Directed subdifferentiable functions and the directed subdifferential without Delta-convex structure
- Authors: Baier, Robert , Farkhi, Elza , Roshchina, Vera
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Optimization Theory and Applications Vol. 160, no. 2 (2014), p. 391-414
- Full Text: false
- Reviewed:
- Description: We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the delta-convex structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions. © 2013 Springer Science+Business Media New York.
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part II: Calculus
- Authors: Baier, Robert , Farkhi, Elza , Roschina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods & Applications Vol. 75, no. 3 (2012), p. 1058-1073
- Full Text: false
- Reviewed:
- Description: We continue the study of the directed subdifferential for quasidifferentiable functions started in [R. Baier, E. Farkhi, V. Roshchina, The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples (this journal)]. Calculus rules for the directed subdifferentials of sum, product, quotient, maximum and minimum of quasidifferentiable functions are derived. The relation between the Rubinov subdifferential and the subdifferentials of Clarke, Dini, Michel–Penot, and Mordukhovich is discussed. Important properties implying the claims of Ioffe’s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained.
The directed and Rubinov subdifferentials of quasidifferentiable functions, Part I: Definition and examples
- Authors: Baier, Robert , Farkhi, Elza , Roschina, Vera
- Date: 2012
- Type: Text , Journal article
- Relation: Nonlinear Analysis: Theory, Methods Applications Vol. 75, no. 3 (2012), p. 1074-1088
- Full Text: false
- Reviewed:
- Description: 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 18–24, 2008, Haifa, Israel, in: AMS Contemp. Math., vol. 513, AMS, Bar-Ilan University, 2010, pp. 27–43], 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-View the MathML source 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.