On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

Robert Baier (), Elza Farkhi () and Vera Roshchina ()
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Robert Baier: University of Bayreuth
Elza Farkhi: Tel Aviv University
Vera Roshchina: Universidade de Évora

A chapter in Variational Analysis and Generalized Differentiation in Optimization and Control, 2010, pp 59-93 from Springer

Abstract: Abstract The Mordukhovich subdifferential, being highly important in variational and nonsmooth analysis and optimization, often happens to be hard to calculate. We propose a method for 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, the Rubinov subdifferential may be constructed as the Mordukhovich one by Painlevé–Kuratowski outer limits of Fréchet subdifferentials. The results are applied to the case of DC functions. Examples illustrating the obtained results are presented. 2010 Mathematics Subject Classification. Primary 49J52; Secondary 26B25, 49J50, 90C26

Keywords: Convex Function; Directional Derivative; Outer Limit; Positive Homogeneity; Sublinear Function (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-1-4419-0437-9_3

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