CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS

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1 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS Abstract. The aim of this paper is to characterize in terms of classical (quasi)convexity of extended real-valued functions the set-valued maps which are K-(quasi)convex with respect to a convex cone K. In particular, we recover some known characterizations of K-(quasi)convex vector-valued functions, given by means of the polar cone of K. 1. Introduction and Preliminaries The classical notions of convexity and quasiconvexity of real-valued functions have been extended to set-valued maps in various ways (see e.g. [2], [4] [6]). Two of them are of special interest and will be studied here. Recall that a set-valued map F : X 2 Y, defined on a vector space X with values in a vector space Y endowed with a convex cone K Y (i.e. K + K R + K K ), is said to be: (a) K-convex, if for all x 1, x 2 X and t [0, 1] we have tf (x 1 ) + (1 t)f (x 2 ) F (tx 1 + (1 t)x 2 ) + K, which means that F has a convex epigraph: epi (F ) = {(x, y) X Y : y F (x) + K}; Date: March 5, Mathematics Subject Classification. Primary 26B25; Secondary 47H04, 90C29. Key words and phrases. Convex and quasiconvex set-valued maps, scalarization, polar cones. 1

2 2 (b) K-quasiconvex, if for all x 1, x 2 X and t [0, 1] we have (F (x 1 ) + K) (F (x 2 ) + K) F (tx 1 + (1 t)x 2 ) + K, which means that for each y Y the following generalized level set is convex: F 1 (y K) = {x X : y F (x) + K}. Obviously, K-convex set-valued maps are K-quasiconvex, since the cone K is convex. Remark that Dom(F ) = {x X : F (x) } is convex whenever F is K-convex, but Dom(F ) is not necessary convex if F is K-quasiconvex. However, if K generates Y, i.e. K K = Y, then Dom(F ) is convex whenever F is K-quasiconvex. Indeed, in this case, Y is directed with respect to K, i.e. (y 1 + K) (y 2 + K) for all y 1, y 2 Y. Note that vector-valued functions may be studied in the same framework. Actually a function f : D Y, defined on a nonempty convex subset D of X, is K-convex (resp. K-quasiconvex) if the set-valued map F : X 2 Y, defined for all x X by {f(x)} if x D (1) F (x) = if x X \ D, is K-convex (resp. K-quasiconvex). Let ϕ : X R = R {, + } be an extended real-valued function. As usual in convex analysis, it is convenient to adopt the following convention: (+ ) +( ) = +, 0 (+ ) = + and 0 ( ) = 0. Recall that ϕ is said to be: (a ) convex, if for all x 1, x 2 X and t [0, 1] we have tϕ(x 1 ) + (1 t)ϕ(x 2 ) ϕ(tx 1 + (1 t)x 2 ),

3 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS 3 which means that ϕ has a convex epigraph: epi (ϕ) = {(x, λ) X R : ϕ(x) λ}; (b ) quasiconvex, if for all x 1, x 2 X and t [0, 1] we have ϕ(tx 1 + (1 t)x 2 ) max{ϕ(x 1 ), ϕ(x 2 )}, which means that for every λ R the following level set is convex: ϕ 1 ([, λ]) = {x X : ϕ(x) λ}. It is known that function ϕ : X R is convex if and only if its strict epigraph, i.e. the set epi s (ϕ) = {(x, λ) X R : ϕ(x) < λ}, is convex. Analogously, function ϕ is quasiconvex if and only if the strict level set ϕ 1 ([, λ[ ) = {x X : ϕ(x) < λ} is convex, for each λ R. By using these characterizations, we can state the following preliminary result, which gives necessary conditions for a set-valued map with values in R to be R + -convex, respectively R + -quasiconvex. Lemma 1.1. Let Φ : X 2 R be a set-valued map, defined on some vector space X, and let ϕ : X R be its marginal function, defined for all x X by ϕ(x) = inf G(x), where inf = +. Then ϕ is convex (resp. quasiconvex) whenever Φ is R + -convex (resp. R + -quasiconvex). Proof. Assume that Φ is R + -convex. In order to prove that ϕ is convex, it suffices to show that its strict epigraph is convex. To this end, let (x 1, λ 1 ), (x 2, λ 2 ) epi s (ϕ) and t [0, 1] be arbitrary. Then, for any numbers µ 1 and µ 2 such that inf Φ(x 1 ) < µ 1 < λ 1 and

4 4 inf Φ(x 2 ) < µ 2 < λ 2, we have µ 1 Φ(x 1 )+R + and µ 2 Φ(x 2 )+R +. Since Φ is R + -convex we have tµ 1 + (1 t)µ 2 tφ(x 1 ) + (1 t)φ(x 2 ) + R + Φ(tx 1 + (1 t)x 2 ) + R +, which yields inf Φ(tx 1 + (1 t)x 2 ) tµ 1 + (1 t)µ 2. Hence ϕ(tx 1 + (1 t)x 2 ) < tλ 1 + (1 t)λ 2, i.e. t(x 1, λ 1 ) + (1 t)(x 2, λ 2 ) epi s (ϕ). Now, assume that Φ is R + -quasiconvex. We will prove that ϕ is quasiconvex, by showing that its lower strict level sets are convex. Consider an arbitrary λ R. Let x 1, x 2 ϕ 1 ([, λ[ ) and let t [0, 1]. Then we have ϕ(x 1 ) < λ and ϕ(x 2 ) < λ, which show that for any number µ R with inf Φ(x 1 ) < µ < λ and inf Φ(x 2 ) < µ < λ we have µ (Φ(x 1 ) + R + ) (Φ(x 2 ) + R + ). Since Φ is R + -quasiconvex, we can deduce that µ Φ(tx 1 + (1 t)x 2 ) + R +, which yields inf Φ(tx 1 + (1 t)x 2 ) µ < λ. Hence ϕ(tx 1 + (1 t)x 2 ) < λ, i.e. tx 1 + (1 t)x 2 ϕ 1 ([, λ[ ). Remark 1.2. The converse assertions in Lemma 1.1 are not true in general. Indeed, if we consider the set-valued map Φ : X = R 2 R, defined by Φ(0) = ]0, + [ and Φ(x) = [0, + [ for all x R \ {0}, then ϕ(x) = 0 for all x R. In this case ϕ is convex hence quasiconvex, but Φ is not R + -quasiconvex and therefore not R + -convex, since the level set Φ 1 (0 R + ) = R \ {0} is not convex. In the next sections we will characterize the K-(quasi)convex set-valued maps F in terms of usual (quasi)convexity of certain extended real-valued functions. In order to get these characterizations, we have to endow the image space Y with a good enough linear topology, and we need to impose some additional hypotheses on the partial order induced by K and also on the structure of the values of F. Supposing that Y is a topological

5 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS 5 vector space, partially ordered by a convex cone K, we denote by K + = {l Y : l(y) 0, y K} the nonnegative polar cone of K in the topological dual Y of Y. By extd K + we denote the set of extreme directions of K +. Recall that l extd K + if and only if l K + \ {0} and for all l 1, l 2 K + such that l = l 1 + l 2 we actually have l 1, l 2 R + l. For any set-valued map F : X 2 Y and for every l Y, we denote by l F : X 2 R the composite set-valued map given for all x X by l F (x) = l(f (x)), and we denote by l F : X R the marginal function of l F, defined for all x X by l F (x) = inf l(f (x)). 2. Characterization of Convex Set-Valued Maps The following preliminary result is a set-valued version of a classical property regarding the composition of convex functions (see e.g. Proposition 6.8 in [6]). Proposition 2.1. Let F : X Y be a set-valued map defined on a vector space X with values in a vector space Y, partially ordered by a convex cone K, and let ϕ : Y R be a real-valued function. If F is K-convex and ϕ is convex and nondecreasing with respect to K, i.e. ϕ(y) ϕ(z) for all y, z Y with z y + K, then the set-valued map ϕ F : X 2 R, defined for all x X by ϕ F (x) = ϕ(f (x)), is R + -convex.

6 6 Proof. We will prove that tϕ(f (x 1 ))+(1 t)ϕ(f (x 2 )) ϕ(f (tx 1 +(1 t)x 2 ))+R + for all x 1, x 2 X and t [0, 1]. Let x 1, x 2 X, t [0, 1] and λ tϕ(f (x 1 )) + (1 t)ϕ(f (x 2 )). Then λ = tϕ(y 1 ) + (1 t)ϕ(y 2 ) for some y 1 F (x 1 ) and y 2 F (x 2 ). Since F is K-convex we have ty 1 + (1 t)y 2 tf (x 1 ) + (1 t)f (x 2 ) F (tx 1 + (1 t)x 2 ) + K, which imply the existence of some y F (tx 1 + (1 t)x 2 ) such that ty 1 + (1 t)y 2 y + K. Taking into account that ϕ is nondecreasing, we can deduce that ϕ(y) ϕ(ty 1 + (1 t)y 2 ). On the other hand, ϕ being convex, we have ϕ(ty 1 + (1 t)y 2 ) tϕ(y 1 ) + (1 t)ϕ(y 2 ). Hence ϕ(y) λ, which yields λ ϕ(y) + R + ϕ(f (tx 1 + (1 t)x 2 )) + R +. The main theorem of this section is an extension of a well-known characterization of K-convex vector-valued functions (see e.g. Proposition 6.2 in [6]). Theorem 2.2. Let X be a vector space and let Y be a locally convex space over reals, partially ordered by a convex cone K. If F : X 2 Y is a set-valued map such that F (x) + K is closed convex for all x X, then the following assertions are equivalent: (C1) F is K-convex; (C2) l F is R + -convex, for every l K + ; (C3) l F is convex, for every l K +. Proof. Since functionals belonging to K + are nondecreasing with respect to K, the implication (C1) = (C2) directly follows by Proposition 2.1. The implication (C2) = (C3) also holds by virtue of Lemma 1.1. In order to prove the implication (C3) = (C1), assume that (C3) holds and suppose to the contrary that tf (x 1 ) + (1 t)f (x 2 ) F (tx 1 + (1 t)x 2 ) + K for some x 1, x 2 X

7 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS 7 and t [0, 1]. Then we can find some y 1 F (x 1 ) and y 2 F (x 2 ) such that (2) ty 1 + (1 t)y 2 F (tx 1 + (1 t)x 2 ) + K. Since y 1 F (x 1 ) and y 2 F (x 2 ), by (C3) we can deduce that (3) inf l(f (tx 1 + (1 t)x 2 ) tl(y 1 ) + (1 t)l(y 2 ) for every l K +. The cone K + being nonempty it follows by (3) that F (tx 1 + (1 t)x 2 ) is nonempty. Recalling (2) and applying the Tukey s Theorem for separating the nonempty disjoint convex sets: {ty 1 + (1 t)y 2 } (which is compact) and F (tx 1 + (1 t)x 2 ) + K (which is closed), we infer the existence of a functional l Y \ {0} such that l(ty 1 + (1 t)y 2 ) < inf l(f (tx 1 + (1 t)x 2 ) + K) = inf l(f (tx 1 + (1 t)x 2 )) + inf l(k). Then it can be easily seen that l K +, and by (3) we infer that inf l(f (tx 1 +(1 t)x 2 ) t l(y 1 )+(1 t) l(y 2 ) = l(ty 1 + (1 t)y 2 ) < inf l(f (tx 1 + (1 t)x 2 )), i.e. a contradiction. Remark 2.3. Note that F (x) + K is closed convex for all x X in the particular case when F has compact convex values and the convex cone K is closed. In this case, the equivalence (C2) (C3) obviously holds, since epi (l F ) = epi (l F ) for all l K Characterization of Quasiconvex Set-Valued Maps In absence of a characterization of K-quasiconvex vector-valued functions similar to Theorem 2.2, some authors have restricted their study to a more restrictive notion, the so-called -quasiconvexity (see e.g. [3] and [4]) or scalarly-quasiconvexity (see e.g. [7]), by taking as definition the following variant of (C3): (Q ) l F is quasiconvex, for every l K +.

8 8 As shown by Kuroiwa in [4], under the hypotheses of Theorem 2.2, any set-valued map F which satisfies condition (Q ) is K-quasiconvex. However, condition (Q ) is very restrictive and far to characterize K-quasiconvexity, essentially because the sum of quasiconvex functions is usually not quasiconvex, in contrast with the convex case. In fact, as we shall see below, for characterizing K-quasiconvexity we just need to relax condition (Q ) by considering only the extreme directions l of the polar cone K + (see condition (Q3) in Theorem 3.3). Such a characterization was firstly established by Dinh The Luc in [6] in the particular case where Y is finite-dimensional and K is a polyhedral cone generated by an algebraic base of Y. Actually, as shown in [1], this characterization is still true if Y is a Banach space and K is a closed convex cone satisfying certain additional assumptions, weaker than polyhedrality. Our aim here is to extend this result for set-valued maps. To this end, we will use the following preliminary result, the proof of which can be found in [1]. It gives a weak lattice property. Lemma 3.1. Let Y be a Banach space, partially ordered by a closed convex cone K which generates Y. Let l extd K + and v 1, v 2 Y be such that l(v 1 ) 0 and l(v 2 ) 0. Then, for each ε > 0, there exists w ε (v 1 + K) (v 2 + K) with l(w ε ) ε. Proposition 3.2. Let X be a vector space and let Y be a Banach space, partially ordered by a closed convex cone K which generates Y. Let F : X 2 Y be a K-quasiconvex set-valued map. Then, for every l extd K +, the set-valued map l F is R + -quasiconvex. Proof. Let l extd K +. Let any λ R. We have to prove that the set (l F ) 1 (λ R + ) is convex. To this end, let x 1, x 2 (l F ) 1 (λ R + ) and t [0, 1] be arbitrary. On

9 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS 9 one hand, there exist some points y 1 F (x 1 ) and y 2 F (x 2 ) such that l(y 1 ) λ and l(y 2 ) λ. On the other hand, we can write λ = l(z) for some z Y since l 0. Hence l(y 1 ) l(z) and l(y 2 ) l(z), which means that l(y 1 z) 0 and l(y 2 z) 0. By applying Lemma 3.1 to v 1 = y 1 z and v 2 = y 2 z, it follows that for every ε > 0 there is some w ε (y 1 z + K) (y 2 z + K) such that l(w ε ) ε. Recalling that y 1 F (x 1 ) and y 2 F (x 2 ), we can deduce that w ε + z (F (x 1 ) + K) (F (x 2 ) + K). Further, by K-quasiconvexity of F, we infer w ε + z F (tx 1 + (1 t)x 2 ) + K. This means that there exists some y F (tx 1 + (1 t)x 2 ) such that w ε + z y + K, i.e. w ε + z y K. Then we have l(w ε + z y) 0, which yields l(y) l(z) + l(w ε ) λ + ε. Since ε > 0 was arbitrary, we can conclude that l(y) λ. So, we have found y F (tx 1 + (1 t)x 2 ) with l(y) λ, which shows that tx 1 + (1 t)x 2 (l F ) 1 (λ R + ). We are now ready to state the main theorem of this section. Theorem 3.3. Let X be a vector space and let Y be a Banach space, partially ordered by a closed convex cone K. Assume that K generates Y and K + is the weak-star closed convex hull of extd K +. Let F : X 2 Y be a set-valued map such that, for all x Dom(F ), there is a smallest element f(x) in F (x) with respect to K, i.e. F (x) f(x) + K. Then the following assertions are equivalent: (Q1) F is K-quasiconvex; (Q2) l F is R + -quasiconvex, for every l extd K + ; (Q3) l F is quasiconvex, for every l extd K +. Proof. The implications (Q1) = (Q2) = (Q3) hold by Proposition 3.2 and Lemma 1.1.

10 10 In order to prove the implication (Q3) = (Q1), assume that (Q3) holds and consider some arbitrary x 1, x 2 X and t [0, 1]. Let any y (F (x 1 ) + K) (F (x 2 ) + K). We need to prove that y F (tx 1 + (1 t)x 2 ) + K. To this end, firstly observe that by (Q3) it follows that for each l extd K + there is some y l F (tx 1 + (1 t)x 2 ) such that l(y l ) max {inf l(f (x 1 )), inf l(f (x 1 ))}. On the other hand, notice that inf l(f (x 1 )) l(y) and inf l(f (x 1 )) l(y) since y (F (x 1 )+K) (F (x 2 )+K). Hence we have l(y l ) l(y) for all l extd K +. Now, taking into account that {y l : l extd K + } F (tx 1 + (1 t)x 2 ), and recalling that f(tx 1 + (1 t)x 2 ) denotes the smallest element of the set F (tx 1 + (1 t)x 2 ), we infer that l(f(tx 1 + (1 t)x 2 )) l(y), i.e. l(y f(tx 1 + (1 t)x 2 )) 0, for all l extd K +. Since K + is the weak-star closed convex hull of extd K +, we can deduce that l(y f(tx 1 +(1 t)x 2 )) 0, for all l K +, which means that y f(tx 1 +(1 t)x 2 ) K ++. Finally, by Bipolar Theorem, we can conclude that y f(tx 1 + (1 t)x 2 ) K, hence y f(tx 1 + (1 t)x 2 ) + K F (tx 1 + (1 t)x 2 ) + K. Remark 3.4. If the interior of K is nonempty, then K + has a bounded hence weak-star compact base and K generates Y. However, these properties may hold even if the interior of K is empty, as in the particular case where Y = l p (1 p < + ) is partially ordered by the cone K = l p + = {(y i ) i N l p : y i 0, i N}. Indeed, in this case we have K + = l+, q where Y = (l p ) is identified, as usual, with l q (1/p + 1/q = 1). Remark 3.5. In the particular case where f : D Y is a vector-valued function, defined on some nonempty convex subset D of X, and the set-valued map F : X 2 Y is given by formula (1), then f(x) is actually the smallest element of F (x), for each x Dom(F ) = D.

11 CHARACTERIZATION OF (QUASI)CONVEX SET-VALUED MAPS 11 For this reason, Theorem 3.3 extends the characterization Theorem 3.1 in [1] and indeed Proposition 6.5 in [6], both of them being given for vector-valued K-quasiconvex functions. Note also that Theorem 3.3 can be derived as a consequence of Theorem 3.1 in [1]. Indeed, it can be easily seen that the vector-valued function f : Dom(F ) Y, which assigns to each x Dom(F ) the smallest element f(x) of F (x), is K-quasiconvex if and only if the set-valued map F is K-quasiconvex. Remark 3.6. The existence of a smallest element in each nonempty value of F is essential for the implication (Q3) = (Q1) in Theorem 3.3, as shown by the following example. Let X = R and let Y = R 2 be partially ordered by K = R 2 +. Then extd K + = a>0 { l 1 a, l 2 a}, where l 1 a (y) = ay 1 and l 2 a(y) = ay 2 for all y = (y 1, y 2 ) Y. Consider the set-valued map F : X 2 Y defined for all x X by F (x) = ( 1, 0) if x < 0 {(t, 1 t) : t [0, 1]} if x = 0 (0, 1) if x > 0. It can be easily seen that, for each real number a > 0, we have a if x < 0 l 1 a F (x) = 0 if x 0 0 if x 0 and l 2 a F (x) = a if x > 0. Obviously, the functions l 1 a F and l 2 a F are quasiconvex for all a > 0, i.e. (Q3) holds. But (Q1) is not true, since for y = 0 the set F 1 (y K) = R \ {0} is not convex.

12 12 References [1] Benoist, J., Borwein, J. M., and Popovici, N.: A Characterization of Quasiconvex Vector-Valued Functions, Simon Fraser University of Burnaby, CECM Preprints 01:170 (2001). [2] Borwein, J. M.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints, Math. Programming 13 (1977), [3] Jeyakumar, V., Oettli, W., and Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl. 179 (1993), [4] Kuroiwa, D.: Convexity for set-valued maps, Appl. Math. Lett. 9 (1996), [5] Li, S. J., Chen, G. Y., and Lee, G. M.: Minimax theorems for set-valued mappings, J. Optim. Theory Appl. 106 (2000), [6] Luc, D. T.: Theory of vector optimization, Springer-Verlag, Berlin, [7] Sach, P. H.: Characterization of scalar quasiconvexity and convexity of locally Lipschitz vector-valued maps, Optimization 46 (1999), LACO, UPRESSA 6090, Department of Mathematics, University of Limoges, Limoges, France address: benoist@unilim.fr Faculty of Mathematics and Computer Science, Babeş-Bolyai University of Cluj, 3400 Cluj-Napoca, Romania address: popovici@math.ubbcluj.ro