Nonconvex Subdifferentials in Nonsmooth Analysis

[function plot of |x<sub>1</sub>| + |x<sub>2</sub>| - 5/4*sqrt(x<sub>1</sub><sup>2</sup> + x<sub>1</sub><sup>2</sup>)] [directed/Rubinov subdifferential of |x<sub>1</sub>| + |x<sub>2</sub>| - 5/4*sqrt(x<sub>1</sub><sup>2</sup> + x<sub>1</sub><sup>2</sup>)]
more visualizations

start of the project: 2008

funding institution: The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel

project members

principal investigator

Dr. Robert Baier and Dr. Elza Farkhi (Tel Aviv University, Israel)

external project members

Dr. Vera Roshchina (Collaborative Research Network (CRN), University of Ballarat, Australia)

aims of the project

In this project, new nonconvex subdifferentials for subclasses of Lipschitz continuous functions are introduced, i.e. the directed subdifferential and its visualization, the Rubinov subdifferential. Subdifferentials are used for the description of necessary and sufficient optimality conditions for non-smooth optimization problems. In these problems, the objective function and the functions for the description of the admissible set given by equations and inequalities are in general only Lipschitz continuous and not continuously differentiable. Therefore, the gradients and the Hesse matrices are not everywhere defined in the admissible set of points.

The embedding of convex, compact sets into vector spaces (e.g., the space of directed sets) is essential for this project. In these vector spaces, a difference of embedded convex sets is available. For important problem classes, in which e.g. the objective function is a difference of convex functions (so-called DC functions), the directed subdifferential could be calculated on the basis of the difference of the (embedded) convex subdifferentials. The visualization of these differences leads to new subdifferentials and emphasizes the directional derivative, which is used to formulate strict optimality conditions and descent directions for optimization methods.

Connections to other subdifferentials (Dini, Michel-Penot, Mordukhovich, Clarke as well as the quasidifferential of Demyanov/Rubinov) are another focus of research. Most of the usual disadvantages of other subdifferentials could be avoided in most cases. Important calculus rules are valid as equality and not only as inclusion and often need weaker assumptions to hold. Applying the calculus leads to the expression of the new subdifferentials of complicated functions by subdifferentials of simpler functions.

The project investigator Dr. Farkhi visited 2007 Bayreuth once again. For 2008 and in April 2011 (together with Vera Roshchina), the second project investigator continues the cooperation with a research stay in the School of Mathematical Sciences in Tel Aviv University. In July 2012 the project investigator visited Dr. Roshchina at the University of Ballarat, in August 2012 Dr. Roshchina come to Bayreuth, to explore new research subjects. One common target is to extent the results onto bigger problem classes, e.g. to the class of quasi-differentiable functions (the directional derivative is a difference of special convex functions) as well as to lower-/upper-Ck and amenable functions.

Minisymposia/Sessions

The project members organized the following minisymposia and sessions on generalized differentiation.

Links

publications

Please go to the list of publications of this project.

Chair -

|  University of Bayreuth -