On computing the Mordukhovich subdifferential using directed sets in two dimensions. (English) Zbl 1216.49014

Baier, Robert; Farkhi, Elza; Roshchina, Vera

Burachik, Regina S. (ed.) et al., Variational analysis and generalized differentiation in optimization and control. In honor of Boris S. Mordukhovich. Berlin: Springer (ISBN 978-1-4419-0436-2/hbk; 978-1-4419-0437-9/ebook). Springer Optimization and Its Applications 47, 59-93 (2010).

Summary: 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 Painleé-Kuratowski outer limits of Fréchet subdifferentials. The results are applied to the case of DC functions. Examples illustrating the obtained results are presented.
For the entire collection see [Zbl 1203.49002].

MSC:

49J52	Nonsmooth analysis
26B25	Convexity of real functions of several variables, generalizations
49J50	Fréchet and Gateaux differentiability in optimization
90C26	Nonconvex programming, global optimization

Keywords:

Mordukhovich subdifferential; variational analysis; nonsmooth analysis; differences of sublinear (DS) functions; Rubinov subdifferential; outer limits of Fréchet subdifferentials

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