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Gateaux Differentiability Revisited Malek Abbasi, Alexander Kruger, Michel Théra Gateaux differentiability revisited Malek Abbasi, Alexander Kruger, Michel Théra To cite this version: Malek Abbasi, Alexander Kruger, Michel Théra. Gateaux differentiability revisited. 2020. hal- 02963967 HAL Id: hal-02963967 https://hal.archives-ouvertes.fr/hal-02963967 Preprint submitted on 12 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Noname manuscript No. (will be inserted by the editor) Gateaux differentiability revisited Malek Abbasi · Alexander Y. Kruger · Michel Thera´ Received: date / Accepted: date Abstract We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and general- ized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. Keywords Gateaux differentiability · Moreau–Rockafellar subdifferential · Convex function · Directional derivative. Mathematics Subject Classification (2000) 49J52 · 49J53 · 90C30 1 Introduction Gateaux derivatives are widely used in the calculus of variations, optimization and physics. According to Laurent Mazliak [7,8], Gateaux differentiability first appeared in Gateaux’s notes [3,4] under the name variation premiere` . Paul Levy´ [6] was probably the first to use the name differentielle´ au sens de Gateaux. Recall that a mapping f : X ! Y between Banach spaces is Gateaux differentiable atx ¯ 2 X if, for each d 2 X, the limit f (x¯+td) − f (x¯) f 0(x¯;d) := lim (1.1) t#0 t exists, and f 0(x¯;·) is a linear continuous functional on X. This functional is called the Gateaux derivative of f atx ¯. If the limit (1.1) is uniform over d on the unit sphere of X, then f is Frechet´ differentiable atx ¯. Gateaux and Frechet´ differentiability coincide for real-valued locally Lipschitz functions on a space in which bounded subsets are relatively compact, in particular, for convex con- tinuous functions in finite dimensions (see, for instance, [12]). Without convexity, the two concepts are different even in finite dimensions. Malek Abbasi Department of Mathematics, University of Isfahan, Isfahan, Iran E-mail: [email protected] Alexander Y. Kruger ORCID 0000-0002-7861-7380 School of Engineering, Information Technology and Physical Science, Federation University Australia, Ballarat, Vic, Australia E-mail: [email protected] Michel Thera´ ORCID 0000-0001-9022-6406 Mathematics and Computer Science Department, University of Limoges E-mail: [email protected] 2 Malek Abbasi et al. Gateaux differentiability is easier to check computationally. Being a weaker concept, Ga- teaux derivative is more universal. For instance, the conventional norm in `1(N) is nowhere Frechet´ differentiable, but there exists an equivalent norm, which is Gateaux differentiable at all non-zero points (see Phelps [10, Example 1.14(c)]. Note that the conventional norms in `1(G ) (with G uncountable) and L¥[0;1] are nowhere Gateaux differentiable. As observed by Moreau [9, Corollary 10.g], if a real-valued convex function is Gateaux differentiable, its Moreau–Rockafellar subdifferential is a singleton. Conversely if a function is finite and continuous at a point and its subdifferential is a singleton, then it is necessarily Gateaux differentiable at this point. The vanishing of the Gateaux derivative is a necessary condition for an extremum. Throughout the paper, X is a real topological vector space. Its topological dual is denoted ∗ by X . Given an extended-real-valued function f : X ! R¥ := R [ f+¥g, the directional derivative of f atx ¯ 2 dom f := fx 2 X : f (x) < +¥g in direction d 2 X is defined by the (possibly infinite) limit (1.1). We say that f is directionally differentiable atx ¯ if the limit exists and is finite for all d 2 X. The mapping d 7! f 0(x¯;d) is positively homogeneous. If it is linear and continuous, f is Gateaux differentiable atx ¯. If f is convex, the limit exists and 1 is finite at every pointx ¯ 2 core(dom f ), i.e. when [l>0l(dom f − x¯) = X, and satisfies f 0(x¯;d) ≤ f (x¯+ d) − f (x¯): (1.2) Furthermore, the function d 7! f 0(x¯;d) is convex. The subdifferential of f atx ¯ 2 X is the set (cf. [5]) ¶ f (x¯) := fx∗ 2 X∗ : hx∗;di ≤ f 0(x¯;d) for all d 2 Xg: (1.3) If f is convex, then the set (1.3) coincides with the Moreau–Rockafellar subdifferential of f atx ¯: ¶ f (x¯) = fx∗ 2 X∗ : hx∗;xi ≤ f (x¯+ x) − f (x¯) 8x 2 Xg: In this paper, we revisit some basic concepts and ideas of the classical differential calcu- lus and convex analysis, extending them to a broader frame. Building partially on the tools developed in [1], we reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. The directional derivative (1.1) is replaced by the following limit: x( f ;A(h;x¯)) − f (x¯) Dx f (x¯)(A) := lim ; h#0 h where A : [0;1]×X ⇒ X is a set-valued mapping satisfying A(0;x) = fxg for all x 2 X, and x is an extended-real-valued function defined on the product space of the families of all functions f : X ! R¥ and all nonempty subsets of X, satisfying x( f ;fxg) = f (x) for all f : X ! R¥ and x 2 X. Different choices of A and x lead to different derivative and subdifferential concepts. Some meaningful examples preserving the key properties of the original notion of derivative are provided in the paper. They include the sup-, average and measure derivatives. In the next section, we give the main notation, terminology and definitions, as well as some examples. Section 3 is devoted to generalized derivatives and subdifferentials. 2 Definitions and Examples In the rest of the paper, F and P stand for the collections of all extended-real-valued func- tions f : X ! R¥ on a real topological vector space X and all nonempty subsets of X, re- spectively. Unless explicitly stated otherwise, linear operations on sets are understood in the Minkowski sense, i.e., if A;B ⊂ X and l 2 R, then lA := fla : a 2 Ag, and A+B := fa+b : a 2 A; b 2 Bg. The constructions defined in this paper employ the following tools: 1 If f is convex and lower semicontinuous, core(dom f ) coincides with the interior of dom f ; see, for instance, [2]. Gateaux differentiability revisited 3 – the family A of all set-valued mappings A : [0;1] × X ⇒ X satisfying A(h;x) 6= /0for all h 2 [0;1] and x 2 X, and A(0;x) = fxg for all x 2 X; – the family X of all functions x : F × P ! R¥ satisfying x( f ;fxg) = f (x) for all f 2 F and x 2 X: Observe that X is a convex set. Convexity of mappings from A is defined in the usual sense. Definition 2.1 A mapping A 2 A is convex if A(lh + (1 − l)t;lx + (1 − l)y) ⊆ lA(h;x) + (1 − l)A(t;y) for all l;h;t 2 [0;1] and x;y 2 X. Concavity is defined similarly with the opposite inclusion in the above definition. Definition 2.2 Let F 0 ⊆ F and P0 ⊆ P. A function x 2 X is increasing (respectively, decreasing) on F 0 × P0 if x( f ;U) ≤ x( f ;V) (respectively; x( f ;U) ≥ x( f ;V)) for all functions f 2 F 0 and all nonempty subsets U;V 2 P0 with U ⊆ V. Definition 2.3 Let A 2 A and x 2 X. A function f 2 F is (i) x-convex if x( f ;lU + (1 − l)V) ≤ lx( f ;U) + (1 − l)x( f ;V) for all nonempty U;V ⊆ X and l 2 [0;1]; (ii) (x;A)-convex if x( f ;A(lh + (1 − l)t;lx + (1 − l)y)) ≤ lx( f ;A(h;x)) + (1 − l)x( f ;A(t;y)) for all x;y 2 X and l;h;t 2 [0;1]. x-concavity and (x;A)-concavity are defined similarly with the opposite inequalities in the above definition. Proposition 2.1 Let f 2 F ,A 2 A and x 2 X. (i) If f is (x;A)-convex, then it is convex; (ii) Suppose A is convex and x is increasing. If f is x-convex, then it is (x;A)-convex. Proof (i) Suppose f is (x;A)-convex. Let x;y 2 X and l 2 [0;1]. Then f (lx + (1 − l)y) = x( f ;flx + (1 − l)yg) = x( f ;A(0;lx + (1 − l)y)) ≤ lx( f ;A(0;x)) + (1 − l)x( f ;A(0;y)) = lx( f ;fxg) + (1 − l)x( f ;fyg) = l f (x) + (1 − l) f (y): (ii) Suppose f is x-convex. Let x;y 2 X and l;h;t 2 [0;1]. Then x( f ;A(lh + (1 − l)t;lx + (1 − l)y)) ≤ x( f ;lA(h;x) + (1 − l)A(t;y)) ≤ lx( f ;A(h;x)) + (1 − l)x( f ;A(t;y)): The proof is completed.
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