Nonconvex subdifferentials in nonsmooth optimization

start of the project: 2008

funding institution: The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel (until 2017)

PROJECT MEMBERS

principal investigator:

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

external project members:

Dr. Vera Roshchina (School of Mathematics and Statistics, University of New South Wales (UNSW), Sydney, Australien)

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 the autumn in 2008 and in April 2011 (together with Vera Roshchina) as well as in October 2012, in February 2014 and in February/March 2017, 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.

• Minisymposium "Generalized Differentiation and Applications"

  conference: ISMP 2012, 21st International Symposium on Mathematical Programming in Berlin

  chair: Robert Baier, Vera Roshchina (Collaborative Research Network, University of Ballarat, Australia)

• Minisymposium "Generalized Differentiation and Applications in Optimization I-II"

  conference: IFIP 2011, 25-th TC7 Conference on System Modeling and Optimization in Berlin, Germany

  chair: Robert Baier, Vera Roshchina (University of Évora, Portugal)

• Minisymposium "Generalized Differentiation and Applications to Control Problems"

  conference: IFIP 2011, 25-th TC7 Conference on System Modeling and Optimization in Berlin, Germany

  chair: Robert Baier, Vera Roshchina (University of Évora, Portugal)

• Invited Session "Generalized differentiation and applications"

  workshop: EUROPT 2010 Workshop "Advances in Continuous Optimization" at the University of Aveiro, Portugal

  chair: Vera Roshchina (University of Évora, Portugal)

  The EUROPT 2010 Workshop was a satellite event of the EURO 2010 Conference in Lisbon, Portugal. Please see also the program of the workshop.