Generalized weak subdifferentials

KÜÇÜK Y., ATASEVER GÜVENÇ İ., Kucuk M.

OPTIMIZATION, vol.60, no.5, pp.537-552, 2011 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 60 Issue: 5
  • Publication Date: 2011
  • Doi Number: 10.1080/02331930903524670
  • Journal Name: OPTIMIZATION
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.537-552
  • Keywords: nonconvex analysis, nonsmooth analysis, generalized weak subdifferentials, generalized lower Lipschitz functions, nonconvex optimization, RADIAL EPIDERIVATIVES
  • Anadolu University Affiliated: Yes

Abstract

In this article, generalized weak subgradient (gw-subgradient) and generalized weak subdifferential (gw-subdifferential) are defined for nonconvex functions with values in an ordered vector space. Convexity and closedness of the gw-subdifferential are stated and proved. By using the gw-subdifferential, it is shown that the epigraph of nonconvex functions can be supported by a cone instead of an affine subspace. A generalized lower (locally) Lipschitz function is also defined. By using this definition, some existence conditions of the gw-subdifferentiability of any function are stated and some properties of gw-subdifferentials of any function are examined. Finally, by using gw-subdifferential, a global minimality condition is obtained for nonconvex functions.