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Semicontinuity of Convex-Valued Multifunctions

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Semicontinuity of Convex-Valued Multifunctions Semicontinuity of Convex-valued Multifunctions Andreas L¨ohne∗ May 2, 2005 Abstract We introduce semicontinuity concepts for functions f with values in the space C(Y ) of closed convex subsets of a finite dimensional normed vector space Y by appropriate notions of upper and lower limits. We characterize the upper semicontinuity of f : X → C(Y ) by ∗ the upper semicontinuity of the scalarizations σf( · )(y ): X → R by the support function. Furthermore, we compare our semicontinuity concepts with well-known concepts. 1 Introduction Working in the framework of convex-valued multifunctions we expect that an appropriate notion of an upper semicontinuous hull produces a convex-valued multifunction being upper semicontinuous. This cannot be ensured by the classical concept of outer semicontinuity [4], 0 as the following examples show. We denote by LIM SUPx0→x f(x ) the outer limit of f at 0 x and by (osc f)(x) = LIM SUPx0→x f(x ) the corresponding outer semicontinuous hull, see Section 2 for the exact definitions. Example 1.1 Let f : R ¶ R, f(x) := {x / |x|} if x 6= 0 and f(0) := {0}. Then the outer semicontinuous hull of f, namely (osc f): R ¶ R, (osc f)(x) = f(x) if x 6= 0 and (osc f)(0) = {−1, 0, 1}, is not convex-valued. This might suggest to redefine the outer semicontinuous hull as follows: (oscf f)(x) := cl conv LIM SUP f(x0). x0→x However, (oscf f) is not necessarily outer semicontinuous as the following example shows. Example 1.2 Let f : R ¶ R, 8 © ª £ ¢ 1 if ∃n ∈ N : x ∈ 2−2n, 2−2n+1 < © x ª £ ¢ f(x) := − 1 if ∃n ∈ N : x ∈ 2−2n+1, 2−2n+2 : x ∅ else. Then the modified outer semicontinuous hull (oscf f) of f is obtained as 8 © ª ¡ ¢ > 1 if ∃n ∈ N : x ∈ 2−2n, 2−2n+1 <> © x ª ¡ ¢ − 1 if ∃n ∈ N : x ∈ 2−2n+1, 2−2n+2 (oscf f)(x) = £ x ¤ > − 1 , 1 if ∃n ∈ N : x = 2−n :> x x ∅ else. ∗Department of Mathematics and Computer Science Martin–Luther–University Halle–Wittenberg 06099 Halle (Saale), Germany, email: [email protected] 1 −n It is easily seen that gr (oscf f) is not closed. Indeed, the sequence (2 , 0)n∈N belongs to the graph of (oscf f), but its limit (0, 0) does not. Hence (oscf f) is not outer semicontinuous. Let us illuminate another aspect. An important idea of Convex Analysis is the relationship p p between a convex set A ⊂ R and its support function σA : R → R. In particular, for closed p convex sets A, B ⊂ R and α ∈ R+ we have the following relationships (in particular, we set −∞ + ∞ = −∞, 0 · ∅ = {0}): µ ¶ A ⊂ B ⇔ σA ≤ σB , σA + σB = σA+B, ασA = σαA. This yields, for instance, that a set-valued map f : Rn ¶ Rp is concave (i.e. graph-convex) ∗ n ∗ p if and only if the functions σf( · )(y ): R → R have the same property for all y ∈ R . For some reasons it could be useful to have a corresponding relationship for continuity properties, too. However, the usual outer and inner semicontinuity is not appropriate for this, as the following example shows. © 1 ª Example 1.3 Let f : R ¶ R, f(x) := x if x 6= 0 and f(0) := {0}. Then f is outer ∗ semicontinuous (in particular at x = 0), but σf( · )(y ) is not upper semicontinuous at x = 0 whenever y∗ 6= 0. Motivated by these examples we introduce semicontinuity concepts such that the corre- sponding upper semicontinuous hull operation yields a convex-valued upper semicontinuous function and such that upper semicontinuity can be described by upper semicontinuity of the ∗ n functions σf( · )(y ): R → R. Our investigations are based on some results on C-convergence, which were recently obtained by C. Z˘alinescuand the author [2], [1]. This paper is organized as follows. In Section 2 we shortly summarize some facts on outer and inner semicontinuity in the sense of Painlev´eand Kuratowski (e.g., see [4]). In the third section, we recall the definition of upper and lower limits for sequences of convex sets, as introduced in [2], and we propose our main tools. In Section 4 we extend these concepts, which leads to our semicontinuity concepts. We show that f : Rn ¶ Rp, having closed convex ∗ values, is upper semicontinuous atx ¯ if and only if σf( · )(y ): X → R is upper semicontinuous atx ¯ for all y∗ belonging to the set ri (0+f(¯x)◦). Section 5 is devoted to a comparison of our semicontinuity concepts with the classical outer and inner semicontinuity. Finally, in Section 6, we discuss the special case of concave (i.e. graph-convex) maps. 2 Preliminaries Throughout the paper we denote by F := F(Y ) the space of closed subsets of a finite dimen- sional normed vector space Y with dimension p ≥ 1. We start with some basic concepts with respect to Painlev´e–Kuratowski convergence (shortly PK-convergence), see also [4]. We frequently use the following notation of [4]: # N∞ := {N ⊂ N| N \ N finite} and N∞ := {N ⊂ N| N infinite} . For a sequence (An)n∈N ⊂ F the outer limit is the set © # N ª LIM SUP An := y ∈ Y | ∃N ∈ N , ∀n ∈ N, ∃yn ∈ An : yn −→ y . n→∞ ∞ 2 and the inner limit is the set © N ª LIM INF An := y ∈ Y | ∃N ∈ N∞, ∀n ∈ N, ∃yn ∈ An : yn −→ y . n→∞ A sequence (An)n∈N ⊂ F is PK-convergent to some A ∈ F if A = LIM SUPn→∞ An = PK LIM INFn→∞ An. Then we write A = LIMn→∞ An or An −→ A. In contrast to [4], we use capital letters in the notation of the (outer and inner) limit, because the notation with small letters is reserved for the (upper and lower) limit in the space C to be defined later on. The following characterization of outer and inner limits (see [4, Exercise 4.2.(b)]) is very important for the considerations in the next section. For a sequence (An)n∈N ⊂ F we have \ [ \ [ LIM SUP An = cl An, LIM INF An = cl An. n→∞ n→∞ N∈N n∈N # n∈N ∞ N∈N∞ Note that (F, ⊂) provides a complete lattice, i.e. every nonempty subset of F has a supremum and an infimum (denoted by SUP A and INF A). Of course, for a nonempty subset A ⊂ F we S T have SUP A = cl {A| A ∈ A} and INF A = {A| A ∈ A}. Further, we set INF ∅ = SUP F and SUP ∅ = INF F. Hence we can write LIM SUP An = INF SUP An, LIM INF An = INF SUP An. n→∞ N∈N n∈N n→∞ # n∈N ∞ N∈N∞ Throughout the paper let X = Rn, although many assertions are also valid in a more S T general context. The notations and stand for the union and intersection over xn→x¯ xn→x¯ all sequences converging tox ¯, respectively. In the following let f : X → F. The outer and inner limits of f atx ¯ ∈ X are defined, respectively, by [ \ LIM SUP f(x) := LIM SUP f(xn) and LIM INF f(x) := LIM INF f(xn). x→x¯ n→∞ x→x¯ n→∞ xn→x¯ xn→x¯ The limit of f atx ¯ exists if the outer and inner limits coincide. Then we write LIM f(x) = LIM SUP f(x) = LIM INF f(x). x→x¯ x→x¯ x→x¯ Note that the outer limit (and obviously also the inner limit) is always a closed subset of Y , see [4, Proposition 4.4]. Hence we can write 0 0 LIM SUP f(x ) = SUP INF SUP f(xn), LIM INF f(x ) = INF INF SUP f(xn). x0→x xn→x N∈N n∈N x0→x xn→x # n∈N ∞ N∈N∞ The function f is said to be outer semicontinuous (osc), inner semicontinuous (isc), contin- uous atx ¯ ∈ X if f(¯x) ⊃ LIM SUPx→x¯ f(x), f(¯x) ⊂ LIM INFx→x¯ f(x), f(¯x) = LIMx→x¯ f(x), respectively. If f is osc, isc, continuous at everyx ¯ ∈ X we just say f is osc, isc, continuous, respectively. The epigraph and the hypograph of f : X → F are defined, respectively, by epi f := {(x, A) ∈ X × F| A ⊃ f(x)} , hyp f := {(x, A) ∈ X × F| A ⊂ f(x)} . Note that, for all x ∈ X, we have (x, ∅) ∈ hyp f and (x, Y ) ∈ epi f. For a characterization of semicontinuity we need to know what is meant by closedness of the epigraph and hypograph. A subset A ⊂ X × F is said to be closed if for every sequence (xn,An)n∈N ⊂ A with 3 PK xn → x¯ ∈ X and An −→ A¯ ∈ F it is true that (¯x, A¯) ∈ A. The closure of a set A ⊂ X × F, denoted by cl A, is the set of all limits (¯x, A¯) ∈ X × F of sequences (xn,An)n∈N ⊂ A. From [4, Exercise 5.6 (c)] and [4, Theorem 5.7 (a)] we obtain the following characterization of outer semicontinuity hyp f is closed ⇔ f is osc ⇔ gr f ⊂ X × Y is closed. Likewise, by [4, Exercise 5.6 (d)], inner semicontinuity of f is equivalent to the closedness of the epigraph. Note that the description by the graph fails in this case, i.e. a function f : X → F that is isc has not necessarily a closed graph, see [4, Fig. 5–3. (b)]. Let us collect some basic properties of the outer semicontinuous hull of f, defined by 0 (osc f): X → F, (osc f)(x) := LIM SUPx0→x f(x ).
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