TY - CHAP
T1 - On computing the mordukhovich subdifferential using directed sets in two dimensions
AU - Baier, Robert
AU - Farkhi, Elza
AU - Roshchina, Vera
N1 - Publisher Copyright:
© Springer Science+Business Media, LLC 2010.
PY - 2010
Y1 - 2010
N2 - 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 via 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.
AB - 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 via 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.
UR - http://www.scopus.com/inward/record.url?scp=84976466634&partnerID=8YFLogxK
U2 - 10.1007/978-1-4419-0437-9_3
DO - 10.1007/978-1-4419-0437-9_3
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AN - SCOPUS:84976466634
T3 - Springer Optimization and Its Applications
SP - 59
EP - 93
BT - Springer Optimization and Its Applications
PB - Springer International Publishing
ER -