Abstract
The most general optimality conditions in quasiconvex programming are expressed in terms of normal cones to the level sets of functions. Then Kuhn-Tucker type conditions are derived by expressing these normal cones in terms of some generalized subdifferentials which often are related to some generalized derivatives.
We study some properties of the Dini-derivatives of quasiconvex and pseudoconvex functions and we show that these derivatives are useful for our purposes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Y. Chabrillac and J. P. Crouzeix, Continuity and differentiability properties of monotone real functions of several variables, Mathematical Programming Study, 30, 1987, pp. 1–16.
J. P. Crouzeix, Some differentiability properties of quasiconvex functions on fin, Optimization and Optimal Control, Springer-Verlag, Lecture Notes in Control and Information Sciences, Vol. 30, 1981, pp. 9–20.
J. P. Crouzeix, About differentiability properties of quasiconvex functions, Journal of Optimization Theory and Applications, 36, 1982, pp. 367–385.
J. P. Crouzeix, Continuity and differentiability properties of quasiconvex functions, Generalized Concavity in Optimization and Economics, Schaible and Ziemba editors, Academic Press, New-York, 1981, pp. 109–130.
W. E. Diewert, Alternative characterization of six kinds of quasiconvex functions, Generalized Concavity in Optimization and Economics, Schaible and Ziemba editors, Academic Press, New-York, 1981, pp. 51–93.
G. Giorgi and S. Komlosi, Dini-derivatives in optimization-Part 1, Rivista A.M.A.S.E.S. 15 n°1, 1992, pp. 3–30.
G. Giorgi and S. Komlosi, Dini-derivatives in optimization-Part 2, Rivista A.M.A.S.E.S. 15 n°2, 1992, pp. 3–24.
G. Giorgi and S. Komlosi, Dini-derivatives in optimization-Part 3, Rivista A.M.A.S.E.S. 18 n°3, 1995, pp. 47–63.
J. A. Gromicho, Quasiconvex Optimization and Location Theory., Dissertation Thesis, Timbergen Institute Research Series, nr. 90, 1995.
S. Komlosi, Some properties of nondifferentiable pseudoconvex functions, Mathematical Programming 26, 1983, pp. 232–237.
S. Komlosi, On a possible generalization of Pshenichnyi’s differentiability, Optimization, 21, 1990, pp. 189–201.
S. Komlosi, On generalized upper quasidifferentiability, Nonsmooth optimization: methods and applications, F. Gianessi editor, Gordon and Breach, London, 1992, pp. 189–201.
S. Komlosi, Quasiconvex first-order approximations and generalized Kuhn-Tucker conditions, European Journal on Operations Research, 65, 1993, pp. 327–335.
J. E. Martinez-Legaz, On lower subdifferentiable functions, Trends in Mathematical functions, Hoffman, Hiriart-Urruty, Lemaréchal and Zowe editors, Birkhäuser-Verlag, Basel, 1988, pp. 197–232.
J. E. Martinez-Legaz, Weak lower subdifferential and applications, Optimization, 21, 1990, pp. 321–341.
F. Plastria, Lower subdifferentiable functions and their minimization by cutting planes, Journal of Optimization Theory and Applications, 46, 1985, pp. 37–53
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Crouzeix, JP. (1998). Some Properties of Dini-Derivatives of Quasiconvex Functions. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds) New Trends in Mathematical Programming. Applied Optimization, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2878-1_5
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2878-1_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-4793-2
Online ISBN: 978-1-4757-2878-1
eBook Packages: Springer Book Archive