On computing the Mordukhovich subdifferential using directed sets in two dimensions.

*(English)*Zbl 1216.49014
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].

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 |