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...

Vietnam J Math DOI 10.1007/s10013-015-0148-9 Approximations of Optimization-Related Problems in Terms of Variational Convergence Huynh Thi Hong Diem1 · Phan Quoc Khanh2 Received: 14 May 2014 / Accepted: 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: equilibrium 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 Department of Mathematics, College of Can Tho, Can Tho, Vietnam Department of Mathematics, International University, Vietnam National University Ho Chi Minh City, Linh Trung, Thu Duc, Ho Chi Minh City, Vietnam H T H Diem, P Q Khanh Introduction Stability of a problem, or more precisely, of solutions of a problem, can be generally understood 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 semicontinuity and continuity 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 estimates/computations of constants/modulus and (Holder) degrees are discussed Quantitative stability is connected to quantitative estimates/computations Then, a more specific terminology “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 contain 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 problem under question Variational convergence is important in optimization-related problems 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 not use an exact definition of “variational convergence”, but mention it only as an idea, while each statement or argument 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 wellposedness 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 considered 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 minimization, 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 Sections and 3) In [12, 19] and some others, the term “approximation” 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” 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 definitions and notations for our later use In Section 2, 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 Section contains the definition of epi/hypo convergence and lopsided convergence They are the basic types of variational convergence of bivariate functions (bifunctions, 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 Approximations of Optimization-Related Problems 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 Sections and 3, in Section 4, we establish convergence 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 Sections and the corresponding convergence properties for multiobjective optimization and multi-player noncooperative games ¯ for the Our notations are consistent with those of [4, 15, 22] We use N, Rn , Rn+ , and R set of the natural numbers, the n-dimensional space, its positive orthant, and the extendedreal line R ∪ {±∞}, respectively (shortly, resp.) For a subset A ⊂ Rn , intA and bdA stand for the interior and boundary, resp., of A ε (ε 0, resp.) means ε > (ε < 0, resp.) ¯ its domain, epigraph, hypograph, and graph are and tends to For a function ψ : Rn → R, defined by dom ψ := {x ∈ Rn | ψ(x) < ∞}, epi ψ := {(x, r) ∈ Rn × R | ψ(x) ≤ r}, hypo ψ := {(x, r) ∈ Rn × R | ψ(x) ≥ r}, gph ψ := {(x, r) ∈ Rn × 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 ψ(x) := lim x→x¯ δ lim sup ψ(x) := lim x→x¯ δ inf x∈B(x,δ) ¯ ψ(x) = sup sup ψ(x) = inf x∈B(x,δ) ¯ inf ¯ δ>0 x∈B(x,δ) δ>0 ψ(x) , sup ψ(x) x∈B(x,δ) ¯ We adopt the notation argminψ(x) := A {x ∈ Rn | ψ(x) = infA ψ(x)} if infA ψ(x) < ∞, ∅ if infA ψ(x) = ∞, ¯ is said to be lower (upper, resp.) and similarly for argmax A function ψ : Rn → R semicontinuous, shortly lsc (usc, resp.), at x¯ if lim infx→x¯ ψ(x) ≥ ψ(x) ¯ (lim supx→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 {Aν }ν∈N in Rn , the lower/inner limit and upper/outer limit are defined by LiminfAν := x ∈ Rn | ∃x ν → x with x ν ∈ Aν , ν LimsupAν := x ∈ Rn | ∃νl (a subsequence), ∃x νl → x with x νl ∈ Aνl ν If Liminfν Aν = Limsupν Aν , one says that Aν tends to A or A = Limν Aν (in the Painlev´e– Kuratowski sense) Variational Convergence of Univariate Functions In this paper, we are concerned with numerical functions defined on nonempty subsets of Rn 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 H T H Diem, P Q Khanh Definition Let C ν , C ⊂ Rn be nonempty and {f ν : C ν → R}ν∈N , f : C → R (i) (ii) (iii) g ([22]) {f ν } is called converging graphically to f , denoted by f ν → f , if gphf ν converges to gphf in the Painlev´e–Kuratowski sense ([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) e ([15]) {f ν } is called epi-converging to f , denoted by f ν → f or f = e-limν f ν , if for all x ν ∈ C ν → x, lim infν 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 supν f ν (x ν ) ≤ f (x) (a) Note that Definition (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 from Rn to R e n n ν Observe that irrespective of concerning fv-fcn(R ) or fcn(R ), f → f if and only if epif ν → epif in the sense of Painlev´e–Kuratowski {f ν } is called hypo-converging to f , h denoted by f ν → f or f = h-limν f ν if {−f ν } epi-converges to −f Proposition Let C ν , C ⊂ Rn be nonempty and {f ν : C ν → R}ν∈N , f : C → R (i) g ([22, Proposition 5.33])f ν →f if and only if the following two conditions hold, for all x ∈ C, (α) for all x ν ∈ C ν → x, there exists a subsequence x νj such that limj f νj (x νj ) = f (x); (β) there exists x ν ∈ C ν → x such that limν f ν (x ν ) = f (x) (ii) (iii) {f ν } graphically converges to f if and only if it both epi- and hypo-converges to f and C ν → C If {f ν } converges continuously to f relative to sequence C ν → C, then it both epiand 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 (Epi-convergence: basic property) Let f ν , f ∈ fv-fcn(Rn ) and f = e-limf ν Then lim sup infν f ν (x) ≤ inf f (x) ν C C for some subsequence {νk }k∈N and xk → x, ¯ then x¯ ∈ Moreover, if xk ∈ argminC νk argminC f and minC νk f νk → minC f f νk Note from Proposition that graphical and continuous convergence also enjoy the properties stated in this theorem The second part of Theorem can be expressed equivalently e as: if f ν → f then Limsup argminf ν ν Cν ⊂ argminf C Approximations of Optimization-Related Problems It is easy to prove the extension that if ε ν then Limsup ε ν -argminf ν Cν ν ⊂ argminf C 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 (Tight epi-convergence) We say that a sequence {f ν }ν∈N epi-converges tightly to f in fv-fcn(Rn ) if it epi-converges and, for all positive ε, there exists a compact set Bε and an index νε such that, for all ν ≥ νε , inf f ν ≤ infν f ν + ε Bε ∩C ν C Theorem (Convergence of infima) Let f ν , f ∈ fv-fcn(Rn ), f = e-limν f ν and infC f be finite Then, the epi-convergence is tight (i) (ii) if and only if infC ν f ν → infC f ; if and only if there exists a sequence ε ν such that ε ν -argminf ν → argminf Epi/hypo Convergence of Bivariate Functions and Variational Properties ¯ was Epi/hypo convergence of a sequence of bivariate functions K ν : Rn × Rm → R 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 introduced 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 bifunctions 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 ⊂ Rn × 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 (Epi/hypo convergence) Bifunctions K ν , ν ∈ N, in fv-biv(Rn ×Rm ) are called epi/hypo converging (shortly e/h-converge) to a bifunction K ∈ fv-biv(Rn × Rm ) if (a) (b) 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; 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 e/ h We denote this convergence by K ν → K or K = e/h-limν K ν Note that if the functions ν K not depend on y, then epi/hypo convergence reduces to epi-convergence, and if they not depend on x, it collapses to hypo-convergence However, note that epi/hypo convergence is not epi-convergence 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 H T H Diem, P Q Khanh not necessary Indeed, K ν in Example below e/h-converges, but it does not hold that e K ν (·, y) → K(·, y) for all y ∈ D 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 (Minsup-lop convergence, [15]) Bifunctions K ν ∈ fv-biv(Rn × Rm ) are said to minsup-lopsided converge (shortly, minsup-lop converge) to K ∈ fv-biv(Rn × Rm ) if (a) (b) 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; 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 is nonsymmetric: the following maxinf-lop convergence is different from minsup-lop convergence: (a) (b) for all x ∈ C and all y ν ∈ D ν → y, there exists x ν ∈ C ν → x such that / D; lim supν K ν (x ν , y ν ) ≤ K(x, y) if y ∈ D or K ν (x ν , y ν ) → −∞ if y ∈ 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 However, 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 definitions 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∈N such 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 Let C ν = D ν = [1/ν, 1], C = D = [0, 1], and K ν (x, y) = if (x, y) ∈ C ν × D ν and x = y, if (x, y) ∈ C ν × D ν and x = y Then K(x, y) = e/ h if (x, y) ∈ [0, 1]2 and x = y, if (x, y) ∈ [0, 1]2 and x = y Clearly, K ν → K We show that condition (b) of Definition of minsup-lop convergence is violated For x = and any x ν ∈ C ν → x, we take y = and y ν ∈ D ν → such that y ν = x ν for all ν Then lim supν K ν (x ν , y ν ) = > = K(x, y) Approximations of Optimization-Related Problems Remark (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 maxinflop convergence (we know already in Section that 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 finitevalued bifunctions were established In particular, [7, Theorem 3] asserted the equivalence of the e/h-convergence of a sequence of finite-valued bifunctions 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 ¯ y) ¯ ∈ sdlK, if, for all x ∈ C and be a saddle point of K ∈ fv-biv(Rn × Rm ), denoted by (x, 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 not Hence, we will prove the following convergence of approximate saddle points The convergence 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(Rn × 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(Rn × Rm ) by, resp., h(·) := sup K(·, y), g(·) := inf K(x, ·) x∈C y∈D We have the following simple relation between approximate solutions Proposition Let K ∈ fv-biv(Rn × Rm ) and h and g be its sup-projection and inf-projection, respectively (i) (ii) If (x, ¯ y) ¯ ∈ ε-sdl K, then x¯ ∈ 2ε-argmin(h) and y¯ ∈ 2ε-argmax(g) If x¯ ∈ ε-argmin(h) and y¯ ∈ ε-argmax(g), then g(y) ¯ ≤ K(x, ¯ y) ¯ ≤ h(x), ¯ ¯ y) ¯ ≤ inf sup K(x, y) + ε sup inf K(x, y) − ε ≤ K(x, y∈D x∈C x∈C y∈D Therefore, if K has a saddle point (x, ˜ y), ˜ then K(x, ˜ y) − ε ≤ K(x, ¯ y) ¯ ≤ K(x, y) ˜ + ε H T H Diem, P Q Khanh Proof (i) We have h(x) ¯ = sup K(x, ¯ y) = K(x, ¯ y) ¯ +ε y∈D = inf K(x, y) ¯ + 2ε ≤ inf sup K(x, y) + 2ε = inf h(x) + 2ε x∈C x∈C y∈D x∈C The corresponding property of g is checked similarly, (ii) It is clear that K(x, ¯ y) ¯ ≥ g(y) ¯ ≥ sup inf K(x, y) − ε y∈D x∈C 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 equivalence class share many common properties This fact should be highlighted, since in many applications, it helps to avoid dealing with whole equivalence classes Theorem (Convergence of approximate saddle points) Let a sequence {K ν }ν∈N e/hconverge to K in fv-biv(Rn × Rm ), ε ν ε ≥ 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) ¯ = ¯ y) ¯ is an ε-saddle point of K and limν∈N (x¯ ν , y¯ ν ) for some subsequence N ⊂ N Then (x, K(x, ¯ y) ¯ = lim K ν x¯ ν , y¯ ν ν∈N ¯ y) ¯ Pick any (x, y) ∈ C × D Any Proof We can assume that actually (x¯ ν , y¯ ν ) → (x, 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 inf K ν x¯ ν , y ν − ε ν ≤ lim inf K ν x¯ ν , y¯ ν ν ν ≤ lim sup K ν x¯ ν , y¯ ν ν ≤ inf {x ν ∈C ν →x} lim sup K ν x ν , y¯ ν + ε ν , ν By the definition of e/h-convergence, one has K(x, ¯ y) − ε ≤ ≤ sup lim inf K ν x¯ ν , y ν − ε ν inf ¯ + ε lim sup K ν x ν , y¯ ν + ε ν ≤ K(x, y) {y ν ∈D ν →y} {x ν ∈C ν →x} ν ν These inequalities mean that (x, ¯ y) ¯ is an ε-saddle point of K To see that K(x, ¯ y) ¯ = limν∈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 inf K ν (x¯ ν , y¯ ν ), ν ν ν Approximations of Optimization-Related Problems 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 supν→∞ K ν (x¯ ν , y¯ ν ) Clearly, by taking ε ν ≡ in the preceding statement, we obtain the following basic result on convergence of saddle points Theorem (Convergence of saddle points) Let a sequence {K ν }ν∈N e/h-converge to K in ¯ y) ¯ be a cluster fv-biv(Rn × Rm ) and (x¯ ν , y¯ ν ) be a saddle point of K ν for all ν ∈ N Let (x, point of this sequence of saddle points, say (x, ¯ y) ¯ = limν∈N (x¯ ν , y¯ ν ) for some subsequence N ⊂ N Then (x, ¯ y) ¯ is a saddle point of K and K(x, ¯ y) ¯ = lim K ν (x¯ ν , y¯ ν ) ν∈N Observe that, in the above two theorems, neither convex-concave conditions nor continuity, 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 and some more insights about convergence properties Example Consider the sequence of bifunctions K ν (x, y) = ln x ln y, defined on C ν × Dν = 1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2 × 1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2 We can check directly that on [1/2, 3/2]2 , K ν converges 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 ν not have saddle points But, they have approximate saddle points Namely, with ε ν = max ln − ν −1 ln 1/2 − ln − ν −1 ln 3/2; ln + ν −1 ln 3/2 − ln + ν −1 ln 1/2 , the set consisting of the vertical intervals {1 − ν −1 } × ([1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2]) and {1 + ν −1 } × ([1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2]) and the horizontal intervals ([1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2]) × {1 + ν −1 } and ([1/2, − ν −1 ] ∪ [1 + ν −1 , 3/2]) × {1 − ν −1 } is ε ν -sdlK ν and converges to sdlK = ({1} × [1/2, 3/2]) ∪ ([1/2, 3/2] × {1}) Though without saddle points, the K ν have arg max Kν = ν ν D C 1/2, − ν −1 ∪ + ν −1 , 3/2 × {1} H T H Diem, P Q Khanh tending to the whole arg max K = [1/2, 3/2] × {1} C D ¯ y) ¯ such that Here, arg(maxD minC )K denotes the set of the points (x, K(x, ¯ y) ¯ = max K(x, y) D C and similarly for arg(minC maxD )K However, max arg K ν = − ν −1 × {1/2} ν ν C D converges to {(1, 1/2)}, which is only a point of arg max K = {1} × [1/2, 3/2] C εν D Theorem can be restated as follows, for the case ε = If K ν e/h-converges to K and 0, then Limsup(ε ν -sdlK ν ) ⊂ sdlK ν To have equality with the full Lim instead of Limsup in the above relation, i.e., to have also Liminf(ε ν -sdlK ν ) ⊃ sdlK, ν we propose new notions of tightness in Definition below Note that these tightness definitions 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 (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 and the following condition are satisfied: (b’-t) (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 is fulfilled together with (a’-t) (iii) h for all x ∈ C, there is x ν ∈ C ν → x such that K ν (x ν , ·) → K(x, ·) and hν (x ν ) → h(x) e for all y ∈ D, there is y ν ∈ D ν → y such that K ν (·, y ν ) → K(·, y) and g ν (y ν ) → g(y) (Tightness) If both (b’-t) and (a’-t) are satisfied, K ν is called e/h-convergent (fully) tightly to K Theorem (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) (ii) sdlK ⊂ ∩ε>0 Liminfν (ε-sdlK ν ) 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., Liminfν (ε ν -sdlK ν ) ⊃ sdlK Approximations of Optimization-Related Problems Proof (i) For each (x, ¯ y) ¯ ∈ sdlK, the tightness ensures the existence of x¯ ν ∈ C ν → x¯ and ν ν y¯ ∈ D → y¯ such that h K ν (x¯ ν , ·) → K(x, ¯ ·) e ¯ K (·, y¯ ) → K(·, y) ν ν and hν (x¯ ν ) → h(x), ¯ (1) and g (y¯ ) → g(y) ¯ (2) ν ν It suffices to show that, for all positive ε and large ν, K ν (x¯ ν , y¯ ν ) ≥ sup K ν (x¯ ν , y) − ε (3) K ν (x¯ ν , y¯ ν ) ≤ infν K ν (x, y¯ ν ) + ε (4) y∈D ν and x∈C Suppose to the contrary to (3) that there are ε0 and a subsequence νk such that K νk (x¯ νk , y¯ νk ) < sup K νk (x¯ νk , y) − ε0 y∈D νk Taking liminf on both sides, (1) and (2) imply that K(x, ¯ y) ¯ ≤ lim inf K νk (x¯ νk , y¯ νk ) ≤ sup K(x, ¯ y) − ε0 , k y∈D which is impossible since (x, ¯ y) ¯ ∈ sdlK Inequality (4) is similarly proved (ii) Given εμ and a fixed μ, by (i) one has a sequence (x¯μν , y¯μν ) ∈ εμ -sdlK ν converging to (x, ¯ y) ¯ Taking the diagonal sequence (x¯νν , y¯νν ), we complete the proof Example Let n = m = and K ν (x, y) = y x on [0, 1]2 for all ν ∈ N, with the convention that 00 = In Example of [7], it was computed that all the bifunctions Ka (x, y) = y x if (x, y) ∈ [0, 1]2 \ {(0, 0)}, a if (x, y) = (0, 0) for any a ∈ [0, 1], are e/h-limits of the sequence {K ν }, i.e., they form the e/h-equivalence e/ h class It is easy to check that K ν → Ka fully tightly Evidently, each point in {(x, 1) | ≤ x ≤ 1} is a saddle point of all K ν and Ka for all ν ∈ N and a ∈ [0, 1] So, the saddle points are preserved under tight e/h-convergence and the saddle points are the same for all limits in the e/h-equivalence class Approximations of Equilibrium Problems Consider the following equilibrium problem find x¯ ∈ C such that K(x, ¯ y) ≤ for all y ∈ D, (EP) where C × D ⊂ Rn × Rm and K : C × D → R This equilibrium problem is also called a Ky Fan inequality, since he studied first the existence of its solution in [10] This famous result was developed later by many authors in connection with fixed-point theorems and other existence theorems in nonlinear and functional analysis, see [5] In [6], the (EP) model was studied as a generalization of constrained minimization and variational inequalities It was shown soon after this starting point that (EP) contains all major optimization-related H T H Diem, P Q Khanh problems and has attracted an increasing number of researchers The (EP) model was shown to be a genuine generalization of many optimization-related problems in [14], by pointing out particular (EP) models which not fit the framework of these problems Note that, in fact, we have not seen papers arguing if a solution of (EP) may be really a physical/technical equilibrium or not But, the economical and social meaning of “equilibrium” has been confirmed by many considerations of practical problems like traffic networks, non-cooperative games, etc Assume that (EP) is subject to perturbation and we have a sequence of approximating problems (EPν ) with K ν : C ν × D ν → R Denote the solution set of (EP) ((EPν ), resp.) by S (Sν , resp.) e/ h e Proposition If K ν → K or K ν → K, then Limsupν Sν ⊂ S, i.e., any cluster point of a sequence of solutions of problems (EPν ) is a solution of (EP) ¯ For Proof Let x¯ ∈ Limsupν Sν , i.e., there exists a sequence {x νj } in Sνj converging to x any (fixed) y ∈ D, by (a) of Definition 3, there is a sequence {y νj }ν∈N in D νj converging to y such that lim infj K νj (x νj , y νj ) ≥ K(x, ¯ y) if x¯ ∈ C Since K νj (x, ¯ y) ≤ for all j ∈ N ν ν ν ν j j j j and y ∈ D , K (x , y ) ≤ for all j ∈ N and hence lim infνj K νj (x νj , y νj ) ≤ Thus, K(x, ¯ y) ≤ for any y ∈ D, i.e., x¯ is in S Suppose now x¯ ∈ C Then, by the mentioned condition (a), K νj (x νj , y νj ) → ∞, which is impossible since K νj (x νj , y νj ) ≤ for all j The proof for the case where K ν epi-converges is similar The above assertion improves Theorem 6.11 of [19], where the e/h-convergence is replaced by the stronger minsup-lop convergence, and it is assumed further that C = D, C ν = D ν are closed, C ν × D ν → C × D and K ν , K are lsc-usc for the case of minsup-lop convergence and lsc for the epi-convergence case The above statement can be extended to the case of approximate solutions as follows Let ξ : Rn → R be continuous such that ξ(x) > if x = Consider (EP) with m = n and C = D Following Definition 7.1 in [19], for ε ≥ 0, a point x¯ ∈ C is called an (ε, ξ )approximate solution of (EP) if K(x, ¯ y) ≤ εξ(x¯ − y) for all y ∈ C Denote the set of all (ε, ξ )-approximate solutions of (EP) by Sε,ξ Proposition If ε ν e/ h e ε ≥ and K ν → K or K ν → K, then LimsupSνεν ,ξ ⊂ Sε,ξ , ν i.e., any cluster point of a sequence of (ε ν , ξ )-solutions of problems (EPν ) is an (ε, ξ )solution of problem (EP) The proof is similar to that of Proposition The above result sharpens Theorem 7.7 of [19], where the e/h convergence is replaced by the stronger minsup-lop convergence under the same additional assumptions as for the mentioned Theorem 6.11 of [19] The preceding two results can be stated in terms of outer continuity as follows For n brevity, look only at Proposition Consider the set-valued map S : fv-biv(Rn ×Rm ) → 2R defined by S (K) = S and the e/h-convergence in fv-biv(Rn × Rm ) (though limits are whole equivalence classes, we deal with one a priori given limit bifunction K) Then, the Limsup inclusion in Proposition can be restated as the outer continuity of S Approximations of Optimization-Related Problems We consider also the dual equilibrium problem, introduced in [17], find y¯ ∈ D such that K(x, y) ¯ ≥ for all x ∈ C (DEP) Observe that (EP) and (DEP) are dual to each other, i.e., the dual of (DEP) is just (EP) To see clearer the essence of duality of these problems, we reformulate them as find x¯ ∈ C solving sup K(x, y) ≤ 0, (EP) find y¯ ∈ D solving max inf K(x, y) ≥ (DEP) x∈C y∈D y∈D x∈C In other words, (EP) is of finding minsup-points of K on C × D, and (DEP) of its maxinfpoints We have the following evident assertion, which is stronger than corresponding statements in many other duality schemes: x¯ is a solution of (EP) and also y¯ is that of (DEP) if and only if (x, ¯ y) ¯ is a saddle point of K in C × D and K(x, ¯ y) ¯ = Hence, we have zero duality gap for any couple of solutions x, ¯ y¯ of (EP) and (DEP), resp Similarly, x¯ε,ξ is an (ε, ξ )-approximate solution of (EP) and y¯ε,ξ is that of (DEP) if and only if (x¯ε,ξ , y¯ε,ξ ) is an (ε, ξ )-approximate saddle point of K in C × D and −εξ(x¯ − y) ¯ ≤ K(x, ¯ y) ¯ ≤ εξ(x¯ − y) ¯ Here, we naturally call (x¯ε,ξ , y¯ε,ξ ) an (ε, ξ )-approximate saddle point of K if K x¯ε,ξ , y − εξ x¯ε,ξ − y ≤ K x, y¯ε,ξ + εξ x − y¯ε,ξ for all (x, y) ∈ C × D Suppose that (EP) and (DEP) are subject to perturbation and the perturbed problems, called (EPν ) and (DEPν ), are stated in terms of K ν and C ν , D ν instead of K and C, D The following stability result is an immediate consequence of Theorems and e/ h Proposition Let K ν → K (i) ε ≥ and x¯ενν ,ξ and y¯ενν ,ξ are an (ε ν , ξ )-approximate solution of (EPν ) and If ε ν ν (DEP ), resp., then the two components of any cluster point (x¯ε,ξ , y¯ε,ξ ) of the sequence {(x¯ενν ,ξ , y¯ενν ,ξ )} are an (ε, ξ )-approximate solution of (EP) and (DEP), resp., and one has −ε ≤ K x¯ε,ξ , y¯ε,ξ = lim K ν x¯ενν ,ξ , y¯ενν ,ξ ≤ ε (ii) In particular, for ε = (i) becomes: if x¯ενν ,ξ and y¯ενν ,ξ are an (ε ν , ξ )-approximate solution of (EPν ) and (DEPν ), resp., then the two components of any cluster point (x, ¯ y) ¯ of the sequence {(x¯ενν ,ξ , y¯ενν ,ξ )} are a solution of (EP) and (DEP), resp., and one has K(x, ¯ y) ¯ = lim K ν x¯ενν ,ξ , y¯ενν ,ξ ν∈N ν∈N Denote the set of the solutions (the (ε, ξ )-approximate solutions, resp.) of (DEP) by DS then (DSε,ξ , resp.) Proposition (ii) can be rephrased as follows: if ε ν Limsup Sνεν ,ξ × DSνεν ,ξ ⊂ S × DS ν To have equality and with the “full” Lim instead of Limsup, i.e., to have additionally Liminf Sνεν ,ξ × DSνεν ,ξ ⊃ S × DS, ν we impose tightness conditions and apply Theorem to obtain the following H T H Diem, P Q Khanh Proposition If K ν e/h-converges fully tightly to K and ε ν 0, then Lim Sνεν ,ξ × DSνεν ,ξ = S × DS ν Approximations of Multiobjective Optimization As above-mentioned, it is well-known that equilibrium models encompass most of optimization-related problems However, in this paper, we restrict ourselves to the particular equilibrium problem (EP), which is a single-valued and scalar problem So, in principle, it contains only single-valued and scalar optimization-related models But, we can apply Propositions 3–6 for our scalar problem (EP) to the following multiobjective minimization problem Let ϕ1 , , ϕk : C ⊂ Rn → R and Rk be ordered partially by Rk+ Our multiobjective minimization problem is find x¯ ∈ C such that ϕ(x) ¯ − ϕ(y) ∈ / intRk+ for all y ∈ C (OP) Such a x¯ is called a weak minimizer (or weakly efficient point) of ϕ on C We can convert (OP) to a special case of (EP) by setting, see, e.g., [14], K(x, y) = (ϕi (x) − ϕi (y)) (5) 1≤i≤k Indeed, taking D = C, we have the three equivalent assertions for all y ∈ C, K(x, ¯ y) ≤ ⇔ there exists i, ϕi (x) ¯ − ϕi (y) ≤ ⇔ ϕ(x) ¯ − ϕ(y) ∈ / intRk+ The dual to (OP) according to the duality scheme for (EP), i.e., problem (DEP) for K(x, y) = min1≤i≤k (ϕi (x) − ϕi (y)), is find y¯ such that ϕi (x) − ϕi (y) ¯ ≥ for all x ∈ C, i = 1, , k, (DOP) Rk+ i.e., ϕ(x) − ϕ(y) ¯ ∈ Such a solution y¯ is called a strong/ideal minimizer (or a strongly efficient point) of ϕ on C This duality scheme is different from the known ones for multiobjective set-constrained minimization From the definition of (OP) and (DOP), we see that x, ¯ y¯ are a solution of (OP) and (DOP), resp., if and only if ¯ + Rk \ (−intRk+ ) ∪ ϕ(y) ¯ + Rk+ C ⊂ x ∈ Rn | ϕ(x) ∈ ϕ(x) By substituting x¯ and y¯ in this inclusion, we obtain ϕi (x) ¯ − ϕi (y) ¯ ∈ bdRk+ Furthermore, y¯ must be unique, but x¯ not Thus, we have a simple geometric explanation in the objective space Rk for (OP) and (DOP) Now, we are interested in approximations of these two dual-to-each-other problems Corresponding to the notion of (ε, ξ )-approximate solutions for (EP) and (DEP), we define (ε, ξ )-approximate solutions for (OP) and (DOP) as follows With e := (1, , 1) ∈ Rk , if ϕ(x¯ε,ξ ) − ϕ(y) − εξ(x¯ε,ξ − y)e ∈ / intRk+ (ϕ(x) − ϕ(y¯ε,ξ ) + εξ(x − y¯ε,ξ )e ∈ Rk+ for all y ∈ C for all x ∈ C, resp.), then x¯ε,ξ (y¯ε,ξ , resp.) is called an (ε, ξ )-approximate weak minimizer (an (ε, ξ )-approximate strong minimizer, resp.) of ϕ on C In fact, a little more general notion called an ε-quasi minimizer was already defined in [13] for multiobjective optimization Here, we formulate the above two (ε, ξ )-approximate solutions as a special case of the corresponding definitions for (EP) and (DEP) and pay attention also on their duality To state consequences of Propositions 3–6 in terms of the data of (OP), we need the following definition Approximations of Optimization-Related Problems Definition (i) A sequence of k functions {ϕ1ν , , ϕkν }ν∈N , defined on C ν , in fv-fcn(Rn ) is said to uniformly epi-converge to k limits ϕ1 , , ϕk , resp., if Definition (iii) is satisfied for all ϕiν , ϕi , and i, with the sequence x ν in (b) being common for all ≤ i ≤ k The definition of uniform hypo-convergence is similar (ii) A sequence of k functions {ϕ1ν , , ϕkν }ν∈N , defined on C ν , in fv-fcn(Rn ) is said to converge uniformly graphically to k limits ϕ1 , , ϕk , resp., if they all graphically converge and the condition (β) in Proposition 1(i) is satisfied with the sequence x ν being common for all ≤ i ≤ k Note that, in [19, 21], there was already the notion defined in Definition (i), but for the special case where C ν → C and these sets are convex We have a relation between Definition and the convergence of K ν defined by the rule (5) as follows For the sake of completeness, we provide also a proof of part (i), which is similar to that of part (a) of Proposition 5.2 in [19], but we not assume that C ν , C are convex and C ν → C Lemma (i) If ϕ1ν , , ϕkν uniformly epi-converge to ϕ1 , , ϕk , then K ν defined by the rule (5) e/h-converges to K (ii) If ϕ1ν , , ϕkν uniformly graphically converge to ϕ1 , , ϕk , then K ν defined by the rule (5) epi-converges to K and C ν → C Proof (i) We check first (a) in Definition (of e/h-convergence) For any x ν ∈ C ν → x, Definition (i)(a) gives, for all i, lim infν ϕiν (x ν ) ≥ ϕi (x) if x ∈ C By Definition (i)(b), for all y ∈ C, there exists y ν ∈ C ν → y such that lim supν ϕiν (y ν ) ≤ ϕi (y) for all i Hence, for all i, lim infν (ϕiν (x ν ) − ϕiν (y ν )) ≥ ϕi (x) − ϕi (y) On the other hand, by the definition of K ν , for all ν, there is iν such that K ν (x ν , y ν ) = ϕiν (x ν ) − ϕiν (y ν ) Since the set of indices i is finite (consisting of k elements), there exists an index i0 such that K ν (x ν , y ν ) = ϕi0 (x ν ) − ϕi0 (y ν ) for all ν (in a subsequence of N) Therefore, lim inf K ν (x ν , y ν ) = lim inf ϕiν0 (x ν ) − ϕiν0 (y ν ) ν ν ≥ ϕi0 (x) − ϕi0 (y) ≥ K(x, y) If x ∈ C, ϕiν (x ν ) → ∞ for all i, and hence K ν (x ν , y ν ) → ∞ Thus, (a) of Definition is checked Now consider (b) For all x ∈ C and all y ν ∈ C ν → y, (i)(a) of Definition yields lim infν ϕiν (y ν ) ≥ ϕi (y) for all i if y ∈ C, and (i)(b) gives x ν ∈ C ν → x with lim supν ϕiν (x ν ) ≤ ϕi (x) for all i Consequently, for all i, lim sup K ν (x ν , y ν ) ≤ lim sup ϕiν (x ν ) − ϕiν (y ν ) ≤ ϕi (x) − ϕi (y) ν ν Consequently, lim supν K ν (x ν , y ν ) ≤ K(x, y) If y ∈ C, ϕiν (y ν ) → ∞ for all i, which implies that K ν (x ν , y ν ) → −∞ Thus, Definition is verified completely (ii) By Proposition (ii), ϕ1ν , , ϕkν both epi- and hypo-converge uniformly to ϕ1 , , ϕk , and C ν → C We check first condition (a) for the epi-convergence of K ν For all (x ν , y ν ) ∈ C ν × C ν → (x, y) and indices i, we have both lim infν ϕiν (x ν ) ≥ ϕ(x) and H T H Diem, P Q Khanh lim supν ϕiν (y ν ) ≤ ϕ(y) Since the number k of indices is finite, there is an index i0 such that K ν (x ν , y ν ) = ϕiν0 (x ν ) − ϕiν0 (y ν ) for all ν up to subsequences of N Hence, lim inf K ν (x ν , y ν ) = lim inf ϕiν0 (x ν ) − ϕiν0 (y ν ) ν ν ≥ ϕi0 (x) − ϕi0 (y) ≥ K(x, y) Consider now condition (b) By the uniform convergence (given by (ii)), for all (x, y) ∈ C × C, there exists (x ν , y ν ) → (x, y) such that, for all indices i, lim sup K ν (x ν , y ν ) ≤ lim sup ϕiν (x ν ) − ϕiν (y ν ) ≤ ϕi (x) − ϕi (y) ν ν and then lim supν K ν (x ν , y ν ) ≤ K(x, y) Denote the set of the weak minimizers and (ε, ξ )-approximate weak minimizers of ϕ on C by WE(ϕ, C) and WEε,ξ (ϕ, C) We have clearly the following consequence of Propoe sition for stability of (OP) (the assumption that K ν → K is more restrictive than the e/ h assumption that K ν → K in the case of (OP) by Lemma 1, and hence the corresponding assertion is omitted) Proposition If ε ν ε and ϕ1ν , , ϕkν uniformly epi-converge to ϕ1 , , ϕk , then Limsupν WEνεν ,ξ (ϕ ν , C ν ) ⊂ WEε,ξ (ϕ, C), i.e., any cluster point of a sequence of (ε ν , ξ )-approximate solutions of problems (OPν ) is an (ε, ξ )-approximate solution of problem (OP) Of course, we also have a consequence of Proposition for (OP), with the corresponding tightness condition, for a “complete” convergence of approximate solution sets of (OP) and (DOP) Note that results similar to Proposition (and other related properties) can be found in the recent literature, for instance, in [20, 21, 31], where epi-convergence of maps (not a scalar functions like here) was considered with applications to approximations of convex multiobjective optimization (but, is was assumed that C ν → C, no duality scheme was considered and hence no strong minimizers either [31] included only the special case ξ(x) ≡ 1) Here, we state Proposition as a direct consequence of Proposition 4, which connects also to duality, and not aim to studying (OP) in length Approximations of Nash Equilibria In this final section, we choose an important practical particular case of equilibrium problems, the Nash equilibrium problem, to illustrate consequences of results in Section Consider a non-cooperative game with m players i ∈ I := {1, , m} Let Ci ⊂ Rn be the set of the available strategies of player i and ri (xi , x−i ) be the return of player i for the choice of strategy xi ∈ Ci , where x−i is the vector of the strategies chosen by the remaining players I \ {i} Then, ri : Ci × j ∈I \{i} Cj → R for i ∈ I The strategies x¯ := (x¯i )i∈I determine a Nash equilibrium point of this game if, for all i ∈ I , x¯i ∈ arg max ri (xi , x¯−i ) xi ∈Ci Approximations of Optimization-Related Problems We denote this game by G := {(Ci , ri ) | i ∈ I } and C := is called the Nikaido–Isoda bifunction: i∈I Ci The following bifunction [ri (xi , x−i ) − ri (yi , x−i )] N (x, y) := i∈I Proposition (i) The following three assertions are equivalent (α) x¯ ∈ C is a Nash equilibrium point; ¯ y) ≥ 0; (β) infy∈C N (x, (γ ) x¯ is a maxinf-point of N with infy∈C N (x, ¯ y) ≥ (ii) The following two assertions are equivalent and each implies the assertions in (i) (δ) supx∈C N (x, x) ¯ ≤ 0; (ε) x¯ is a minsup-point of N with supx∈C N (x, x) ¯ ≤ Proof The formula of N clearly yields that N (x, x) = 0, infy∈C N (x, y) ≤ for all x ∈ C, and supx∈C N (x, y) ≥ for all y ∈ C (i) “(α) implies (β)” This is obvious by the definition of Nash equilibria and the formula of N “(β) implies (α)” infy∈C N (x, ¯ y) ≥ means that, for all y ∈ C, ri (x¯i , x¯−i ) − ri (yi , x¯−i ) ≥ i∈I For any fixed k ∈ I and yk ∈ Ck , substituting y = (yk , x¯−k ) into this inequality we obtain ri (x¯k , x¯−k ) − ri (yk , x¯−k ) + rj (x¯j , (x¯−j,k , yk )) − rj (x¯j , (x¯−j,k , yk )) ≥ j ∈I \{k} (x¯−j,k is the collection of the m − components of x = (x¯i , x¯−i ), except x¯j and xk ), i.e., ri (x¯k , x¯−k )−ri (yk , x¯−k ) ≥ By the arbitrariness of k and yk , x¯ is a Nash equilibrium point “(β) is equivalent with (γ )” The assertion (β) together with the above observation that N (x, x) = and infy∈C N (x, y) ≤ for all x ∈ C implies evidently that x¯ is a maxinf-point ¯ y) = 0) The converse is trivial (and in fact, infy∈C N (x, (ii) “(δ) is equivalent with (ε)” We need to show only “(δ) implies (ε)” Clearly (δ) together with the observation for N that supx∈C N (x, y) ≥ for all y ∈ C ensure that x¯ is a minsup-point of N “(δ) implies (α)” supx∈C N (x, x) ¯ ≤ means that, for all x ∈ C, ri (xi , x−i ) − ri (x¯i , x−i ) ≤ i∈I In particular, for any given k and xk ∈ Ck , substituting x = (xk , x¯−k ) into this inequality yields rk (xk , x¯−k ) − rk (x¯k , x¯−k ) + rj (x¯j , (x¯−j,k , xk )) − rj (x¯j , (x¯−j,k , xk )) ≤ j ∈I \{k} (x¯−j,k is the collection of the m − components of x = (xk , x¯−k ), except x¯j and xk ) or, equivalently, ri (x¯i , x¯−i ) ≥ ri (xi , x¯−i ) for all i and xi ∈ Ci , i.e., x¯ is a Nash equilibrium point of G H T H Diem, P Q Khanh Remark From the above proposition, we have the following (a) The game G can be restated as the equilibrium problem of finding x¯ ∈ C such that N (x, ¯ y) ≥ for all y ∈ C, (b) (c) by the equivalence of (α) and (β) in Proposition Any Nash equilibrium point x¯ is a maxinf-point of the Nikaido–Isoda bifunction N with maxx∈C infy∈C N (x, y) = The dual game of G , defined according to the duality framework of equilibrium problem (EP), which is the problem of finding y¯ ∈ C such that N (x, y) ¯ ≤ for all x ∈ C, (d) is stronger than G Namely, its solutions y¯ are always Nash equilibrium points Furthermore, these y¯ are minsup-points of N with miny∈C supx∈C N (x, y) = So, if the dual game of G has solutions, then the duality gap is zero, and then, for any solution x¯ of G , (x, ¯ y) ¯ is a saddle point of N x¯ is a Nash equilibrium point if and only if infy∈C N (x, ¯ y) = The dual game of G has a solution y¯ if and only if supx∈C N (x, y) ¯ = Now assume that our game is perturbed and we have a sequence of approximating games G ν := {(Ciν , riν )| i ∈ I } Denote the solution set of G ν and G by S ν and S , resp Applying Proposition 3, we obtain the following approximation result Proposition For all i ∈ I , let riν converge continuously to ri relative to sequence Ciν → Ci or riν converge graphically to ri Then Limsupν S ν ⊂ S Proof Assume first that riν converges continuously to ri relative to sequence Ciν → Ci Clearly −riν also converges continuously to −ri relative to sequence Ciν → Ci Hence, it h/e is easy to see that N ν → N (as the product of Ciν , i ∈ I , C ν → C, by [22, Exercise 4.29]) Thus, Proposition gives the first assertion of this proposition Now let riν converge graphically Using Proposition (ii), we easily see that N ν hypo-converges to N Again by Proposition 3, we are done Note that here, we simply derive this statement as a consequence of the study in Section 4, including attention on duality properties Approximations of Nash equilibria in terms of variational convergence was investigated in more detail in [12], but without duality and epi/hypo convergence discussions Acknowledgements This research was supported by Vietnam National University Hochiminh City (VNUHCM) under grant number B2015-28-03 The second author is very grateful to Professor Roger J.-B Wets for the suggestion of the topic and many important discussions during his work He thanks also Professor Alejandro Jofr´e for helpful discussions The authors are deeply indebted to the two anonymous referees for their valuable remarks and suggestions, which have helped them much in completing the final version of this paper References Attouch, H., Wets, R.J.-B In: Mangasarian, O., Meyer, R., Robinson, S (eds.) Approximation and Convergence in Nonlinear Optimization, pp 367–394 Academic Press, New York (1981) Approximations of Optimization-Related Problems Attouch, H., Wets, R.J.-B.: Convergence des points min/sup et de points fixes Comptes Ren Acad Sci Paris 296, 657–660 (1983) Attouch, H., Wets, R.J.-B.: A convergence for bivariate functions aimed at the convergence of saddle values In: Cecconi, J.P., Zolezzi, T (eds.) Mathematical Theory Optimization, pp 1–42 Springer, Berlin–Heidelberg (1983) Attouch, H., Wets, R.J.-B.: A convergence theory for saddle functions Trans Am Math Soc 280, 1–41 (1983) Aubin, J.-P., Frankowska, H.: Set-Valued Analysis Birkhăauser, Boston (1990) Blum, B., Oettli, W.: From optimization and variational inequalities to equilibrium problems Math Stud 63, 123–145 (1994) Diem, H.T.H., Khanh, P.Q.: Criteria for epi/hypo convergence of finite-valued bifunctions Vietnam J, Math (2015) doi:10.1007/s10013-015-0139-x Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems Springer, Berlin (1993) Dupaˇcov´a, J., Wets, R.J.-B.: Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems Ann Stat 16, 1517–1549 (1988) 10 Fan, K.: A minimax inequality and applications In: Shisha, O (ed.) Inequality III, pp 103–113 Academic Press, New York (1972) 11 Flores-Baz´an, F., Lopez, R.: Asymptotic, existence and sensitivity results for a class of complementarity problems ESAIM Control Optim Cal Var 12, 271293 (2006) 12 Găurkan, G., Pang, J.P.: Approximations of Nash equilibria Math Program Ser B 117, 223–253 (2009) 13 Guti´errez, C., Lopez, R., Novo, V.: Generalized ε-quasi solutions in multiobjective optimization problems: existence results and optimality conditions Nonlinear Anal 72, 4331–4346 (2010) 14 Iusem, A., Sosa, W.: New existence results for equilibrium problems Nonlinear Anal 52, 621–635 (2003) 15 Jofr´e, A., Wets, R.J.-B.: Variational convergence of bivariate functions: lopsided convergence Math Program Ser B 116, 275–295 (2009) 16 Jofr´e, A., Wets, R.J.-B.: Variational convergence of bifunctions: motivating applications SIAM J Optim 24, 1952–1979 (2014) 17 Konnov, I.V., Schaible, S.: Duality for equilibrium problems under generalized monotonicity J Optim Theory Appl 104, 395–408 (2000) 18 Lignola, M.B., Morgan, J.: Generalized variational inequalities with pseudomonotone operators under perturbations J Optim Theory Appl 101, 213–220 (1999) 19 Lop´ez, R.: Approximations of equilibrium problems SIAM J Control Optim 50, 1038–1070 (2012) 20 Lop´ez, R.: Variational convergence for vector-valued functions and its applications to convex multiobjective optimization Math Meth Oper Res 78, 1–34 (2013) 21 Lop´ez, R., Vera, C.: On the set of weakly efficient minimizers for convex multiobjective programming Oper Res Lett 36, 651–655 (2008) 22 Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd edition, Grundlehren der Mathematischen Wissenschatt, vol 317 Springer, Berlin (2009) 23 Soueycatt, M.: Stabilit´e quantitative des fonctions convexes-concaves Topol e´ pi/hypo-distance S´eminaire d’Anal Conv., Univ Montpelier II 20, 2.1–2 54 (1990) 24 Soueycatt, M.: Analyse e´ pi/hypo-graphique des problemes de points-selles, PhD thesis, Univ Montpelier (1991) 25 Soueycatt, M., Abdulfattar, S.: Analyse e´ pi/hypo-graphique S´eminaire d’Anal Conv., Univ, Montpelier II 21, 13.1–13 49 (1991) 26 Volle, M.: Contributions a` la Dualit´e en Optimisation et a` l’Epi-convergence PhD thesis, Univ Pau, France (1986) 27 Walkup, D.W., Wets, R.J.-B.: Continuity of some convex-cone-valued mappings Proc Am Math Soc 18, 229–235 (1967) 28 Wijman, R.A.: Convergence of sequences of convex sets, cones and functions Bull Am Math Soc 70, 186–188 (1964) 29 Wijman, R.A.: Convergence of sequences of convex sets, cones and functions II Trans Am Math Soc 123, 32–45 (1966) 30 Wright, S.E.: Consistency of primal-dual approximations for convex optimal control problems SIAM J Control Optim 33, 1489–1509 (1995) 31 Zeng, J., Li, S.J., Zhang, W.Y., Xue, X.W.: Stability results for convex vector-valued optimization problems Positivity 15, 441–453 (2011) ... optimization-related problems included also variational convergence of solutions of approximating problems to that of the true problem under question Variational convergence is important in optimization-related. .. appropriate for investigations of a saddle point, the basic object one is often interested in, when dealing with bivariate functions involved in optimization-related Approximations of Optimization-Related. .. state consequences of Propositions 3–6 in terms of the data of (OP), we need the following definition Approximations of Optimization-Related Problems Definition (i) A sequence of k functions {ϕ1ν