Geodesic
Second-order sufficient condition for $\tilde\ell$-stable functions
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 46 (2007) no. 1, pp. 7-18 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The aim of our article is to present a proof of the existence of local minimizer in the classical optimality problem without constraints under weaker assumptions in comparisons with common statements of the result. In addition we will provide rather elementary and self-contained proof of that result.
The aim of our article is to present a proof of the existence of local minimizer in the classical optimality problem without constraints under weaker assumptions in comparisons with common statements of the result. In addition we will provide rather elementary and self-contained proof of that result.
Classification : 26B05, 49J52, 49K10, 90C30
Keywords: second-order derivative; $C^{1, 1}$ function; stable function; isolated minimizer of order 2 
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Bednařík, Dušan; Pastor, Karel. Second-order sufficient condition for $\tilde\ell$-stable functions. Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, Tome 46 (2007) no. 1, pp. 7-18. http://geodesic.mathdoc.fr/item/AUPO_2007_46_1_a0/

[BP1] Bednařík D., Pastor K.: Elimination of strict convergence in optimization. SIAM J. Control Optim. 43, 3 (2004), 1063–1077. | MR | Zbl

[BP2] Bednařík D., Pastor K.: On second-order conditions in unconstrained optimization. Math. Programming, in print; online: http://www.springerlink.com/content/tt7g83q36243l144/ | Zbl

[BP3] Bednařík D., Pastor K.: Erratum to Elimination of strict convergence in optimization. SIAM J. Control Optim. 45 (2006), 382–387. | MR

[BZ] Ben-Tal A., Zowe J.: Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47 (1985), 483–490. | MR | Zbl

[CC] Cominetti R., Correa R.: A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28 (1990), 789–809. | MR | Zbl

[GGR] Ginchev I., Guerraggio A., Rocca M.: From scalar to vector optimization. Appl. Math. 51 (2006), 5–36. | MR | Zbl

[HSN] 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 (1984), 43–56. | MR

[KT] Klatte D., Tammer K.: On second-order sufficient optimality conditions for $C^{1,1}$ optimization problems. Optimization 19 (1988), 169–179. | MR

[LK] Liu L., Křížek M.: The second order optimality conditions for nonlinear mathematical programming with $C^{1,1}$ data. Appl. Math. 42 (1997), 311–320. | MR

[Q1] Qi L.: Superlinearly convergent approximate Newton methods for $LC^1$ optimization problem. Math. Programming 64 (1994), 277–294. | MR | Zbl

[Q2] Qi L.: $LC^1$ functions and $LC^1$ optimization. Operations Research and its applications (D.Z. Du, X.S. Zhang and K. Cheng eds.), World Publishing, Beijing, 1996, pp. 4–13. | Zbl

[TR] Torre D. L., Rocca M.: Remarks on second order generalized derivatives for differentiable functions with Lipschitzian jacobian. Applied Mathematics E-Notes 3 (2003), 130–137. | MR | Zbl

[Zo] Zorich V. A.: Mathematical Analysis. : Springer-Verlag, Berlin. 2004.