\(\ell \)-stable functions are continuous. (English) Zbl 1158.49022

Summary: It has been shown that \(\ell \)-stability of the function at some point implies continuity near the point. Then the previous second-order optimality condition introduced by the authors can be stated under weak assumptions.


49K10 Optimality conditions for free problems in two or more independent variables
26B05 Continuity and differentiation questions
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[1] Auslender, A., Stability in mathematical programming with nondifferentiable data, SIAM J. control optim., 22, 239-254, (1984) · Zbl 0538.49020
[2] Bednařík, D.; Pastor, K., Elimination of strict convergence in optimization, SIAM J. control optim., 43, 3, 1063-1077, (2004) · Zbl 1089.49023
[3] Bednařík, D.; Pastor, K., On second-order conditions in unconstrained optimization, Math. program., 113, 2, 283-298, (2008) · Zbl 1211.90276
[4] D. Bednařík, K. Pastor, Differentiability properties of functions that are \(\ell\)-stable at a point, Nonlinear Anal., in print, doi:10.1016/j.na.2007.09.006
[5] Ben-Tal, A.; Zowe, J., Directional derivatives in nonsmooth optimization, J. optim. theory appl., 47, 483-490, (1985) · Zbl 0556.90074
[6] Cominetti, R.; Correa, R., A generalized second-order derivative in nonsmooth optimization, SIAM J. control optim., 28, 789-809, (1990) · Zbl 0714.49020
[7] Diewert, W.E., Alternative characterizations of six kinds of quasiconcavity in the nondifferentiable cse with applications to nonsmooth programming, () · Zbl 0539.90088
[8] Ginchev, I.; Guerraggio, A.; Rocca, M., From scalar to vector optimization, Appl. math., 51, 5-36, (2006) · Zbl 1164.90399
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