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Generalized Convexities

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

Convexity plays a crucial role in mathematical optimization theory. Especially, in duality theory and in constructing optimality conditions, convexity has been the most important concept since the basic reference by Rockafellar was published. Different types of generalized convexities have proved to be the main tool when constructing optimality conditions, particularly sufficient conditions for optimality. In this chapter, we analyze the properties of the generalized pseudo- and quasiconvexities for nondifferentiable locally Lipschitz continuous functions. The treatment is based on the Clarke subdifferentials and generalized directional derivatives.

Suggested Citation

  • Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Generalized Convexities," Springer Books, in: Introduction to Nonsmooth Optimization, edition 127, chapter 0, pages 139-168, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-08114-4_5
    DOI: 10.1007/978-3-319-08114-4_5
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