On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

R. Baier, E. Farkhi, V. Roshchina: On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions
in: Special Volume "Variational Analysis and Generalized Differentiation in Optimization and Control: In Honor of Boris S. Mordukhovich". The International Symposium on "Variational Analysis and Optimization", held on November 28-30, 2008 at the Departme, Springer Optimization and Its Applications 47, New York-Dordrecht-Heidelberg-London, 2010, 59 - 93

DOI: 10.1007/978-1-4419-0437-9_3
MR Nummer: 2766755
Zentralblattnummer: 1216.49014
Keywords: non-convex subdifferentials and superdifferentials (basic subdifferentials; Rubinov subdifferential); Frêchet subdifferential; difference of convex (DC) functions; differences of sets
Mathematics Subject Classification Code: 49J52 (26B25 49J50 90C26)
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Abstract:

The Mordukhovich subdifferential is highly important in the variational and non-smooth analysis and optimization, but it may often be hard to calculate it. Here we propose a method of computing the Mordukhovich subdifferential of differences of sublinear (DS) functions applying the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations.
The equivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential), to the Rubinov subdifferential (the visualization of the directed subdifferential), is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identified as parts of the Rubinov subdifferential. In addition it is possible to construct the directed subdifferential in a way similar to the Mordukhovich one by considering outer limits of Fréchet subdifferentials. The results are extended to the case of DC functions. Examples illustrating the obtained results are presented.

Contents:

1. Introduction
2. Preliminaries - Some Known Subdifferentials
2. Preliminaries
3. The Mordukhovich and the Directed Subdifferential in |R^2
4. Examples
5. Conclusions

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