 (l h K ν ).
Note that this notion is weaker than the (usual) e/h-convergence and the interval of extended e/h-limits is larger than that of e/h-limits.
To establish the bicontinuity of the partial Legendre-Fenchel transform, we introduce the concept of upper modulated sequences A sequence {f ν }ν∈ N in the space of functions fv-fcn(R n ) is defined as upper modulated if there exists a bounded sequence {x ν }ν∈ N such that x ν belongs to C ν and the limit superior of f ν (x ν ) as ν approaches infinity is finite This condition is commonly met; for example, if the sequence {f ν }ν∈ N exhibits epi-convergence, it qualifies as upper modulated according to Definition 2.2.1.
Lemma 2.5.1 (Theorem 3.7, [5]) If a sequence {f ν }ν∈ N of closed convex functions in fv-fcn(R n ) is upper modulated, then
Theorem 2.5.1 establishes the bicontinuity of the partial Legendre-Fenchel transform for closed, convex-concave functions K ν and K in the space fv-biv(R n × R m), along with their corresponding convex parents F ν and F It asserts that if at least one of the sequences {F ν }ν∈ N or {(F ν ) ∗ }ν∈ N is upper modulated, then two key assertions regarding their properties are equivalent.
(ii) K ν epi/hypo converges to K in the extended sense; for all K ν and K in their minimax equivalence classes, i.e., for all K ν ∈ [K ν , K ν ], all K ∈ [K, K], and at all involved (x, y), one has cl 1 (l e K ν )(x, y)≤(ηK)(x, y)≤cl 2 (l h K ν )(x, y).
Proof (i)⇒(ii) We first prove the left inequality in (ii) Note that, if we consider an involved pointx, then we have
Consider a pair (¯x,y) such that (ηK)(¯¯ x,y)¯ is less than α, where α is in the range of (−∞,∞] According to equation (2.12), there exists a vector ¯v in the domain of F(¯x, ) such that the difference F(¯x,v¯)− h¯y,¯vi is less than α For any y ν in the domain D ν leading to y, and for a real number β where F(¯x,v)¯ is less than β, as F ν approaches e F, there exist pairs (x ν , v ν ) in the domain of F ν converging to (¯x,v¯), which implies x ν is in C ν Consequently, it follows that the limit superior of K ν (x ν , y ν ) is less than or equal to the limit superior of {F ν (x ν , v ν )− hy ν , v ν i} as ν approaches infinity.
≤β− h¯y,vi.¯ Passingβ &F(¯x,v¯), one obtains limsup ν K ν (x ν , y ν )≤F(¯x,¯v) − h¯y,v¯i< α.
If (ηK)(¯x,y) =¯ ∞, this inequality is clearly satisfied Therefore, l e K ν ≤ηK.
Since cl 1 (ηK) =ηK, this implies that cl 1 (l e K ν )≤cl 1 (l e K ν )≤cl 1 (ηK) =ηK ≤ηK,and we get the required left inequality.
Now we take care of the right inequality (of (ii)) We have ηK(x, y) = sup u∈ R n
Applying the previously-proved result to−ηK,−K ν (x ν , y ν ), (F ν ) ∗ (u ν , y ν ),y ν ,x ν in the place ofηK,K ν (x ν , y ν ),F ν (x ν , v ν ),x ν ,y ν , we see that, for (¯x,y) with¯ ηK(¯x,y)¯ >
−α∈[−∞,∞) and all x ν ∈C ν →x, there exists¯ y ν ∈D ν →y¯satisfying limsup ν (−K ν (x ν , y ν ))≤ −ηK(¯x,y).¯
If ηK(¯x,y) =¯ −∞, this inequality is evidently fulfilled Consequently, l h K ν ≥ ηK. From this we have cl 2 (l h K ν )≥cl 2 (l h K ν )≥cl 2 (ηK) = ηK ≥ηK, as required.
(ii)⇒(i) We have to prove that, at all involved points, e−ls ν F ν ≤ηF ≤e−li ν F ν (2.13) Let us verify first the right inequality We have, for all involved (¯x,¯v) and (¯x, y), ηF(¯x,v) = sup¯ y∈ R m
The inequality arises from (ii), and the final equality is valid since condition 2 can be omitted when determining the supremum According to the definition of lhK ν, for every y involved and for all α in the range [−∞,∞) where α is less than (l h K ν)(¯x, y), it follows that (l h K ν)(¯x, y) is within the interval (−∞,∞] Moreover, for any x ν in C ν approaching x, there exists a sequence ¯y ν in D ν converging to y, such that the limit inferior of ν K ν (x ν, y ν) plus the inner product hy, vi is greater than or equal to ¯α plus hy, vi.
On the other hand, for (x ν , v ν )∈domF ν ,
Hence, for all v ν →v with (x ν , v ν )∈domF ν , liminfνF ν (x ν , v ν )≥liminfνK ν (x ν , y ν ) +hy,¯vi.
Letting α%(l h K ν )(¯x, y), one gets liminf ν F ν (x ν , v ν )≥(l h K ν )(¯x, y) +hy,¯vi (2.15)
This inequality obviously holds if (l h K ν )(¯x, y) = −∞ Combining (2.14) and (2.15) yields (ηF)(¯x,¯v)≤(e−li ν F ν )(¯x,¯v), i.e., the right inequality in (2.13).
Now we check the left one We have, for all involved (¯u,y) and (x,¯ y),¯
(the inequality is due to (ii)) By the definition of l e K ν , for all involved (¯x,y) and¯ α ∈ (−∞,∞] with α > (l e K ν )(¯x,y) and all¯ y ν ∈ D ν → y, there exists¯ x ν ∈ C ν → x¯ such that limsup ν K ν (x ν , y ν )− h¯x,ui ≤¯ α − h¯x,ui.¯
Therefore, for all (u ν , y ν )∈C (F ν) ∗ →(¯u,y),¯ liminf ν (F ν ) ∗ (u ν , y ν )≥liminf ν {−K ν (x ν , y ν ) +hx ν , u ν i}
Letting α&(l e K ν )(¯x,y), one gets¯ liminf ν (F ν ) ∗ (u ν , y ν )≥ −(l e K ν )(¯x,y) +¯ h¯x,ui.¯
If (l e K ν )(¯x,y) =¯ ∞, this inequality is evidently satisfied Hence, we have e−li ν (F ν ) ∗ ≥ηF ∗ (2.16)
Combining the result of the first part of the proof of (ii)⇒(i) that e−li ν F ν ≥ηF with Lemma 2.5.1 , we see that
(e−li ν F ν ) ∗ = e−ls ν (F ν ) ∗ ≤ηF ∗ Finally, this and (2.16) imply that (F ν ) ∗ → e F ∗ , which is equivalent toF ν → e F
This chapter explores the definitions of epi and hypo convergence for finite-valued bifunctions on two sets, highlighting their relationship with lopsided convergences We establish comprehensive characterizations of these convergences, proving their equivalence with the epi and hypo convergence of proper bifunctions, which align with extended real-valued bifunctions Additionally, we provide a geometric characterization, revealing that epi and hypo limits form entire intervals of bifunctions, with specific formulas for their endpoints Unlike these limits, minsup-lop and maxinf-lop limits are unique Finally, we demonstrate that the epi/hypo convergence of convex-concave bifunctions corresponds to the epi-convergence of their convex parents, derived from their Legendre-Frechel transforms.
This chapter provides basic tools for the study of variational properties of conver- gence of bifunctions and approximations/stability of optimization problems in the next two chapters.
Variational properties of epi/hypo convergence and approximations of optimization problems
This chapter explores the variational properties of variational convergence, specifically from epi-convergence to epi/hypo convergence, highlighting how these properties preserve optimal values, solutions, and critical points such as minimax and maximin values We demonstrate that if a sequence of bifunctions K ν epi/hypo converges to a bifunction K, under certain conditions, a cluster point of the minimax or maximin values from the sequence will correspond to the minimax or maximin value of the limit bifunction K Similar preservation holds for other critical points like minsup and maxinf points While we do not provide a strict definition of "variational convergence," we acknowledge its prevalent use in related literature, particularly regarding stability studies A more precise definition can be found in [27, pages 121-124], and in [58, Definition 4], where "variational convergence" is specifically associated with epi-convergence, regarded as the fundamental type of variational convergence.
This article explores the application of variational properties to optimization problem approximations by reformulating them as finding minsup, maxinf, or saddle points of bifunctions It examines a sequence of problems that converge to the equivalent problem through epi/hypo convergence, leading to the convergence of solution-related objects Previous studies have utilized various forms of convergence: epi-convergence for scalar minimization, graphical convergence for complementarity problems, and lop-convergence for different models Epi/hypo convergence has been extensively studied and applied in several works, highlighting its significance in the stability of optimization solutions For definitions of these convergence types, refer to Chapter 2.
Variational properties of epi/hypo convergence and approximations
Variational properties of epi-convergence
In this section, we are concerned with the class fv-fcn(R n ) of finite-valued functions on
R n First, recall the definition of several types of convergence.
Definition 3.2.1 Let C ν , C ⊂R n be nonempty and {f ν :C ν →R} ν∈ N ,f :C →R. (i) ([72]) {f ν } is called convergent graphically to f, denoted by f ν → g f, if gphf ν converges to gphf in the Painlev´e-Kuratowski sense.
(ii) ([72]) {f ν } is said to converge continuously to f relative to sequence C ν → C if, for any sequence x ν ∈C ν →x∈C, f ν (x ν )→f(x).
(iii) ([42]){f ν } is called epi-convergent tof, denoted byf ν → e f orf = e-lim ν f ν if (a) for all x ν ∈C ν →x, liminf ν f ν (x ν ) ≥f(x) when x∈ C and f ν (x ν )→ ∞ when x /∈C;
(b) for all x∈C, there exists x ν ∈C ν →xsuch that limsup ν f ν (x ν )≤f(x).
Note that Definition 3.2.1(i) and (ii) are classic The part (iii) is taken from Chapter
2 Recall that irrespective of concerning fv-fcn(R n ) or fcn(R n ), f ν → e f if and only if epif ν → e epif in the sense of Painlev´e-Kuratowski {f ν } is called hypo-converging to f, denoted by f ν → h f orf = h-lim ν f ν , if {−f ν } epi-converges to−f.
Proposition 3.2.1 Let C ν , C ⊂R n be nonempty and {f ν :C ν →R}ν∈ N, f :C →R. (i) ([72], Proposition 5.33) f ν → g f if and only if the following two conditions hold, for all x∈C,
(α) for allx ν ∈C ν →xthere exists a subsequencex ν j such thatlim j f ν j (x ν j ) =f(x);(β) there exists x ν ∈C ν →x such that lim ν f ν (x ν ) =f(x).
(ii) {f ν }graphically converges to f if and only if it both epi- and hypo-converges to f and C ν →C.
(iii) If {f ν } converges continuously to f relative to sequence C ν →C, then it both epi- and hypo-converges to f.
Proof (ii) Using the characterization by conditions (α) and (β) in (i) for graphical convergence, the proof is immediate.
Now we recall basic variational properties of epi-convergence, see [42].
Theorem 3.2.1 (epi-convergence: basic property) Let f ν , f ∈ fv-fcn(R n ) and f e-limf ν Then, limsup ν (inf C ν f ν (x))≤inf C f(x).
Moreover, if x k ∈ argmin C νkf ν k for some subsequence {ν k }k∈ N and x k → x, then¯ ¯ x∈argmin C f and minC νk f ν k →minCf.
Note from Proposition 3.2.1 that graphical and continuous convergences also en- joy the properties stated in this theorem The second part of Theorem 3.2.1 can be reformulated as: if f ν → e f then
It is easy to prove the extension that if ν &0 then
To guarantee the equality in this relation with the full Lim instead of Limsup and also the convergence of the infimal values, we need the following tightness notion.
Definition 3.2.2 (tight epi-convergence) We say that a sequence{f ν }ν∈ Nepi-converges tightly tof in fv-fcn(R n ) if it epi-converges and, for all positive, there exist a compact setB and an index ν such that, for allν ≥ν , infB ∩C ν f ν ≤infC ν f ν +.
Theorem 3.2.2 (convergence of infima) Let f ν , f ∈ fv-fcn(R n ), f = e-lim ν f ν and inf C f be finite Then, the epi-convergence is tight
(i) if and only if inf C ν f ν →inf C f;
(ii) if and only if there exists a sequence ν &0such that ν -argminf ν →argminf.
Variational properties of epi/hypo convergence
We first recall several facts about the definitions already given in Chapter 2, Section 3.2.
Continuous convergence of bifunctions K ν (ã,ã) from C ν × D ν to R, as the sequence converges to C × D, ensures various forms of convergence, including e/h-, minsup-lop, and maxinf-lop convergence As established in Section 3.2, continuous convergence also guarantees both epi- and hypo-convergence of K ν (ã,ã), indicating that it qualifies as a form of variational convergence However, this type of convergence is notably strong, making it challenging to achieve.
The limits of an e/h-convergent sequence are not unique, resulting in a classification known as an e/h-equivalence class of bifunctions Despite this lack of uniqueness, it is important to note that nearly all variational properties remain consistent across all limit bifunctions within a given equivalence class.
The naturally anticipated variational properties of e/h-convergence are associated with saddle points due to the symmetry of this convergence A point (¯x,y)¯ in C×D is classified as a saddle point of K, which belongs to the set of fv-biv(R n ×R m ), denoted as (¯x,y)¯ ∈ sdlK, if it satisfies the condition for all x in C and y in D.
In various applications, approximate saddle points can be present even in the absence of true saddle points This article aims to demonstrate the convergence of these approximate saddle points, which will subsequently imply the convergence of actual saddle points It is important to note that a point \((\bar{x}, \bar{y})\) in the set \(C \times D\) is classified as an \(-\)saddle point of the function \(K \in fv-biv(\mathbb{R}^n \times \mathbb{R}^m)\) if it satisfies the condition for all \(x \in C\) and \(y \in D\).
Let us define the sup-projection and inf-projection of a bifunctionK ∈fv-biv(R n ×
R m ) by, resp, h(.) := sup y∈D K(., y), g(.) := infx∈CK(x, ).
We have the following simple relation between approximate solutions.
Proposition 3.3.1 Let K ∈ fv-biv(R n ×R m ) and h and g be its sup-projection and inf-projection, resp.
(i) If (¯x,y)¯ ∈-sdlK, then x¯∈2-argmin(h) and y¯∈2-argmax(g).
(ii) If x¯∈-argmin(h) and y¯∈-argmax(g), then g(¯y)≤K(¯x,y)¯ ≤h(¯x), sup y∈D x∈Cinf K(x, y)−≤K(¯x,y)¯ ≤ inf x∈Csup y∈D
K(x, y) +. Therefore, if K has a saddle point (˜x,y), then˜
The corresponding property ofg is checked similarly,
(ii) It is clear that
The two right inequalities are proved similarly.
This section explores the variational properties of an arbitrary e/h-limit under specific conditions, revealing that all e/h-limits within an equivalence class possess numerous shared characteristics This observation is significant, as it simplifies many applications by allowing us to bypass the complexities of handling entire equivalence classes.
Theorem 3.3.1 (convergence of approximate saddle points) Let a sequence {K ν }ν∈ N e/h-converge to K in fv-biv(R n ×R m ), ν & ≥ 0 and, for all ν ∈ N, (¯x ν ,y¯ ν ) ∈ ν - sdlK ν Let (¯x,y)¯ be a cluster point of this sequence of approximate saddle points, say (¯x,y) = lim¯ ν∈N(¯x ν ,y¯ ν )for some subsequence N ⊂N Then, (¯x,y)¯ is an -saddle point of K and
Proof We can assume that actually (¯x ν ,y¯ ν )→(¯x,y) Pick any (x, y)¯ ∈ C×D Any sequences x ν ∈C ν →x and y ν ∈D ν →y satisfy
K ν (¯x ν , y ν )− ν ≤K ν (¯x ν ,y¯ ν )≤K ν (x ν ,y¯ ν ) + ν These inequalities imply that sup
By the definition of e/h-convergence, one has
These inequalities mean that (¯x,y) is an¯ -saddle point of K.
To see that K(¯x,y) = lim¯ ν∈NK ν (¯x ν ,y¯ ν ), simply observe that the e/h-convergence and ¯x ν →x¯ensure the existence of a sequence y ν ∈D ν →y¯satisfying
The relationship K(¯x,y)¯ is bounded by the limit inferior of K ν (¯x ν , y ν ), specifically, K(¯x,y)¯ ≤ liminfνK ν (¯x ν , y ν ) ≤ liminfν(K ν (¯x ν , y¯ ν ) + ν) = liminfνK ν (¯x ν , y¯ ν ) A similar analysis, reversing the roles of the x-variable and y-variable, leads to the conclusion that K(¯x,y)¯ is also constrained by the limit superior, yielding K(¯x,y)¯ ≥ lim sup ν →∞ K ν (¯x ν , y¯ ν ).
Clearly, by taking ν ≡0 in the preceding statement, we obtain the following basic result on convergence of saddle points.
Theorem 3.3.2 (convergence of saddle points) Let a sequence {K ν }ν∈ N e/h-converge to K in fv-biv(R n ×R m ) and (¯x ν ,y¯ ν ) be a saddle point of K ν for all ν ∈N Let (¯x,y)¯ be a cluster point of this sequence of saddle points, say(¯x,y) = lim¯ ν∈N(¯x ν ,y¯ ν )for some subsequence N ⊂N Then, (¯x,y)¯ is a saddle point of K and
The theorems discussed do not impose conditions such as convex-concave properties, continuity, compactness, or closedness; they solely rely on epi/hypo convergence This makes epi/hypo convergence an ideal concept for analyzing saddle points or approximate saddle points.
The following example illustrates Theorem 3.3.2 and gives some insights about variational properties of the mentioned kinds of convergence.
Example 3.3.1 Consider the sequence of bifunctions
We can check directly that on [1/2,3/2] 2 , K ν converges to
K(x, y) = lnxlny in the sense of all kinds of epi/hypo-, minsup-lop and maxinf-lop-convergence.
The K ν functions do not exhibit saddle points; however, they possess approximate saddle points defined by ν, which is calculated as the maximum of two logarithmic expressions The relevant vertical intervals are {1−ν −1 } ×([1/2,1−ν −1 ]∪[1 +ν −1 ,3/2]) and {1 +ν −1 } ×([1/2,1−ν −1 ]∪[1 +ν −1 ,3/2]), while the horizontal intervals are ([1/2,1−ν −1 ]∪[1 +ν −1 ,3/2])× {1 +ν −1 } and ([1/2,1−ν −1 ]∪[1 +ν −1 ,3/2])× {1−ν −1 } This collection forms the ν -sdlK ν set, which converges to the sdlK set represented as ({1} ×[1/2,3/2])∪([1/2,3/2]× {1}).
Though without saddle points, the K ν have arg(max D νmin C ν )K ν = ([1/2,1−ν −1 ]∪[1 +ν −1 ,3/2])× {1} tending to the whole arg(max D min C )K = [1/2,3/2]× {1}.
The expression arg(max D min C )K represents the set of points (¯x,y) where ¯K(¯x,y) equals max¯ D min C K(x, y) Similarly, the notation arg(min C max D )K is defined in the same context Notably, the set arg(min C νmax D ν )K ν, which equals {1−ν −1 } × {1/2}, converges to the single point {(1,1/2)} This point is distinct from the set arg(min C max D )K, which is defined as ={1} ×[1/2,3/2].
Theorem 3.3.2 can be restated as follows, for the case = 0 If K ν e/h-converges toK and ν &0, then
To have equality with the full Lim instead of Limsup in the above relation, i.e., to have also
In Definition 3.3.1, we introduce new concepts of tightness, highlighting the symmetric roles of x and y in the context of symmetric e/h-convergence, as discussed following Definition 2.3.2 These proposed tightness definitions differ from the previously established nonsymmetric notions found in [42].
Definition 3.3.1 (i) (x-ancillary tightness) K ν is called e/h-convergent x-ancillary tightly to K in fv-biv(R n ×R m ) if (a) of Definition 2.3.2 and the following condition are satisfied:
(b’-t) for all x ∈ C, there is x ν ∈ C ν → x such that K ν (x ν , ) → h K(x, ) and h ν (x ν )→h(x).
(ii) (y-ancillary tightness) K ν is said to e/h-converge y-ancillary tightly to K in fv-biv(R n ×R m ) if (b) of Definition 2.3.2 is fulfilled together with
(a’-t) for all y ∈ D, there is y ν ∈ D ν → y such that K ν (., y ν ) → e K(., y) and g ν (y ν )→g(y).
(iii) (tightness) If both (b’-t) and (a’-t) are satisfied, K ν is called e/h-convergent (fully) tightly to K.
Theorem 3.3.3 (convergence of approximate saddle points to any given saddle point) Suppose that K ν e/h-converges (fully) tightly to K in fv-biv(R n ×R m ) Then, the following statements hold.
(ii) Therefore, for each ν &0, (¯x,y)¯ ∈sdlK and large ν, there exists (¯x ν ,y¯ ν )∈ ν - sdlK ν such that (¯x ν ,y¯ ν )→(¯x,y), i.e.,¯ Liminf ν ( ν -sdlK ν )⊃ sdlK.
Proof (i) For each (¯x,y)¯ ∈sdlK, the tightness ensures the existence of ¯x ν ∈C ν → ¯ x and ¯y ν ∈D ν →y¯such that
It suffices to show that, for all positive and large ν,
K ν (¯x ν ,y¯ ν )≤ inf x∈C ν K ν (x,y¯ ν ) + (3.4) Suppose to the contrary to (3.3) that there is 0 and a subsequence ν k such that
Taking liminf on both sides, (3.1) and (3.2) imply that
K(¯x, y)− 0 , which is impossible since (¯x,y)¯ ∈sdlK.
(ii) Given à & 0 and a fixed à, by (i) one has a sequence (¯x ν à ,y¯ ν à ) ∈ à -sdlK ν convergent to (¯x,y) Taking the diagonal sequence (¯¯ x ν ν ,y¯ ν ν ), we complete the proof
Example 3.3.2 Let n = m = 1 and K ν (x, y) = y x on [0,1] 2 for all ν ∈ N, with the convention that 0 0 = 1 In Example 2.3.2 of Chapter 2, it was computed that all the bifunctions
The sequence {K ν} forms an e/h-equivalence class, with limits defined as K(x, y) = y/x for (x, y) ∈ [0,1] \ {(0,0)} and K(0,0) = a, where a ∈ [0,1] It is straightforward to verify that K ν converges to K a fully tightly Notably, every point along the line segment {(x,1) | 0 ≤ x ≤ 1} serves as a saddle point for all K ν and K a across all ν ∈ N and a ∈ [0,1] This indicates that saddle points are maintained under tight e/h-convergence, and the saddle points remain consistent for all limits within the e/h-equivalence class.
Approximations of equilibrium problems
Consider the following equilibrium problem
The equilibrium problem seeks to find a point ¯x in the set C such that the function K(¯x, y) is less than or equal to zero for all y in the set D, where C and D are subsets of R^n and R^m, respectively This problem is known as the Ky Fan inequality, named after the mathematician who first explored its solution Subsequent research has expanded on this significant result, linking it to fixed-point theorems and various existence theorems within the fields of nonlinear and functional analysis.
The (EP) model, introduced as a generalization of constrained minimization and variational inequalities, encompasses all major optimization-related problems, leading to a surge of interest among researchers Subsequent studies demonstrated that the (EP) model genuinely generalizes various optimization challenges, highlighting specific (EP) instances that fall outside traditional problem frameworks.
While there is a lack of research discussing whether a solution to (EP) represents a true physical or technical equilibrium, the economic and social implications of "equilibrium" have been validated through various practical issues, such as traffic networks and non-cooperative games.
Assume that (EP) is subject to perturbation and we have a sequence of approx- imating problems (EP ν ) with K ν : C ν ×D ν → R Denote the solution set of (EP) ((EP ν ), resp) by S (S ν , resp).
Limsup ν S ν ⊂S, i.e., any cluster point of a sequence of solutions of problems (EP ν ) is a solution of (EP).
Proof Let ¯x ∈ Limsupν S ν , i.e., there exists a sequence {x ν j } in S ν j converging to ¯ x For any (fixed) y ∈ D, by (a) of Definition 2.3.2, there is a sequence {y ν j }ν∈ N in D ν j converging to y such that liminfjK ν j (x ν j , y ν j ) ≥ K(¯x, y) if ¯x ∈ C Since
K ν j (¯x, y) ≤ 0 for all j ∈ N and y ∈ D ν j , K ν j (x ν j , y ν j ) ≤ 0 for all j ∈ N and hence liminfν jK ν j (x ν j , y ν j )≤0 Thus,K(¯x, y)≤0 for any y∈D, i.e., ¯xis in S Suppose now ¯ x6∈C Then, by the mentioned condition (a),K ν j (x ν j , y ν j )→ ∞, which is impossible since K ν j (x ν j , y ν j )≤0 for all j.
The proof for the case of K ν epi-convergence follows a similar approach This assertion enhances Theorem 6.11 from reference [57], as it substitutes e/h-convergence with the more robust minsup-lop convergence, while also adding further assumptions.
C =D, C ν =D ν are closed, C ν ×D ν →C×D and K ν , K are lsc-usc for the case of minsup-lop convergence and lsc for the epi-convergence case.
The concept of approximate solutions can be applied to the case where ξ : R n → R is a continuous function, ensuring that ξ(x) > 0 for all x ≠ 0 In this context, consider the optimization problem (EP) with m = n and C = D According to Definition 7.1 in [57], a point ¯x ∈ C is referred to as an (, ξ)-approximate solution of (EP) if it satisfies the condition K(¯x, y) ≤ ξ(¯x−y) for every y within the set C The collection of all such (, ξ)-approximate solutions is denoted as S ,ξ.
Proposition 3.4.2 If ν &≥0 and K ν e/h → K or K ν → e K, then
Limsup ν S ν ν ,ξ ⊂S ,ξ , i.e., any cluster point of a sequence of ( ν , ξ)-solutions of problems (EP ν ) is an (, ξ)- solution of problem (EP).
The proof closely resembles that of Proposition 3.4.1, enhancing Theorem 7.7 from [57] by substituting e/h convergence with the more robust minsup-lop convergence, while maintaining the same additional assumptions outlined in Theorem 6.11 of [57].
The preceding two results can be stated in terms of outer continuity as follows For brevity, look only at Proposition 3.4.1 Consider the set-valued map S : fv-biv(R n ×
The function S(K) = S establishes a mapping from R^m to R^n, with a focus on e/h-convergence in the space of fv-biv(R^n × R^m) Although limits represent entire equivalence classes, our analysis is centered on a predetermined limit bifunction K Consequently, the Limsup inclusion outlined in Proposition 3.4.1 can be reinterpreted to reflect the outer continuity of the function S.
We consider also the dual equilibrium problem, introduced in [48],
(DEP) find ¯y∈D such thatK(x,y)¯ ≥0 for all x∈C.
The Equilibrium Problem (EP) and the Dual Equilibrium Problem (DEP), commonly referred to as the Stampacchia and Minty problems, respectively, are dual in nature, meaning that the dual of the DEP is the EP This relationship highlights the fundamental concept of duality inherent in these problems.
(EP) find ¯x∈C solving minx∈Csupy∈DK(x, y)≤0,
(DEP) find ¯y∈D solving maxy∈Dinfx∈CK(x, y)≥0.
The article discusses the relationship between the minimization problem (EP) and the maximization problem (DEP) within the context of a saddle point framework It asserts that a solution (¯x) for (EP) and (¯y) for (DEP) exists if and only if the pair (¯x, ¯y) constitutes a saddle point of the function K in the product space C × D, with K(¯x, ¯y) equating to zero, indicating a zero duality gap between the two solutions Furthermore, it highlights that (¯x, ξ) serves as an (ε, ξ)-approximate solution for (EP) and (¯y, ξ) for (DEP) if the quadruple (¯x, ξ, ¯y, ξ) represents an (ε, ξ)-approximate saddle point of K, satisfying the inequalities −ξ(¯x−¯y) ≤ K(¯x, ¯y) ≤ ξ(¯x−¯y).
When considering the effects of perturbation on the problems (EP) and (DEP), we denote the perturbed versions as (EP ν) and (DEP ν), which utilize K ν and C ν, D ν in place of K, C, and D This leads to a significant stability result that directly follows from Theorems 3.3.1 and 3.3.2.
If ν ≥ 0 and x̄νν,ξ and ȳνν,ξ represent an (ν, ξ)-approximate solution of (EPν) and (DEPν), respectively, then the components of any cluster point (x̄, ξ, ȳ, ξ) from the sequence {(x̄νν,ξ, ȳνν,ξ)} serve as an (ξ)-approximate solution of (EP) and (DEP), respectively.
For the case where \( \nu = 0 \), if \( \bar{x}_{\nu, \xi} \) and \( \bar{y}_{\nu, \xi} \) serve as \( (\nu, \xi) \)-approximate solutions to the problems \( (EP_{\nu}) \) and \( (DEP_{\nu}) \), respectively, then the components of any cluster point \( (\bar{x}, \bar{y}) \) from the sequence \( \{(\bar{x}_{\nu, \xi}, \bar{y}_{\nu, \xi})\} \) will be solutions to \( (EP) \) and \( (DEP) \).
Denote the set of the solutions (the (, ξ)-approximate solutions, resp) of (DEP) by
DS (DS ,ξ , resp) Proposition 3.4.3 (ii) can be rephrased as follows: if ν &0 then
To have equality and with the “full” Lim instead of Limsup, i.e., to have additionally
Liminfν(S ν ν ,ξ ×DS ν ν ,ξ)⊃S×DS, we impose tightness conditions and apply Theorem 3.3.3 to obtain the following. Proposition 3.4.4 If K ν e/h-converges fully tightly to K and ν &0, then
Approximations of multi-objective optimization
Equilibrium models are widely recognized for addressing optimization-related issues, but this paper focuses specifically on the equilibrium problem (EP), which is defined as a single-valued and scalar problem Consequently, it primarily involves single-valued and scalar optimization models Nonetheless, we can leverage Propositions 3.4.1-3.4.4 from our scalar problem (EP) to tackle a multi-objective minimization problem, where ϕ 1, , ϕ k: C ⊂ R n → R and R k is partially ordered by R k +.
(OP) find ¯x∈C such thatϕ(¯x)−ϕ(y)∈/ intR k + for all y∈C.
Such an ¯x is called a weak minimizer (or weakly efficient point) of ϕ on C We can convert (OP) to a special case of (EP) by setting, see, e.g., [40],
Indeed, taking D=C, we have the three equivalent assertions, for all y ∈C,
The dual to (OP) according to the duality scheme for (EP), i.e., problem (DEP) for K(x, y) = min 1≤i≤k (ϕ i (x)−ϕ i (y)), is
In the context of optimization problems, we seek to find a point ¯y such that the inequality ϕ i (x)−ϕ i (¯y)≥0 holds for all x in the set C, where i ranges from 1 to k This condition implies that the difference ϕ(x)−ϕ(¯y) belongs to the non-negative orthant of R k + A point ¯y satisfying this criterion is referred to as a strong or ideal minimizer, also known as a strongly efficient point, of the function ϕ on the set C This approach represents a distinct duality scheme compared to existing methods for multi-objective set-constrained minimization Notably, the points ¯x and ¯y serve as solutions for the optimization problems (OP) and (DOP), respectively, under the specified definitions.
By substituting ¯xand ¯yin this inclusion, we obtainϕ i (¯x)−ϕ i (¯y)∈bdR k + Furthermore, ¯ y must be unique, but ¯x not Thus, we have a simple geometric explanation in the objective spaceR k for (OP) and (DOP).
We are now focusing on the approximations of two dual problems In line with the concept of (, ξ)-approximate solutions for the (EP) and (DEP) problems, we introduce the definition of (, ξ)-approximate solutions for the (OP) and (DOP) problems.
Withe:= (1, ,1)∈R k , if ϕ(¯x,ξ)−ϕ(y)−ξ(¯x,ξ −y)e /∈intR k +for all y∈C
In the context of optimization, if the expression \(ϕ(x)−ϕ(¯y,ξ) +ξ(x−y¯,ξ)e ∈R^k +\) holds for all \(x∈C\), then \(¯x\) and \(ξ(¯y,ξ)\) are identified as an \((, ξ)\)-approximate weak minimizer and an \((, ξ)\)-approximate strong minimizer of \(ϕ\) on \(C\) Additionally, a broader concept known as an \(n\)-quasi minimizer has been previously established for multi-objective optimization This article presents the two \((, ξ)\)-approximate solutions as specific instances of the definitions associated with the extended programming (EP) and dual extended programming (DEP), while also emphasizing their duality.
To state consequences of Propositions 3.4.1-3.4.4 in terms of the data of (OP), we need the following definition.
Definition 3.5.1 (i) A sequence of k functions {ϕ ν 1 , , ϕ ν k } ν∈ N , defined on C ν , in fv- fcn(R n ) is said to uniformly epi-converge to k limits ϕ 1 , , ϕ k , resp, if Definition 3.2.1 (iii) is satisfied for all ϕ ν i , ϕ i and i, with the sequence x ν in (b) being common for all
1≤i≤k The definition of uniform hypo-convergence is similar.
A sequence of k functions {ϕ ν 1 , , ϕ ν k } defined on C ν in the space of fv-fcn(R n ) is considered to converge uniformly graphically to k limits ϕ 1 , , ϕ k if all functions exhibit graphical convergence Additionally, this convergence must satisfy condition (β) outlined in Proposition 3.2.1 (i), ensuring that the sequence x ν is consistent across all indices 1≤i≤k.
The article discusses the relationship between Definition 3.5.1 and the convergence of K ν, as outlined in rule (3.5) It highlights that while the notion was previously defined in references [57] and [59] for convex sets, the current context does not require C ν and C to be convex, nor does it assume that C ν converges to C Additionally, the article includes a proof of item (i), which parallels the proof of item (a) from Proposition 5.2 in [57].
Lemma 3.5.1 (i)Ifϕ ν 1 , , ϕ ν k uniformly epi-converge to ϕ 1 , , ϕ k , thenK ν defined by the rule (3.5) e/h-converges to K.
(ii) If ϕ ν 1 , , ϕ ν k uniformly graphically converge to ϕ 1 , , ϕ k , then K ν defined by the rule (3.5) epi-converges to K and C ν →C.
Proof (i) We check first (a) in Definition 2.3.2 (of e/h-convergence) For any x ν ∈
According to Definition 3.5.1(i)(a), for any index i, the limit inferior of ν ϕ ν i (x ν) is greater than or equal to ϕ i (x) if x is in the set C Additionally, Definition 3.5.1(i)(b) states that for every y in C, there exists a corresponding y ν in C ν → y such that the limit superior of ν ϕ ν i (y ν) is less than or equal to ϕ i (y) for all i Consequently, this leads to the conclusion that for every index i, the limit inferior of the difference (ϕ ν i (x ν) - ϕ ν i (y ν)) is at least equal to the difference (ϕ i (x) - ϕ i (y)) Furthermore, based on the definition of K ν, there exists an index i ν for which K ν (x ν, y ν) equals ϕ i ν (x ν) - ϕ i ν (y ν) for all ν Given that the set of indices is finite, it follows that there exists an index i 0 such that K ν (x ν, y ν) consistently equals ϕ i 0 (x ν) - ϕ i 0 (y ν) across a subsequence of N Therefore, the limit inferior of K ν (x ν, y ν) can be expressed as the limit inferior of (ϕ ν i).
If \( x \notin C \), then \( \phi_i(x_\nu) \to \infty \) for all \( i \), leading to \( K_\nu(x_\nu, y_\nu) \to \infty \), thereby satisfying condition (a) of Definition 2.3.2 For condition (b), when \( x \in C \) and \( y_\nu \) converges to \( y \) in \( C \), Definition 3.5.1 implies that \( \liminf_\nu \phi_i(y_\nu) \geq \phi_i(y) \) for all \( i \), while the convergence \( x_\nu \to x \) ensures \( \limsup_\nu \phi_i(x_\nu) \leq \phi_i(x) \) for all \( i \) Therefore, it follows that \( \limsup_\nu K_\nu(x_\nu, y_\nu) \leq \limsup_\nu (\phi_i(x_\nu) - \phi_i(y_\nu)) \leq \phi_i(x) - \phi_i(y) \) for all \( i \).
If y 6∈ C, ϕ ν i (y ν ) → ∞ for all i, which implies that K ν (x ν , y ν ) → −∞ Thus, Definition 2.3.2 is verified completely.
According to Proposition 3.2.1(ii), the sequences ϕ ν 1, , ϕ ν k exhibit both epi- and hypo-convergence uniformly to ϕ 1, , ϕ k, while C ν converges to C To verify the first condition (a) for the epi-convergence of K ν, we consider pairs (x ν, y ν) in C ν × C ν approaching (x, y) For each index i, we observe that the limit inferior of ϕ ν i (x ν) is greater than or equal to ϕ(x), and the limit superior of ϕ ν i (y ν) is less than or equal to ϕ(y) Given that the number of indices k is finite, there exists an index i 0 such that K ν (x ν, y ν) equals ϕ ν i.
0(y ν ) for all ν up to subsequences of N Hence, liminf ν K ν (x ν , y ν ) = liminf ν (ϕ ν i 0 (x ν )−ϕ ν i 0 (y ν ))
Consider now condition (b) By the uniform convergence (given by (ii)), for all (x, y)∈
In the context of C×C, there exist points (x ν, y ν) converging to (x, y) such that for all indices i, the limit superior of K ν (x ν, y ν) is less than or equal to the limit superior of the difference between φ ν i (x ν) and φ ν i (y ν), which in turn is less than or equal to φ i (x) minus φ i (y) Consequently, this leads to the conclusion that the limit superior of K ν (x ν, y ν) is less than or equal to K(x, y) The sets of weak minimizers and (ξ)-approximate weak minimizers of φ on C are denoted as WE(φ, C) and WE ,ξ (φ, C), respectively This establishes a clear implication from Proposition 3.4.2 regarding the stability of the optimization problem (OP) Notably, the condition that K ν converges to e K is more stringent than the condition that K ν e/h converges to K, as indicated by Lemma 3.5.1, thus the associated assertion is not included.
Proposition 3.5.1 If ν & and ϕ ν 1 , , ϕ ν k uniformly epi-converge to ϕ 1 , , ϕ k , then
Limsup ν WE ν ν ,ξ(ϕ ν , C ν )⊂WE ,ξ (ϕ, C), i.e., any cluster point of a sequence of ( ν , ξ)-approximate solutions of problems (OP ν ) is an (, ξ)-approximate solution of problem (OP).
Proposition 3.4.4 establishes a consequence for the optimization problem (OP), highlighting the necessary tightness condition for achieving complete convergence between the approximate solution sets of (OP) and its dual (DOP) Notably, similar findings to Proposition 3.5.1 and other related properties are documented in recent literature, including references [58] and [59].
In reference [88], the focus was on the epi-convergence of maps in the context of convex multi-objective optimization, although it did not explore duality due to the assumption that C ν converges to C, and thus strong minimizers were not addressed The study specifically examined the case where ξ(x) is constant and presented Proposition 3.5.1 as a direct result of Proposition 3.4.2, which also relates to duality, without an extensive analysis of (OP).
Approximations of Nash equilibria
In this section, we focus on the Nash equilibrium problem as a significant example of equilibrium issues, demonstrating the implications of the findings presented in Section 3.4 We examine a non-cooperative game involving m players, denoted as i∈I :={1, , m}.
In a game theory context, let \( C_i \subset R^n \) represent the set of available strategies for player \( i \), and let \( r_i(x_i, x_{-i}) \) denote the return for player \( i \) based on their strategy choice \( x_i \in C_i \), with \( x_{-i} \) being the vector of strategies selected by the other players \( I \setminus \{i\} \) The function \( r_i : C_i \times \prod_{j \in I \setminus \{i\}} C_j \rightarrow R \) is defined for each player \( i \in I \) A Nash equilibrium is reached when the strategies \( \bar{x} = (\bar{x}_i)_{i \in I} \) satisfy the condition that for all players \( i \in I \), the strategy \( \bar{x}_i \) is in the set of best responses, specifically \( \bar{x}_i \in \text{argmax}_{x_i \in C_i} r_i(x_i, \bar{x}_{-i}) \).
We denote this game by G := {(C i , r i )|i ∈ I} and C := Q i∈IC i The following bifunction is called the Nikaido-Isoda bifunction:
Note that, following the traditional setting of Nash equilibria, we state the definition of a Nash equilibrium point with maximizing, different from format in the other places of this thesis.
Proposition 3.6.1 (i) The following three assertions are equivalent
(γ) ¯x is a maxinf point of N with infy∈CN(¯x, y)≥0.
(ii) The following assertions are equivalent and each implies the assertions in (i) (δ) supx∈CN(x,x)¯ ≤0;
() ¯y is a minsup point of N with supx∈CN(x,x)¯ ≤0.
Proof The formula of N clearly yields that N(x, x) = 0, infy∈CN(x, y) ≤ 0 for all x∈C, and sup x∈C N(x, y)≥0 for all y∈C.
(i) “(α) implies (β)” This is obvious by the definition of Nash equilibria and the formula ofN.
“(β) implies (α)” infy∈CN(¯x, y)≥0 means that, for ally ∈C, Σ i∈I [r i (¯x i ,x¯ −i )−r i (y i ,x¯ −i )]≥0.
For any fixed k ∈ I and y k ∈ C k , substituting y = (y k ,x¯−k) into this inequality we obtain r i (¯x k ,x¯−k)−r i (y k ,x¯−k) + Σj∈I\{k}[r j (¯x j ,(¯x−j,k, y k ))−r j (¯x j ,(¯x−j,k, y k ))]≥0
(¯x−j,k is the collection of them−2 components ofx= (¯x i ,x¯−i), except ¯x j andx k ), i.e., r i (¯x k ,x¯−k)−r i (y k ,x¯−k)≥0 By the arbitrariness ofk and y k , ¯xis a Nash equilibrium point.
The statement "(β) is equivalent to (γ)" indicates that, based on the observation that N(x, x) = 0 and infy∈CN(x, y) ≤ 0 for all x ∈ C, it is clear that ¯x serves as a maxinf-point, with infy∈CN(¯x, y) equating to 0 The reverse implication is also evident.
(ii) “(δ) is equivalent with ()” We need to show only “(δ) implies ()” Clearly (δ) together with the observation for N that supx∈CN(x, y)≥0 for all y∈C ensures that ¯ x is a minsup-point ofN.
“(δ) implies (α)” supx∈CN(x,x)¯ ≤0 means that, for all x∈C, Σi∈I[r i (x i , x−i)−r i (¯x i , x−i)]≤0.
In particular, for any givenk and xk∈Ck, substitutingx= (xk,x¯−k) into this inequal- ity yields r k (x k ,x¯ −k )−r k (¯x k ,x¯ −k ) + Σ j∈I\{k} [r j (¯x j ,(¯x −j,k , x k ))−r j (¯x j ,(¯x −j,k , x k ))]≤0
(¯x−j,k is the collection of them−2 components ofx= (x k ,x¯−k), except ¯x j and x k ) or, equivalently, r i (¯x i ,x¯ −i )≥r i (x i ,x¯ −i ) for all i and x i ∈C i , i.e., ¯x is a Nash equilibrium point of G
Remark 3.6.1 From the above proposition, we have the following.
(a) The game G can be restated as the equilibrium problem of finding ¯x∈C such thatN(¯x, y)≥0 for ally ∈C, by the equivalence of (α) and (β) in Proposition 3.6.1
(b) Any Nash equilibrium point ¯xis a maxinf-point of the Nikaido-Isoda bifunction
The dual game of G, framed within the duality of the equilibrium problem (EP), demonstrates a stronger structure than G itself, as its solutions, denoted as ¯y, are consistently Nash equilibrium points Additionally, these points are characterized as minsup-points of N, satisfying the condition miny∈Csupx∈CN(x, y) = 0 Consequently, when the dual game of G yields solutions, the duality gap is zero, indicating that for any solution ¯x of G, the pair (¯x, ¯y) forms a saddle point of ¯N.
(d) ¯x is a Nash equilibrium point if and only if infy∈CN(¯x, y) = 0 The dual game of G has a solution ¯y if and only if supx∈CN(x,y) = 0.¯
In the context of our perturbed game, we define a sequence of approximating games G ν, represented as {(C i ν, r ν i) | i ∈ I} The solution sets for these games are denoted as S ν for G ν and S for G By applying Proposition 3.4.1, we can derive a significant approximation result.
Proposition 3.6.2 For alli∈I, letr i ν converge continuously tor i relative to sequence
C i ν →C i or r ν i converge graphically to r i Then,
Assuming that \( r_{\nu i} \) converges continuously to \( r_i \) with respect to the sequence \( C_i \nu \to C_i \), it follows that \( -r_{\nu i} \) also converges continuously to \( -r_i \) Consequently, \( N_{\nu} h/e \) converges to \( N \) as the product of \( C_i \nu \) converges to \( C \) This establishes the first assertion of the proposition as outlined in Proposition 3.4.1 When \( r_{\nu i} \) converges graphically, Proposition 3.2.1 (ii) indicates that \( N_{\nu} \) hypo-converges to \( N \), and again, Proposition 3.4.1 confirms the proof This statement is derived from the analysis in Section 3.4, which includes duality properties Further investigations into approximations of Nash equilibria through variational convergence are discussed in [35], although they do not cover duality or epi/hypo convergence.
Conclusions
This chapter focuses on the variational properties of epi/hypo convergence and their applications in approximating variational problems in optimization Given the symmetry of epi/hypo convergence, the discussion primarily addresses saddle points, approximate saddle points, and associated minimax values An equilibrium problem serves as the central variational model for our approximation study The findings are then applied to two significant cases: multiobjective optimization and Nash equilibria The subsequent chapter will expand the concepts of epi/hypo and lopsided convergence to bifunctions on nonrectangular domains, facilitating the exploration of approximations in quasivariational problems.
Variational convergence of bifunctions on nonrectangular domains and approximations of quasiequilibrium problems
Variational convergence has evolved over the past fifty years, with the initial four decades focusing on extended real-valued functions and bifunctions In 2009, the concept of lopsided convergence for finite-valued bifunctions defined on rectangles was introduced, highlighting its applications in the approximation and stability of variational problems Subsequent developments included the epi/hypo convergence of finite-valued bifunctions, which provided essential characterizations and variational properties relevant to specific variational issues However, the existing framework of variational convergence for these bifunctions is not applicable to quasi-variational models, where constraint sets vary with problem variables, underscoring the need for further research in this critical area.