DSpace at VNU: Approximations of Optimization-Related Problems in Terms of Variational Convergence

DSpace at VNU: Approximations of Optimization-Related Problems in Terms of Variational Convergence tài liệu, giáo án, bà...

DOI 10.1007/s10013-015-0148-9

Approximations of Optimization-Related Problems

in Terms of Variational Convergence

Huynh Thi Hong Diem 1 · Phan Quoc Khanh 2

Received: 14 May 2014 / Accepted: 2 December 2014

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Abstract In this paper, we first adjust the known definition of epi/hypo convergence of

bivariate extended-real-valued functions for finite-valued ones defined on product sets Then, we develop basic variational properties of this type of convergence Using these results, we consider approximations of optimization-related problems in terms of epi/hypo convergence and related types of convergence, for some selected important cases: equilib-rium problems, multiobjective optimization, and Nash equilibria Our approximation results can be viewed also as contributions to qualitative stability (as the terminology “stability” was also used instead of “approximation” for similar results in the literature)

Keywords Epi-convergence· Epi/hypo convergence · Saddle points · Variational

properties· Approximations · Equilibrium problems · Multiobjective optimization · Nash equilibria· Dual problems

Mathematics Subject Classification (2010) 90C31· 91A10

Professor Phan Quoc Khanh was Plenary speaker at the Vietnam Congress of Mathematicians 2013.

 Phan Quoc Khanh

pqkhanh@hcmiu.edu.vn

Huynh Thi Hong Diem

hthdiem@gmail.com

1 Department of Mathematics, College of Can Tho, Can Tho, Vietnam

2 Department of Mathematics, International University, Vietnam National University Ho Chi Minh City, Linh Trung, Thu Duc, Ho Chi Minh City, Vietnam

1 Introduction

Stability of a problem, or more precisely, of solutions of a problem, can be generally under-stood as both qualitative stability and quantitative stability However, the splitting is rather relative (and some authors use the general word “stability” for both) We can have several related observations as follows In (qualitative) stability studies, various types of semicon-tinuity and consemicon-tinuity with respect to parameters are considered Such studies may include also calmness, Lipschitz, or Holder continuity of solution sets if no (or few) results on esti-mates/computations of constants/modulus and (Holder) degrees are discussed Quantitative stability is connected to quantitative estimates/computations Then, a more specific termi-nology “sensitivity” often replaces “stability” So, usually, in sensitivity analysis, various (generalized) derivatives or derivative-like objects (with respect to perturbation parameters)

of marginal functions, i.e., optimal-valued functions, or optimal solutions are computed or estimated Studies of calmness, Lipschitz, or Holder continuity of solution maps which con-tain estimates/computations of constants/modulus and (Holder) degrees are often classified

as quantitative stability

Recently, many stability investigations in optimization-related problems included also variational convergence of solutions of approximating problems to that of the true prob-lem under question Variational convergence is important in optimization-related probprob-lems

In this paper, by variational convergence we mean generally types of convergence which preserve, to some extent, the so-called variational properties such as being optimal values, optimal solutions, minimax values, saddle points, etc (we do not use an exact definition

of “variational convergence”, but mention it only as an idea, while each statement or argu-ment is for a particular exactly-defined type of convergence) This “vague” meaning is a kind of implicit acceptance in a major part the related literature, e.g., in references below for studies of stability in terms of variational convergence A more precise definition of

“variational convergence” was given in [8, pages 121–124] from the view point of well-posedness studies In [20, Definition 4], the term “variational convergence” was even used just for epi-convergence (while in this paper and many others, epi-convergence is consid-ered a basic type of variational convergence) For studies of stability in terms of variational convergence, we can observe that epi-convergence is used in [22,26] for scalar minimiza-tion, graphical convergence is applied in [11] for complementarity problems and in [18] for variational inequalities, and lop-convergence is the tool in [15,16,19] for various models Epi/hypo convergence is studied and applied in [3,15,23–25,30] (for the definition of these types of convergence, see Sections2and3) In [12,19] and some others, the term “approx-imation” was used instead of “stability” Though, in this context, these two terms are close

in meaning, in this paper, we use “approximation”, since “approximating problems” do not coincide in meaning with “perturbed problems” In [9], the word “estimators” was used for solutions of approximating problems when these solutions epi/hypo converge to the true problem

The layout of our paper is as follows The rest of this section is devoted to giving defi-nitions and notations for our later use In Section2, we present various types of variational convergence of univariate functions (unifunctions, for short), some of their relations, and basic variational properties of epi-convergence, which is regarded as the basic variational convergence of unifunctions Section3contains the definition of epi/hypo convergence and lopsided convergence They are the basic types of variational convergence of bivariate func-tions (bifuncfunc-tions, for short) Here and later on, we focus on epi/hypo convergence, since

it is symmetric and appropriate for investigations of a saddle point, the basic object one is often interested in, when dealing with bivariate functions involved in optimization-related

problems We show that any cluster point of a sequence of (approximate) saddle points of bivariate functions which epi/hypo converge is an (approximate) saddle point of any limit bivariate function (epi/hypo limits are not unique, see, e.g., [7]), without any additional assumption Then, we define notions of ancillary tightness and (full) tightness to ensure the Painlev´e–Kuratowski convergence of the whole set of saddle points of epi/hypo convergent bivariate functions Applying results of Sections2and3, in Section4, we establish conver-gence results for a sequence of approximating equilibrium problems and their dual problems

in terms of variational convergence From these statements for equilibrium problems, we derive in the subsequent two Sections5and6the corresponding convergence properties for multiobjective optimization and multi-player noncooperative games

Our notations are consistent with those of [4,15,22] We useN, Rn,Rn

+, and ¯R for the

set of the natural numbers, the n-dimensional space, its positive orthant, and the

extended-real lineR ∪ {±∞}, respectively (shortly, resp.) For a subset A ⊂ R n , intA and bdA stand for the interior and boundary, resp., of A ε  0 (ε  0, resp.) means ε > 0 (ε < 0, resp.) and tends to 0 For a function ψ: Rn→ ¯R, its domain, epigraph, hypograph, and graph are

defined by dom ψ := {x ∈ R n | ψ(x) < ∞}, epi ψ := {(x, r) ∈ R n × R | ψ(x) ≤ r}, hypo ψ := {(x, r) ∈ R n × R | ψ(x) ≥ r}, gph ψ := {(x, r) ∈ R n × R | ψ(x) = r}, resp lim inf ψ and lim sup ψ designate the lower and upper limits of ψ as x tends to ¯x, defined,

resp., by

lim inf

x → ¯x ψ(x):= lim

δ0

 inf

x ∈B( ¯x,δ) ψ(x)



= sup

δ>0

 inf

x ∈B( ¯x,δ) ψ(x)



,

lim sup

x → ¯x ψ(x):= lim

δ0

 sup

x ∈B( ¯x,δ) ψ(x)



= inf

δ>0

 sup

x ∈B( ¯x,δ) ψ(x)



.

We adopt the notation

argmin

A

ψ(x):=



{x ∈ R n | ψ(x) = inf A ψ(x)} if infA ψ(x) < ∞,

and similarly for argmax A function ψ : Rn → ¯R is said to be lower (upper, resp.) semicontinuous, shortly lsc (usc, resp.), at ¯x if lim inf x → ¯x ψ(x) ≥ ψ( ¯x) (lim sup x → ¯x ψ(x)

≤ ψ( ¯x)) It is known that ψ is lsc (usc) at ¯x if and only if the epigraph (hypograpgh, resp.)

of ψ is closed at ( ¯x, ψ( ¯x)) Hence, a lower semicontinuous function is called also a (lower)

closed function (similarly for a usc function) For a sequence of subsets{A ν}ν∈NinRn, the lower/inner limit and upper/outer limit are defined by

Liminf

ν A ν := x∈ Rn | ∃x ν → x with x ν ∈ A ν

,

Limsup

ν

A ν := x∈ Rn | ∃ν l (a subsequence), ∃x ν l → x with x ν l ∈ A ν l

.

If Liminfν A ν= Limsupν A ν , one says that A ν tends to A or A= Limν A ν(in the Painlev´e– Kuratowski sense)

2 Variational Convergence of Univariate Functions

In this paper, we are concerned with numerical functions defined on nonempty subsets ofRn

and having the finite values We call them finite-valued univariate functions, or unifunctions

and denote the class of all such functions by fv-fcn(Rn ) First, recall the definition of several types of convergence

Definition 1 Let C ν , C⊂ Rnbe nonempty and{f ν : C ν→ R}ν∈N, f : C → R.

(i) ([22]) {f ν } is called converging graphically to f , denoted by f ν → f , if gphf g ν

converges to gphf in the Painlev´e–Kuratowski sense.

(ii) ([22]){f ν } is said to converge continuously to f relative to sequence C ν → C if, for any sequence x ν ∈ C ν → x ∈ C, f ν (x ν ) → f (x).

(iii) ([15]){f ν } is called epi-converging to f , denoted by f ν → f or f = e-lim e ν f ν, if (a) for all x ν ∈ C ν → x, lim inf ν f ν (x ν ) ≥ f (x) when x ∈ C and f ν (x ν ) → ∞

when x / ∈ C;

(b) for all x ∈ C, there exists x ν ∈ C ν → x such that lim sup ν f ν (x ν ) ≤ f (x).

Note that Definition 1 (i) and (ii) are classic The part (iii) is adjusted from the epi-convergence defined in [27–29] for the class fcn(Rn )of the unifunctions fromRn to ¯R

Observe that irrespective of concerning fv-fcn(Rn ) or fcn(Rn ) , f ν → f if and only if e epif ν → epif in the sense of Painlev´e–Kuratowski {f ν } is called hypo-converging to f , denoted by f ν → f or f = h-lim h ν f νif{−f ν } epi-converges to −f

Proposition 1 Let C ν , C⊂ Rn be nonempty and {f ν : C ν → R}ν∈N, f : C → R.

(i) ([22, Proposition 5.33])f ν →f if and only if the following two conditions hold, for g all x ∈ C,

(α) for all x ν ∈ C ν → x, there exists a subsequence x ν j such that lim j f ν j (x ν j )=

f (x);

(β) there exists x ν ∈ C ν → x such that lim ν f ν (x ν ) = f (x).

(ii) {f ν } graphically converges to f if and only if it both epi- and hypo-converges to f and C ν → C.

(iii) If {f ν } converges continuously to f relative to sequence C ν → C, then it both epi-and hypo-converges to f

Proof (ii) Using the characterization by conditions (α) and (β) in (i) for graphical

convergence, the proof is immediate

(iii) This is clear

In the rest of this section, we recall basic variational properties of epi-convergence, see [15]

Theorem 1 (Epi-convergence: basic property) Let f ν , f ∈ fv-fcn(R n ) and f = e-limf ν Then

lim sup

ν

inf

C ν f ν (x) ≤ inf

C f (x).

Moreover, if x k ∈ argminC νk f ν k for some subsequence {ν k}k∈Nand x k → ¯x, then ¯x ∈

argminC f andminC νk f ν k → minC f

Note from Proposition 1 that graphical and continuous convergence also enjoy the prop-erties stated in this theorem The second part of Theorem 1 can be expressed equivalently

as: if f ν → f then e

Limsup

ν

argmin

ν

f ν ⊂ argminf.

It is easy to prove the extension that if ε ν 0 then

Limsup

ν

ε ν-argmin

C ν

f ν ⊂ argmin

C

f.

To guarantee the equality in this relation with the full Lim instead of Limsup and also the convergence of the infimal values, we need the following tightness notion

Definition 2 (Tight epi-convergence) We say that a sequence{f ν}ν∈Nepi-converges tightly

to f in fv-fcn(Rn ) if it epi-converges and, for all positive ε, there exists a compact set B ε

and an index ν ε such that, for all ν ≥ ν ε,

inf

B ε ∩C ν f ν≤ inf

C ν f ν + ε.

Theorem 2 (Convergence of infima) Let f ν , f ∈ fv-fcn(R n ), f = e-limν f ν and inf C f

be finite Then, the epi-convergence is tight

(i) if and only if inf C ν f ν→ infC f ;

(ii) if and only if there exists a sequence ε ν  0 such that ε ν -argminf ν → argminf

3 Epi/hypo Convergence of Bivariate Functions and Variational

Properties

Epi/hypo convergence of a sequence of bivariate functions K ν : Rn× Rm → ¯R was proposed in [1] and rigorously treated in [4] In [2], a modified and stronger form than epi/hypo convergence, called lopsided convergence (in short, lop-convergence), was intro-duced Epi/hypo and lopsided convergence have been recognized as the main types of variational convergence for extended-real-valued bifunctions The class of all such

bifunc-tions is denoted by biv(Rn× Rm ) In [15], lop-convergence was adjusted and considered

for finite-valued bifunctions defined on subsets of the form C × D ⊂ R n× Rm The motivation is that important bifunctions met in applications like Lagrangians in constraint optimization, payoff functions in zero-sum games, or Hamiltonians in variational calculus and optimal control are finite-valued bifunctions defined on product sets The collection of

the finite-valued bifunctions, denoted by fv-biv(Rn× Rm ), is just the object of our study in this paper Following the way of dealing with finite-valued bifunctions in [15], we propose the following natural definition of epi/hypo convergence

Definition 3 (Epi/hypo convergence) Bifunctions K ν , ν ∈ N, in fv-biv(R n×Rm )are called

epi/hypo converging (shortly e/h-converge) to a bifunction K ∈ fv-biv(R n× Rm )if (a) for all y ∈ D and all x ν ∈ C ν → x, there exists y ν ∈ D ν → y such that

lim infν K ν (x ν , y ν ) ≥ K(x, y) if x ∈ C or K ν (x ν , y ν ) → ∞ if x /∈ C;

(b) for all x ∈ C and all y ν ∈ D ν → y, there exists x ν ∈ C ν → x such that

lim supν K ν (x ν , y ν ) ≤ K(x, y) if y ∈ D or K ν (x ν , y ν ) → −∞ if y /∈ D.

We denote this convergence by K ν e/ h → K or K = e/h-lim ν K ν Note that if the functions

K ν do not depend on y, then epi/hypo convergence reduces to epi-convergence, and if they

do not depend on x, it collapses to hypo-convergence However, note that epi/hypo con-vergence is not epi-concon-vergence of the K ν ( ·, y) to K(·, y) for all y and hypo-convergence

of the K ν (x, ·) to K(x, ·) for all x This is a sufficient condition for e/h-convergence, but

not necessary Indeed, K ν in Example 1 below e/h-converges, but it does not hold that

K ν ( ·, y) → K(·, y) for all y ∈ D e

It should be noticed also that the definition of epi/hypo convergence is symmetric To see that this symmetry is an important feature of epi/hypo convergence, let us recall the following

Definition 4 (Minsup-lop convergence, [15]) Bifunctions K ν ∈ fv-biv(R n× Rm )are said

to minsup-lopsided converge (shortly, minsup-lop converge) to K ∈ fv-biv(R n× Rm )if (a) for all y ∈ D and all x ν ∈ C ν → x, there exists y ν ∈ D ν → y such that

lim infν K ν (x ν , y ν ) ≥ K(x, y) if x ∈ C or K ν (x ν , y ν ) → ∞ if x /∈ C;

(b) for all x ∈ C, there exists x ν ∈ C ν → x such that, for all y ν ∈ D ν → y,

lim supν K ν (x ν , y ν ) ≤ K(x, y) if y ∈ D or K ν (x ν , y ν ) → −∞ if y /∈ D.

Observe that Definition 4 is nonsymmetric: the following maxinf-lop convergence is different from minsup-lop convergence:

(a) for all x ∈ C and all y ν ∈ D ν → y, there exists x ν ∈ C ν → x such that

lim supν K ν (x ν , y ν ) ≤ K(x, y) if y ∈ D or K ν (x ν , y ν ) → −∞ if y /∈ D;

(b) for all y ∈ D, there exists y ν ∈ D ν → y such that, for all x ν ∈ C ν → x,

lim infν K ν (x ν , y ν ) ≥ K(x, y) if x ∈ C or K ν (x ν , y ν ) → ∞ if x /∈ C.

This nonsymmetric notion was motivated by many applications in practical models discussed in [16] Namely, in many cases, one is interested in either minsup-points or maxinf-points, not in both, and hence needs only one-sided notion of convergence How-ever, a variational convergence for bifunctions should largely be aimed at the convergence of saddle points, which correspond to both minsup-points and maxinf-points Namely, one of the components of a saddle point is a minsup-point and the other is a maxinf-point Because

of its symmetry, epi/hypo convergence is the appropriate type of convergence for saddle points as we will see later in this work

Lopsided convergence clearly implies e/h-convergence Indeed, condition (a) of the def-initions are the same, whereas condition (b) of lop-convergence is clearly stronger than

(b) of epi/hypo convergence To see this, simply observe that, if for all x ∈ C one

can find a common sequence{x ν ∈ C ν}ν∈Nsuch that lim supν K ν (x ν , y ν ) ≤ K(x, y) or

K ν (x ν , y ν ) → −∞ depending on y belonging or not to D as lop-convergence requires,

then certainly (b) for epi/hypo convergence is satisfied, since one can even choose such a

sequence x ν → x to depend on y ν → y However, the converse does not hold as shown by

the following

Example 1 Let C ν = D ν = [1/ν, 1], C = D = [0, 1], and

K ν (x, y)=



1 if (x, y) ∈ C ν × D ν and x = y,

0 if (x, y) ∈ C ν × D ν and x = y.

Then

K(x, y)=



1 if (x, y) ∈ [0, 1]2 and x = y,

0 if (x, y) ∈ [0, 1]2 and x = y.

Clearly, K ν e/ h → K We show that condition (b) of Definition 4 of minsup-lop convergence

is violated For x = 0 and any x ν ∈ C ν → x, we take y = 0 and y ν ∈ D ν → 0 such that

y ν = x ν for all ν Then lim sup K ν (x ν , y ν ) = 1 > 0 = K(x, y).

Remark 1

(i) It is clear that continuous convergence of bifunctions K ν ( ·, ·) : C ν ×D ν→ R relative

to the sequence C ν × D ν → C × D implies all kinds of e/h-, minsup-lop and

maxinf-lop convergence (we know already in Section2that continuous convergence implies

also both epi- and hypo-convergence of K ν ( ·, ·)) So, continuous convergence is a

variational convergence too But, this convergence is very strong and hence difficult

to be satisfied

(ii) Limits of an e/h-convergent sequence are not unique The limits form a class of bifunctions, called an e/h-equivalence class, see, e.g., [7] However, as we will see below, fortunately almost all variational properties are the same for all limit bifunctions in an equivalence class

(iii) In [7], characterizations of e/h-convergence and lop-convergence of finite-valued bifunctions were established In particular, [7, Theorem 3] asserted the equivalence of the e/h-convergence of a sequence of finite-valued bifunc-tions and the e/h-convergence of the corresponding proper extended-real-valued bifunctions

Naturally expected variational properties of e/h-convergence are those related to saddle

points, since this convergence is symmetric Recall that a point ( ¯x, ¯y) ∈ C × D is said to

be a saddle point of K ∈ fv-biv(R n× Rm ) , denoted by ( ¯x, ¯y) ∈ sdlK, if, for all x ∈ C and

y ∈ D,

K( ¯x, y) ≤ K( ¯x, ¯y) ≤ K(x, ¯y),

or equivalently, K( ¯x, y) ≤ K(x, ¯y) for all (x, y) ∈ C × D.

In applications, approximate saddle points often exist even when saddle points do not Hence, we will prove the following convergence of approximate saddle points The

conver-gence of saddle points will follow immediately Recall that, for a non-negative ε, a point ( ¯x, ¯y) ∈ C × D is said to be an ε-saddle point of K ∈ fv − biv(R n× Rm ), denoted by

( ¯x, ¯y) ∈ ε-sdl K, if, for all x ∈ C and y ∈ D,

K( ¯x, y) − ε ≤ K( ¯x, ¯y) ≤ K(x, ¯y) + ε,

or equivalently, K( ¯x, y) − ε ≤ K(x, ¯y) + ε for all (x, y) ∈ C × D.

Let us define the sup-projection and inf-projection of a bifunction K ∈ fv-biv(R n× Rm )

by, resp.,

h( ·) := sup

y ∈D K( ·, y), g(·) := inf

x ∈C K(x, ·).

We have the following simple relation between approximate solutions

Proposition 2 Let K ∈ fv-biv(R n × Rm ) and h and g be its sup-projection and inf-projection, respectively.

(i) If ( ¯x, ¯y) ∈ ε-sdl K, then ¯x ∈ 2ε-argmin(h) and ¯y ∈ 2ε-argmax(g).

(ii) If ¯x ∈ ε-argmin(h) and ¯y ∈ ε-argmax(g), then

g( ¯y) ≤ K( ¯x, ¯y) ≤ h( ¯x),

sup

y ∈D xinf∈C K(x, y) − ε ≤ K( ¯x, ¯y) ≤ inf

x ∈C ysup∈D K(x, y) + ε.

Therefore, if K has a saddle point ( ˜x, ˜y), then

K( ˜x, y) − ε ≤ K( ¯x, ¯y) ≤ K(x, ˜y) + ε.

Proof (i) We have

h( ¯x) = sup

y ∈D K( ¯x, y) = K( ¯x, ¯y) + ε

= inf

x ∈C K(x, ¯y) + 2ε ≤ inf

x ∈C ysup∈D K(x, y) + 2ε = inf

x ∈C h(x) + 2ε.

The corresponding property of g is checked similarly,

(ii) It is clear that

K( ¯x, ¯y) ≥ g( ¯y) ≥ sup

y ∈D xinf∈C K(x, y) − ε.

The two right inequalities are proved similarly

Finally, if ( ˜x, ˜y) ∈ sdlK, then

K( ˜x, y) − ε ≤ K( ˜x, ˜y) − ε ≤ K( ¯x, ¯y) ≤ K( ˜x, ˜y) + ε ≤ K(x, ˜y) + ε.

In the remaining part of this section, we investigate variational properties of an arbitrary e/h-limit under some additional conditions We will see that all e/h-limits in an equiva-lence class share many common properties This fact should be highlighted, since in many applications, it helps to avoid dealing with whole equivalence classes

Theorem 3 (Convergence of approximate saddle points) Let a sequence {K ν}ν∈N e/h-converge to K in fv-biv(Rn× Rm ), ε ν  ε ≥ 0 and, for all ν ∈ N, ( ¯x ν , ¯y ν ) ∈ ε ν -sdlK ν Let ( ¯x, ¯y) be a cluster point of this sequence of approximate saddle points, say ( ¯x, ¯y) =

limν ∈N ( ¯x ν , ¯y ν ) for some subsequence N ⊂ N Then ( ¯x, ¯y) is an ε-saddle point of K and

K( ¯x, ¯y) = lim

ν ∈N K

ν

¯x ν , ¯y ν

.

Proof We can assume that actually ( ¯x ν , ¯y ν ) → ( ¯x, ¯y) Pick any (x, y) ∈ C × D Any sequences x ν ∈ C ν → x and y ν ∈ D ν → y satisfy

K ν

¯x ν , y ν

− ε ν ≤ K ν

¯x ν , ¯y ν

≤ K ν

x ν , ¯y ν

+ ε ν

These inequalities imply that

sup

{y ν ∈D ν →y}lim infν

K ν

¯x ν , y ν

− ε ν

≤ lim infν K ν

¯x ν , ¯y ν

≤ lim sup

ν

K ν

¯x ν , ¯y ν

{x ν ∈C ν →x}lim supν

K ν

x ν , ¯y ν

+ ε ν

,

By the definition of e/h-convergence, one has

K( ¯x, y) − ε ≤ sup

{y ν ∈D ν →y}lim infν

K ν

¯x ν , y ν

− ε ν

{x ν ∈C ν →x}lim supν

K ν

x ν , ¯y ν

+ ε ν

≤ K(x, ¯y) + ε.

These inequalities mean that ( ¯x, ¯y) is an ε-saddle point of K.

To see that K( ¯x, ¯y) = lim ν ∈N K ν ( ¯x ν , ¯y ν ), simply observe that the e/h-convergence and

¯x ν → ¯x ensure the existence of a sequence y ν ∈ D ν → ¯y satisfying

K( ¯x, ¯y) ≤ lim inf K ν ( ¯x ν , y ν )≤ lim inf(K ν ( ¯x ν , ¯y ν ) + ε ν )= lim infK ν ( ¯x ν , ¯y ν ),

where the second inequality follows from the approximate saddle point inequalities With

the role played by the x-variable and the y-variable reversed, a similar argument gives K( ¯x, ¯y) ≥ lim sup ν→∞K ν ( ¯x ν , ¯y ν )

Clearly, by taking ε ν≡ 0 in the preceding statement, we obtain the following basic result

on convergence of saddle points

Theorem 4 (Convergence of saddle points) Let a sequence {K ν}ν∈Ne/h-converge to K in fv-biv(Rn× Rm ) and ( ¯x ν , ¯y ν ) be a saddle point of K ν for all ν ∈ N Let ( ¯x, ¯y) be a cluster point of this sequence of saddle points, say ( ¯x, ¯y) = lim ν ∈N ( ¯x ν , ¯y ν ) for some subsequence

N ⊂ N Then ( ¯x, ¯y) is a saddle point of K and

K( ¯x, ¯y) = lim

ν ∈N K

ν ( ¯x ν , ¯y ν ).

Observe that, in the above two theorems, neither convex-concave conditions nor con-tinuity, nor compactness, nor even closedness are imposed We assume only epi/hypo convergence So, this convergence is a very suitable notion for considering saddle (or approximate saddle) points

The following example illustrates Theorem 4 and some more insights about convergence properties

Example 2 Consider the sequence of bifunctions

K ν (x, y) = ln x ln y,

defined on

C ν × D ν= 1/2, 1 − ν−1] ∪ [1 + ν−1, 3/2

× 1/2, 1 − ν−1] ∪ [1 + ν−1, 3/2

.

We can check directly that on[1/2, 3/2]2, K νconverges to

K(x, y) = ln x ln y

in the sense of all kinds of epi/hypo-, minsup-lop- and maxinf-lop-convergence

Clearly, these K νdo not have saddle points But, they do have approximate saddle points Namely, with

ε ν = maxln 1− ν−1ln 1/2− ln 1− ν−1ln 3/2;

ln 1+ ν−1

ln 3/2− ln 1+ ν−1

ln 1/2

,

the set consisting of the vertical intervals{1 − ν−1} × ([1/2, 1 − ν−1] ∪ [1 + ν−1, 3/2 ])

and{1 + ν−1} × ([1/2, 1 − ν−1] ∪ [1 + ν−1, 3/2 ]) and the horizontal intervals ([1/2, 1 −

ν−1] ∪ [1 + ν−1, 3/2 ]) × {1 + ν−1} and ([1/2, 1 − ν−1] ∪ [1 + ν−1, 3/2 ]) × {1 − ν−1} is

ε ν -sdlK νand converges to

sdlK = ({1} × [1/2, 3/2]) ∪ ([1/2, 3/2] × {1}).

Though without saddle points, the K νhave

arg

max

ν K ν = 1/2, 1 − ν−1

∪1+ ν−1, 3/2

× {1}

tending to the whole

arg

max

D min

C K = [1/2, 3/2] × {1}.

Here, arg(max DminC )K denotes the set of the points ( ¯x, ¯y) such that

K( ¯x, ¯y) = max

D min

C K(x, y) and similarly for arg(min CmaxD )K However,

arg

min

C ν max

D ν K ν =1− ν−1

× {1/2}

converges to{(1, 1/2)}, which is only a point of

arg

min

C max

D K = {1} × [1/2, 3/2].

Theorem 3 can be restated as follows, for the case ε = 0 If K ν e/h-converges to K and

ε ν 0, then

Limsup

ν

(ε ν -sdlK ν ) ⊂ sdlK.

To have equality with the full Lim instead of Limsup in the above relation, i.e., to have also

Liminf

ν (ε ν -sdlK ν ) ⊃ sdlK,

we propose new notions of tightness in Definition 5 below Note that these tightness

def-initions reflect the symmetric roles of x and y in the symmetric e/h-convergence (cf.

discussions after Definition 3) They are different from the known notions of tightness in [15] which are nonsymmetric

Definition 5

(i) (x-ancillary tightness) K ν is called e/h-convergent x-ancillary tightly to K in fv-biv(Rn× Rm )if (a) of Definition 3 and the following condition are satisfied: (b’-t) for all x ∈ C, there is x ν ∈ C ν → x such that K ν (x ν , ·) → K(x, ·) and h

h ν (x ν ) → h(x).

(ii) (y-ancillary tightness) K ν is said to e/h-converge y-ancillary tightly to K in fv-biv(Rn× Rm )if (b) of Definition 3 is fulfilled together with

(a’-t) for all y ∈ D, there is y ν ∈ D ν → y such that K ν ( ·, y ν ) → K(·, y) and e

g ν (y ν ) → g(y).

(iii) (Tightness) If both (b’-t) and (a’-t) are satisfied, K ν is called e/h-convergent (fully)

tightly to K.

Theorem 5 (Convergence of approximate saddle points to any given saddle point) Suppose

that K ν e/h-converges (fully) tightly to K in fv-biv(Rn×Rm ) Then the following statements hold.

(i) sdlK⊂ ∩ε>0Liminfν (ε -sdlK ν ).

(ii) Therefore, for each ε ν  0, ( ¯x, ¯y) ∈ sdlK and large ν, there exists ( ¯x ν , ¯y ν ) ∈

ε ν -sdlK ν such that ( ¯x ν , ¯y ν ) → ( ¯x, ¯y), i.e., Liminf ν (ε ν -sdlK ν ) ⊃ sdlK.

on convergence of saddle points

Theorem (Convergence of saddle points) Let... considering saddle (or approximate saddle) points

The following example illustrates Theorem and some more insights about convergence properties

Example Consider the sequence of bifunctions... that on[1/2, 3/2]2, K νconverges to

K(x, y) = ln x ln y

in the sense of all kinds of epi/hypo-, minsup-lop- and maxinf-lop-convergence