Exhausters and subdifferentials in non-smooth analysis

Abstract

Non-smooth analysis manifested itself in the 1960s of the last century and is still gaining momentum developing new tools and harnesses and covering new areas of application. One of the notions of non-smooth analysis is that of the exhauster. The exhauster represents a dual construction in Nonsmooth Analysis. The relationships between upper and lower exhausters and various subdifferentials of non-smooth functions are discussed in this article. It is shown that exhausters are closely related to other non-smooth tools, such as the Michel–Penot, Clarke, Gâteaux and Fréchet subdifferentials. Formulae for computing these subdifferentials by means of exhausters are obtained. The discovered relations are all in the form of equalities, i.e. a calculus for computing the mentioned subdifferentials by means of exhausters is provided.

Keywords:

• upper and lower exhausters
• positively homogeneous function; Clarke
• Fréchet
• Gâteaux and Michel–Penot subdifferentials

Mathematics Subject Classifications 2000 :

• 26B25
• 49J52
• 49K99
• 58C20
• 90C46

Acknowledgement

The authors are thankful to the anonymous Referee for his valuable remarks. The work was supported by the Russian Foundation for Fundamental Studies (RFFI) under Grant No. 06-01-00276.