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second-order derivative; $C^{1,1}$ function; $\ell $-stable function; $\tilde {\ell }$-stability
The notion of $\tilde {\ell }$-stability is defined using the lower Dini directional derivatives and was introduced by the authors in their previous papers. In this paper we prove that the class of $\tilde {\ell }$-stable functions coincides with the class of C$^{1,1}$ functions. This also solves the question posed by the authors in SIAM J. Control Optim. 45 (1) (2006), pp.\ 383--387.
[1] Bednařík, D., Pastor, K.: Second-order sufficient condition for $\tilde{\ell}$-stable functions. Acta Univ. Palacki. Olomuc., Fac. Rer. Nat., Mathematica 46 (2007), 7-18. MR 2387488
[2] Bednařík, D., Pastor, K.: Errata: Elimination of strict convergence in optimization. SIAM J. Control Optim. 45 (2006), 383-387. DOI 10.1137/050636309 | MR 2225311
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[5] Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization. Appl. Math. 51 (2006), 5-36. DOI 10.1007/s10492-006-0002-1 | MR 2197320 | Zbl 1164.90399
[6] Thompson, B. S.: Real Analysis. Springer, Berlin (1985).
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