Weak directional closedness and generalized subdifferentials

Spingarn introduced the notion of a submonotone operator and showed that the Clarke subdifferential is submonotone if and only if it is semismooth (in the sense of Mifflin) and regular (in the sense of Clarke). In this article a property of operators referred to as weak directional closedness (WDC) is introduced. The WDC property is used to extend Spingarn's result to a broad class of generalized subdifferentials for locally Lipschitz functions. Two members of this class of subdifferentials are the Clarke subdifferential, which is always WDC, and the Michel-Penot subdifferential, which may or may not be WDC. A subdifferential that is WDC and is contained in the Clarke subdifferential is constructed. It is shown that this subdifferential coincides with the Michel-Penot subdifferential whenever the Michel-Penot subdifferential is WDC and submonotone.