Argonne National Laboratory

Relating Lexicographic Smoothness and Directed Subdifferentiabilty

TitleRelating Lexicographic Smoothness and Directed Subdifferentiabilty
Publication TypeReport
Year of Publication2015
AuthorsKhan, KA
Other NumbersANL/MCS-P5411-1015

Lexicographic derivatives developed by Nesterov and directed subdifferen- tials developed by Baier, Farkhi, and Roshchina are both essentially nonconvex generalized derivatives for nonsmooth nonconvex functions and satisfy strict calculus rules and mean-value theorems. This article aims to clarify the relationship between the two generalized derivatives. In particular, for scalar-valued functions that are locally Lipschitz continuous, lexicographic smoothness and directed subdifferentiability are shown to be equivalent, along with the necessary optimality conditions corresponding to each. For such functions, the visualization of the directed subdifferential—the Rubinov subdifferential—is shown to include the lexicographic subdifferential, and is also shown to be included in its convex hull. Implications of these results are discussed.