PUC Rio | Departamento de Economia

Dear DrProf RogerJ-B WetsHere are the proofs of your article

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6amp7 5th Street Radhakrishnan Salai Chennai Tamil Nadu India ndash 600004Re Mathematical Programming DOI101007s10107-007-0122-8

Variational convergence of bivariate functions lopsided convergenceAuthors Alejandro Jofreacute middot RogerJ-B Wets

I Permission to publishDear Springer Correction TeamI have checked the proofs of my article andq I have no corrections The article is ready to be published without changes

q I have a few corrections I am enclosing the following pagesq I have made many corrections Enclosed is the complete article

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Journal Mathematical Programming101007s10107-007-0122-8

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RogerJ-B Wets RogerJ-B WetsDepartment of MathematicsUniversity of CaliforniaDavis USA

Department of MathematicsUniversity of CaliforniaDavis USA

q q

Metadata of the article that will be visualized in OnlineFirst

ArticleTitle Variational convergence of bivariate functions lopsided convergenceArticle Sub-Title

Journal Name Mathematical Programming

Corresponding Author Family Name WetsParticle

Given Name Roger J-BSuffix

Division Department of Mathematics

Organization University of California

Address Davis USA

Email rjbwetsucdavisedu

Author Family Name JofreacuteParticle

Given Name AlejandroSuffix

Division Ingeneria Matematica

Organization Universidad de Chile

Address Santiago Chile

Email ajofredimuchilecl

Schedule

Received 11 September 2005

Revised

Accepted 3 April 2006

Abstract We explore convergence notions for bivariate functions that yield convergence and stability results for theirmaxinf (or minsup) points This lays the foundations for the study of the stability of solutions to variationalinequalities the solutions of inclusions of Nash equilibrium points of non-cooperative games and Walraseconomic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions ofconvex optimization problems and of zero-sum games These applications will be dealt with in a couple ofaccompanying papers

Keywords (separated by -) Lopsided convergence - Maxinf-points - Ky Fan functions - Variational inequalities - Epi-convergence

Mathematics SubjectClassification (2000)(separated by -)

65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45

Footnote Information Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical proceduresResearch supported in part by grants of the National Science Foundation and Fondap-Matematicas AplicadasUniversidad de Chile

Journal 10107 Article 122 Dear Author During the process of typesetting your article the following queries have arisen Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proofonline grid or in the lsquoAuthorrsquos responsersquo area provided below

Author Query Form

Query Details required Authorrsquos response 1 Please check the author name

ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)

unco

rrec

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Math Program Ser BDOI 101007s10107-007-0122-8

FULL LENGTH PAPER

Variational convergence of bivariate functions lopsidedconvergence

Alejandro Jofreacute middot Roger J-B Wets

Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007

Abstract We explore convergence notions for bivariate functions that yield1

convergence and stability results for their maxinf (or minsup) points This lays the2

foundations for the study of the stability of solutions to variational inequalities the3

solutions of inclusions of Nash equilibrium points of non-cooperative games and4

Walras economic equilibrium points of fixed points of solutions to inclusions the5

primal and dual solutions of convex optimization problems and of zero-sum games6

These applications will be dealt with in a couple of accompanying papers7

Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8

inequalities middot Epi-convergence9

Mathematics Subject Classification (2000) 65K10 middot 90C31 middot 91A10 middot 47J20 middot10

47J30 middot 49J4511

Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical procedures

Research supported in part by grants of the National Science Foundation and Fondap-MatematicasAplicadas Universidad de Chile

A JofreacuteIngeneria Matematica Universidad de Chile Santiago Chilee-mail ajofredimuchilecl

RJ-B Wets (B)Department of Mathematics University of California Davis USAe-mail rjbwetsucdavisedu

123Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21

unco

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A Jofreacute R J -B Wets

1 Variational convergence of bivariate functions12

A fundamental component of lsquoVariational Analysisrsquo is the analysis of the properties13

of bivariate functions For example the analysis of the Lagrangians associated with14

an optimization problem of the Hamiltonians associated with Calculus of Variations15

and Optimal Control problems the reward functions associated with cooperative or16

non-cooperative games and so on In a series of articles we deal with the stability17

of the solutions of a wide collection of problems that can be re-cast as finding the18

maxinf-points of such bivariate functions19

So more explicitly given a bivariate function F C times D rarr R we are interested20

in finding a point say x isin C that maximizes with respect to the first variable x 21

the infimum of F inf yisinD F(middot y) with respect to the second variable y We refer to22

such a point x as a maxinf-point In some particular situations for example when the23

bivariate function is concavendashconvex such a point can be a saddle point but in many24

other situation its just a maxinf-point or a minsup-point when minimizing with respect25

to the first variable the supremum of F with respect to the second variable To study the26

stability and the existence of such points and the sensitivity of their associated values27

one is lead to introduce and analyze convergence notion(s) for bivariate functions that28

in turn will guarantee the convergence either of their saddle points or of just their29

maxinf-points30

This paper is devoted to the foundations Two accompanying papers deal with the31

motivating examples [1011] variational inequalities fixed points Nash equilibrium32

points of non-cooperative games equilibrium points of zero-sum games etc We make33

a distinction between the situations when the bivariate function is generated from a34

single-valued mapping [11] or when the mapping can also be set-valued [10]35

The major tool is the notion of lopsided convergence that was introduced in [2]36

but is modified here so that a wider class of applications can be handled The major37

adjustment is that bivariate functions are not as in [2] no longer defined on all of38

Rn timesR

m with values in the extended reals but are now only finite-valued on a specific39

product C times D with C D subsets of Rn and R

m Dealing with lsquogeneralrsquo bivariate40

functions defined on the full product space was in keeping with the elegant work41

of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42

the subsequent work [3] on the epihypo-convergence of saddle functions However43

our present analysis actually shows that notwithstanding its esthetic allurement one44

should not cast bivariate functions even in the convexndashconcave case in the general45

extended-real valued framework In some way this is in contradiction with the uni-46

variate case where the extension by allowing for the values plusmninfin of functions defined47

on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48

convergence and differentiation theory [514] We shall show that some of this can be49

recovered but one must first make a clear distinction between max-inf problems and50

min-sup ones and only then one can generate the appropriate extensions after all51

also in the univariate case one makes a clear distinction when extending a function in52

a minimization setting or a maximization setting53

In order to be consistent in our presentation and to set up the results required54

later on we begin by a presentation of the theory of epi-convergence for real-valued55

univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56

123

Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21

unco

rrec

ted

proo

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Variational convergence of bivariate functions

actually derived although a revised formulation is required We make the connection57

with the standard approach ie when these (univariate) functions are extended real-58

valued We then turn to lopsided convergence and point out the shortcomings of an59

lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60

a extension of Ky Fan inequality [7] to situations when the domain of definition of the61

bivariate function is not necessarily compact62

2 Epi-convergence63

One can always represent an optimization problem involving constraints or not as64

one of minimizing an extended real-valued function In the case of a constrained-65

minimization problem simply redefine the objective as taking on the value infin out-66

side the feasible region the set determined by the constraints In this framework the67

canonical problem can be formulated as one of minimizing on all of Rn an extended68

real-valued function f Rn rarr R Approximation issues can consequently be stud-69

ied in terms of the convergence of such functions This has lead to the notion of70

epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71

ing with a maximization problem it is hypo-convergence the convergence of the72

hypographs that is the appropriate convergence notion73

Henceforth we restrict our development to the lsquominimization settingrsquo but at the74

end of this section we translate results and observations to the lsquomaximizationrsquo case75

As already indicated in Variational Analysis one usually deals with76

fcn(Rn) = f R

n rarr R

77

the space of extended real-valued functions that are defined on all of Rn even allowing78

for the possibility that they are nowhere finite-valued Definitions properties limits79

etc generally do not refer to the domain on which they are finite For reasons that will80

become clearer when we deal with the convergence of bivariate functions we need to81

depart from this simple and very convenient paradigm Our focus will be on82

f v-fcn(Rn) = f D rarr R

∣∣ for some empty = D sub R

n83

the class of all finite-valued functions with non-empty domain D sub Rn It must be84

understood that in this notation Rn does not refer to the domain of definition but to85

the underlying space that contains the domains on which the functions are defined86

The epigraph of a function f is always the set of all points in Rn+1 that lie on87

or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88

f D rarr R belongs to f v-fcn(Rn) then89

epi f = (x α) isin D times R

∣∣ α ge f (x)

sub Rn+190

1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]

123


unco

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ted

proo

f


and if f belongs to fcn(Rn) then91

epi f = (x α) isin R

n+1∣∣α ge f (x)

92

A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93

Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294

So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95

ndash if x isin D liminfν f (xν) ge f (x) and96

ndash if x isin cl D D f (xν) rarr infin97

In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98

closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99

possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100

D but always contained in cl D101

Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102

Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103

ndash (a-set) all cluster points of a sequence

xν isin CννisinIN belong to C 104

ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105

When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν

νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107

Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108

[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109

capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν

νisinIN110

consists of convex sets its inner limit and its limit if it exists are also convex [14111

Proposition 415]112

Definition 1 (epi-convergence) A sequence of functions

f ν ν isin IN whose113

domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114

of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115

f ν rarre f 116

Figure 1 provides an example of two functions f and f ν that are close to each117

other in terms of the distance between their epigraphsmdashie the distance between the118

location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119

to the infin-normmdashie the size of the jumps120

Let f ννisinIN be a sequence of functions with domains in Rn When121

ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122

functions f ν 123

ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124

functions f ν 125

Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126

2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology

3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D

123


unco

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proo

f


x

f

f

Fig 1 f and f ν epigraphically close to each other

Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127

domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128

all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129

epi-limit if it exists130

Proof Follows immediately from the properties of set-limits 131

The last proposition implies in particular that the family of lsc functions is closed132

under epi-convergence133

The definition of epi-convergence for families of functions in fcn(Rn) is the usual134

one [14 Chap 7 Sect B] with all the implications concerning the convergence of135

the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136

definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137

its for this class of functions that we need to know the conditions under which one can138

claim convergence of the minimizers and infimums We chose to make the presentation139

self-contained although as will be shown later one could also embed f v-fcn(Rn) in140

a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141

requires that the non-initiated reader plows through a substantial amount of material142

When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143

limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144

necessarily closed Simply think of the collection of functions f ν = f for all ν with145

D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146

function f on D with closed epigraph but not with closed domain147

Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148

of a sequence

f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149

sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150

Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151

liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152

to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153

the line x times R would have to lie in epi f contradicting the assumption that f the154

epi-limit of the f ν belongs to f v-fcn(Rn) 155

123


unco

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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R

νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩

minusν2x if 0 le x le νminus1

ν2x minus 2ν if νminus1 le x le 2νminus1

0 for x ge 2νminus1

158

Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159

and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160

functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161

extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162

function in f v-fcn(R) 163

In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164

characterization of epi-converging sequences in f v-fcn(Rn)165

Proposition 2 [epi-convergence in f v-fcn(Rn)] Let

f D rarr R f ν Dν rarr166

R ν isin IN

be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167

following conditions are satisfied168

(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169

(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170

(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171

Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172

not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173

(x α) is a cluster point of(xν f ν(xν)) isin epi f ν

νisinIN and thus belongs to epi f 174

ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175

valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176

and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177

a(xν αν) isin epi f ν

νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178

αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179

Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180

or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181

with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182

not dealing with a converging sequence of points (in Rn+1) and there is no need to183

consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184

and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185

this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186

existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187

can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188

condition for the set-convergence of epi f ν rarr epi f 189

Theorem 1 (epi-convergence basic properties) Consider a sequence

f ν Dν rarr190

R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191

4 means non-decreasing and converging to ie not necessarily monotonically

123


unco

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f


lim supνrarrinfin

(inf f ν) le inf f192

Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193

x isin argminD f and minDνk f νk rarr minD f 194

If argminD f is a singleton then every convergent subsequence of minimizers con-195

verges to argminD f 196

Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197

one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198

for all ν inf f ν le f ν(xνl) it follows that for all l199

lim supν

(inf f ν) le lim supν

f ν(xνl) le f (xl)200

and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201

For the sequence xk isin Dνk rarr x from the above and 2(a)202

inf f ge lim supk

f νk (xk) ge lim infk

f νk (xk) ge f (x)203

ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204

Finally since every convergent subsequence of minimizers of the functions f ν205

converges to a minimizer of f it follows that it must converge to the unique minimizer206

when argminD f is a singleton 207

In most of the applications we shall rely on a somewhat more restrictive notion208

than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209

Definition 2 (tight epi-convergence) The sequence210

f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211

f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212

dex νε such that213

forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214

Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215

be a sequence of functions that epi-converges to the function f D rarr R also in216

f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217

(a) if and only if inf Dν f ν rarr inf D f 218

(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219

Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220

imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν

f ν) + ε le lim supν

(infDν

f ν) + ε le infD

f + ε lt infin222

123


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ted

proo

f


If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223

follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224

point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225

with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226

it follows that there is no such sequences with κ arbitrarily negative In other words227

excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228

from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229

a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230

Hence231

inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232

In combination with our first string of inequalities and the fact that ε gt 0 can be233

chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234

Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235

enough given any δ gt 0 to exhibit a compact set B such that lim supν

(inf BcapDν f ν

) le236

inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237

f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238

Let B be any compact set large enough to contain all the points xν Then inf B f ν le239

f ν(xν) for all ν so B has the desired property240

We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241

tion and consequently for ν large enough also αν is finite Since convergence of242

the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243

find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244

εν = αν minus αν 245

For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246

f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247

because f ν rarre f one obtains248

inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249

le lim infν(inf f ν) le limsupν(inf f ν) le inf f250

where the last inequality comes from Theorem 1 251

Remark 1 (convergence of domains) Although epi-convergence essentially implies252

convergence of the level sets [14 Proposition 77] it does not follow that it implies253

the convergence of their (effective) domains Indeed consider the following sequence254

f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255

function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256

This vigorously argues against the temptation of involving the convergence of their257

domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258

This concludes the presentation of the results that will be used in the sequel As259

indicated earlier its also possible to derive these results from those for extended real-260

valued functions To do so one identifies f v-fcn(Rn) with261

pr -fcn(Rn) = f isin fcn(Rn)

∣∣ minus infin lt f equiv infin

262

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f


the subset of proper functions in fcn(Rn) in a minimization context a function f is263

said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264

domain265

dom f = x isin R

n∣∣ f (x) lt infin

266

There is an one-to-one correspondence a bijection5 denoted η between the elements267

of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268

Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269

uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270

of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271

observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272

and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273

Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274

defined in terms of the convergence of the epigraphs there is really no need to verify275

that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276

However for completeness sake and to highlight the connections we go through the277

details of an argument278

Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN

be a collection of functions in f v-fcn(Rn)280

Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281

Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282

(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283

(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284

Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285

(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286

elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287

(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288

yield (aη)289

If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290

xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291

finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292

η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293

As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294

observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295

convergence related to the convergence of infimums and minimizers apply equally296

well to functions in f v-fcn(Rn) and not just those featured here In particular if one297

takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298

is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299

cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300

5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below

123


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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301

that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302

of the infimal values303

Finally in a maximization setting one can simply pass from f to minus f or one can304

repeat the previous arguments with the following changes in the terminology min to305

max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306

(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1

307

that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308

is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309

the maximization setting cl f denotes the function whose hypograph is the closure310

relative to Rn+1 of hypo f its also called its usc regularization311

A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312

equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313

tightly to minus f And consequently if the sequence hypo-converges tightly to f with314

supD f finite then supDν f ν rarr supD f 315

When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316

of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317

follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318

if it exists are all usc Moreover if the functions f ν are concave so is the lower319

hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320

functions is closed under hypo-convergence321

3 Lopsided convergence322

Lopsided convergence for bivariate functions was introduced in [2] we already relied323

on this notion to formalize the convergence of pure exchange economies and to study324

the stability of their Walras equilibrium points [9] Its aimed at the convergence of325

maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326

convergence However our present more comprehensive analysis has lead us to adjust327

the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328

and values like those depicted in Fig 2 would essentially be excluded ie could not329

be included in (lopsided or) lop-convergent families And these are precisely the class330

of functions that needs to be dealt with in many applications Moreover like in Sect331

2 the main focus will not be on extended real-valued functions but on finite-valued332

bivariate functions that are only defined on a product of non-empty sets rather than333

on extended real-valued functions defined on the full product space The motivation334

for proceeding in this manner again coming from the applications But this time its335

not just one possible approach its in fact mandated by the underlying structure of the336

class of bivariates that are of interest in the applications We shall however like in the337

previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338

was used in [2]339

The definition of lop-convergence is necessarily one-sided One is either inter-340

ested in the convergence of maxinf-points or minsup-points but not both In general341

the maxinf-points are not minsup-points and vice-versa When they identify the342

same points such points are saddle-points In this article our concern is with the343

123


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f

Variational convergence of bivariate functionsB

ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus

Fig 2 Partition of the domain of a proper bivariate function maxinf framework

lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344

of the article345

Definitions and results can be stated either in terms of the convergence of maxinf-346

points or minsup-points with some obvious adjustments for signs and terminology347

However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348

work and this is in keeping with the (plain) univariate case where one has to focus349

on either minimization or maximization Because most of the applications we are350

interested in are more naturally formulated in terms of maxinf-problems thats the351

version that will be dealt with in this section We provide at the end of the section352

the necessary translations required to deal with minsup-problems353

Here the term bivariate function always refers to functions defined on the product354

of two non-empty subsets of Rn and R

m respectively6 We write355

biv(Rn+m) = F R

n times Rm rarr R

356

for the class of bivariate functions that are extended real-valued and defined on all of357

Rn times R

m and358

f v-biv(Rn+m) = F C times D rarr R

∣∣ empty = C sub R

n empty = D sub Rm

359

for the class of bivariate functions that are real-valued and defined on the product360

C times D of non-empty subsets of Rn and R

m respectively here its understood that361

Rn+m does not refer to the domain of definition but to the (operational) product space362

that includes C times D363

6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies

123


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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364

point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366

its a minsup-point if367

x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R

m are not excluded369

Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370

with the maxinf case371

Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)

Fν Cν times372

Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373

also in f v-biv(Rn+m) if374

(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375

with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376

(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377

yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378

(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379

yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380

point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381

Although a number of properties can be immediately derived from this convergence382

notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383

however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384

This (partial) y-tightness condition is new it was inspired by the work of Bagh385

[6] on approximation for optimal control problems A more conventional condition386

that implies y-tightness would be the following (b) holds and there is a compact set387

B sub Rm such that388

forallx isin Rn B sup

y∣∣ Fν(x y) lt infin

389

This last condition suggested in [2] is too restrictive in many applications Moreover390

the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391

section that can be exploited in situations when the domain of definition of the bivariate392

function is not compact393

Now lets turn to the convergence of the marginal functions394

gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395

in the extended real-valued framework one can find a number of related results in the396

literature see in particular [12]397

123


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f


Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν

νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399

as follows400

(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401

liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402

to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403

the sequence xν rarr x such that for all ν larger than some νε404

inf DνcapBε Fν(xν middot) le inf Dν Fν(xν middot) + ε405

Let gν = inf yisinDν Fν(middot y) g = inf yisinD F(middot y) Then gν rarrh g in f v-fcn(Rn)406

assuming that their domains are non-empty ie Cνg =

x isin Cν∣∣ gν(x) gt minusinfin

407

and Cg = x isin C

∣∣ g(x) gt minusinfin

are non-empty sets except possibly for a finite408

number of indexes ν409

Proof The functions gν and g never take on the value infin so the proof does not have410

to deal with that possibility This means that gν and g belong to f v-fcn(Rn) whenever411

they are defined on non-empty sets Note however that in general the function gν and412

g are not necessarily finite-valued on all of Cν and C since they can take on the value413

minusinfin implying that Cνg =

x∣∣ gν(x) gt minusinfin

and Cg = x


could be414

strictly contained in Cν and C even potentially empty this later instance however415

has been excluded by the hypotheses416

We need to verify the conditions of Proposition 2 Lets begin with (a) and (ainfin)417

Suppose xν isin Cνg rarr x isin Cg So g(x) isin R and yε isin ε- argminD F(x middot) for ε gt 0418

By 3(a) one can find yν isin Dν rarr yε such that limsupν Fν(xν yν) le F(x yε)419

Hence420

limsupν gν(xν) le limsupν Fν(xν yν) le F(x yε) le g(x) + ε421

Since this holds for arbitrary ε gt 0 it follows that limsupν gν(xν) le g(x) When422

g(x) = minusinfin which means that x isin Cg If x isin C for any κ lt 0 there is a yκ isin D423

such that F(x yκ ) lt κ and 2(a) then yields a sequence yν isin Dν rarr yκ such that424

limsupν gν(xν) le limsupν Fν(xν yν) le F(x yκ ) lt κ425

Since this holds for κ arbitrarily negative it follows that limsupν gν(xν) = minusinfin426

When x isin C one appeals directly to 3(ainfin) to arrive at the same implication427

Lets now turn to the second condition 2(b) for hypo-convergence for all x isin Cg428

there exists xν isin Cνg rarr x such that liminfν gν(xν) ge g(x) The inequality would429

clearly be satisfied if g(x) = minusinfin but then x isin Cg and that case does not need to430

concern us So when g(x) isin R let xν isin Cν rarr x be a sequence predicated by431

condition (bndasht) for y-tight lop-convergence It follows that the functions Fν(xν middot) 432

Dν rarr R epi-converge to F(x middot) D rarr R Hence one can apply Theorem 2 since433

g(x) = inf D F(x middot) is finite and the condition on tight epi-convergence is satisfied as434

immediate consequence of (partial) y-tight lop-convergence Hence gν(xν) rarr g(x)435

436

123


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f


Theorem 4 (convergence of maxinf-points f v-biv) Let FννisinIN and F be a family437

of bivariate functions that satisfy the assumptions of Theorem 3 so in particular438

the Fν lop-converge to F and the condition (bndasht) on y-tightness is satisfied For439

all ν large enough let xν be a maxinf-point of Fν and x any cluster point of the440

sequence xν ν isin IN then x is a maxinf-point of the limit function F Moreover441

with xν ν isin N sub IN the (sub)sequence converging to x 442

limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443

ie there is also convergence of the lsquovaluesrsquo of these maxinf-points444

Proof Theorem 3 tells us that with445

gν(x) = inf yisinDν Fν(x y) g(x) = inf yisinD F(x y)446

the functions gν hypo-converge to g Maxinf-points of Fν and F are then maximizers447

of the corresponding functions gν and g The assertions now follow immediately from448

the convergence of the argmax of hypo-converging sequences cf Theorem 1 translated449

to the lsquomaximizationrsquo framework 450

However a number of approximation results require lsquofull tightnessrsquo of the451

converging sequence not just y-tightness452

Definition 4 (tight lopsided convergence f v-biv) A sequence of bivariate functions453 Fν Cν times Dν rarr R

νisinIN in f v-biv(Rn+m) lop-converges tightly to a function454

F C times D rarr R also in f v-biv(Rn+m) if they lop-converge and in addition the455


(andasht) for all ε gt 0 there is a compact set Aε such that for all ν large enough457

supxisinCνcapAεinf yisinDν Fν(x y) ge supxisinCν inf yisinDν Fν(x y) minus ε458

(bndasht) for x isin C and the corresponding sequence xν isin Cν rarr x identified in459

condition 3(b) for any ε gt 0 one can find a compact set Bε possibly depending on460

the sequence xν rarr x such that for all ν larger enough461


Theorem 5 (approximating maxinf-points) Suppose the sequence of bivariate func-463

tions

Fν Cν times Dν rarr RνisinIN in f v-biv(Rn+m) lop-converges tightly to a function464

F C times D rarr R also in f v-biv(Rn+m) Moreover suppose the inf-projections465

gν = inf yisinDν Fν(middot y) and g = inf yisinD F(middot y) are finite-valued on Cν and C respec-466

tively with sup g = supx inf y F(x y) finite Then467

supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


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f


and if x is a maxinf point of F one can always find sequencesεν 0 xν isin469

εν-argmaxx (inf y Fν)νisinIN such that xν rarrN x Conversely if such sequences exist then470

supx (inf y Fν)rarrN inf y F(x middot)471

Proof Tightness in particular condition (bndasht) implies that gν rarrh g see Theorem 3472

From (andasht) it then follows that they hypo-converge tightly The assertions then proceed473

directly from Theorem 2 474

Lets now turn to the situation when our bivariate functions are extended real-valued475

and defined on all of Rn times R

m keeping in mind that we remain in the maxinf setting476

To define convergence we cannot proceed as in Sect 2 where we tied the convergence477

of functions with that of their epigraphs Here there is no easily identifiable (unique)478

geometric object that can be associated with a bivariate function479

Recall that biv(Rn+m) is the family of all extended-real valued functions defined480

on Rn times R

m In our maxinf case as in [13] the effective domain dom F of a bivariate481

function F Rn+m rarr R is482

dom F = domx F times domy F483

where484

domx F = x

∣∣ F(x y) lt infin forall y isin R

m485

domy F = y∣∣ F(x y) gt minusinfin forall x isin R

n486

Thus F is finite-valued on dom F it does not exclude the possibility that F might be487

finite-valued at some points that do not belong to dom F 488

In the lsquomaxinfrsquo framework the term proper is reserved for bivariate functions with489

non-empty domain and such that490

F(x y) = infin when x isin domx F491

F(x y) = minusinfin when x isin domx F but y isin domy F492

see Fig 2 If F is proper we write F isin pr -biv(Rn+m) a sub-collection of biv(Rn+m)493

Definition 5 (lopsided convergence biv) A sequence of bivariate functions

Fν ν isin494

IN sub biv(Rn+m) lop-converges to a function F R

n times Rm rarr R if495

(a) forall (x y) isin Rn+m xν rarr x exist yν rarr y limsupν Fν(xν yν) le F(x y)496

(b) forallx isin domx F exist xν rarr x liminfν Fν(xν yν) ge F(x y) forall yν rarr y isin Rm 497

Observe that when the functions Fν and F do not depend on x they lop-converge if498

and only if they epi-converge and that if they do not depend on y they converge lop-499

sided if and only if they hypo-converge This later assertion follows from Proposition 3500

Moreover if for all (x y) the functions Fν(x middot)rarre F(x middot) and Fν(middot y)rarrh F(middot y)501

then the functions Fν lop-converge to F however one should keep in mind that this502

is a sufficient condition but by no means a necessary one503

123


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f


Remark 2 (rsquo83 versus new definition) The definition of lop-convergence in [2]504

required condition 5(b) to hold not just for all x isin domx F but for all x isin Rn The505

implication is that then lop-convergent families must be restricted to those converging506

to a function F with domx F = Rn 507

Detail Indeed consider the following simple example For all ν isin IN 508

Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩

0 if (x y) isin [01] times [0 1]minusinfin if y isin (0 1) x isin [0 1]infin elsewhere

509

Then in terms of Definition 5 the Fν lop-converge to F but not if we had insisted510

that condition 5(b) holds for all x isin Rn Indeed there is no way to find a sequence511

xν rarr minus1 for example such that for all yν rarr 0 liminfν Fν(xν yν) ge F(minus1 0) =512

infin simply consider yν = 1ν rarr 0 513

As in Sect 2 we set up a bijection also denoted η between the elements of514

f v-fcn(Rn+m) and the (max-inf) proper bivariate functions pr -biv(Rn+m) For F isin515

f v-biv(Rn+m) set516

ηF(x y) =

⎧⎪⎨

⎪⎩

F(x y) when (x y) isin C times D

infin when y isin D

minusinfin when y isin D but x isin C

517

ie ηF extends F to all of Rn timesR

m Then for F isin pr -biv ηminus1 F will be the restriction518

of F to its domain of finiteness namely domx F times domy F 519

Proposition 4 (lop-convergence in f v-biv and biv) A sequence520

Fν Cν times Dν rarr R ν isin IN

sub f v-biv(Rn+m)521

converges lopsided to F C times D rarr R if and only the corresponding sequence of522

extended real-valued bivariate functions523

ηFν R

n+m rarr R ν isin IN sub pr -biv(Rn+m)524

lop-converges (Definition 5) to ηF Rn times R

m rarr R where η is the bijection between525

f v-biv(Rn+m) and pr -biv(Rn+m) defined above526

Proof To show conditions (a) (ainfin) and (b) of Definition 3 for the sequence Fνinfinν=1527

imply and are implied by the conditions (a) and (b) of 5 for the sequence ηFνinfinν=1 in528

pr -biv(Rn+m) lets denote these later conditions (ηa) and (ηb)529

We begin with the implications involving (ηa) and (a) (ainfin) Suppose (ηa) holds530

(x y) isin C times D and xν isin Cν rarr x then there exists yν rarr y such that531

lim supν ηFν(xν yν) le ηF(x y) = F(x y) Necessarily for ν sufficiently large532

yν isin Dν since otherwise lim supν ηFν(xν yν) = infin contradicting the possibility of533

123


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f


having this upper limit less than or equal to F(x y) that is finite This takes care of (a)534

If xν isin Cν rarr x isin C y isin D this implies ηF(x y) = minusinfin and consequently there535

exists yν rarr y such that lim supν ηFν(xν yν) = minusinfin Again this sequence yνinfinν=1536

cannot have a subsequence with yν isin Dν since otherwise this upper limit would be537

infin This yields (ainfin)538

Now suppose (a) and (ainfin) hold As long as y isin D ηF(x y) = infin the inequality539

in (ηa) will always be satisfied henceforth we consider only the case when y isin540

D If xν isin Cν rarr x isin Rn then (a) or (ainfin) directly guarantee the existence of a541

sequence yνinfinν=1 such that lim supν ηFν(xν yν) le ηF(x y) Finally consider the542

case when xν rarr x but xν isin Cν for a subsequence N sub IN there is no loss of543

generality in actually assuming that N = IN Pick any sequence yν isin Dν rarr y544

hence lim supν ηFν(xν yν) = minusinfin will certainly be less than or equal to ηF(x y)545

So (ηa) holds also trivially in this situation546

When (ηb) holds and (x y) isin C times D hence ηF(x y) = F(x y) isin R We547

only have to consider sequences yν isin Dν rarr y isin D and for all such sequences548

exist xν rarr x such that lim infν ηFν(xν yν) ge F(x y) This sequence xν rarr x cannot549

have a subsequence whose elements do not belong to the corresponding sets Cν since550

otherwise the lower limit of ηFν(xν yν)infinν=1 would be minusinfin lt F(x y) isin R This551

means that for this sequence xν rarr x the xν isin Cν for ν sufficiently large Hence (b)552

holds when y isin D When x isin C y isin D ηF(x y) = infin For any yν isin Dν rarr y there553

is a sequence xν rarr x such that lim infν ηFν(xν yν) = infin Since yν isin Dν xν isin Cν554

would imply ηFν(xν yν) = minusinfin this sequence xν rarr x must be such that xν isin Cν555

for ν sufficiently large and consequently one must have Fν(xν yν) rarr infin which556

means that (b) is also satisfied when y isin D557

In the other direction that (b) yields (ηb) is straightforward If y isin D and x isin C then558

lim infν ηFν(yν xν) = infin = F(x y) for the sequence xν isin Cν rarr x predicated by559

(b) irrespective of the sequence yν rarr y Finally if (x y) isin C times D then (b) foresees560

a sequence xν isin Cν rarr x such that the inequality in (ηb) is satisfied as long as the561

sequence yν rarr y is such that all ν or at least for ν sufficiently large the yν isin Dν 562

But if they do not ηFν(xν yν) = infin and these terms will certainly contribute to563

making lim infν ηFν(xν yν) ge F(x y) 564

Lop-convergence y-tightly is also the key to the convergence of the maxinf-points565

of extended real-valued bivariate functions566

Definition 6 (lop-convergence y-tightly biv) A sequence of bivariate functions in567

biv(Rn+m) lop-converges y-tightly if it converges lopsided and for all x isin C the568

following augmented condition of 5(b) holds569

(bndasht) not only exist xν rarr x such that forall yν rarr y liminfν Fν(xν yν) ge F(x y)570

but also for any ε gt 0 one can find a compact set Bε possibly depending on the571

sequence xν rarr x such that for all ν larger than some νε572

inf Bε Fν(xν middot) le inf Fν(xν middot) + ε573

Proposition 5 (y-tight lop-convergence in f v-biv and biv) A sequence574


sub f v-biv(Rn times Rm)575

123


unco

rrec

ted

proo

f


lop-converges y-tightly to F C times D rarr R in f v-biv(Rn times Rm) if and only if the576

corresponding sequence577

ηFν R

n times Rm rarr R ν isin IN

sub pr -biv(Rn times Rm)578

lop-converges y-tightly to ηF Rn times R

m rarr R where η is the bijection from579

f v-biv(Rn times Rm) onto pr -biv(Rn times R

m) defined earlier580

Proof We already showed that lop-convergence in f v-biv and pr -biv are equivalent581

cf Proposition 4 so there only remains to verify the lsquotightlyrsquo condition But thats582

immediate because in both cases it only involves points that belong to C times Rm =583

domx ηF times Rm and sequences converging to such points 584

Theorem 6 (biv hypo-convergence of the inf-projections) Suppose the sequence585 Fν

νisinIN sub biv(Rn+m) lop-converges y-tightly to F and let gν = inf yisinDν Fν(middot y)586

g = inf yisinD F(middot y) Then assuming that g lt infin gν rarrh g in fcn(Rn)587

Proof The proof is the same as that of Theorem 3 with the obvious adjustments when588

the sequences do not belong to dom Fν and the limit point does not lie in dom F 589

Theorem 7 (biv convergence of maxinf-points) Let FννisinIN and F be a family of590

bivariate functions that satisfy the assumptions of Theorem 6 so in particular the Fν591

lop-converge y-tightly to F Then if for all ν xν is a maxinf-point of Fν and x is592

any cluster point of the sequence xν ν isin IN then x is a maxinf-point of the limit593

function F Moreover with xν ν isin N sub IN the (sub)sequence converging to x 594

limν rarrN infin[

infyisinRm

Fν(xν y)] = inf [ sup

yisinRmF(x y) ]595

ie there is also convergence of the lsquovaluesrsquo of the maxinf-points596



the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599

of the corresponding functions gν and g The assertions now follow immediately600

from the convergence of the argmax of hypo-converging sequences cf Theorem [14601

Theorem 731] translated to the lsquomaximizationrsquo framework 602

To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603

inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604

the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605

functions606

123


unco

rrec

ted

proo

f


4 Ky Fanrsquos Inequality extended607

The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608

is the class of concave usc functions [14 Theorem 717] A class of functions that609

is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610

We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611

to claim existence of maxinf-points in situations when the domain of definition of the612

Ky Fan function is not necessarily compact613

Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614

f v-biv(Rn+m) is called a Ky Fan function if615

(a) forall y isin D x rarr F(x y) is usc on C 616

(b) forall x isin C y rarr F(x y) is convex on D617

Note that the sets C or D are not required to be compact618

Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619

sequence

Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620

Ky Fan function621

Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622

forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623

of 3(a) we can choose sequences

y0ν isin Dν rarr y0


such that624

Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625

λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626

certainly converges to yλ For all ν one has627

Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628

Taking lininf on both sides yields629

F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630

that establishes the convexity of F(x middot)631

To prove the upper semicontinuity of F with respect to x-variable we show that632

for y isin D633

hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634

where the limit is with respect to all sequences yν isin DννisinIN converging to y and635

ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636

closed and a function is usc if and only if its hypograph is closed We have to show637

that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638

hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639

since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640

Given a Ky Fan function with compact domain and non-negative on the diagonal641

we have the following important existence result642

123


unco

rrec

ted

proo

f


Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643

a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644

subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645

C F(x middot) ge 0 on C646

One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647

Inequality to the case when it is not possible to apply it directly because one of the648

conditions is not satisfied for example the compactness of the domain However we are649

able to approach the bivariate function F by a sequence FννisinIN defined on compact650

sets Cν This procedure could be useful in many situation where the original maxinf-651

problem is unbounded and then the problem is approached by a family of truncated652

maxinf-problems Such is the case for example when we consider as variables in653

the original problem the multipliers associated to inequality constraints or when the654

original problem is a Walras equilibrium with a positive orthant as consumption set655

in [8] one is precisely confronted with such situations Another simple illustrative656

example follows the statement of the theorem657

Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658

bivariate function defined on CtimesC Suppose one can find sequences of compact convex659

setsCν sub R

n

and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660

converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661

maxinf-points of the Fν is a maxinf-point of F662

Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663

non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664

maxinf-points will be a maxinf-point of F 665

Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666

F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667

inf yisin[0infin) F(x y) = sin x668

and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669

the domain of F is not compact the function F(middot y) is not even sup-compact670

Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671

domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672

inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673

that converges pointwise and hypo- to sin x and674

argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ

∣∣ k isin IN

675

Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676

xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677

678

123


unco

rrec

ted

proo

f


References679

1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680

Pitman London (1984)681

2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682

Paris 296 657ndash660 (1983)683

3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684

1ndash41 (1983)685

4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686

5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687

6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688

(1999)689

7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690

Academic Dublin (1972)691

8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692

equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693

applications pp 553ndash578 Springer New York (2005)694

9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695

243 (2002)696

10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697

(Manuscript) (2006)698

11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699


12 Lignola M Morgan J Convergence of marginal functions with dependent con-701

straints Optimization 23 189ndash213 (1992)702

13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703

14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704

123


Fax to +44 870 622 1325 (UK) or +44 870 762 8807 (UK)From Springer Correction Team

6amp7 5th Street Radhakrishnan Salai Chennai Tamil Nadu India ndash 600004Re Mathematical Programming DOI101007s10107-007-0122-8

Variational convergence of bivariate functions lopsided convergenceAuthors Alejandro Jofreacute middot RogerJ-B Wets

I Permission to publishDear Springer Correction TeamI have checked the proofs of my article andq I have no corrections The article is ready to be published without changes

q I have a few corrections I am enclosing the following pagesq I have made many corrections Enclosed is the complete article

II Offprint orderq Offprint order enclosed q I do not wish to order offprintsRemarks

Date signature ______________________________________________________________________________

III Copyright Transfer Statement (sign only if not submitted previously)The copyright to this article is transferred to Springer-Verlag (respective to owner if other than Springer and for USgovernment employees to the extent transferable) effective if and when the article is accepted for publication Theauthor warrants that hisher contribution is original and that heshe has full power to make this grant The author signsfor and accepts responsibility for releasing this material on behalf of any and all co-authors The copyright transfercovers the exclusive right to reproduce and distribute the article including reprints translations photographicreproductions microform electronic form (offline online) or any other reproductions of similar nature

An author may self-archive an author-created version of hisher article on hisher own website and hisherinstitutionrsquos repository including hisher final version however heshe may not use the publisherrsquos PDF versionwhich is posted on httpwwwspringerlinkcom Furthermore the author may only post hisher version providedacknowledgement is given to the original source of publication and a link is inserted to the published article onSpringerrsquos website The link must be accompanied by the following text ldquoThe original publication is available athttpwwwspringerlinkcomrdquo

The author is requested to use the appropriate DOI for the article (go to the Linking Options in the article then toOpenURL and use the link with the DOI) Articles disseminated via httpwwwspringerlinkcom are indexedabstracted and referenced by many abstracting and information services bibliographic networks subscriptionagencies library networks and consortia

After submission of this agreement signed by the corresponding author changes of authorship or in the order ofthe authors listed will not be accepted by Springer

Date Authorrsquos signature ______________________________________________________________________











Bank code






q q








Address Davis USA








Schedule


Revised





65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45



Author Query Form



unco

rrec

ted

proo

f


FULL LENGTH PAPER




















unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123












Bank code






q q








Address Davis USA








Schedule


Revised





65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45



Author Query Form



unco

rrec

ted

proo

f


FULL LENGTH PAPER




















unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123









Address Davis USA








Schedule


Revised





65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45



Author Query Form



unco

rrec

ted

proo

f


FULL LENGTH PAPER




















unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123



Author Query Form



unco

rrec

ted

proo

f


FULL LENGTH PAPER




















unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f


FULL LENGTH PAPER




















unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f




















maxinf-points30









Rn timesR




















123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f








2 Epi-convergence63














fcn(Rn) = f R

n rarr R

77








n83








∣∣ α ge f (x)

sub Rn+190


123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f





92




















νisinIN110


Proposition 415]112





f ν rarre f 116







functions f ν 123


functions f ν 125




123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f


x

f

f
























of a sequence








123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f



νisinIN with157

f ν(x) =

⎧⎪⎨

⎪⎩




158










R ν isin IN



























f ν Dν rarr190



123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f


lim supνrarrinfin









lim supν





inf f ge lim supk




















imply221

lim infν

( infDνcapBε

f ν) le lim infν

(infDν


(infDν

f ν) + ε le infD

f + ε lt infin222

123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f










Hence231






(inf BcapDν f ν

) le236




























262

123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f




domain265

dom f = x isin R


266























yield (aη)289













123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f









307
































was used in [2]339





123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f


ampW

prin

t

οοο ο ο ο

ο ο

ο ο

ο ο ο ο

οοminus

C

D F(xy)minus



of the article345












biv(Rn+m) = F R

n times Rm rarr R

356


Rn times R

m and358



n empty = D sub Rm

359







123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f



point if365

x isin argmaxxisinC

[infyisinD

F(x y)]366


x isin argminxisinC

[supyisinD

F(x y)]368

C = Rn and D = R





Fν Cν times372



















389









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f




as follows400









407

and Cg = x isin C










and Cg = x


could be414






Hence420
















436

123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f








limν rarrN infin

[inf

yisinDνFν(xν y)

] = infyisinD

F(x y) ]443





















tions





supx

infy

Fν(x y) rarr supx

infy

F(x y)468

123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f















on Rn times R




where484

domx F = x


m485


n486









Fν ν isin494











123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f







Fν(x y) = F(x y) =

⎧⎪⎨

⎪⎩


509








ηF(x y) =

⎧⎪⎨

⎪⎩


infin when y isin D


517









ηFν R












123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f













































123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f




ηFν R





















limν rarrN infin[

infyisinRm


yisinRmF(x y) ]595











functions606

123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f















sequence


Ky Fan function621






such that624









for y isin D633










123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f



















setsCν sub R

n

















∣∣ k isin IN

675



678

123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


unco

rrec

ted

proo

f


References679




Paris 296 657ndash660 (1983)683


1ndash41 (1983)685




(1999)689







243 (2002)696









123


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PUC Rio | Departamento de Economia