## 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.

### MSC:

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

 [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
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