TY - JOUR
T1 - The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I
T2 - Definition and examples
AU - Baier, Robert
AU - Farkhi, Elza
AU - Roshchina, Vera
N1 - Funding Information: The authors would like to thank Boris Mordukhovich who encouraged them to extend the directed subdifferential to amenable and lower- functions. The first two authors were partially supported by The Hermann Minkowski Center for Geometry at Tel Aviv University.
PY - 2012/2
Y1 - 2012/2
N2 - We extend the definition of the directed subdifferential, originally introduced in [R. Baier, E. Farkhi, The directed subdifferential of DC functions, in: A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A.J. Zaslavski (Eds.), Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, June 1824, 2008, Haifa, Israel, in: AMS Contemp. Math., vol. 513, AMS, Bar-Ilan University, 2010, pp. 2743], for differences of convex functions (DC) to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upper-Ck functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: non-uniqueness and "inflation in size" of the two convex sets representing the quasidifferential after applying calculus rules. The Rubinov subdifferential is defined as the visualization of the directed subdifferential.
AB - We extend the definition of the directed subdifferential, originally introduced in [R. Baier, E. Farkhi, The directed subdifferential of DC functions, in: A. Leizarowitz, B.S. Mordukhovich, I. Shafrir, A.J. Zaslavski (Eds.), Nonlinear Analysis and Optimization II: Optimization. A Conference in Celebration of Alex Ioffe's 70th and Simeon Reich's 60th Birthdays, June 1824, 2008, Haifa, Israel, in: AMS Contemp. Math., vol. 513, AMS, Bar-Ilan University, 2010, pp. 2743], for differences of convex functions (DC) to the wider class of quasidifferentiable functions. Such generalization efficiently captures differential properties of a wide class of functions including amenable and lower/upper-Ck functions. While preserving the most important properties of the quasidifferential, such as exact calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential: non-uniqueness and "inflation in size" of the two convex sets representing the quasidifferential after applying calculus rules. The Rubinov subdifferential is defined as the visualization of the directed subdifferential.
KW - Amenable and lower- C functions
KW - Differences of sets
KW - Directed sets
KW - Directed subdifferential
KW - Quasidifferentiable functions
KW - Subdifferentials
UR - http://www.scopus.com/inward/record.url?scp=82155185208&partnerID=8YFLogxK
U2 - 10.1016/j.na.2011.04.074
DO - 10.1016/j.na.2011.04.074
M3 - مقالة
SN - 0362-546X
VL - 75
SP - 1074
EP - 1088
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
IS - 3
ER -