Generalized weak subdifferentials

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.