Stability results for convergence of convex sets and functions in nonreflexive spaces

Publicacions Matem`atiques, Vol 40 (1996), 67–83. STABILITY RESULTS FOR CONVERGENCE OF CONVEX SETS AND FUNCTIONS IN NONREFLEXIVE SPACES J. Lahrache Abstract Let Γ(X) be the convex proper lower semicontinuous functions on a normed linear space X. We show, subject to Rockafellar’s constraints qualifications, that the operations of sum, episum and restriction are continuous with respect to the slice topology that reduces to the topology of Mosco convergence for reflexive X.We show also when Xis complete that the epigraphical difference is continuous. These results are applied to convergence of convex sets. 1. Introduction Let C(X) be the family of the closed nonempty convex sets of a normed linear space (X,·). U. Mosco [22], [23] defined a notion of convergence of sequences on C(X) and on Γ(X) the convex proper lower semicontinuous functions via the identification of a function with its epigraph. This notion of convergence has many applications in reflexive spaces in the study of variational inequalities, laws of large numbers and Banach space geometry (see [1], [11], [14], [22], [29]). In fact without reflexivity the Mosco topology is not operational since it is not separated, the Legendre-Fenchel is not bicontinuous and the polarity is not continuous [9]. G. Beer gives a topology on C(X) which he called the slice topology and which agrees with the Mosco topology if and only if Xis reflexive [7, Theorem 5.7] (see also Y. Sonntag and C. Zˇalinascu [30] and J. L. Joly [18]). For this topology the Legendre-Fenchel is bicontinuous [8, Theorem 4.2], the polarity is continuous [8, Theorem 4.4] and C(X) is separated when Xis a normed linear space. Moreover Attouch’s theorem is extended in any Banach spaces by H. Attouch and G. Beer [4, Theorem 4.2] using the slice topology. This 68 J. Lahrache result is central in the study of continuity of the epigraphical difference in the last section. The goal of this article is to show that the epigraphical operations are continuous with respect to the slice topology which has many applications in optimization problems. More precisely we produce some results about continuity of operations of classical analysis and epigraphical analysis like sum, restriction, episum and epimultiplication with respect to the slice topology. Using Wijsman convergence in finite dimensional spaces, McLinden and C. Bergstrˆom [20] have studied some of these operations, corresponding results for Mosco convergence have been obtained by H. Attouch, D. Aze and R. Wets [3] and H. Riahi [27]; see also J. P. Penot [25]. G. Beer and R. Lucchetti [10] have studied these operations with respect to Attouch-Wets topology in any normed linear space. The last part of this paper is reserved for the study of continuity of the epigraphical difference as already mentioned. 2. Preliminary In the sequel, (X,·) will be a normed linear space with continuous dual (X∗,· ∗) and the value of the functional y∈X∗at x∈X will be denoted by x, y. The closed unit ball and the origin of X will be denoted by Uand Θ. We denote the closed (resp. closed and bounded) convex subsets of Xby C(X) (resp. CB(X)) and the weak∗ closed (resp. weak∗closed and bounded) convex subsets of X∗by C∗(X∗) (resp. CB∗(X∗)). We now review some standard constructions from convex analysis; for further information the reader may consult [17], [26]. A function f: X→(−∞,+∞] is called convex (resp. lower semicontinuous) provided its epigraph epi f={(x, α)∈X×R,f(x)≤α} is a convex (resp. closed) subset of X×R. Furthermore, fis called proper if its epigraph is nonempty and we denote clf the lower semicontinuous regularization of f. Again, Γ(X) will denote the convex proper lower semicontinuous functions on Xinto (−∞,+∞], Γ∗(X∗) will denote the convex proper weak∗lower semicontinuous functions defined on X∗into (−∞,+∞]. The convex conjugate (or Legendre-Fenchel transform) of f∈Γ(X) is the function f∗∈Γ∗(X∗) given by f∗(y) = sup{x, y−f(x),x∈X}. The episum of fand g:X→R∪{+∞} is the function defined by f+ eg(x) = inf{f(u)+g(x−u),u∈X}. Stability for convergence of convex functions 69 We note that f+ egdoes not belong in general to Γ(X). The epimultiplication of λ>0 and fis the function λ∗fdefined for every x∈Xby (λ∗f)(x)=λf(x/λ). The term epimultiplication was chosen because epi(λ∗f)=λepi f.If A⊆X,δ(·,A) is the indicator function of A, it is equal to 0 on Aand +∞elswhere; s(·A) is its support function, s(y,A) = sup{x, y,x∈A} for y∈X∗. The restriction of fto Ais the function f|A=f+δ(·,A). For x∈Xand Aa nonempty subset of X, we write d(x, A) for inf{x−a,a∈A}and for A,Bnonempty subsets of X, we write D(A, B) for inf{d(a, B), b∈B}. We say that a net (Aν) of closed subsets of Xis convergent in the Kuratowski-Painleve sense to the closed subset A[1], [12]ifA= lim inf Aν= lim sup Aν, where lim inf Aν={x∈X, ∀ε>0∃ν0:(x+εU)∩Aν=∅for all ν≥ν0}, lim sup Aν={x∈X, ∀ε>0∀ν∃ν≥ν:(x+εU)∩Aν=∅}. The slice topology τson C(X) is the weak topology determined by the family of gap functionals {D(B,·},B∈CB(X)}; it has as sub-base all sets of the form (see [8, Theorem 5.3]) V−={A∈C(X),A∩V=∅},V norm open (Bc)={A∈C(X):D(A, B)>0},B∈CB(X). The weak topology on C(X) determined by the family of gap functionals {D(K, ·), Kweakly compact of X}is the Mosco topology τM which induces the Mosco convergence in any Banach space. The topology of uniform convergence of distance functions on bounded sets is the Attouch-Wets topology also called the bounded Hausdorff topology, and noted τAW ,[5], [10], [24], [27]. We shall work with another topology on C∗(X∗), the dual slice topology τ∗ s. It has as sub-base all the sets of the form V−={A∈C∗(X∗),C∩V=∅},V norm open in X∗ (Bc)++ ={A∈C∗(X∗):D(A, B)>0},B∈CB∗(X∗). We shall use a diagonalization method, Corollary 3, and a result characterizing slice convergence of nets, Proposition 6 in the appendix, which generalize Corollary 1.11 [1] and Corollary 3.6 [8] respectively. 70 J. Lahrache 3. Principal result Theorem 3.1. Let Xbe a normed linear space. Suppose ϕ,(ϕν),ψ, (ψν)are nets in Γ(X)and there exists x0∈dom ϕ,ρ0>0such that sup ν ψν(ξ)≤M<+∞,∀ξ∈B(x0,ρ 0).Ifϕ=τs−lim ϕνand ψ=τs− lim ψν, then eventually ϕν+ψνis proper and ϕ+ψ=τs−lim(ϕν+ψν). In duals spaces we have the same result: Theorem 3.1(b). Let Xbe a normed linear space. Suppose ϕ,(ϕν), ψ,(ψν)are nets in Γ(X∗)and there exists y0∈dom ϕ,ρ>0such that sup ν ψν(ξ)≤M<+∞,∀ξ∈B(y0,ρ 0).Ifϕ=τ∗ s−lim ϕνand ψ=τ∗ s− lim ψν, then eventually ϕν+ψνis proper and ϕ+ψ=τ∗ s−lim(ϕν+ψν). Proof of Theorem 3.1: (3.1) From Proposition 6(ii), it is easy to see that Mmajories ψon the open ball B(x0,ρ 0) and then ψis continuous on this ball. Since ϕνand ψνare proper, −∞ <ϕ ν+ψν. The same reference above implies that there exists (xν) convergent strongly to x0such that ϕ(x0) = lim ϕν(xν). (3.2) We note that the assumptions easily imply a uniform Lipschitz estimate for ψνthat, jointly with slice convergence, implies pointwise convergence. Then we have ψ(x0) = lim ψν(xν) and for νsufficiently large ϕν(xν)+ ψν(xν)<+∞; for such νϕ ν+ψνis proper. Observe that the Remark (3.1) above implies that ϕ+ψis proper and x0∈dom(ϕ+ψ). Now we show that conditions (i) and (ii) of Proposition 6 characterizing slice convergence of nets are met. Let (xν) be a bounded net in Xand (y,η)∈epi(ϕ+ψ)∗with η> (ϕ+ψ)∗(y). We are going to show that (3.3) (ϕν+ψν)(xν)>xν,y−ηeventually. From (ϕ+ψ)∗(y)=ϕ∗+ eψ∗∗∗ (y) = lim inf z→yϕ∗+ eψ∗(z), η>(ϕ+ ψ)∗(y) implies that ∃v∈X∗,v∗<r 0/2 sup xn,∃ζy∈X∗such that η>ϕ ∗(ζy)+ψ∗(y+v−ζy)+r0, Stability for convergence of convex functions 71 where r0=(η−(ϕ+ψ)∗(y))/2. From the bicontinuity property of the Legendre-Fenchel transform [8, Theorem 4.2], we have ϕ∗=τ∗ s−lim ϕ∗ νand ψ∗=τ∗ s−lim ψ∗ ν. Then, using Proposition 6, there exists (ζν) and (ξν) elements of X∗ convergent strongly respectively to ζyand y+v−ζysuch that ϕ∗(ζy) = lim ϕ∗ ν(ζν) and ψ∗(y+v−ζy) = lim ψ∗ ν(ξν). Then for νsufficiently large we have η>ϕ ∗ ν(ζν)+ψ∗ ν(ξν)+r0/2. Definition of the conjugate gives ϕ∗ ν(ζν)≥ζν,x ν−ϕν(xν), ψ∗ ν(ξν)≥ξν,x ν−ψν(xν). Combining these last three inequalities yields η>ζν+ξν,x ν−(ϕν+ψν)(xν)+r0/2 η>y,xν−(ϕν+ψν)(xν)+ζν+ξν−y,xν+r0/2. Which implies for νsufficiently large η>y,xν−(ϕν+ψν)(xν)+r0/2−ζν+ξν−yxν. Because (xν) is bounded and ζν+ξν−ygoes to v∗, the above inequality shows η>y,xν−(ϕν+ψν)(xν) eventually. Which is exactly (3.3). Let us verify the second assertion of characterization of τs-convergence. Let x∈dom ϕ∩dom ψ, otherwise there is nothing to prove. For integer p≥1, pose xp=(p−1)/px +1/px0, then from convexity of ϕand ψ we have xp∈dom ϕ∩dom ψ(since we showed above x0∈dom(ϕ+ψ)). As ϕ=τs−lim ϕν, using Proposition 6 again, there exists a net (xν p) convergent strongly to xpsuch that ϕ(xp) = lim ϕν(xν p). By Proposition 6, choose xν→xwith ψν(xν)→ψ(x). Now ψν(xν)≤Ksome Kand all large ν.Forpfixed, choose ν0such that conv(xν,B(x0,ρ 0)) contains a neighborhood of xpsay B(xp,δ p) for all ν≥ν0.NowonB(xp,δ p/2), ψν 72 J. Lahrache for ν≥ν0are uniformly Lipschitz with Lipschitz constant L. And we obtain ψ(xp) = lim ψν(xν p) from Remark (3.2) and the inequalities |ψν(xν p)−ψ(xp)|≤|ψν(xν p)−ψν(xp)|+|ψν(xp)−ψ(xp)| ≤Lxν p−xp+|ψν(xp)−ψ(xp)|. Since ϕ+ψis continuous on segments of its domain, we have (ϕ+ψ)(x)= lim p→+∞(ϕ+ψ)(xp). Then we have (xp,(ϕ+ψ)(xp)) = ·×|·|−lim ν(xν p,(ϕν+ψν)(xν p)), (x, (ϕ+ψ)(x)) = ·×|·|−lim p(xp,(ϕ+ψ)(xp)). By diagonalization, Corollary 3, there exists an increasing mapping ν→ p(ν) such that: (x, (ϕ+ψ)(x)) = ·×|·|−lim ν(xν p(ν),(ϕν+ψν)(xν p(ν)). Denoting xν=xν p(ν),wehave: x= lim xνand (ϕ+ψ)(x) = lim(ϕν+ψν)(xν). Which completes the proof of Theorem 3.1. The proof of Theorem 3.1(b) is the same. Remark 3.2. a) The fact that the (ψν) are uniformly bounded above on the open ball centered at a point of dom ϕin Theorem 3.1 is superfluous to have condition (i) of Proposition 6. b) If in Theorem 3.1 we have ψν≡ψ, then the implication (ϕ= τs−lim ϕν)⇒(ϕν+ψare eventually proper and ϕ+ψ=τs−lim(ϕν+ψ)) is satisfied if ψis continuous and finite valued at some point of dom ϕ. Corollary 3.3. Let Xbe a normed linear space and let (ϕν,A ν)be a net in (Γ(X),τ s)×(C(X),τ s)convergent to (ϕ, A). Suppose that either (ϕν)are uniformly bounded above on the ball centered at some point of Aor dom ϕ∩int ∩ νAν=∅. Then ϕν|Aνis τs-convergent to ϕ|A. Proof: Apply Theorem 3.1 with ψν=δ(·,A ν) and ψ=δ(·,A). Stability for convergence of convex functions 73 Corollary 3.4. Let Xbe a normed linear space. Suppose (Aν)and (Bν)are nets in C(X)with A=τs−lim Aνand B=τs−lim Bν. Suppose further that int ∩ νAν∩B=∅. Then A∩B=τs−lim Aν∩Bν. Proof: For each ν, let ϕν=δ(·,B ν) and let ϕ=δ(·,B). Then for each ν,ϕν|Aν=δ(·,A ν∩Bν). By Corollary 3.3 and the fact that the mapping C→δ(·,C) is is an embedding of (C(X),τ s) into (Γ(X),τ s) [8, Theorem 3.1] we obtain A∩B=τs−lim Aν∩Bν. Corollary 3.5. Let Xbe a normed linear space. Suppose f,(fν)and g,(gν)are nets in Γ(X)and there exists y0∈dom f∗,ρ0>0such that sup g∗ ν(ξ)≤M<+∞for all ξ∈B(y0,ρ 0). Then f=τs−lim fνand g=τs−lim gνimplies that f+ egis proper and fν+ egνis proper for ν sufficiently large. Furthermore, we have cl f+ eg=τs−lim cl fν+ egν. Proof: From the bicontinuity property of the Legendre-Fenchel transform with respect to slice topology [8, Theorem 4.2] we have f∗=τ∗ s−lim f∗ νand g∗=τ∗ s−lim g∗ ν. Hence from Theorem 3.1(b) we deduce that f∗+g∗and f∗ ν+g∗ νare proper for νsufficiently large and f∗+g∗=τ∗ s−lim(f∗ ν+g∗ ν). Which it means f+ eg∗ =τ∗ s−lim fν+ egν∗ . Finally, the conclusion is obtained again from Theorem 4.2 [8]. Take gν≡g∈Γ(X), then Corollary 3.5 is reduced to the following corollary. Corollary 3.6. Let Xbe a normed linear space. Suppose g,f,(fν) a net in Γ(X)and g∗is continuous and finite valued at some point of dom f∗. Then f=τs−lim fνimplies that clf+ eg=τs−lim clfν+ eg. Corollary 3.7. Let Xbe a normed linear space. Suppose A,(Aν) and B,(Bν)are nets in C(X)such that sup ν s(y,Bν)≤M<∞for all 74 J. Lahrache y∈B(y0,ρ 0)where ρ0>0and y∈dom s(·,A). Then A=τs−lim Aν and B=τs−lim Bνimplies that cl(A+B)=τs−lim cl(Aν+Bν). Proof: From Theorem 3.1 [8], it is equivalent to show that δ(·,cl(A+B)) = τs−lim δ(·,cl(Aν+Bν)). But from the same reference we have δ(·,A)=τs−lim δ(·,A ν) and δ(·,B)=τs−lim δ(·,B ν). Corollary 3.5 implies that cl δ(·,A)+ eδ(·,B)=τs−lim cl δ(·,A ν)+ eδ(·,B ν) which means that cl(δ(·,A+B)) = τs−lim cl(δ(·,A ν+Bν)). And this is exactly the desired equality since we have cl(δ(·,G)) = δ(·,clG) for G⊆X. Examples 3.8. The following examples are well known. They show that results of Theorem 3.1 fail in general without equi-upperboundness of (ψν) (although ψis finite and continuous at some point of domϕ(see [10])). a) X=R,ψn≡ψ≡ϕ≡δ(·,{Θ}) and ϕn(x)=n|x−1/n|. b) X=l2(N){en,n ≥1}its hilbertian base. Take An={en+1/n} and Bn={x, Σx, ei2≤1 and x, ei= 0 for i>n}. Denotant ϕn=δ(·,A n), ϕ=δ(·,{Θ}), ψn=δ(·,B n) and ψ=δ(·,U) we have (ϕn) (resp. (ψn)) Mosco converges to ϕ(resp. ψ). But ϕn+ψnisn’t proper for all nsince An∩Bn=∅. In [5], using Kenmochi’s conditions, H. Attouch and R. Wets gave a module of continuity of the operation (λ, f)→λ∗fwith respect to the ρHausdorff distance when the functions are quadratically minorized. Here we prove that this operation is continuous from ((0,+∞)×Γ(X),|·|×τs) into (Γ(X),τ s). Theorem 3.9. Let Xbe a normed linear space. Then the mapping (λ, f)→λ∗fis continuous from ((0,+∞)×Γ(X),|·|×τs)into (Γ(X),τ s). Similary, the mapping (α, h)→α∗his continuous from ((0,+∞)× Γ∗(X∗),|·|×τ∗ sinto (Γ∗(X∗),τ∗ s). Proof: Suppose fνconverges slice to fand λν→λ∈(0,+∞). Now let Wbe any bounded closed convex set in X×R. Then D(W, epi(λν∗ Stability for convergence of convex functions 75 fν)) = D(W, λνepi fν)=λνD(W/λν,epi fν). Now |λνD(W/λν,epi fν)− λD(W/λ, epi fν)|→0. From this it follows that λνD(W/λν,epi fν)→ λD(W/λ, epi f) since fνconverges slice to f. Hence D(W, λνepi fν)→ D(W, λ epi f) which shows that λν∗fνconverges slice to λ∗f. The same argument clearly works for other forms of gap convergence, in particular for w∗-slice convergence. We know that when X∗is strongly separable then (C(X),τ s) is metrisable [7, Theorem 5.10]. So continuity can be proved using sequences, Corollary 3.6 [8] permits us to have another proof of Theorem 3.9, we left it to the reader. On the other hand if f=δ(·,C) we have λ∗f=δ(·,λC), then using Theorem 3.9 above and Theorem 3.1 [8] we obtain the following corollary. Corollary 3.10. Let Xbe a normed linear space. Then the mapping (λ, C)→λC is continuous from ((0,+∞)×C(X),|·|×τs)into (C(X),τ s). Similary, in the dual space, the mapping (λ, C)→λC is continuous from ((0,+∞)×C∗(X∗),|·|×τ∗ s)into (C∗(X∗),τ∗ s). 4. Epigraphical difference and slice topology The role of the epigraphical sum is well known in optimization and in the study of variational problems. Now we turn to the inverse operation which is the main concern of this section. It can be formulated as follows: given gand htwo real extended valued functions defined on X, find fa real extended valued function such that f+ eg=h. When we talk about indicator functions the problem has the following geometric interpretation: given Band Ctwo subsets of a linear space Y, find a subset Asuch that A+B=C. In general these problems have no solutions. Existence and uniqueness of such solutions have been studied by J. B. Hiriart-Urruty and M. L. Mazure [16] and M. L. Mazure and M. Volle [21]. We shall write (when it is uniquely defined) f=h− egas the epigraphical difference of hand g. In this section, we study the continuity properties of the inverse mapping h→f=h− eg and we extend some results of H. Attouch, D. Aze and G. Beer [2] and B. E. Ghali [13] (see also [19]). In the end, we write the recession function as an epigraphical difference. We shall use the following result which is a net version of AttouchBeer’s theorem ([4, Theorem 4.2]). The proof of this proposition is identical as for Theorem 4.2 [4]. In fact it reposes on Theorem 3.1 [4] and himself can easily extended to nets by using Corollary 3 and Proposition 6 in the appendix. 82 J. Lahrache 16. J. B. Hiriart-Urruty and M. L. Mazure, Formulation variationnelle de l’addition parall`ele et da la soustraction parall`ele d’op´erateurs semi-d´efinis positifs, C. R. Acad. Sci. Paris 302 (1986), 527–530. 17. R. Holmes,“A course in optimization and best approximation,” Lecture notes in Mathematics 257, Springer-Verlag, New York, 1972. 18. J. L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarit´e est bicontinue, J. Math. Pures Appl. 52 (1973), 421–441. 19. J. Lahrache, Topologies sur l’ensemble des convexes ferm´es d’un espace vectoriel norm´e, stabilit´e et analyse unilat´erale, Th`ese Doctorat Universit´e Montpellier II (1992). 20. L. McLinden and R. C. Bergstrˆ om, Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268(1) (1981), 127–142. 21. M. L. Mazure and M. Volle, Equations inf-convolutives et conjugaison de Moreau-Fenchel, Publication du s´eminaire d’analyse num´erique, Univ. Paul Sabatier, Toulouse (1988). 22. U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3(1969), 510–585. 23. U. Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. 24. A. Moudafi, Algorithmes num´eriques associ´es `a des op´erateurs maximaux monotones, Analyse quantitative et stabilit´e, Th`ese, Univ. de Montpellier (1991). 25. J. P. Penot, Preservation of persistance and stability under intersection and operations, J. Optim. Theory Appl. 79(3) (1993), I: 525–550, II: 551–561. 26. R. Phelps,“Convex functions, monotones operators and differentiability,” Lect. Notes in Math. 1364, Springer, Berlin, 1989. 27. R. Riahi, Quelques r´esultats de stabilit´e en analyse ´epigraphique: Approche topologique et quantitative, Th`ese, Univ. de Montpellier (1989). 28. R. T. Rockafellar,“Convex analysis,” Princeton Univ. Press, Princeton, 1970. 29. Y. Sonntag, Convergence au sens de Mosco: Th´eorie et applications `a l’approximation des solutions d’in´equations, Th`ese d’´etat, Univ. de Provence, Marseille (1982). Stability for convergence of convex functions 83 30. Y. Sonntag and C. Zˇ alinascu, Set convergence, An attempt of classification, Trans. Amer. Math. Soc. 340(1) (1993), 199–226. Keywords. Convex function, infconvolution, episum, epigraphical difference, slice topology, Mosco-convergence, bounded Hausdorff topology. Universit´e C. Doukkali Fac. Sciences, Math´ematiques B.P. 20 24000 El Jadida MAROC Primera versi´o rebuda el 9 de Febrer de 1995, darrera versi´o rebuda el 25 d’Octubre de 1995