Positivity 6: 413–432, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
The Compatibility with Order of Some
Département de Mathématiques, Faculté des Sciences, Av. de l’Université 64000 PAU, France
Received 20 July 1999; accepted 20 May 2001
Abstract. It is well known that elementary subdifferentials which are the simplest and the most
precise among known subdifferentials do not enjoy good calculus rules, whereas more elaborated
subdifferentials do have calculus rules but are not as precise and, in particular, do not preserve order.
This paper explores an order preservation property for the subdifferentials of the second category.
This property concerns the case in which a distance function is involved. It emphasizes the crucial
role played by such functions in nonsmooth analysis. The result enables one to get in a simple,
uniﬁed way the passage from the properties of subdifferentials for Lipschitzian functions to the same
properties for the case of lower semicontinuous functions.
Dedicated to Alexander Ioffe, on the occasion of his 60th birthday
It is elementary to observe that if two functions f , g are differentiable at some
point x of a normed space X and if f g, f(x) = g(x) then their derivat-
ives at x coincide. Given a subdifferential ∂ on a class F (X) of functions on X
(viewed as a mapping from F (X) × X to the set of subsets of X
; see Section
4 for details), we say that ∂ is homotone if for any x ∈ X and any functions f ,
g satisfying the relations f g, f(x) = g(x) ∈ R one has ∂f (x) ⊃ ∂g(x).
Viewed in geometrical terms, this property corresponds to an antitone property of
the normal cone associated with ∂. It is easy to verify that this property holds for
the elementary subdifferentials (proximal subdifferential, Fréchet subdifferential,
Hadamard (or Dini-Hadamard or contingent) subdifferential, viscosity subdiffer-
ential, etc.). However these subdifferentials satisfy fuzzy calculus rules, but do not
enjoy good exact calculus rules. On the other hand, subdifferentials which satisfy
good calculus rules do not enjoy such a property. For instance, it is well known
that the Clarke’s subdifferential is not compatible with order in this sense: one may
have two Lipschitzian functions f , g on a normed space X satisfying f g,
f(x) = g(x) for some x ∈ X with ∂
g(x) not included in ∂
denotes the Clarke’s subdifferential of f at x. The same observation can be made