Calculating the Nonconvex Rubinov Subdifferential for Generalized Quasidifferentiable Functions Without a D.C. Representation

Robert Baier, Elza Farkhi, Vera Roshchina

The Rubinov subdifferential was introduced as the (usually nonconvex) visualization of the directed subdifferential, first for differences of convex (D.C.) and later for quasidifferentiable functions introduced by Demyanov/Rubinov. The directed subdifferential for both function classes is based on a certain arithmetic difference of the corresponding convex subdifferentials or quasisub-/quasisuperdifferential. Here, the convex sets are embedded in the Banach space of directed sets which allows an extension of the usual scalar multiplication and Minkowski sum of convex sets and offers a nonconvex visualization of differences of embedded convex sets.

Recently, the directed subdifferential is extended to the function class of directed subdifferentiable functions for which the function and certain restrictions to recursively defined orthogonal hyperplanes are uniformly directionally differentiable. This function class contains quasidifferentiable, amenable, lower-$C^k$ and locally Lipschitz, definable functions on o-minimal structures. For a special subclass of D.C. functions in $R^2$ the Rubinov subdifferential coincides with the basic subdifferential of Mordukhovich offering thus a geometric calculation of the latter.

The directed subdifferential enjoys as good calculus rules as the quasidifferential, e.g., exact formulas for the sum or the maximum of functions. Due to properties of the visualization of directed sets, the Rubinov subdifferential contains further information on the subdifferentials of Dini, Michel-Penot, Clarke and Mordukhovich.

For simple examples with nonconvex visualizations, comparisons of this subdifferential with other known convex/nonconvex subdifferentials are shown in the context of necessary and sufficient optimality conditions of unconstrained optimization problems.