Are Generalized Derivatives Sseful for Generalized Convex Functions?

Abstract

We present a review of some ad hoc sub differentials which have been devised for the needs of generalized convexity such as the quasi-sub differentials of GreenbergPierskalla, the tangential of Crouzeix, the lower sub differential of Plastria, the infradifferential of Gutierrez, the subdifferentials of Martinez-Legaz-Sach, Penot-Volle, Thach. We complete this list by some new proposals. We compare these specific subdifferentials to some all-purpose sub differentials used in nonsmooth analysis. We give some hints about their uses. We also point out links with duality theories.

Duality methods for the study of Hamilton–Jacobi equations

“…When S is convex, both cones coincide with the polar cone to the tangent cone and T (S, z) = cl T r (S, z) = cl(R + (S − z)). Therefore, for a quasiconvex function f , ∂ f (z) and ∂ ν f (z) coincide with the subdifferentials studied in [93], [94]:…”

Generalized Affine Functions and Generalized Differentials

“…Moreover, in such a case, one sees @f and @f are pseudomonotone (in the sense of [26]; see also [20], [25]). Thus we get a multivalued generalization of the concept of PPM map which has been used in [3] to study variational inequalities.…”

“…X and f : C ! R. Our purpose here is limited: since optimality conditions for (C) and mathematical programming problems using the concepts of the previous sections are dealt with in [13], [14], [15], [20], [22], [23], we are just concerned with characterizations of solution sets. Let S be the set of solutions to (C) and let @f : C X be a generalized di¤erential of f:…”

Generalized convex functions and generalized differentials

“…The de…nitions of @f -pseudoconvexity and @f -quasiconvexity we adopt here for a generalized subdi¤erential @f of f are similar to the ones used for a subdi¤erential by several authors; see [29], [34] and the references therein. We also introduce de…nitions of @f -protoconvexity and strict @f -pseudoconvexity of f as natural variants of the two preceding concepts.…”

“…Among the tools used to de…ne or study these notions are the various subdi¤erentials of nonsmooth analysis ( [1], [17], [16], [24], [29], [32], [34], [39], [35]...), the convexi…cators of [11], the pseudo-di¤erentials of Jeyakumar and Luc ([18]), the normal cones to sublevel sets ( [2], [4], [5], [6]) and the generalized directional derivatives ( [19], [20], [22], [40]). In the present paper we use a concept of generalized derivative which can encompass all these notions but the last one.…”