Lionel Thibault

Local differentiability of distance functions

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C. Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of dC being locally of class C 1+ or such that dC + σ| · |2 is convex around x for some σ > 0. Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x, and leads to a formula for PC by way of NC . The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.

Sweeping process with regular and nonregular sets

Abstract Differential inclusions involving the normal cone to a moving set are investigated. A special attention is paid to the sweeping process associated with sets for which no regularity assumption is required.

Subsmooth sets: Functional characterizations and related concepts

Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist (C;.), or the local uniqueness of the projection mapping, but also because in the case where C is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C 2 property) of the function. In this paper we provide an adapted geometrical concept, called sub smoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C 1 property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

Relaxation of an optimal control problem involving a perturbed sweeping process

We establish first, in the setting of infinite dimensional Hilbert space, a result concerning the existence of solutions for perturbed sweeping processes whose perturbations are Lipschitz single-valued maps. Then we use this result to extend to the infinite dimensional setting a relaxation result concerning optimal control problems involving such processes.

Sub differential Monotonicity as Characterization of Convex Functions

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.

Approximate subdifferential and metric regularity: the finite-dimensional case

The notion of graphical metric regularity is introduced and conditions ensuring this kind of regularity for systems of finite-dimensional multifunctions are given in terms of partial approximate subdifferentials.

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Fréchet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=−g(x)+D. We apply our results to the study of the limiting Fréchet subdifferential of a composite function defined on a Banach space.

Tangent cones and quasi-interiorly tangent cones to multifunctions

R. T. Rockaiellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions. Introduction. Rockafellar has introduced in [23] the very important notion of directionally lipschitzian behaviour for extended real-valued functions, and with the aid of this notion he has proved in [24] a number of rules of subgradient calculus of nonconvex functions. If / is a function from a topological vector space E into R U {-oo, +00} with/(x) G R and if /(epi /; x, fix)) denotes the interiorly tangent cone to epi/={(x,^)G£XR:/(x) 0 such that X n epi /+ ]0, e[VC epi /, then the proof of Theorem 3 of [23, p. 268] shows that / is directionally lipschitzian at x if and only if /(epi /; x, fix)) i= 0. However, if g is a mapping from E into an ordered topological vector space H, then the interior of the cone of positive elements of H must be nonempty whenever /(epi g; x, g(x)) is nonempty. This very unsatisfactory state of affairs has led us to introduce in [29 and 30] the quasi-interiorly tangent cone ß(epi g; x, g(x)). With the help of this cone we have established in [29] rules of subdifferential calculus for nonconvex vector-valued functions. The aim of the present paper is to study the properties of Clarke tangent cones and quasi-interiorly tangent cones to the graphs of multifunctions following the way opened by Rockafellar. In §1 we recall Rockafellars definition of Clarke tangent cone and we give an interpretation in terms of generalized sequences which proves that Rockafellars definition is the same as the one we have given in [27]. Connection with strictly compactly lipschitzian vector-valued mappings is also made. Received by the editors March 19, 1982 and, in revised form, May 10, 1982. 1980 Mathematics Subject Classification. Primary 90C30, 90C48; Secondary 58C06.

Prox-regular sets and epigraphs in uniformly convex Banach spaces: Various regularities and other properties

We continue the study of prox-regular sets that we began in a previous work in the setting of uniformly convex Banach spaces endowed with a norm both uniformly smooth and uniformly convex (e.g., L p , W m,p spaces). We prove normal and tangential regularity properties for these sets, and in particular the equality between Mordukhovich and proximal normal cones. We also compare in this setting the proximal normal cone with different Holderian normal cones depending on the power types s, q of moduli of smoothness and convexity of the norm. In the case of sets that are epigraphs of functions, we show that J-primal lower regular functions have prox-regular epigraphs and we compare these functions with Poliquins primal lower nice functions depending on the power types s, q of the moduli. The preservation of prox-regularity of the intersection of finitely many sets and of the inverse image is obtained under a calmness assumption. A conical derivative formula for the metric projection mapping of prox-regular sets is also established. Among other results of the paper it is proved that the Attouch-Wets convergence preserves the uniform r-prox-regularity property and that the metric projection mapping is in some sense continuous with respect to this convergence for such sets.

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sém Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset