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Nonconvex Analysis

In: Introduction to Nonsmooth Optimization

Author

Listed:
  • Adil Bagirov

    (School of Information Technology and Mathematical Sciences, University of Ballarat)

  • Napsu Karmitsa

    (University of Turku)

  • Marko M. Mäkelä

    (University of Turku)

Abstract

In this chapter, we generalize the convex concepts defined in the previous chapter to nonconvex locally Lipschitz continuous functions. Since the classical directional derivative does not necessarily exist for locally Lipschitz continuous functions, we first define a generalized directional derivative. Then we generalize the subdifferential analogously. We use the approach of Clarke in a finite dimensional case. However, in addition to the Clarke subdifferential, many different generalizations of the concept of a subdifferential for nonconvex nonsmooth functions exist. At the end of this chapter we briefly recall some of them. More specifically we give definitions of the quasidifferential, the codifferential, the basic (limiting) and the singular subdifferentials.

Suggested Citation

  • Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Nonconvex Analysis," Springer Books, in: Introduction to Nonsmooth Optimization, edition 127, chapter 0, pages 61-116, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-08114-4_3
    DOI: 10.1007/978-3-319-08114-4_3
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