Limiting behaviour of the approximate first order and second order directional derivatives for a convex function. (English) Zbl 0536.26007

Given \(\epsilon>0\), the following limit \[ f''_{\epsilon}(x_ 0;d,d)=\lim_{\alpha \to 0^+}\frac{f'_{\epsilon}(x_ 0+\alpha d;d)-f'_{\epsilon}(x_ 0;d)}{\alpha} \] is defined for all convex functions f, all \(x_ 0\) and all directions d. The aim of the paper is to study the behaviour of \(f''_{\epsilon}(x_ 0;d,d)\) when \(\epsilon\) goes to \(O^+\). It is proved that \(f''_{\epsilon}(x_ 0;d,d)\) does have a limit when \(\epsilon \to 0^+;\) more precisely: \[ \lim_{\epsilon \to 0^+}f''_{\epsilon}(x_ 0;d,d)=D''_ rf(x_ 0;d)=2V_ r''f(x_ 0;d), \] where \(D''_ rf(x_ 0;d)\) is the second right Dini derivative and \(V''_ rf(x_ 0;d)\) is the second right de la Vallée Poussin derivative. The following asymptotic development has also been derived: \[ f'_{\epsilon}(x_ 0;d)=f'(x_ 0;d)+[2\epsilon D''_ rf(x_ o;d)]^{1/2}+o(\epsilon^{1/2}). \] An example where such results are useful is the so-called discrete minimax problem, \(f=\max(f_ 1,...,f_ m),\) where the \(f_ i\) are \(C^ 2\) convex functions.


26B05 Continuity and differentiation questions
90C25 Convex programming
Full Text: DOI


[1] Auslender, A., Differential properties of the support function of the ε-subdifferential of a convex function, C. r. hebd. Séanc. Acad. Sc. Paris, 292, 221-224 (1981), & Math. Programming (to appear).
[2] Bourbaki, N., Fonctions d’une variable réelle (1958), Fascicule IX, Livre IV, Chapitres 1-3. Hermann
[3] Busemann, H., Convex Surfaces, Interscience Tracts in Pure and Applied Mathematics (1958) · Zbl 0196.55101
[4] Clarke, F. H., Generalized gradients and applications, Trans. Am. math. Soc., 205, 247-262 (1975) · Zbl 0307.26012
[5] Clarke, F. H., On the inverse function theorem, Pacific J. Math., 64, 97-102 (1976) · Zbl 0331.26013
[6] Clarke, F. H., Generalized gradients of Lipschitz functionals, Advances in Mathematics, 40, 52-67 (1981) · Zbl 0463.49017
[7] Dem’yanov, V. F.; Malozemov, V. N., Introduction to Minimax (1974), John Wiley: John Wiley New York · Zbl 0781.90079
[8] Hiriart-Urruty, J.-B., Gradients généralisés de fonctions composées, applications, C. r. hebd. Séanc. Acad. Sc. Paris, 285, 781-784 (1977) · Zbl 0385.90095
[9] Hiriart-Urruty, J.-B., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Operations Res., 4, 79-97 (1979) · Zbl 0409.90086
[10] Hiriart-Urruty, J.-B., Lipschitz \(r\)-continuity of the approximate subdifferential of a convex function, Math. Scand., 47, 123-134 (1980) · Zbl 0426.26005
[11] Hiriart-Urruty, J.-B., Un concept récent pour l’analyse et l’optimisation de fonctions non différentiables: le gradient généralisé (1980), Publications de l’I.R.E.M. de Clermont-Ferrand
[12] Hiriart-Urruty, J.-B., ε-Subdifferential calculus, (Proceedings of the Colloquium “Convex Analysis and Optimization” (28-29 February 1980), Imperial College: Imperial College London), To appear. · Zbl 0426.26005
[14] Lemaréchal, C.; Nurminskii, E. A., Sur la différentiabilité de la fonction d’appui du sous-différentiel approché, C. r. hebd. Séanc. Acad. Sci. Paris, 290, 855-858 (1980) · Zbl 0459.49011
[15] Nurminskii, E. A., On ε-differential mapping and their applications in nondifferentiable optimization, Working paper 78-58 (December 1978), I.I.A.S.A.
[16] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0229.90020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.