Abstract
The implicit and inverse function theorems provide an essential component of classical differential calculus, and for this reason many attempts have been made to extend these theorems to nonsmooth analysis (see, for example, the work of F. Clarke, H. Halkin, J.-B. Hiriart-Urruty, A.D. Ioffe, B.H. Pourciau, J. Warga). In this paper, we consider the case of quasidifferentiable functions. It is shown that to obtain nontrivial results it is necessary to study a directional implicit function problem (it turns out that in some directions there are several functions, while in others there are none).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983).
V.F. Demyanov, ed., “Nonsmooth problems of control theory and optimization” (Leningrad University Press, Leningrad, 1982).
V.F. Demyanov and A.M. Rubinov, “On quasidifferentiable functionals”, Doklady Akademii Nauk SSSR 250 (1980) 21–25 (translated in Soviet Mathematics Doklady 21 (1980) 14–17).
V.F. Demyanov and A.M. Rubinov, “On quasidifferentiable mappings”, Mathematische Operationsforschung und Statistik, Series Optimization 14 (1) (1983) 3–21.
V.F. Demyanov and L.N. Polyakova, “Minimization of a quasidifferentiable function on a quasidifferentiable set” (in Russian), Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fisiki 20 (4) (1980) 849–856 (translated in the USSR Computational Mathematics and Mathematical Physics 20 (4) (1981) 34–43).
V.F. Demyanov and L.V. Vasiliev, Nondifferentiable optimization (in Russian) (Nauka, Moscow, 1981).
H. Halkin, “Interior mapping theorem with set-valued derivative”, Journal d’Analyse Mathematique 30 (1976) 200–207.
J.-B. Hiriart-Urruty, “Tangent cones, generalized gradients and mathematical programming in Banach spaces”, Mathematics of Operations Research 4: (1979) 79–97.
A.D. Ioffe, “Nonsmooth analysis: Differential calculus of nondifferentiable mappings”, Transactions of the American Mathematical Society 266 (1), (1981) 1–56.
S. Kakutani, “A generalization of Brower’s fixed point theorem”, Duke Mathematical Journal 8 (3) (1941) 457–459.
L.V. Kantorovich and G.P. Akilov, Functional analysis, (Nauka, Moscow, 1977).
L.N. Polyakova, “Necessary conditions for an extremum of quasidifferentiable functions” (in Russian), Vestnik Leningradskogo Universiteta 13 (1980) 57–62 (translated in Vestnik Leningrad University Mathematics 13 (1981) 241–247.
B.H. Pourciau, “Analysis and optimization of Lipschitz continuous mappings”, Journal of Optimization Theory and Applications 22 (1977) 311–351.
J. Warga, “An implicit function theorem without differentiability”, Proceedings of the American Mathematical Society 69 (1978) 65–69.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1986 The Mathematical Programming Society, Inc.
About this chapter
Cite this chapter
Demidova, V.A., Demyanov, V.F. (1986). A directional implicit function theorem for quasidifferentiable functions. In: Demyanov, V.F., Dixon, L.C.W. (eds) Quasidifferential Calculus. Mathematical Programming Studies, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121140
Download citation
DOI: https://doi.org/10.1007/BFb0121140
Received:
Revised:
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00928-0
Online ISBN: 978-3-642-00929-7
eBook Packages: Springer Book Archive