Differentiability properties of functions that are \(\ell \)-stable at a point. (English) Zbl 1146.49017

Summary: The class of the \(\ell \)-stable functions is introduced in previous papers of the authors as a generalization of the class of \(C^{1,1}\) functions. The importance of this class is that minimization problems with \(\ell \)-stable data admit second-order optimality conditions in terms of suitable directional derivatives similar to those for ones with \(C^{1,1}\) data. The present paper studies differentiability properties of \(\ell \)-stable functions in normed spaces and generalizes to infinite-dimensional spaces some results of the authors.


49K10 Optimality conditions for free problems in two or more independent variables
26B05 Continuity and differentiation questions
Full Text: DOI


[1] Bednařík, D.; Pastor, K., Elimination of strict convergence in optimization, SIAM J. control optim., 43, 3, 1063-1077, (2004) · Zbl 1089.49023
[2] D. Bednařík, K. Pastor, On second-order conditions in unconstrained optimization, Math. Program., in press (doi:10.1007/s10107-007-0094-8)
[3] Ben-Tal, A.; Zowe, J., Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems, Math. program., 24, 70-91, (1982) · Zbl 0488.90059
[4] Ben-Tal, A.; Zowe, J., Directional derivatives in nonsmooth optimization, J. optim. theory appl., 47, 483-490, (1985) · Zbl 0556.90074
[5] Cominetti, R.; Correa, R., A generalized second-order derivative in nonsmooth optimization, SIAM J. control optim., 28, 789-809, (1990) · Zbl 0714.49020
[6] Chan, W.L.; Huang, L.R.; Ng, K.F., On generalized second-order derivatives and Taylor expansions in nonsmooth optimization, SIAM J. control optim., 32, 591-611, (1994) · Zbl 0801.49016
[7] Ginchev, I.; Guerraggio, A.; Rocca, M., From scalar to vector optimization, Appl. math., 51, 5-36, (2006) · Zbl 1164.90399
[8] Georgiev, P.G.; Zlateva, N.P., Second-order subdifferentials of \(C^{1, 1}\) functions and optimality conditions, Set-valued anal., 4, 101-117, (1996) · Zbl 0864.49012
[9] J.B. Hiriart-Urruty, Contributions a la programmation mathematique: Deterministe et stocastique, Doctoral Thesis, Univ. Clermont-Ferrand, 1977
[10] Hiriart-Urruty, J.B.; Strodiot, J.J.; Nguyen, V.H., Generalized Hessian matrix and second-order optimality conditions for problems with \(C^{1, 1}\) data, Appl. math. optim., 11, 43-56, (1984) · Zbl 0542.49011
[11] Kawasaki, H., An envelope-like effect of infinite many inequality constraints on second-order necessary conditions for minimization problems, Math. program., 41, 73-96, (1988) · Zbl 0661.49012
[12] Klatte, D., Upper Lipschitz behavior of solutions to perturbed \(C^{1, 1}\) programs, Math. program. (ser B), 88, 285-311, (2000) · Zbl 1017.90111
[13] Phelps, R.R., Convex functions, monotone operators and differentiability, (1993), Springer-Verlag Berlin · Zbl 0921.46039
[14] Preiss, D., Fréchet derivatives of Lipschitz functions, J. funct. anal., 91, 312-345, (1990) · Zbl 0711.46036
[15] Qi, L., Superlinearly convergent approximate Newton methods for \(L C^1\) optimization problem, Math. program., 64, 277-294, (1994) · Zbl 0820.90102
[16] Rockafellar, R.T., Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. oper. res., 14, 462-484, (1989) · Zbl 0698.90070
[17] Rockafellar, R.T.; Wets, J.B., Variational analysis, (1998), Springer-Verlag New York · Zbl 0888.49001
[18] Torre, D.L.; Rocca, M., Remarks on second order generalized derivatives for differentiable functions with lipschitzian Jacobian, Appl. math. E-notes, 3, 130-137, (2003) · Zbl 1057.49016
[19] Yang, X.Q., On second-order directional derivatives, Nonlinear anal., 26, 55-66, (1996) · Zbl 0839.90138
[20] Yang, X.Q., On relations and applications of generalized second-order directional derivatives, Nonlinear anal., 36, 595-614, (1999) · Zbl 0990.49016
[21] Diewert, W.E., Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable case with applications to nonsmooth programming, () · Zbl 0539.90088
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.