Abstract
We show the link between different concepts of (open) tangent cones and give characterizations of different regularity properties of these cones, including the lower semicontinuity of the open tangent cone set-valued mapping. This paper continues work done by the author on closed tangent cones.
Preview
Unable to display preview. Download preview PDF.
References
J.P. Aubin, “Micro-cours: Gradients généralisés de Clarke,” Annales des Sciences Mathématiques du Québec 2 (1978) 197–252.
J.P. Aubin, “Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,” in: L. Nachbin, ed., Advances in Mathematics (Academic Press, New York, 1981) pp. 159–229.
J.P. Aubin and A. Cellina, Differential Inclusions (Springer-Verlag, Berlin, 1984).
P. Beato, “The existence of marginal cost pricing equilibria with increasing returns,” The Quarterly Journal of Economics XCVII (1982) 669–688.
C. Berge, Espace Topologiques, Fonctions Multivoques (Dunod, Paris, 1966).
J.M. Bony, “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés,” Annales de l’Institut Fourier de Grenoble 19 (1969) 277–304.
G. Bouligand, “Sur l’existence des demi-tangentes à une courbe de Jordan,” Fundamenta Mathematicae 15 (1930) 215–218.
F. Clarke, “Generalized gradients and applications,” Transactions of the American Mathematical Society 205 (1975) 247–262.
F. Clarke, Optimization and Nonsmooth Analysis (John Wiley, New York, 1983).
B. Cornet, “Regular properties of tangent and normal cones,” Chapter 1 of Ph.D. Thesis, Université Paris IX Dauphine (Paris, 1981).
B. Cornet, “Existence of slow solutions for a class of differential inclusions,” Journal of Mathematical Analysis and Applications 96 (1983) 130–147.
B. Cornet, “Existence of equilibria in economies with increasing returns,” Technical Report 82-311, Department of Economics, University of California (Berkeley, CA, 1982).
B. Cornet and G. Haddad, “Theorèmes de viabilité pour les inclusions differentièlles du second ordre,” Israel Journal of Mathematics (1987), to appear.
A.J. Dubovickii and A.A. Miljutin, “Extremum problems in presence of restrictions,” USSR Computational Mathematics and Mathematical Physis 5 (1965) 1–80.
R. Guesnerie, “Pareto optimality in non-convex economies,” Econometrica 43 (1975) 1–29.
G. Haddad, “The role of tangent and normal cones in the viability theory of differential inclusions,” Mathematical Programming Study 30 (1987) 34–44 (this volume).
J.B. Hiriart-Urruty, “Tangent cones, generalized gradients and mathematical programming in Banach spaces,” Mathematics of Operations Research 4 (1979) 79–97.
K. Kuratowski, Topology, Volume I (Academic Press, New York, 1966).
J.P. Laurent, Approximation et Optimisation (Hermann, Paris, 1972).
M. Nagumo, “Über die Lage der Integralkurven gewöhnlicher differential Gleichungen,” Proceedings of the Physical and Mathematical Society of Japan 24 (1942) 551–559.
J.P. Penot, “A characterization of tangential regularity,” Nonlinear Analysis, Theory, Methods and Applications 6 (1981) 625–643.
S. Robinson, “Generalized equations,” in: A. Bachem, M. Grötschel and B. Korte, eds., Mathematical Programming: The State of the Art Bonn, 1982 (Springer-Verlag, Berlin, 1983) pp. 346–367.
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar, “Clarke’s tangent cones and the boundaries of closed sets in ℝn,” Nonlinear Analysis, Theory Methods and Applications 3 (1979a) 145–154.
R.T. Rockafellar, “Directionally Lipschitzian functions and subdifferential calculus,” Proceedings of the London Mathematical Society 3 (1979b) 331–335.
J.S. Treiman, “A new characterization of Clarke’s tangent cone and its applications to subgradient analysis and optimization,” Ph.D. Thesis, University of Washington (Seattle, WA, 1983).
C. Ursescu, “Tangent sets’ calculus and necessary conditions for extremality,” SIAM Journal on Control and Optimization 20 (1982) 563–574.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
Cornet, B. (1987). Regularity properties of open tangent cones. In: Cornet, B., Nguyen, V.H., Vial, J.P. (eds) Nonlinear Analysis and Optimization. Mathematical Programming Studies, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121152
Download citation
DOI: https://doi.org/10.1007/BFb0121152
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00930-3
Online ISBN: 978-3-642-00931-0
eBook Packages: Springer Book Archive