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Second-order moderate derivatives. (English) Zbl 0810.49017
For a real-valued function $$f$$ defined on a normed vector space, this paper introduces two kinds of second-order directional derivatives: $f^{[2]}(x, u,v)= \sup_{w,z\in X} \limsup_{s,t\to 0_ +} \Delta^ 2_{s,t} f(x,u,v,w,z),$ where $$\Delta^ 2_{s,t} f(x,u,v,w,z)= {1\over st}[f(y+ su+ tv)- f(y+ su)- f(y+ tv)- f(y)]$$, with $$y= x+ sw+ tz$$ and $f^{\square\square}(x,u,v)= \sup_{w\in X} \limsup_{t\to 0_ +} \textstyle{{1\over t}} [f^ \square (x+ tw+ tv,u)- f^ \square(x+ tw,u)],$ where $$f^ \square(x,v)= \sup_{w\in X} \limsup_{t\to 0_ +} {1\over t} [f(x+ tw+ tv)- f(x+ tw)]$$. It is shown in this paper that if $$f$$ is Gâteaux differentiable and [2]-semi- differentiable, that is, $f''(x,u,v)= \lim_{s,t\to 0_ +}\textstyle{{1\over st}} [f(x+ su+ tv)- f(x+ su)- f(x+ tv)+ f(x)],$ exists, then $f^{[2]}(x, u,v)= f^{\square\square}(x,u,v)= f''(x, u,v).\tag{1}$ It would be interesting to find a more general class of functions where the first equality in (1) still holds. Calculus rules, such as chain rules and Taylor expansions, are established using these generalized second-order directional derivatives. Applications are given to derive optimality conditions of optimization problems.

##### MSC:
 49J52 Nonsmooth analysis
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##### References:
 [1] Michel, P.; Penot, J.-P., Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes, C. r. acad. sci. Paris, 298, 269-272, (1984) · Zbl 0567.49008 [2] Michel, P.; Penot, J.-P., A generalized derivative for calm and stable functions, Diff. integral eqns, 5, 2, 433-454, (1992) · Zbl 0787.49007 [3] Cominetti, R.; Correa, R., Sur une dérivée du second ordre en analyse non différentiable, C. r. acad. sci. Paris serie I, 303, 17, 861-864, (1986) · Zbl 0604.46049 [4] Milosz, T., Thesis, (1988), Institute of Mathematics, Polish Academy of Sciences, (in Russian). [5] Cominetti, R.; Correa, R., A generalized second-order derivative in nonsmooth optimization, SIAM J. control optim., 28, 4, 789-809, (1990) · Zbl 0714.49020 [6] Michel, P.; Penot, J.-P., Dérivées secondes modérées de fonctions non dérivables, C. r. acad. sci. Paris, serie I, 316, 995-998, (1993) · Zbl 0789.46034 [7] Dieudonne, J., Foundations of modern analysis, (1960), Academic Press New York · Zbl 0100.04201 [8] Frankowska, H., The adjoint differential inclusions associated to a minimal trajectory of a differential inclusion, Cahiers math. de la Décision, IX, (1983), Université de Paris, No. 8315 [9] Giner, E., Ensembles et fonctions étoilées: application à l’optimisation et au calcul différentiel généralisé, (1981), Toulouse (preprint) [10] {\scMazure} M. L. & {\scVolle} M., Equations inf-convolutives et conjugaison de Moreau-Fenchel (to appear). · Zbl 0760.49013 [11] Penot, J.-P., Variations on the theme of nonsmooth analysis: another sub-differential, (), 41-54, Lecture Notes in Economics and Mathematics [12] Chang, W.L.; Huang, L.R.; Ng, K.F., On generalized second-order derivatives and Taylor expansions in nonsmooth optimization, (1992), Chinese Univ. of Hong Kong, (preprint) [13] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, Lecture notes in mathematics, Vol. 580, (1977), Springer Berlin · Zbl 0346.46038 [14] Rockafellar, R.T., Conjugate duality and optimization, Cbms-nsf no. 16., (1974), SIAM Philadelphia · Zbl 0326.49008 [15] Vainberg, M.M., Variational methods for the study of nonlinear operators, (1964), Holden Day San Francisco · Zbl 0122.35501 [16] Mordukhovich, B.S., On variational analysis of differential inclusions, (), 199-213, Pitman Research Notes in Mathematics · Zbl 0761.49003
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