Lopsided Convergence: an Extension and its Quantification

Transcription

1 Lopsided Convergence: an Extension and its Quantification Johannes O. Royset Operations Research Department Naval Postgraduate School Roger J-B Wets Department of Mathematics University of California, Davis Abstract. Much of the development of lopsided convergence for bifunctions defined on product spaces was in response to motivating applications. A wider class of applications requires an extension that would allow for arbitrary domains, not only product spaces. This leads to an extension of the definition and its implications that include the convergence of solutions and optimal values of a broad class of minsup problems. In the process we relax the definition of lopsided convergence even for the classical situation of product spaces. We now capture applications in optimization under stochastic ambiguity, Generalized Nash games, and many others. We also introduce the lop-distance between bifunctions, which leads to the first quantification of lopsided convergence. This quantification facilitates the study of convergence rates of methods for solving a series of problems including minsup problems, (Generalized) Nash games, and various equilibrium problems. Keywords: lopsided convergence, lop-convergence, lop-distance, aw-distance epi-convergence, hypo-convergence, minsup problems, Generalized Nash games AMS Classification: 49M99, 65K10, 90C15, 90C33, 90C47 Date: August 22, Introduction The notion of lopsided convergence of bifunctions (= bivariate functions defined on a product space) was introduced in [3] for extended real-valued bifunctions. The focus on extended realvalued functions was motivated by the incentive to keep the development in concordance with the elegant duality results of Rockafellar [17, Chapters 33-37] and the subsequent convergence theory for saddle functions [4, 2, 14]. But this paradigm turned out to become unmanageable when confronted with a series of applications that required dealing with bifunctions that were 1

2 not of the convex-concave type. Eventually, this led to restricting the convergence theory for bifunctions to real-valued bivariate functions (only) defined on specific subsets of the product space, cf. [15, 19] and especially [16]. Lopsided convergence is emerging as a central tool in the study of linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, noncooperative games, mathematical programs with equilibrium constraints, optimality conditions, Walras and Nash equilibrium problems, optimization under stochastic ambiguity, and robust optimization; see the recent developments in [15, 19, 20]. Already, Aubin and Ekeland [7, Chapter 6] brought to the fore the ineluctable connections between some of these applications when dealing with existence issues. Prior studies deal exclusively with bifunctions defined on a product space, but applications in Generalized Nash games, robust optimization, stochastic optimization with decision-dependent measures, and Generalized semi-infinite programming require extensions to bifunctions for which the second variable s domain depend on the first variable. For example, in a minsup problem this corresponds to the situation when the inner maximization has a feasible region that depends on the outer minimization variable, in a Generalized Nash game, the need arises when the set of feasible actions for any given agent depends on the actions of the other agents. We extend the definition of lopsided convergence to deal with these situations and establish an array of results addressing this wider setting. Specifically, we show that for this extended notion of lopsided convergence, optimal solutions and optimal values of approximate minsup problems tend to those of an original minsup problem. We also relax the definition of lopsided convergence for bifunctions defined on a product space and, therefore, broaden the area of application even in this classical situation. In the process, we recast, and in a couple of instances refine, the fundamental implications of epi-convergence in the present framework, i.e., for finite-valued functions defined on a subset of a metric space. For the first time we quantify lopsided convergence by defining the lop-distance. The lopdistance between two bifunctions is given in terms of the Attouch-Wets distance between the sup-projections of the bifunctions with respect to the second variable. Thus, we place firmly the emphasis on the outer minimization in a minsup problem at the expense of the inner maximization. This imbalance indeed motivated the terminology lopsided. In the context of minsup problems, Generalized Nash game, and many other situations, this perspective is reasonable as the inner problem is certainly secondary as illustrated below; the solution of the outer minimization being primary. For example, this leads to estimates of the rate of convergence of that outer solution as demonstrated in [20]. We note that the point of view differs from that of epi/hypo-convergence [4] and analysis of the convex/concave case [17, Chapters 33-37]. There the focus is on finding saddle-point pairs, which implies a certain balance between the inner and outer problems and indeed symmetry in the convex/concave case. Of course, our new viewpoint remains applicable in the convex/concave case and it yields somewhat sharper results not discussed here. 2

3 The article proceeds in 2 with a couple of motivating examples. In 3, we give the new definition, provide sufficiency conditions, and also discuss foundations related to epi-convergence. Consequences of lopsided convergence are established in 4 and the lop-distance is introduced in 5. 2 Motivation Throughout the article, we let (X, d X ) and (Y, d Y ) be two metric spaces and consider finitevalued bifunctions defined on subsets of X Y. In contrast to this article predecessors, e.g., [19], that deal with bifunctions of the form F : C D X Y IR with D a (fixed) subset of Y, we consider bifunctions with arbitrary domains. That is, the set D Y of permissible values for the second variable might depend on x X, the first variable. This extension is essential to deal with some applications as illustrated next. 2.1 Optimization under Stochastic Ambiguity Consider the minsup problem min x C sup y D(x) F (x, y), where D : C Y, C = dom D = { x : D(x) } X, and Y = P is the space of distribution functions defined on IR m, with an appropriate metric. It identifies an optimization model with stochastic ambiguity where C is a collection of feasible decisions, D(x) an ambiguity set of probability distributions, and F a bifunction that depends both on the decision and the distribution. For example, F (x, P ) = IE P [φ(x, ξ)], where ξ is a random vector with distribution function P and the expectation is therefore taken with respect to a distribution function that is determined by the inner maximization problem. In applications, it is sometimes crucial to allow the ambiguity set to depend in a nontrivial manner on the decision x to capture situations where the decision maker affects the uncertainty as modeled here via the set D(x); see [20] for a more comprehensive exposition as well as [21, 11, 13] for related models. 2.2 Generalized Nash Games As we shall see, the study of Generalized Nash games naturally leads to the study of bifunctions that are defined on a (proper) subset of a product space. An equilibrium of a Generalized Nash game with a finite set A of agents, is a solution x = ( x a, a A) that satisfies x a argmax xa D a ( x a ) r a (x a, x a ), for all a A, where x a = (x a : a A\{a}), r a is the payoff function for agent a, and D a ( x a ) is the set of available strategies for agent a; this means that the set of available strategies for agent a depends 3

4 on the choice of strategies by the other agents. Equilibria of such games are fully characterized by the minsup-points of the Nikaido-Isoda bifunction: F (x, y) = [ ] r a (x a, x a ) r a (y a, x a ) for x C, y D(x) a A with C = { } x : x a D a (x a ) for all a A and D(x) = D a(x a ). a A Clearly, the (effective) domain of this bifunction might be rather involved. 2.1 Proposition x C is an equilibrium if and only if it is a maxinf-point of F with nonnegative maxinf-value, i.e., x argmax x C inf y D(x) F (x, y) and inf y D( x) F ( x, y) 0. Proof. If x is a Nash equilibrium, then r a ( x a, x a ) r a (y a, x a ) for all y a D a ( x a ), a A. Thus, F ( x, y) 0 for all y D( x). For any x C, x D(x) and therefore inf y D(x) F (x, y) 0. In particular, inf y D( x) F ( x, y) = F ( x, x) = 0. Consequently, x argmax x C inf y D(x) F (x, y). For the converse, let x be a maxinf-point of F. Then, 0 inf y D( x) F ( x, y) = [ ] r a ( x a, x a ) sup ya D a A a( x a ) r a (y a, x a ). The lower bound of zero and the fact that x a D a ( x a ) for all a A imply that each term in the sum must be zero and the conclusion follows. 3 Lopsided Convergence As already mentioned in the Introduction, to encompass the family of applications sketched out in 2 and others, we need to extend the definition of lopsided convergence, and the resulting theory, to a larger class of bifunctions than those considered in earlier work. The domain, dom F, of a finite-valued bifunction F will no longer be restricted to a product subset of X Y but could be any subset. The family of all such bifunctions is denoted bfcns(x, Y ) := { F : dom F IR : = dom F X Y }. It is on this family we introduce the new definition of lopsided convergence. 4

5 3.1 Definition The (first) x-variable of a bifunction takes a primary role in our development leading us to the following description of the domain of a bifunction. We associate with a bifunction F, the set C := { x X : y Y such that (x, y) dom F } and the set-valued mapping D : C Y such that D(x) := { y Y : (x, y) dom F } for x C. Thus, dom F = { (x, y) X Y : x C, y D(x) }, with dom D = { x X : D(x) } = C; when dom F is a product set it agrees with having D a constant mapping. Figure 1 illustrates the case with a product set (left portion) and the general case (right portion). Throughout, we let C and D be the set and set-valued mapping, respectively, associated with a bifunction F. With a slight abuse of notation, occasionally we simply write D instead of D(x) when the set-valued mapping is constant on its domain. ( ) dom ( ) dom Figure 1: Domains of bifunctions: product set (left) and general (right). In applications, a bifunction might be defined on a large subset of X Y, possibly everywhere, but the context requires restrictions to some smaller subset for example dictated by constraints imposed on the variables. In this case, C and D would represent these restrictions and F in our notation would become the original bifunction restricted to the set {(x, y) X Y : x C, y D(x)}. In other applications, F might be the only problem data and then C and D would be specified by dom F. These differences are immaterial to the following development as both are captured by considering bfcns(x, Y ). With IN := {1, 2, 3,...}, every bifunction in the collection {F ν } ν IN bfcns(x, Y ) will analogously to the situation for F be associated with a slicing of its domain in terms of a set C ν X and a set-valued mapping D ν : C ν Y such that dom F ν = { (x, y) X Y : x C ν, y D ν (x) }. The extended definition of lopsided convergence takes the following form. 5

6 3.1 Definition (lopsided convergence) Let {F, F ν, ν IN} bfcns(x, Y ). The bifunctions F ν converge lopsided, or lop-converge, to F, written F ν lop F, when (a) x ν C ν x C, y D(x), y ν D ν (x ν ) y such that liminf ν F ν (x ν, y ν ) F (x, y) and x ν C ν x / C, y ν D ν (x ν ) such that F ν (x ν, y ν ) ; (b) x C, x ν C ν x such that y ν D ν (x ν ) y Y, limsup ν F ν (x ν, y ν ) F (x, y) if y D(x) and F ν (x ν, y ν ) otherwise. Lop-convergence does not have a direct geometric interpretation. However, as discussed in 4, it is intimately tied to epi- and hypo-convergence, which are easily understood in terms of the convergence of epigraphs and hypographs; see 3.4. We can therefore understand, in part, lopconvergence through these geometric interpretations. A preview of the conclusions reached in 4 builds intuition at this stage: If the bifunctions F ν and F do not depend on y, then lopconvergence collapses to epi-convergence. Under mild assumptions, F ν lop F implies that sup y D ν ( ) F ν (, y) epi-converges to sup y D( ) F (, y). We also have that at every x C, F ν (x ν, ) hypo-converges to F (x, ) for some x ν x. We note that even in the case when dom F and dom F ν are product sets for all ν, our definition represents a relaxation of the requirements in prior definitions as {y ν } ν IN in condition (a) of Definition 3.1 for the case x C is not required to converge to y. Contrast with earlier definition for product sets. Suppose that {F, F ν, ν IN} bfcns(x, Y ) have D, D ν Y so that F : C D IR and F ν : C ν D ν IR. In [16, 19], lopconvergence is defined to take place when (a ) x ν C ν x X, y D, y ν D ν y such that liminf ν F ν (x ν, y ν ) F (x, y) if x C and F ν (x ν, y ν ) otherwise; and (b ) x C, x ν C ν x such that y ν D ν y Y, limsup ν F ν (x ν, y ν ) F (x, y) if y D and F ν (x ν, y ν ) otherwise. The condition (b ) is exactly condition (b) of Definition 3.1 for the case of product sets. However, condition (a ) is stronger than condition (a) of Definition 3.1 as illustrated by the following trivial example. Let X = Y = IR, C ν = C = (0, 1], D ν = D = [0, 1], and F ν (x, y) = F (x, y) = 1/(x + y) for (x, y) in the their domains. It is clear that (a ) fails for y = 1, x = 0, and x ν = 1/ν as there is no y ν y such that F ν (x ν, y ν ). However, condition (a) of Definition 3.1 holds as one can take y ν = 1/ν in the case x C. Then, F ν (x ν, y ν ) = ν/2. If x C, then one can take y ν = y and obtain that F ν (x ν, y ν ) = F (x ν, y) F (x, y) by continuity of F. For condition (b) of Definition 3.1, one can take x ν = x and only be concerned with y Y due to the closedness of Y. Continuity of F then allows us to conclude that F ν lop F. In this case, with F ν = F, convergence is indeed natural and Definition 3.1 addresses this situation. 3.2 About Sufficiency One can come up with a wide collection of sufficient conditions for lop-convergence in terms of the way the components of a sequence of bifunctions F ν converge to those of the limiting 6

7 bifunction F ; note that it will be necessary to broaden convergence notions for functions and mappings to take into account the fact that the domains of these bifunctions are generally not identical. The following results are only meant to illustrate the possibilities and might not be as taut as possible and certainly not necessary. The more interesting ones come from specific applications such as those laid out in [15, 12] involving bifunctions whose domains are product sets and the more general family, when the domains are not restricted to product sets, such as the examples in 2 and those described in [20]. In the following, convergence of sequences in X Y are always in the sense of the product topology, i.e., (x ν, y ν ) X Y (x, y) X Y if max{d X (x ν, x), d Y (y ν, y)} 0. Convergence of sets are always in the sense of Painlevé-Kuratowski. Specifically, in a metric space, the outer limit of a sequence of sets {A ν } ν IN, denoted by Limsup A ν, is the collection of points x to which a subsequence of {x ν } ν IN, with x ν A ν, converges. The inner limit, denote by Liminf ν A ν, is the points to which a sequence of {x ν } ν IN, with x ν A ν, converges. If both limits exist and are identical to A, we say that A ν (set-)converges to A, which is denoted by A ν A; see [9, 18]. 3.2 Theorem (sufficiency when C = C ν ) For bifunctions { F, F ν, ν IN } bfcns(x, Y ), F ν lop F when (a) C = C ν, ν IN, are closed; (b) the mappings D ν continuously converge to D, relative to C, i.e., x ν C x C, D ν (x ν ) D(x); and (c) the bifunctions F ν continuously converge to F, relative to their domains, i.e., (x ν, y ν ) dom F ν (x, y) dom F, F ν (x ν, y ν ) F (x, y). Proof. Since C is closed, given any x ν C x it always entails x C. Moreover, continuous convergence of the mappings D ν to D, relative to C, implies D ν (x ν ) D(x) which, in turn, implies that for any y D(x) one can find y ν D ν (x ν ) converging to y. From (b), the continuous convergence of the bifunctions F ν to F, implies F (x ν, y ν ) F (x, y) which immediately yields condition (a) of Definition 3.1. To verify condition (b) of Definition 3.1, given any x C, by choosing the sequence {x ν = x, ν IN}, D ν (x) D(x) follows from continuous convergence of the mappings. Thus, whenever y ν D ν (x) y, y D(x). In turn, this means that we only have to check if limsup ν F ν (x, y ν ) F (x, y) which, of course, is satisfied since F ν (x, y ν ) F (x, y) in view of assumption (b). Next, we deal with the situation when the domains of the bifunctions are product sets or, equivalently, when the mappings D ν are constant on C. In fact the next statement can be viewed as a refinement of the earlier theorem. Below, we recall that a function f : C IR, with C X, with X any metric space, is lower semicontinuous (lsc) when, for all x ν C x X, liminf ν f(x ν ) f(x) if x C and f(x ν ) otherwise. 7

8 3.3 Proposition (sufficiency under product sets) Suppose that {F, F ν, ν IN} bfcns(x, Y ) has C = C ν closed and D ν Y D Y, i.e., the set-valued mappings are constant. If in terms of some lsc bifunction F : X Y IR, F = F on C D and F ν = F on C D ν, then F ν lop F, provided that, for all x C, F (x, ) is usc. Proof. First, consider condition (a) of Definition 3.1, which now simplifies to finding, for every y D and x ν C ν x C, a sequence y ν D ν y such that liminf ν F (x ν, y ν ) F (x, y). This condition follows from the set-convergence of D ν to D and lower semicontinuity of F. Second, for condition (b) of Definition 3.1, we select x ν = x C for all ν. Since D ν D, any sequence y ν D ν y, implies y D. Thus, the condition simplifies to limsup ν F (x, y ν ) F (x, y) which holds in view of the last assumption of the proposition. Finally, we record a result for the case when there is approximation in the set controlling the (first) x-variable. 3.4 Proposition (sufficiency when C C ν ) Suppose that {F, F ν, ν IN} bfcns(x, Y ) has C ν C and for some continuous set-valued mapping D : X Y, D = D on C and D ν = D on C ν and for some continuous bifunction F : X Y IR, Then, F ν lop-converges to F. F = F on dom F and F ν = F on dom F ν. Proof. Since C ν C, every x ν C ν x must have x C. Thus, condition (a) of Definition 3.1 simplifies to finding, for every x ν C ν x C and y D(x), a sequence y ν D(x ν ) y with liminf ν F (x ν, y ν ) F (x, y). Since D is continuous, such a sequence exists and the inequality therefore follows from the continuity of F. We next turn to condition (b) of Definition 3.1. Since D is continuous, y ν D(x ν ) y implies that y D(x) whenever x ν x. Thus, the condition simplifies to finding, for every x C, a sequence x ν C ν x such that limsup ν F (x ν, y ν ) F (x, y) for all y ν D(x ν ) y D(x). Since C ν C, there is certainly a sequence x ν C ν x for all x C. The inequality then holds in view of the continuity of F. It is easy to find generalizations of the preceding results, for example rather than requiring continuous convergence of the F ν in Theorem 3.2 one could be satisfied with some semicontinuous convergence complemented with a pointwise upper semicontinuity condition. At this point, we shall not get involved in all the possibilities as eventually one is bound to be mostly interested in conditions that apply in specific applications. However, we caution that certain natural conditions are not sufficient as exemplified next. 8

9 Failure of lop-convergence under graphical convergence. Consider the following situation where F ν = F for all ν IN, with C = C ν = IR and D ν (x) = D(x) = [ 1, 1] if x 0, and {0} otherwise. Certainly the mappings D ν graphically converge to D since they are identical. However, when considering condition (a) of Definition 3.1 with x ν > 0 x = 0 and y = 1/2 D(x), there are no y ν D ν (x ν ) y and lop-convergence fails. We note that in this case pointwise convergence D ν (x) D(x) holds for all x IR and, consequently, the mappings are equi-osc [18, Theorem 5.40]. We next give a more involved example where pointwise convergence again holds, but now for problems with different solutions. Failure of lop-convergence under pointwise set-convergence. Suppose that C = C ν = [0, 1], D(x) = D ν (x) = {0} for x [0, 1), D ν (1) = [0, 1 + 1/ν], and D(1) = [0, 1]. Moreover, let F (x, y) = F ν (x, y) = 0 if x [0, 1), and F (1, y) = F ν (1, y) = 2 + y if y 1 and F ν (1, y) = 1 if y > 1. Clearly, for every x [0, 1], D ν (x) D(x). However, lop-convergence of F ν to F fails as for x = 1, x ν = 1 1/ν, and y = 1, there exists no sequence {y ν } ν IN, with y ν D ν (x ν ) = {0} that converges to y as required by condition (a) of Definition 3.1. Here, sup y D ν (x) F ν (x, y) = 0 if x [0, 1) and sup y D ν (x) F ν (x, y) = 1 if x = 1, and sup y D(x) F (x, y) = 0 if x [0, 1) and sup y D(x) F (1, y) = 1. Thus, the optimal value of min x C ν sup y D ν (x) F ν (x, y), which is 0, does not converge to the optimal value of min x C sup y D(x) F (x, y), which is 1. Since a main purpose of a notion of variational convergence of bifunctions is to ensure convergence of such optimal values, it is clear that pointwise set-convergence is not strong enough. 3.3 Tightness A slight strengthening of lop-convergence that amounts to a relaxed compactness assumption becomes beneficial in 5 when deriving consequences. 3.5 Definition (ancillary-tight lop-convergence) The lop-convergence of {F ν } ν IN bfcns(x, Y ) to F bfcns(x, Y ) is ancilliary-tight if Definition 3.1 holds and for every ε > 0 and sequence x ν x selected in condition (b) of Definition 3.1, there exists a compact set B ε Y and an integer ν ε such that sup F ν (x ν, y) sup F ν (x ν, y) ε for all ν ν ε. y D ν (x ν ) B ε y D ν (x ν ) As usual, we interpret the supremum over an empty subset of IR as. The added requirement for ancillary-tightness is satisfied if all D ν (x ν ) are contained in a compact set. If ancillarytightness is combined with a similar condition for the outer minimization, we obtain a further strengthening of the notion. 3.6 Definition (tight lop-convergence) The lop-convergence of {F ν } ν IN bfcns(x, Y ) to the bifunction F bfcns(x, Y ) is tight if Definition 3.5 holds and for any ε > 0 one can find a compact set A ε X and an integer ν ε such that inf x C ν A ε sup y D ν (x) F ν (x, y) inf x C ν sup y D ν (x) F ν (x, y) + ε for all ν ν ε. 9

10 The infimum of an empty subset of IR is interpreted as. This further strengthening of the requirements would be satisfied if all C ν are contained in a compact set, but this is certainly not a necessity. 3.4 Epi- and Hypo-Convergence: A Brief Overview Before we develop consequences of lop-convergence, we give some background facts about epiand hypo-convergence for (univariate) functions; see [1, 8, 18] for comprehensive treatments. We present results for real-valued functions defined on (nonempty) subsets of (X, d X ); in many ways, this is just a variant of the more traditional framework that considers extended real-valued functions, cf. [8, 18]. Our focus will, thus, be on fcns(x) = { f : C IR : for some C X. } In this section, we let C = dom f and, similarly, C ν = dom f ν for f ν fcns(x). The epigraph of f, epi f X IR, consists of all points that lie on or above the graph of f; it is lsc if epi f is closed and, provided X is a linear space 1, it is convex if its epigraph is convex. The hypograph of f, hypo f X IR, consists of all points that lie on or below the graph of f; it is upper semicontinuous (usc) if hypo f is closed and, provided X is a linear space, it is concave if its hypograph is convex. A sequence of functions {f ν } ν IN fcns(x) epi-converges to a function f fcns(x), written f ν e f, when the epigraphs epi f ν set-converge to epi f; similarly, they hypo-converge, written f ν h f, if the hypographs hypo f ν set-converge to hypo f. Equivalently, epi-convergence can also be defined as follows: 3.7 Definition (epi- and hypo-convergence) A sequence of functions {f ν } ν IN fcns(x) epiconverges to a function f fcns(x) if and only the following conditions are satisfied: (a) x ν C ν x, liminf ν f ν (x ν ) f(x) if x C and f ν (x ν ) otherwise, (b) x C, x ν C ν x such that limsup ν f ν (x ν ) f(x). The functions f ν are said to epi-converges tightly to f when f ν e find a compact set B ε X and an index ν ε such that f and for all ε > 0, one can ν ν ε : inf x C ν B ε f ν (x) inf x C ν f ν (x) + ε. Moreover, f ν h f if and only if f ν e f and they hypo-converge tightly if the functions f ν epi-converge tightly to f. As follows immediately from the properties of set-limits, an epi-limit is always lsc and, provided that X is linear, it is convex whenever the functions f ν are convex. Moreover, a hypo-limit is 1 Statements about convexity/concavity are the only ones that require a linear space in this paper. 10

11 always usc and, provided that X is linear, it is concave whenever the functions f ν are concave. The topology induced by epi-convergence is metrizable, a property we leverage in 5. For f fcns(x), optimal values are denoted by inf f := inf{f(x) : x C} and sup f := sup{f(x) : x C}, and, with ε 0, (near-)optimal solutions by ε- argmin f := {x C : f(x) inf f + ε} and ε- argmax f := {x C : f(x) sup f ε}. Since C = dom f is nonempty for f fcns(x), inf f <. Moreover, inf f = implies that argmin f =. Convergence of optimal solutions and optimal values are summarized in the next theorem. The result and proof are mostly the same as those of [18, Theorem 7.31] and [16, Theorems 2.5 and 2.8], which consider X = IR n, but stated here for completeness with some clarification and improvements, especially regarding the role of finiteness of inf f and convergence of near-optimal solutions. We refer to [4] for early results of this kind. 3.8 Theorem (epi- and hypo-convergence: basic properties) Consider {f, f ν, ν IN} fcns(x). If f ν e f, then the following hold: (a) limsup ν (inf f ν ) inf f and {ε ν 0, ν IN}, Limsup ν ( ε ν -argmin f ν) argmin f. (b) If x is a cluster point of a sequence { x ν argmin f ν, ν IN }, i.e., the limit of a subsequence {x ν k }k IN, then lim k (inf f ν k ) = inf f. (c) inf f ν inf f > f ν e f tightly. (d) inf f ν inf f and ε > 0 = Liminf ν (ε- argmin f ν ) argmin f. (e) inf f ν inf f and X is separable 2 = {ε ν 0, ν IN} such that ε ν -argmin f ν argmin f. (f) {ε ν 0, ν IN} such that ε ν -argmin f ν argmin f = inf f ν inf f >. If f ν h f, then liminf ν (sup f ν ) sup f and (a)-(f) hold with min/inf replaced by max/sup, > by <, and tight epi-convergence by tight hypo-convergence. Proof. For part (a), we first suppose that inf f is finite and let ε > 0. There exists x C such that f(x) inf f + ε and also, by condition (b) of Definition 3.7, x ν C ν x such that limsup ν f ν (x ν ) f(x). Thus, limsup ν (inf f ν ) limsup ν f ν (x ν ) f(x) inf f + ε. Second, suppose that inf f = and let δ > 0. Then, there exists x C such that f(x) δ and also, by condition (b) of Definition 3.7, x ν C ν x such that limsup ν f ν (x ν ) f(x). Thus, limsup ν (inf f ν ) limsup ν f ν (x ν ) f(x) δ. Since ε and δ are arbitrary, the first result of 2 We deduce from a counterexample in [10] that the separability assumption cannot be relaxed. 11

12 part (a) is established. For the second result, suppose that x Limsup ν (ε ν -argmin f ν ). Then, there exists {x k ε ν k -argmin f ν k, k IN} x. Thus, limsup k f ν k (x k ) limsup k (inf f ν k + ε ν k ) inf f <. In view of condition (a) of Definition 3.7, this implies that x C and also f( x) liminf k f ν k (x k ) limsup k f ν k (x k ) inf f. Hence, x argmin f and the proof of part (a) is complete. For part (b), we observe in view of part (a) that x argmin f and also limsup ν (inf f ν ) inf f. Condition (a) of Definition 3.7 implies that liminf k (inf f v k ) = liminfk (f v k (x v k)) f( x) = inf f and the conclusion holds. Part (c), necessity. Suppose that inf f ν inf f > and let ε > 0. Then, there exist ν 1 IN such that inf f inf f ν + ε/3 for all ν ν 1 and also x C such that f( x) inf f + ε/3. In view of condition (b) of Definition 3.7, there exist x ν C ν x and ν 2 ν 1 such that f ν (x ν ) f( x) + ε/3 for all ν ν 2. Let B X be a compact set containing {x ν } ν IN. Thus, for ν ν 2, inf x C ν B f ν (x) f ν (x ν ) f( x) + ε/3 inf f + 2ε/3 inf f ν + ε. Part (c), sufficiency. For the sake of a contradiction let inf f =. Then, (a) implies that inf f ν. Since the epi-convergence is tight, there exists a compact set B X such that inf x C ν B f ν (x) and therefore also a sequence {x ν C ν B, ν IN} such that f ν (x ν ). The compactness of B implies that for some subsequence {x ν k }k IN and x B, lim k x ν k = x. In view of condition (a) of Definition 3.7, liminfk f ν k (x ν k) f(x) IR if x C and f ν k (x ν k) otherwise. However, both cases contradict f ν (x ν ) and thus inf f >. We next show that liminf ν (inf f ν ) inf f, which, together with (a) completes the proof of (c). We start by showing that for any compact set B X, {inf x C ν B f ν (x), ν IN}, except possibly for a finite number of indexes, is bounded away from. For the sake of a contradiction, suppose that for some subsequence {ν k } k IN we have inf x C ν k B f ν k (x) < inf f 1. Then, there exists {x k C ν k B, k IN} such that f ν k(x k ) < inf f 1. Since B is compact, there is a cluster point x B of {x k } k IN. By condition (a) of Definition 3.7 and the boundedness of {f ν k (x k )} k IN, x C. Then, the same condition gives that liminf k f ν k (x k ) f( x) inf f, which is a contradiction. Hence, {inf x C ν B f ν (x), ν IN}, except possibly for a finite number of indexes, is bounded away from. Next, for ε > 0, let the compact set B ε X and ν ε be such that inf x C ν B ε f ν (x) inf f ν + ε for all ν ν ε, which holds by the tightness assumption. Since {inf x C ν B f ν (x), ν IN} is eventually bounded away from, there exist ν ε ν ε and x ν C ν B ε such that f ν (x ν ) inf C ν B ε f ν + ε for all ν ν ε. Thus, ν ν ε, f ν (x ν ) inf x C ν B ε f ν (x) + ε inf f ν + 2ε. 12

13 In view of (a), we conclude that {f ν } ν IN is bounded from above. The compactness of B ε implies that there exist a subsequence {x ν k }k IN and x B ε such that lim k x ν k = x. In view of condition (a) of Definition 3.7, x C because {f ν k (x ν k)}k IN is bounded from above. The same condition then implies that liminf k (inf f ν k ) + 2ε liminf k (inf x C ν k Bε f ν k (x)) + ε liminf k f ν k (x ν k ) f( x) inf f. Since this argument holds not only for {f ν } ν IN but also for all subsequences, liminf ν (inf f ν ) + 2ε inf f. We reach the conclusion of part (c) after recognizing that ε is arbitrary. For part (d), let x argmin f. Condition (b) of Definition 3.7 implies that there exist x ν C ν x and ν 1 IN such that f ν (x ν ) f( x) + ε/2 for all ν ν 1. Since inf f is finite, there is also ν 2 ν 1 such that inf f inf f ν + ε/2 for all ν ν 2. Thus, f ν (x ν ) f( x) + ε/2 = inf f + ε/2 inf f ν + ε for all ν ν 2 and we conclude that x Liminf ν (ε- argmin f ν ). Part (e). From (a), Limsup ν (ε ν -argmin f ν ) argmin f for any ε ν 0. Thus it suffices to show that Liminf ν (ε ν -argmin f ν ) argmin f for some ε ν 0. If argmin f =, then the inclusion is automatic. Thus, we can assume that argmin f and inf f is finite. Theorem 3.1 in [10] states the following result: For any collection {h, h ν, ν IN} of extended real-valued lsc functions on a separable metric space, if h ν e h and α IR, then there exists α ν α such that {x : h ν (x) α ν } {x : h(x) α}. To apply this theorem here, let lsc f ν be the lscregularization of f ν on X, i.e., the highest lsc function on X not exceeding f ν extended to the whole of X by assigning it the value outside its domain. Since lsc f ν e f, which already is lsc and can be extended in the same manner, it follows that there exists α ν inf f such that {x X : (lsc f ν )(x) α ν } {x X : f(x) inf f} = argmin f. Set {ε ν = 1/ν + max[0, α ν inf f ν ], ν IN}. Let x argmin f. In view of the above set convergence, there exists a sequence {x ν } ν IN, with (lsc f ν )(x ν ) α ν, converging to x. The definition of lsc-regularization implies that there exists {y ν C ν, ν IN} such that d X (x ν, y ν ) 1/ν and f ν (y ν ) (lsc f ν )(x ν ) + 1/ν. Thus, f ν (y ν ) (lsc f ν )(x ν ) + 1/ν α ν + 1/ν ε ν + inf f ν and therefore y ν ε ν -argmin f ν. Consequently, x Liminf ν (ε ν -argmin f ν ), which implies that Liminf ν (ε ν -argmin f ν ) argmin f. Since inf f ν inf f, ε ν 0 and the conclusion holds. For part (f), let x argmin f. The assumption permits the selection of ν IN and a sequence {x ν ε ν -argmin f ν, ν ν} x. Thus, by condition (a) of Definition 3.7, inf f = f( x) liminf ν f ν (x ν ) liminf ν (inf f ν + ε ν ) = liminf ν (inf f ν ). In view of part (a), the conclusion follows. 13

14 4 Consequences of Lopsided Convergence The main consequence of lop-convergence of F ν to F is the convergence of solutions of the approximate minsup problems min x C ν sup y D ν (x) F ν (x, y) to those of an actual minsup problem min x C sup y D(x) F (x, y). We give detailed results below, but start with fundamental properties associated with lopsided convergence. 4.1 Basic Properties For degenerate bifunctions that only depend on their first argument, lopsided convergence reduces to epi-convergence, a direct consequence of the definitions, as stated next. 4.1 Proposition (lop-convergence reduces to epi-convergence) Suppose that the bifunctions {F, F ν, ν IN} bfcns(x, Y ) has dom F = C Y, dom F ν = C ν Y, F (x, y) = F (x, y ) for all x C and y, y Y, and F ν (x, y) = F ν (x, y ) for all x C ν and y, y Y. Then, for any y Y, F ν lop-converges to F F ν (, y) : C ν IR epi-converges to F (, y) : C IR. Lop-convergence implies hypo-convergence for certain functions as seen next. 4.2 Proposition (hypo-convergence of slices) For {F, F ν, ν IN} bfcns(x, Y ), F ν lop F implies that for all x C, there exists x ν C ν x such that the functions F ν (x ν, ) : D ν (x ν ) IR hypo-converge to F (x, ) : D(x) IR. Proof. From condition (b) of Definition 3.1 there exists x ν C ν x such that the functions { F ν (x ν, ) : D ν (x ν ) IR} ν IN and F (x, ) : D(x) IR satisfy condition (a) of Definition 3.7. From condition (a) of Definition 3.1, for any y D(x) and x ν C ν x one can find y ν D ν (x ν ) y such that condition (b) of Definition 3.7 holds. The sup-projection (in y) of F bfcns(x, Y ), henceforth denoted by f, is given as f(x) := sup y D(x) F (x, y) whenever x C and sup y D(x) F (x, y) <. This means that the domain of f, dom f C; this inclusion could be strict. Conceivably, dom f could be empty in which case the sup-projection of F does not exist. Of course, there is no practical loss to exclude this pathological case, which in the case of a minsup problem corresponds to infeasibility. Since D(x) for x C, f(x) > and, thus, f : dom f IR. Similarly, the sup-projection of F ν bfcns(x, y) is denoted by f ν. It is clear that f ν might not exist even if f does and F ν lop F as the following example shows. Absence of sup-projection. Let C = C ν = [0, 1], D = D ν = {y IR : y 0}, F (x, y) = yx for (x, y) C D, and F ν (x, y) = y(x + 1/ν) for (x, y) C ν D ν. Clearly, the sup-projection of F has dom f = {0}. However, for F ν there exists no sup-projection for all ν IN. This 14

15 situation takes place even though on can show that F ν lop F. A consequence of lopsided convergence for the epigraphs of sup-projection is given next. 4.3 Theorem (containment of sup-projections) For bifunctions {F, F ν, ν IN} bfcns(x, Y ), with corresponding sup-projections {f, f ν, ν IN}, F ν lop F implies that Limsup ν ( epi f ν ) epi f. Proof. Suppose that (x, α) limsup ν (epi f ν ). Then there exists a sequence {(x ν, α ν )} ν N, with N a subsequence of IN, x ν C ν, sup y D ν (x ν ) F ν (x ν, y) α ν, x ν N x, and α ν N α. If x C, then construct a sequence y ν D ν (x ν ) y such that F ν (x ν, y ν ) N, which exists by condition (a) of Definition 3.1. However, α ν sup y D ν (x ν ) F ν (x ν, y) F ν (x ν, y ν ), ν N, implies a contradiction since α ν N α IR. Thus, x C. If sup y D(x) F (x, y) =, then there exists y D(x) such that F (x, y) α + 1. Condition (a) of Definition 3.1 ensures that there exists a sequence y ν D ν (x ν ) y such that liminf ν F ν (x ν, y ν ) F (x, y). Consequently, α = liminf ν N α ν liminf ν N sup y D ν (x ν ) F ν (x ν, y) liminf ν N F ν (x ν, y ν ) F (x, y) α + 1, which is a contradiction. Hence, it suffices to consider the case with sup y D(x) F (x, y) finite. Given any ε > 0 arbitrarily small, pick y ε D(x) such that F (x, y ε ) sup y D(x) F (x, y) ε. Then condition (a) of Definition 3.1 again yields y ν D ν (x ν ) y ε such that liminf ν N sup y D ν (x ν ) F ν (x ν, y) liminf ν N F ν (x ν, y ν ) F (x, y ε ) sup y D(x) F (x, y) ε, implying liminf ν N sup y D ν (x ν ) F ν (x ν, y) sup y D(x) F (x, y). Since the conclusion follows. α = liminf ν N α ν liminf ν N sup y D ν (x ν ) F ν (x ν, y) sup y D(x) F (x, y), It is clear from the following example that lop-convergence does not guarantee that epi f ν epi f holds in general. Absence of tightness. For {F, F ν, ν IN} bfcns(x, Y ), let C = C ν = X = IR, D = D ν = Y = IR, F (x, y) = 0 for (x, y) X Y, and F ν (x, y) = 1 if x X and y = ν, and zero otherwise. It is easy to show that F ν lop F. The sup-projection of F ν has f ν (x) = 1 for all x X and that of F has f(x) = 0 for all x X. Thus, the inclusion in Theorem 4.3 is strict. We observe that ancillary-tightness fails in this instance and that property is indeed key to eliminating such possibilities. 15

16 4.4 Theorem (epi-convergence of sup-projections) Suppose the bifunctions {F, F ν, ν IN} bfcns(x, Y ) have corresponding sup-projections {f, f ν, ν IN}. If F ν lop-converges ancillarytightly to F, then epi f ν epi f or, equivalently, f ν e f. Proof. Let x dom f. Now, choose x ν C ν x such that F ν (x ν, ) : D ν (x ν ) IR hypoconverge to F (x, ) : D(x) IR, cf. Proposition 4.2. In fact, these functions hypo-converge tightly as an immediate consequence of ancillary-tightness. Thus, sup F ν (x ν, y ν ) sup F (x, y), y D ν (x ν ) y D(x) via Theorem 3.8. This fact together with Theorem 4.3 establish the first conclusion. The second conclusion is then immediate; see 4. It is clear that lop-convergence is a stronger condition than epi-convergence of the corresponding sup-projections. For example, if C = C ν = [0, 1], D = D ν = [0, 1], and F (x, y) = 1, F ν (x, y) = y for all (x, y) C D, then the corresponding sup-projections are identical and, certainly, f ν e f. However, condition (a) of Definition 3.1 fails and therefore F ν does not lop-converge to F. We end this subsection by listing consequences for semicontinuity, concavity, and convexity. 4.5 Proposition (usc and concavity) For bifunctions {F, F ν, ν IN} bfcns(x, Y ) and any x C, F ν lop F implies that the (univariate) function F (x, ) : D(x) IR is usc. When Y is a linear space and for all x ν C ν x, F ν (x ν, ) : D ν (x ν ) IR is concave, it also implies that F (x, ) is concave. Proof. In view of Proposition 4.2, for every x C, there exists x ν C ν x such that F ν (x ν, ) hypo-converges to F (x, ), which therefore must be usc. If F ν (x ν, ) are concave, its hypo-limit must also be concave, which establishes the conclusions. 4.6 Proposition (lsc and convexity) Suppose the bifunctions {F, F ν, ν IN} bfcns(x, Y ) have corresponding sup-projections {f, f ν, ν IN}. If F ν lop-converges ancillary-tightly to F, then f is lsc and also convex provided that X is a linear space, D ν is constant on C ν, and F ν (, y) is convex for all y D ν. Proof. In view of Theorem 4.4, f is an epi-limit and thus lsc. Convexity is guaranteed if f ν = sup y D ν ( ) F ν (, y), ν IN, are convex, which holds by the stated assumptions. 4.2 Consequence for Minsup Problems We have now reached the main results of the paper. The minsup-value of a bifunction F bfcns(x, Y ) is defined as minsup F := inf x C sup y D(x) F (x, y), 16

17 which clearly is the optimal value of the minsup problem min x C sup y D(x) F (x, y). The corresponding ε-optimal solutions, referred to as ε-minsup-points of F, are given by ε- argminsup F := { x C : sup y D(x) F (x, y) minsup F + ε }, for ε 0. If ε = 0, we simply refer to such points as minsup-points. 4.7 Theorem (bounds on minsup-value) Suppose that {F, F ν, ν IN} bfcns(x, Y ) has sup-projections, F ν lop F, and {x ν argminsup F ν, ν IN} exist. Then, the following hold: (a) If {x ν } ν IN has a cluster point, then liminf ν (minsup F ν ) minsup F. (b) If lop-convergence is ancillary-tight, then limsup ν (minsup F ν ) minsup F and {ε ν 0, ν IN}, Limsup ν (ε ν - argminsup F ν ) argminsup F. (c) If lop-convergence is ancillary-tight and x is a cluster point of {x ν } ν IN, i.e., the limit of a subsequence {x ν k }k IN, then lim k (minsup F ν k ) = minsup F. Proof. We first consider (a): Let {x ν k }k IN be a convergent subsequence of {x ν } ν IN tending to x. If x dom f, then by Theorem 4.3 liminf ν (minsup F ν ) = liminf ν f ν (x ν ) f( x) minsup F. If x dom f, then also by Theorem 4.3 liminf ν (minsup F ν ) = liminf ν f ν (x ν ) and the conclusion holds. Second, consider (b) and (c): Theorem 4.4 implies that f ν e f and thus the conclusion is a direct application of Theorem 3.8. Tight lop-convergence implies the following strengthening of the result for minsup-points and values. 4.8 Theorem (approximating minsup-points) Suppose that {F, F ν, ν IN} bfcns(x, Y ) have sup-projections and F ν lop F tightly. Then, (a) minsup F ν minsup F, which is finite; (b) for ε > 0, Liminf ν (ε- argminsup F ν ) argminsup F ; (c) for X separable, {ε ν 0, ν IN} such that ε ν - argminsup F ν argminsup F. Proof. The assumptions imply those of Theorem 4.4 and thus f ν e f. In view of Definition 3.6, the epi-convergence is tight; see Definition 3.7. The conclusion is then a direct application of Theorem 3.8. It is well-known that the infimum of a lsc function with a domain contained in a compact set is attained. Consequently, if the sup-projection f for some bifunction F bfcns(x, Y ) is lsc, which holds under mild assumptions (cf. Proposition 5.1), and C is contained in some compact set (and thus also dom f), then there exists a minimizer of f and thus also a minsup-point of F. We next state a result that relaxes the compactness requirement. 17

18 4.9 Theorem (existence of minsup-points) Suppose that {F, F ν, ν IN} bfcns(x, Y ) have sup-projections, F ν lop F ancillary-tightly, {f ν, ν IN} are lsc, and there are compact sets {B ν X, ν IN} such that C ν B ν. Then, {x ν argminsup F ν, ν IN} exist and every cluster point of {x ν } ν IN is a minsup-point of F. Proof. The discussion prior to the theorem ensures the existence of minsup-points of F ν for every ν. The result is then a consequence of Theorem 4.7. Theorem 4.9 does not ensure the existence of a cluster point of {x ν } ν IN. Still, it provides an approach for establishing the existence of a minsup-point of F : first construct a sequence {F ν } ν IN, with the required properties, that lop-converges ancillary-tightly to F and, second, prove that {x ν } ν IN has a cluster point. 5 Quantification of Lopsided Convergence We next turn to a definition of distance between bifunctions that in some sense characterizes lop-convergence. We would like the distance between F ν and F to tend to zero if and only if F ν lop-converges to F. In this section, we develop for the first time such a distance, which essentially characterizes lop-convergence after passing to certain equivalence classes of bifunctions. We begin with establishing a foundation regarding a distance between (univariate) functions. 5.1 Attouch-Wets Distance between Functions We limit the scope to the subset of fcns(x) consisting of lsc functions and let lsc-fcns(x) := {h fcns(x) : h lsc}. This set is equipped with the Attouch-Wets (aw) distance dl aw, which is given for a fixed but arbitrary point 3 x X. Specifically, the aw-distance is defined for any h, g lsc-fcns(x) as dl aw (h, g) := where the ρ-aw-distance, for any ρ 0 is given by 0 dl aw ρ (h, g)e ρ dρ, dl aw ρ (h, g) := sup { dist ( (x, α), epi h ) dist ( (x, α), epi g ) : d X (x, x) ρ, α ρ } and dist ( (x, α), epi h ) := inf {max{d X (x, x ), α α } : (x, α ) epi h}, 3 Although the topology induced by the aw-distance is the same for any point selected, the value of the distance will depend on this choice; see [5] and [18, Section 7.J] for details. 18

19 with a similar expression pertaining to epi g. We observe that x serves as the center of a ball on which dl aw ρ is computed. It is immediate that dl aw is a metric on lsc-fcns(x). We deduce from [9, Theorem 3.1.7] that for h ν, h lsc-fcns(x), dl aw (h ν, h) 0 = h ν e h. If (X, d X ) is a proper 4 metric space, then the converse also holds in view of [9, Theorem 3.1.7] and [5, Theorem 4.2, Lemma 4.3]. The latter results are stated only for IR n, but the arguments hold for any proper metric space. 5.2 Lop-Distance Parallel to the restriction to lsc-fcns(x) in the definition of dl aw, we focus on bifunctions with lsc sup-projections. Specifically, let lsc-bfcns(x, Y ) := { F bfcns(x, Y ) with lsc sup-projection }. In view of Theorem 4.6, the sup-projection of a bifunction to which a sequence of bifunctions lop-converge ancillary-tightly must be lsc. In fact, it is well-known that sup-projections are lsc under mild assumptions. Sufficient conditions are given next, a result essentially in [6, Theorem 2, Section 2.5.2] but proved here for completeness. 5.1 Proposition (lsc sup-projection) When it exists, the sup-projection f of a bifunction F bfcns(x, Y ) is lsc if either (a) D is inner semicontinuous 5 and F is lsc; or (b) D Y, i.e., the set-valued mapping is constant, and F (, y) is lsc for all y D. Proof. Let x ν dom f x dom f. Then sup y D(x) F (x, y) is finite and for every ε > 0 there is a y ε D(x) such that sup y D(x) F (x, y) F (x, y ε ) + ε. Moreover, the inner semicontinuity of D implies that there exists a sequence y ν ε D(x ν ) y ε. Since F is lsc, there exists ν such that for all ν ν, F (x ν, y ν ε ) F (x, y ε ) ε. Thus, for ν ν, sup y D(x ν ) F (x ν, y) F (x ν, y ν ε ) F (x, y ε ) ε sup y D(x) F (x, y) 2ε. Since ε is arbitrary, liminf ν sup y D(x ν ) F (x ν, y) sup y D(x) F (x, y). Next, suppose that x ν dom f x dom f. We consider two cases. First, let x C. Then, there exist y ν D(x ν ) such that F (x ν, y ν ). Thus, sup y D(x ν ) F (x ν, y). Second, let x C, but sup y D(x) F (x, y) =. Then, for every M < there is a y M D(x) such that F (x, y M ) > M. Following a similar argument as earlier, we find that liminf ν sup y D(x ν ) F (x ν, y) > M. Since M is arbitrary, the first conclusion follows. In the second case with D constant, the sequence {yε ν } can be selected to coincide with y ε for all ν and lsc in x only suffices. 4 A metric space is proper if all its closed balls are compact. 5 D : C Y is inner semicontinuous if for every x ν C x C, Liminf ν D(x ν ) D(x). 19

20 5.2 Definition (lop-distance) For any two bifunctions F, F lsc-bfcns(x, Y ), with sup-projections f, f, the lop-distance dl lop( F, F ) := dl aw( f, f ). It is immediately clear that the lop-distance is only a pseudometric on lsc-bfcns(x, Y ) as there are nonidentical bifunctions with the same sup-projection. Thus, we pass to equivalence classes. 5.3 Definition (equivalence classes of bifunctions) Bifunctions F, F lsc-bfcns(x, Y ) are equivalent, denoted by F F, if their sup-projections exist and are identical. We note that F F might hold even if C C as long as every x C that is not in C has sup y D(x) F (x, y) = and every x C that is not in C has sup y D (x) F (x, y) =. Thus, dom f = dom f and F F holds provided that the sup-projections exist (i.e., dom f = dom f ) and coincide on the domain. It is clear that dl lop is a metric on the quotient set of lsc-bfcns(x, Y ). 5.4 Theorem (quantification of lop-convergence) For bifunctions {F, F ν, ν IN} lsc-bfcns(x, Y ), the following hold. (a) If dl lop (F ν, F ) 0, then there exist F ν, F lsc-bfcns(x, Y ) such that F ν F ν, F F, and F ν lop F. (b) If F ν lop F ancillary-tightly and (X, d X ) is proper, then dl lop (F ν, F ) 0. Proof. For part (a), set C = dom f and D(x) = Y for all x C. Let F (x, y) = f(x) for x C and y Y. Since F lsc-bfcns(x, Y ), f exists. Thus, C is nonempty and F : C Y IR. By construction, for x C, sup y Y sup y D(x) F (x, y ) = f(x) <. Consequently, f, the sup-projection of F, exists and dom f = C. Moreover, f = f on C. where f is the sup-projection of F. Thus, F lsc-bfcns(x, Y ) and F F. An identical construction with C ν = dom f ν, Dν (x) = Y for all x C ν, and F ν : C ν Y IR having F ν (x, y) = f ν (x) for x C ν and y Y results in F ν lsc-bfcns(x, Y ) and F ν F ν. We then obtain for any y Y that ( dl aw F ν (, y), F ) ( ) (, y) = dl aw (f ν, f) = dl lop F ν, F 0. Since dl aw ( F ν (, y), F (, y)) 0 implies that F ν (, y) : C ν IR epi-converges to F (, y) : C IR, it follows from Proposition 4.1 that F ν lop-converges to F. For part (b) we conclude from Theorem 4.4 that f ν epi-converges to f. Thus, dl aw (f ν, f) 0 under the additional assumption that (X, d X ) is proper and the result follows from the definition of the lop-distance. 20

21 We recall that ancillary-tightness is ensured, for example, by some compactness property on the sets over which the inner maximization is taking place. Thus, a restriction to bifunctions satisfying such properties would ensure that the lop-distance fully characterizes lopsided convergence on this class provided that (X, d X ) is proper. Thus, the lop-distance generates what we define as the lop-topology on the space of equivalence classes of such bifunctions. Acknowledgement. This work in supported in parts by DARPA under grant HR References [1] H. Attouch. Variational Convergence for Functions and Operators. Applicable Mathematics Sciences. Pitman, [2] H. Attouch, D. Azé, and R. Wets. Convergence of convex-concave saddle functions: continuity properties of the Legendre-Fenchel transform with applications to convex programming and mechanics. Annales de l Institut H. Poincaré: Analyse Nonlinéaire, 5: , [3] H. Attouch and R. J-B Wets. Convergence des points min/sup et de points fixes. Comptes Rendus de l Académie des Sciences de Paris, 296: , [4] H. Attouch and R. J-B Wets. A convergence theory for saddle functions. Transactions of the American Mathematical Society, 280(1):1 41, [5] H. Attouch and R. J-B Wets. Quantitative stability of variational systems: I. the epigraphical distance. Transactions of the American Mathematical Society, 328(2): , [6] J.-P. Aubin. Mathematical Methods of Game and Economic Theory. North-Holland, [7] J.-P. Aubin and I. Ekeland. Applied Nonlinear Analysis. Issue 1237 of Pure and applied mathematics. Wiley, [8] J.-P. Aubin and H. Frankowska. Set-Valued Analysis. Birkhäuser, [9] G. Beer. Topologies on Closed and Closed Convex Sets, volume 268 of Mathematics and its Applications. Kluwer, [10] G. Beer, R. T. Rockafellar, and R. Wets. A characterization of epi-convergence in terms of convergence of level sets. Proceedings of the American Mathematical Society, 116(3): , [11] A. Ben-Tal, L. El Ghaoui, and A. Nemirovski. Robust Optimization. Princeton University Press,