Characterizations of Nonsmooth Robustly Quasiconvex Functions

Characterizations of Nonsmooth Robustly Quasiconvex arXiv:1804.06983v1 [math.FA] 19 Apr 2018 Functions Hoa T. Bui∗, Pham Duy Khanh†,Tran Thi Tu Trinh‡ April 20, 2018 Abstract Two criteria for the robust quasiconvexity of lower semicontinuous functions are established in terms of Fréchet subdifferentials in Asplund spaces. Keywords Quasiconvexity, robust quasiconvexity, quasimonotone, Fréchet subdifferential, approximate mean value theorem Mathematics Subject Classification (2010) 26A48, 26A51, 49J52, 49J53 1 Introduction The question of characterizing convexity and generalized convexity properties in terms of subdifferentials receives tremendous attention in optimization theory and variational analysis. For decades, there have been received many significant contributions devoted to this question such as [8, 9, 11, 14, 16] for convex functions, [2, 3, 4, 6, 12, 13] for quasiconvex functions and [6] for robustly quasiconvex functions. This paper follows this stream of research. Our aim is to establish the first-order characterizations for the robust quasiconvexity of lower semicontinuous functions in Asplund spaces. First, some existing results regarding to the properties of subdifferential operators of convex, quasiconvex functions are recalled in Section 2, where the definitions and some basic results are given as well. Besides, necessary and sufficient first-order conditions for a lower semicontinuous function to be quasiconvex are reconsidered. Those characterizations moreover could be used to characterize the Asplund property of the given space. Second, two criteria for the robust quasiconvexity of lower semicontinuous functions in Asplund spaces are obtained by using Fréchet subdifferentials in Section 3. Each criterion corresponds to each type of analogous conditions for quasiconvexity. The first one is based on the zero and first order condition for quasiconvexity (see Theorem 2.2(b) in Section 2). It extends [6, Proposition 5.3] from finite dimensional spaces to Asplund spaces. Moreover, its proof also overcomes a glitch in the proof of the sufficient condition of [6, Proposition 5.3]. The second criterion is totally new. It is settled from the equivalence of the quasiconvexity of lower semicontinuous functions and the quasimonotonicity of their subdifferential operators (see Theorem 2.2(c) in Section 2). ∗ Centre for Informatics and Applied Optimization, Faculty of Science and Technology, Federation University Australia, POB 663, Ballarat, Vic, 3350, Australia. E-mail: [email protected] † Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam and Center for Mathematical Modeling, Universidad de Chile, Beauchef 851, Edificio Norte, Piso 7, Santiago, Chile. E-mails: [email protected]; [email protected] ‡ Department of Mathematics and Statistics, Oakland University, 318 Meadow Brook Rd, Rochester, MI 48309, USA. Email: [email protected] 1 2 Preliminaries Let X be a Banach space and X ∗ its dual space. X is called an Asplund space, or has the Asplund property, if every separable subspace Y of X has separable continuous dual space Y ∗ . The duality pairing on X × X ∗ is denoted by h., .i. In what follows, R :=] − ∞, ∞]; Br (x) is the open ball of radius r > 0 centered at x ∈ X and B∗ ⊂ X ∗ is the closed ball of radius 1 centered at 0X ∗ . The extended real-valued function ϕ : X → R considered mostly is proper lower semicontinuous (l.s.c), i.e. ϕ is not identically +∞, and the lower level sets ϕ≤ α := {x ∈ X : ϕ(x) ≤ α} are closed for all α ∈ R. As usual domϕ stands for the domain of ϕ, defined as domϕ := {x ∈ X : ϕ(x) < ∞}. For a set-valued mapping A : X ⇒ X ∗ , the domain of A is written domA := {x ∈ X : A(x) 6= ∅}. The graphs of ϕ and A are respectively defined as graphϕ := {(x, α) ∈ X × R : ϕ(x) = α}, graphA := {(x, x∗ ) ∈ X × X ∗ : x∗ ∈ A(x)}. A subset U of X is convex if it contains all closed segments connecting two points in U . The function ϕ is said to be convex if the domain of ϕ is convex and for any α ∈ [0, 1], x, y ∈ domϕ we always have the inequality ϕ(αx + (1 − α)y) ≤ αϕ(x) + (1 − α)ϕ(y). As usual, the Fréchet subdifferential of a proper lower semicontinuous function ϕ is the set-valued mapping b : X ⇒ X ∗ defined by ∂ϕ   ϕ(y) − ϕ(x) − hx∗ , y − xi ∗ ∗ b ≥ 0 , for all x ∈ domϕ. ∂ϕ(x) := x ∈ X : lim inf y→x ky − xk When ϕ is convex, the Fréchet subdifferential reduces to the convex analysis subdifferential b ∂ϕ(x) = ∂ϕ(x) := {x∗ ∈ X ∗ : hx∗ , y − xi ≤ ϕ(y) − ϕ(x)}, for all x ∈ domϕ. An operator A is monotone if for all x, y ∈ domA, one has hx∗ − y ∗ , x − yi ≥ 0 with x∗ ∈ A(x), y ∗ ∈ A(y). It b is monotone [16]. The inverse implication also holds is well-known that when ϕ is convex, the operator ∂ϕ in Asplund space [11, Theorem 3.56]; but it is not true in general Banach spaces. The reader is referred to the proof of the reverse implication in [10, Theorem 2.4] for a counter-example. Let us recall some notions of generalized convex functions. Definition 2.1 A function ϕ : X → R is 1. quasiconvex if ∀x, y ∈ X, λ ∈]0, 1[, f (λx + (1 − λ)y) ≤ max{f (x), f (y)}. (1) 2. α-robustly quasiconvex with α > 0 if, for every v ∗ ∈ αB∗ , the function ϕv∗ : x 7→ ϕ(x) + hv ∗ , xi is quasiconvex. Clearly, ϕ is α-robustly quasiconvex iff the function ϕv∗ is quasiconvex for all v ∗ ∈ X ∗ such that kv ∗ k < α. Tracing back to the original definition of robustly quasiconvex functions, they were first defined in [15] under the name “s-quasiconvex” or “stable quasiconvex”, and then renamed “robustly quasiconvex” in [5]. 2 This class of functions holds a notable role, as many important optimization properties of generalized convex functions are stable when disturbed by a linear functional with a sufficiently small norm (for instance, all lower level sets are convex, each minimum is global minimum, each stationary point is a global minimizer). For interested readers, we refer to [15] again, and further related works [1, 5]. Definition 2.2 An operator A : X ⇒ X ∗ is quasimonotone if for all x, y ∈ X and x∗ ∈ A(x), y ∗ ∈ A(y) we have min{hx∗ , y − xi, hy ∗ , x − yi} ≤ 0. Significant contributions concerning dual criteria for quasiconvex functions are in [2, 4]. Those characterizations are applicable for a wide range of subdifferentials, for instance Rockafellar-Clarke subdifferentials in Banach spaces, and Fréchet subdifferentials in reflexive spaces. These results are still unclear for Fréchet subdifferentials in Asplund spaces. Below, we give a short proof to clarify this. Our proof relies on the proof scheme of [2] and the following approximate mean value theorem [11, Theorem 3.49]. Theorem 2.1 Let X be an Asplund space and ϕ : X → R be a proper lower semicontinuous function finite at two given points a = 6 b. Consider any point c ∈ [a, b) at which the function ψ(x) := ϕ(x) − ϕ(b) − ϕ(a) kx − ak ka − bk ϕ b attains its minimum on [a, b]; such a point always exists. Then, there are sequences xk → c and x∗k ∈ ∂ϕ(x k) satisfying ϕ(b) − ϕ(a) lim inf hx∗k , b − xk i ≥ kb − ck, (2) k→∞ ka − bk lim inf hx∗k , b − ai ≥ ϕ(b) − ϕ(a). (3) lim hx∗k , b − ai = ϕ(b) − ϕ(a). (4) k→∞ Moreover, when c 6= a one has k→∞ Theorem 2.1 allows us to deduce the following three-points lemma which is similar to [3, Lemma 3.1]. Lemma 2.1 Let ϕ : X → R be a proper, lower semicontinuous function on an Asplund space X. Let b u, v, w ∈ X such that v ∈ [u, w], ϕ(v) > ϕ(u) and λ > 0. Then, there are x̄ ∈ domϕ and x̄∗ ∈ ∂ϕ(x̄) such that x̄ ∈ Bλ ([u, v]) and hx̄∗ , w − x̄i > 0, where Bλ ([u, v]) := {x ∈ X : ∃y ∈ [u, v] such that kx − yk < λ}. We are in position to establish characterizations of quasiconvexity in terms of Fréchet subdifferentials in Asplund spaces. Theorem 2.2 Let ϕ : X → R be a proper lower semicontinuous function on an Asplund space X. The following statements are equivalent (a) ϕ is quasiconvex; b (b) If there are x, y ∈ X such that ϕ(y) ≤ ϕ(x), then hx∗ , y − xi ≤ 0 for all x∗ ∈ ∂ϕ(x). b is quasimonotone. (c) ∂ϕ 3 b Proof. (a)⇒(b) Assume that x, y ∈ X, ϕ(x) ≥ ϕ(y), and x∗ ∈ ∂ϕ(x). Consider Sx := {u ∈ X : ϕ(u) ≤ ϕ(x)}. Since ϕ is quasiconvex, then Sx is a convex set. Thus, we have the function f := δSx + ϕ(x) is convex, where δSx is equal to 0 for u ∈ Sx and to ∞ otherwise. On the other hand, f (x) = ϕ(x) and f (u) ≥ ϕ(u) b b (x). By the definition of convex subdifferential, since x∗ ∈ ∂ϕ(x) b b (x), for all u ∈ X, thus ∂ϕ(x) ⊂ ∂f ⊂ ∂f ∗ we have hx , y − xi ≤ 0. b b (b)⇒(c) Assume that there are x, y ∈ X and x∗ ∈ ∂ϕ(x), y ∗ ∈ ∂ϕ(y) such that hx∗ , x − yi < 0 and hy ∗ , x − yi > 0. Then, by (b), ϕ(x) < ϕ(y) and ϕ(y) < ϕ(x), which is a contradiction. (c)⇒(a) By using Lemma 2.1, the proof of this assertion is similar to one in [2, Theorem 4.1]. Remark 2.1 Observe that the implications (a) ⇒ (b) and (b) ⇒ (c) hold in Banach spaces while (c) ⇒ (a) only holds in Asplund spaces. In fact, the equivalence of these statements actually can characterize the Asplund property in the sense that if X is not an Asplund space, then there is a function ϕ whose Fréchet subdiferential satisfies (b) and (c) but is not quasiconvex. Such a function ϕ can be found in [10, Theorem 2.4]. 3 Characterizations of Robustly Quasiconvex Functions A zero and first order characterization of robust convexity was given in [6, Proposition 5.3] for finite dimensional spaces. We remark that there is an oversight in the proof given there; although the function f is only assumed to be lower semicontinuous, the existence of z in the second paragraph actually requires continuity. Here we show that this conclusion is still correct not only when f is assumed just to be lower semicontinuous, but also when X is only assumed to be an Asplund space. To derive this generalization, we need the following lemmas, revealing that quasiconvex functions have certain nice properties which resemble those of convex functions. Lemma 3.1 If ϕ : X → R is a quasiconvex and lower semicontinuous function, and u, v ∈ X are such that ϕ(v) ≥ ϕ(u) then lim ϕ(v + t(u − v)) = ϕ(v). (5) t↓0 Proof. Suppose that u, v ∈ X and that ϕ(v) ≥ ϕ(u). Since ϕ is quasiconvex, for all t ∈]0, 1[, we have ϕ(v + t(u − v)) ≤ max{ϕ(v), ϕ(u)} = ϕ(v). It follows that lim supt↓0 ϕ(v + t(u − v)) ≤ ϕ(v). Combining the latter with the lower semicontinuity of ϕ we get (5). ✷ Lemma 3.2 Let ϕ : X → R be a quasiconvex function and u, v, w ∈ X such that v ∈]u, w[, ϕ(u) ≤ ϕ(w). Suppose that there exist v ∗ ∈ X ∗ and z ∈]u, v[ such that ϕv∗ (z) > max{ϕv∗ (u), ϕv∗ (w)}. Then ϕ(u) < ϕ(z) ≤ ϕ(v) ≤ ϕ(w). (6) Proof. Since z ∈]u, v[⊂]u, w[, ϕ(u) ≤ ϕ(w) and ϕ is quasiconvex we have ϕ(z) ≤ max{ϕ(u), ϕ(w)} = ϕ(w). Hence, the latter and the inequality ϕv∗ (z) > ϕv∗ (w) implies that hv ∗ , zi > hv ∗ , wi. Again, z ∈]u, w[ implies hv ∗ , zi < hv ∗ , ui. Therefore, the inequality ϕv∗ (u) < ϕv∗ (z) yields ϕ(u) < ϕ(z). Since z ∈]u, v[ and v ∈]z, w[, we deduce ϕ(z) ≤ ϕ(v) ≤ ϕ(w) from the latter inequality and the quasiconvexity of ϕ. Hence, (6) holds. ✷ Lemma 3.3 Let ϕ : X → R be a quasiconvex, proper, and lower semicontinuous function, and v ∗ ∈ X ∗ . If ϕv∗ is not quasiconvex then there exist u, v, w ∈ X such that v ∈]u, w[ and ϕ(w) ≥ ϕ(v) > ϕ(u), (7) ϕv∗ (v) > max{ϕv∗ (u), ϕv∗ (w)}, (8) ∀γ > 0, ∃vγ ∈ Bγ (v)∩]v, w[ : ϕv∗ (v) > ϕv∗ (vγ ). (9) 4 Proof. Since ϕv∗ is not quasiconvex, there exist u, w ∈ X such that u 6= w, ϕ(u) ≤ ϕ(w) and v0 ∈]u, w[ such that ϕv∗ (v0 ) > max{ϕv∗ (u), ϕv∗ (w)}. Applying Lemma 3.1, we get limt↓0 ϕ(w + t(u − w)) = ϕ(w), and so limt↓0 ϕv∗ (w + t(u − w)) = ϕv∗ (w). Since ϕv∗ (w) < ϕv∗ (v0 ), there exists t0 ∈]0, 1[ such that ϕv∗ (w + t(u − w)) < ϕv∗ (v0 ), ∀t ∈]0, t0 [. (10) Consider the set L := {z ∈]u, w[: ϕv∗ (z) ≥ ϕv∗ (v0 )}. Clearly, L 6= ∅ and for each z ∈ L we have kz − wk ≥ t0 ku − wk by (10). It follows that r := inf{kz − wk : z ∈ L } ∈ [t0 ku − wk, ku − wk[ ⊂ ]0, ku − wk[, v := w + r u−w ∈ ]u, w[. ku − wk We will show that v ∈ L and so (8) holds. Suppose on the contrary that v ∈ / L . Then v0 ∈]u, v[ and we get ϕ(u) < ϕ(v0 ) ≤ ϕ(v) ≤ ϕ(w) by Lemma 3.2. Applying Lemma 3.1, we get limt↓0 ϕ(v + t(u − v)) = ϕ(v), and so limt↓0 ϕv∗ (v + t(u − v)) = ϕv∗ (v). By the definition of r, there exists a sequence (zn ) ⊂ L such that kzn − wk → r and kzn − wk > r for all n ∈ N. Therefore, ϕv∗ (v) = lim ϕv∗ (v + t(u − v)) t↓0   kzn − wk − r lim ϕv∗ v + (u − v) ku − vk   r kzn − wk (u − v) + (u − v) lim ϕv∗ v − ku − vk ku − vk   kzn − wk r (u − w) + (u − w) lim ϕv∗ v − ku − wk ku − wk   kzn − wk lim ϕv∗ w + (u − w) ku − wk lim ϕv∗ (zn ) ≥ ϕv∗ (v0 ), = = = = = which is a contradiction. Now we show that v satisfies (9). Let γ be any positive real number and vγ := w + r − rγ (u − w) with rγ := min{r/2, γ/2} > 0. ku − wk Since 0 < r − rγ < r < ku − w|, it implies that vγ ∈]v, w[ \ L . Therefore, ϕv∗ (vγ ) < ϕv∗ (v0 ) ≤ ϕv∗ (v). Furthermore, kvγ − vk = w+ r − rγ u−w (u − w) − w − r = rγ < γ. ku − wk ku − wk Hence, v satisfies (9). ✷ Theorem 3.1 Let ϕ : X → R be a proper lower semicontinuous function on a Banach space X, and α > 0. Consider the following statements (a) ϕ is α−robustly quasiconvex; (b) For every x, y ∈ X b ϕ(y) ≤ ϕ(x) =⇒ hx∗ , y − xi ≤ − min {αky − xk, ϕ(x) − ϕ(y)} , ∀x∗ ∈ ∂ϕ(x). 5 (11) Then (a)⇒(b). Additionally, if X is an Asplund space, then (b)⇒(a). Proof. Suppose that ϕ is α−robustly quasiconvex, and x, y ∈ X satisfy ϕ(y) ≤ ϕ(x). Assume that ∗ b x ∈ ∂ϕ(x). We will prove hx∗ , y − xi ≤ − min {αky − xk, ϕ(x) − ϕ(y)} . If x = y, the above inequality is trivial. Otherwise, we consider two cases: Case 1. αky − xk ≤ ϕ(x) − ϕ(y) We then need to prove that hx∗ , y − xi ≤ −αky − xk. (12) By the Hahn-Banach theorem, there exists v ∗ ∈ X ∗ , kv ∗ k = 1 such that hv ∗ , y − xi = ky − xk. Consider the function f : X → R given by f (z) = ϕ(z) + αhv ∗ , z − xi ∀z ∈ X. Then f (x) = ϕ(x), and f (y) = ϕ(y) + αhv ∗ , y − xi = ϕ(y) + αky − xk ≤ ϕ(x) = f (x), i.e., max{f (x), f (y)} = f (x). Since ϕ is α−robustly quasiconvex, f is quasiconvex. Therefore for each t ∈ [0, 1], we always have ϕ(x) = f (x) = max{f (x), f (y)} ≥ f (x + t(y − x)) = ϕ(x + t(y − x)) + tαhv ∗ , y − xi = ϕ(x + t(y − x)) + tαky − xk, which implies that ϕ(x) − tαky − xk ≥ ϕ(x + t(y − x)). b Since x∗ ∈ ∂ϕ(x), for any γ > 0, there exists a number r > 0 such that ϕ(z) ≥ ϕ(x) + hx∗ , z − xi − γkz − xk ∀z ∈ Br (x). (13) (14) Let t ∈]0, 1[ such that x + t(y − x) ∈ Br (x). It follows from (13) and (14) that ϕ(x) − tαky − xk ≥ ϕ(x) + thx∗ , y − xi − tγky − xk, and so hx∗ , y − xi ≤ −αky − xk + γky − xk. On taking limit on both sides of the above inequality as γ → 0+ , we get (12). Case 2. αky − xk > ϕ(x) − ϕ(y) We have ᾱky − xk = ϕ(x) − ϕ(y), where ᾱ := ϕ(x) − ϕ(y) ∈]0, α[. ky − xk Since ϕ is ᾱ−robustly quasiconvex, we derive from Case 1 that hx∗ , y − xi ≤ −ᾱky − xk = ϕ(y) − ϕ(x) = − min {αky − xk, ϕ(x) − ϕ(y)} . 6 (15) Conversely, assume that X is Asplund, and (b) holds. It follows from Theorem 2.2 that ϕ is quasiconvex. Suppose that ϕ is not α−robustly quasiconvex, i.e., there exists v ∗ ∈ X ∗ \ {0}, kv ∗k < α such that ϕv∗ is not quasiconvex. By Lemma 3.3, there are u, w ∈ X and v ∈]u, w[ satisfying (7),(8), and (9). Since ϕv∗ (v) > ϕv∗ (u), there exists δ > 0 such that v̄ ∗ := (1 + δ)v ∗ satisfies kv̄ ∗ k < α and ϕv̄∗ (v) > ϕv̄∗ (u). Thus, we have ϕ(v) > ϕ(u), ϕv∗ (v) > ϕv∗ (u), ϕv̄∗ (v) > ϕv̄∗ (u) and the lower semicontinuity of ϕ, ϕv∗ , and ϕv̄∗ . This implies the existence of γ > 0 satisfying ϕ(z) > ϕ(u), ϕv∗ (z) > ϕv∗ (u), ϕv̄∗ (z) > ϕv̄∗ (u) ∀z ∈ Bγ (v). (16) By the assertion (9), there is vγ ∈ Bγ (v)∩]v, w[ such that ϕv∗ (v) > ϕv∗ (vγ ). Then, vγ can be written as    γ . vγ := v + λ(w − v) with λ ∈ 0, min 1, kw − vk Since ϕv∗ (v) > ϕv∗ (w) and ϕ(v) ≤ ϕ(w), we have hv ∗ , w − vi < 0 and so ϕv̄∗ (vγ ) − ϕv̄∗ (v) = = ϕv∗ (vγ ) − ϕv∗ (v) + δhv ∗ , vγ − vi ϕv∗ (vγ ) − ϕv∗ (v) + δλhv ∗ , w − vi < 0. b v̄∗ (x) Applying Lemma 2.1 for ϕv̄∗ , v ∈ [vγ , u] with ϕv̄∗ (v) > ϕv̄∗ (vγ ), there exist x ∈ domϕv̄∗ and x∗ ∈ ∂ϕ such that x ∈ [vγ , v] + (r − kvγ − vk)B and hx∗ , u − xi > 0. (17) Then x ∈ Bγ (v) and so ϕ(x) > ϕ(u) by (16). By the assumption (b) and the second inequality of (17), −hv̄ ∗ , u − xi < hx∗ − v̄ ∗ , u − xi ≤ − min{αku − xk, ϕ(x) − ϕ(u)}. Since hv̄ ∗ , u − xi ≤ kv̄ ∗ kku − xk < αku − xk, the above inequality implies that hv̄ ∗ , u − xi > ϕ(x) − ϕ(u), i.e., ϕv̄∗ (x) < ϕv̄∗ (u) and this contradicts (16). ✷ We next construct a completely new characterization for the robust quasiconvexity. It is based on the equivalence of the quasiconvexity of a lower semicontinuous function and the quasimonotonicity of its subdifferential operator. Theorem 3.2 Let ϕ : X → R be proper, lower semicontinuous on an Asplund space X and α > 0. Then, b we have ϕ is α−robustly quasiconvex if and only if for any (x, x∗ ), (y, y ∗ ) ∈ graph ∂ϕ, min{hx∗ , y − xi, hy ∗ , x − yi} > −αky − xk =⇒ hx∗ − y ∗ , x − yi ≥ 0. Proof. that (18) b such Suppose that ϕ is α−robustly quasiconvex and that there exist (x, x∗ ), (y, y ∗ ) ∈ graph ∂ϕ min{hx∗ , y − xi, hy ∗ , x − yi} > −αky − xk. b is quasimonotone by Theorem 2.2. It follows that Since ϕ is quasiconvex, ∂ϕ min{hx∗ , y − xi, hy ∗ , x − yi} ≤ 0. Combining (19) and (20), we have 0 ≤ − min     y−x x−y x∗ , , y∗, < α. ky − xk kx − yk 7 (19) (20) Without loss of generality, we may assume       y−x y−x x−y ∗ ∗ ∗ x , = min x , , y , . ky − xk ky − xk kx − yk Let r > 0 be such that   y−x ∗ − x , < r ≤ α. ky − xk (21) By the Hahn-Banach theorem, there exists v ∗ ∈ X ∗ satisfying hv ∗ , y − xi = rky − xk and kv ∗ k = r ≤ α. It follows that hx∗ , y − xi + hv ∗ , y − xi > −rky − xk + rky − xk = 0. (22) b v∗ (u) = ∂ϕ(u) b Consider ϕv∗ : X → R given by ϕv∗ (u) = ϕ(u) + hv ∗ , ui for any u ∈ X. Then, we have ∂ϕ + v∗ for u ∈ domϕ. Hence, by the quasiconvexity of ϕv∗ and by Theorem 2.2, we have min{hx∗ , y − xi + hv ∗ , y − xi, hy ∗ , x − yi + hv ∗ , x − yi} ≤ 0. Combining with (22), it implies hy ∗ , x − yi + hv ∗ , x − yi ≤ 0, i.e., hy ∗ , x − yi ≤ hv ∗ , y − xi = rkx − yk. E D y−x , we obtain hy ∗ , x − yi ≤ hx∗ , x − yi and thus (18) holds. Letting r → − x∗ , ky−xk b b Conversely, assume that (18) holds for all x, y ∈ X and x∗ ∈ ∂ϕ(x), y ∗ ∈ ∂ϕ(y). Taking any v ∗ in αB∗ , we ∗ next prove that ϕv∗ : X → R, defined by ϕv∗ (u) = ϕ(u) + hv , ui for any u ∈ X, is quasiconvex by showing b b v∗ (y), then x∗ − v ∗ ∈ ∂ϕ(x), b v∗ (x), y ∗ ∈ ∂ϕ b v∗ . Taking any x, y ∈ X and x∗ ∈ ∂ϕ the quasimonotonicity of ∂ϕ ∗ ∗ b y − v ∈ ∂ϕ(y). We then consider two cases. Case 1. min{hx∗ − v ∗ , y − xi, hy ∗ − v ∗ , x − yi} ≤ −αky − xk Without loss of generality, assume that hx∗ − v ∗ , y − xi = min{hx∗ − v ∗ , y − xi, hy ∗ − v ∗ , x − yi}. Since kv ∗ k ≤ α, we have min{hx∗ , y − xi, hy ∗ , x − yi} ≤ hx∗ , y − xi = hx∗ − v ∗ , y − xi + hv ∗ , y − xi ≤ −αky − xk + kv ∗ kky − xk ≤ 0. Case 2. min{hx∗ − v ∗ , y − xi, hy ∗ − v ∗ , x − yi} > −αky − xk Since (18) is satisfied, we have h(x∗ − v ∗ ) − (y ∗ − v ∗ ), x − yi ≥ 0, i.e., hx∗ − y ∗ , x − yi ≥ 0. It implies that 2 min{hx∗ , y − xi, hy ∗ , x − yi} ≤ hx∗ , y − xi + hy ∗ , x − yi ≤ 0. b v∗ is quasimonotone and thus ϕv∗ is quasiconvex for any v ∗ ∈ αB∗ by Theorem 2.2. This yields Hence, ∂ϕ the α-robust quasiconvexity of ϕ. ✷ 8 4 Conclusions Using Fréchet subdifferentials, we have obtained two first-order characterizations for the robust quasiconvexity of lower semicontinuous functions in Asplund spaces. The first one is a generalization of [6, Proposition 5.3] from finite dimensional spaces to Asplund spaces and its proof also overcomes a glitch in the proof of the sufficient condition of [6, Proposition 5.3]. The second criterion is totally new and it is settled from the equivalence of the quasiconvexity of lower semicontinuous functions and the quasimonotonicity of their subdifferential operators. 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