Key Words
- convex functions
- ɛ-subdifferential
- approximate first-order directional derivative
- approximate second-order directional derivative
- generalized second derivatives
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.D. ALEXANDROV. The existence almost everywhere of the second differential of a convex function and some associated properties of convex surfaces (in Russian), Učenye Zapiski Leningr. Gos. Univ. Ser. Mat. 37 no6 (1939), 3–35.
E. ASPLUND and R.T. ROCKAFELLAR, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467.
A. AUSLENDER, Differential properties of the support function of the ɛ-subdifferential or a convex function, Note aux Comptes Rendus Acad. Sc. Paris, t. 292 (1981), 221–224 & Math. Programming, to appear.
A. AUSLENDER, Stability in mathematical programming with nondifferentiable data; second-order directional derivative for lower-C 2 functions. Preprint 1981.
M.L. BALINSKI and P. WOLFE, editors, Nondifferentiable Optimization, Math. Programming Study 3, North-Holland (1975).
A. BRØNDSTED and R.T. ROCKAFELLAR, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611.
H. BUSEMANN, Convex Surfaces, Interscience Tracts in Pure and Applied Mathematics, 1958.
F.H. CLARKE, Generalized gradients and applications, Trans. Amer. Math. Soc. 205, (1975), 247–262.
F.H. CLARKE, Generalized gradients of Lipschitz functionals, Advances in Mathematics 40, (1981), 52–67.
F.H. CLARKE, Nonsmooth Analysis and Optimization, John Wiley & Sons, book to appear in 1983.
J.-P. CROUZEIX, Contributions à l’étude des fonctions quasiconvexes, Thèse de Doctorat Es Sciences Mathématiques, Université de Clermont-Ferrand II, (1977).
V.F. DEM’YANOV and V.N. MALOZEMOV, Introduction to Minimax, John Wiley & Sons, 1974.
R.M. DUDLEY, On second derivatives of convex functions, Math. Scand. 41 (1977), 159–174 & 46 (1980), 61.
J.-B. HIRIART-URRUTY, Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), 123–134.
J.-B. HIRIART-URRUTY, ɛ-subdifferential calculus, in Proceedings of the Colloquium "Convex Analysis and Optimization", Imperial College, London (28–29 February 1980), to appear in 1982.
J.-B. HIRIART-URRUTY, Approximating a second-order directional derivative for nonsmooth convex functions, SIAM J. on Control and Optimization, to appear in 1982.
J.-B. HIRIART-URRUTY, Limiting behaviour of the approximate first-order and second-order directional derivatives for a convex function, Nonlinear Analysis: Theory, Methods & Applications, to appear in 1982.
J.-B. HIRIART-URRUTY, Calculus rules on the approximate second-order directional derivative of a convex function, in preparation.
B. JESSEN, Om konvekes Kurvers Krumning, Mat. Tidsskr. B (1929), 50–62.
S.S. KUTATELADZE, Convex ɛ-programming. Soviet Math. Dokl. 20 (1979), 391–393.
S.S. KUTATELADZE, ɛ-subdifferentials and ɛ-optimization (in Russian), Sibirskii Matematicheskii Journal (1980), 120–130.
C. LEMARECHAL and R. MIFFLIN, editors, Nonsmooth Optimization, I.I.A.S.A. Proceedings Series, Pergamon Press (1978).
C. LEMARECHAL, Extensions Diverses des Méthodes de Gradient et Applications, Thèse de Doctorat Es Sciences Mathématiques, Paris (1980).
C. LEMARECHAL and E.A. NURMINSKII, Sur la différentiabilité de la fonction d’appui du sous-différentiel approché, Note aux Comptes Rendus Acad. Sc. Paris, t. 290 (1980), 855–858.
C. LEMARECHAL, Some remarks on second-order methods for convex optimization, Meeting "Optimization: Theory & Algorithms" Confolant, 16–20 March 1981.
C. LEMARECHAL, personal communication (March 1981).
F. MIGNOT, Contrôle dans les inéquations variationnelles elliptiques, J. of Functional Analysis, Vol. 22 (1976), 130–185.
F. MIGNOT, personal communication (February 1981).
E. A. NURMINSKII, on ɛ-differential mapping and their applications in nondifferentiable optimization, Working paper 78–58, I.I.A.S.A., December 1978.
R.-T. ROCKAFELLAR, Convex Analysis, Princeton University Press, 1970.
R.-T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control & Optimization 14 (1976), 877–898.
J.-J. STRODIOT, NGUYEN VAN HIEN and N. HEUKEMES, ɛ-optimal solutions in nondifferentiable convex programming and some related questions. Department of Mathematics, University of Namur, preprint 1980.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Hiriart-Urruty, J.B. (1983). The approximate first-order and second-order directional derivatives for a convex function. In: Cecconi, J.P., Zolezzi, T. (eds) Mathematical Theories of Optimization. Lecture Notes in Mathematics, vol 979. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066253
Download citation
DOI: https://doi.org/10.1007/BFb0066253
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11999-9
Online ISBN: 978-3-540-39473-0
eBook Packages: Springer Book Archive