On the theory of subdifferentials

By Alexander D. Ioffe. 2012

59 citations recorded by the source corpus.

This paper is an attempt to present the theory of subdifferentials of nonsmooth and nonconvex functions on infinite dimensional Banach spaces in a unified and systematic way. The necessity to study such “bad” functions and mappings was mainly determined by the needs of optimization and optimal control theories. The first subdifferentials for such functions were introduced in mid-70s FHC73,FHC75,BM76,JPP74 [10, 11, 43, 50], more appeared shortly afterwards CL83,DMT80,AI81,AK81,KM80 [15, 17, 25, 38, 40] and since then have developed into a powerful instrument with a wide range of applications. The crucial role of smooth variational principles of Borwein-Preiss and Deville-Godefroy-Zizler in these developments should be emphasized. Multiplicity of subdifferentials used now in variational analysis is necessitated by complexity of objects, on the one hand, and differencies in geometries of different classes of Banach spaces, so that certain constructions that work well in some spaces are not very useful in others. The idea to find a common viewpoint for the variety of available and utilizable subdifferentials is by no means new. We can mention e.g. ACL95,ACL,CCT09,AI87,IP96,BM,TZ [2, 4, 14, 30, 33, 46, 55] where one or another generalizing concept was offered, each time, however, to embrace properties relating to some specific topic or group of results. Interconnections between various types of subdifferentials have been thoroughly studied in many publications since the very beginning, but no real attempt has been made to develop a sufficiently comprehesive theory and/or unified techniques that could be equally applied to various subdifferentials. The reader now has to turn to different sources to learn about one or another subdifferential. A sufficiently complete account of the finite dimensional theory is contained in RW [54]. The calculus of Clarke’s generalized gradients is presented in FHC83,CLSW [12, 13] For acquaintance with Frechet subdifferential one can go to BZ [9], an exaustive treatment of limiting Frechet subdifferentials can be found in BM [46], information about approximate subdifferentials is scattered in many non-monographic publications. The two main new elements that have made the unification possible are a (rather simple) observation that a more accurate fuzzy calculus of β-subdifferential ( associated with bornologies) is available with the level of fuzziness gradually varying from fuzziness up to a norm neighborhood of zero for the Frechet subdifferential to fuzziness up to a weak∗neighborhood for the subdifferential associated with the simple (Gâteaux) bornology and a new series of subdifferentials, called here “metric modifications” that includes the limiting Frechet subdifferential and the approximate G-subdifferential at its opposite poles. The paper consists of two parts. The first called “General theory” is devoted to general principles and results that are equally applied to all subdifferentials. The second part, “Special theory” deals with a number of specific subdifferential that are studied

Subfields: Computational Theory and Mathematics, Geometry and Topology, Management Science and Operations Research

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