Decrease of $C^{1,1}$ property in vector optimization

Bednařík, Dušan ; Pastor, Karel

RAIRO - Operations Research - Recherche Opérationnelle, Tome 43 (2009) no. 4, pp. 359-372.

Résumé

In the paper we generalize sufficient and necessary optimality conditions obtained by Ginchev, Guerraggio, Rocca, and by authors with the help of the notion of $\ell$-stability for vector functions.

Détail

DOI : 10.1051/ro/2009023

Classification : 49K10, 49J52, 49J50, 90C29, 90C30
Keywords: $C^{1,1}$ function, ${\ell }$-stable function, generalized second-order directional derivative, Dini derivative, weakly efficient minimizer, isolated minimizer of second-order

Bibliographie

[1] J.P. Aubin and A. Cellina, Differential inclusions, Springer Verlag, Berlin (1984). | Zbl | MR

[2] D. Bednařík and K. Pastor, On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57 (2004) 85-97. | Zbl | MR

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

[4] D. Bednařík and K. Pastor, Using the Peano derivative in unconstrained optimization. Math. Program. 113 (2008) 283-298. | Zbl

[5] D. Bednařík and K. Pastor, Differentiability properties of functions that are $\ell$-stable at a point. Nonlinear Anal. 69 (2008) 3128-3135. | Zbl | MR

[6] D. Bednařík and K. Pastor, $\ell$-stable functions are continuous. Nonlinear Anal. 70 (2009) 2317-2324. | Zbl | MR

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

[8] W.L. Chan, L.R. Huang and K.F. Ng, On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32 (1994) 591-611. | Zbl | MR

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

[10] P.G. Georgiev and N.P. Zlateva, Second-order Subdifferentials of $C^{1,1}$ Functions and Optimality Conditions. Set-Valued Anal. 4 (1996) 101-117. | Zbl | MR

[11] I. Ginchev, Higher order optimality conditions in nonsmooth optimization. Optimization 51 (2002) 47-72. | Zbl | MR

[12] I. Ginchev, A. Guerraggio and M. Rocca, Second order conditions for $C^{1,1}$ constrained vector optimization. Math. Program. Ser. B 104 (2005) 389-405. | Zbl | MR

[13] I. Ginchev, A. Guerraggio and M. Rocca, From scalar to vector optimization. Applications of Mathematics 51 (2006) 5-36. | Zbl | MR

[14] A. Guerraggio and D.T. Luc, Optimality conditions for $C^{1,1}$ vector optimization problems. J. Optim. Theory Appl. 109 (2001) 615-629. | Zbl | MR

[15] J.J. Hiriart-Urruty, J.J. Strodiot and V.H. Nguyen, Generalized Hessian matrix and second order optimality conditions for problems with $C^{1,1}$ data. Appl. Math. Optim. 11 (1984) 169-180. | Zbl | MR

[16] L.R. Huang and K.F. Ng, On lower bounds of the second-order directional derivatives of Ben-Tal-Zowe and Chaney. Math. Oper. Res. 22 (1997) 747-753. | Zbl | MR

[17] J. Jahn, Vector optimization, Springer Verlag, New York (2004). | Zbl | MR

[18] B. Jiménez and V. Novo, First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284 (2003) 496-510. | Zbl | MR

[19] B. Jiménez and V. Novo, First order optimality conditions in vector optimization involving stable functions. Optimization 57 (2008) 449-471. | MR

[20] P.Q. Khanh and N.D. Tuan, Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133 (2007) 341-357. | Zbl | MR

[21] D. Klatte, Upper Lipschitz behavior of solutions to perturbed $C^{1,1}$ programs. Math. Program. (Ser B) 88 (2000) 285-311. | Zbl | MR

[22] L. Liu, The second-order conditions of nondominated solutions for $C^{1,1}$ generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 4 (1991) 128-138. | Zbl | MR

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

[24] L. Liu, P. Neittaanmäki and M. Křížek, Second-order optimality conditions for nondominated solutions of multiobjective programming with $C^{1,1}$ data. Appl. Math. 45 (2000) 381-397. | Zbl | MR

[25] Y. Maruyama, Second-order necessary conditions for nonlinear optimization problems in Banach spaces and their application to an optimal control problem. Math. Oper. Res. 15 (1990) 467-482. | Zbl | MR

[26] K. Pastor, Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal. 60 (2005) 547-555. | Zbl | MR

[27] K. Pastor, Fréchet approach to generalized second-order differentiability. to appear in Studia Scientiarum Mathematicarum Hungarica 45 (2008) 333-352.

[28] R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton (1970). | Zbl | MR

[29] R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer Verlag, New York (1998). | Zbl | MR