Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T07:31:07.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Decrease of C 1,1 property in vector optimization

Published online by Cambridge University Press:  08 October 2009

Dušan Bednařík  and
Karel Pastor
Dušan Bednařík
Affiliation:
Institute of Applied Physics and Mathematics, University of Pardubice, Studentská 95, 532 10 Pardubice, Czech Republic; and Department of Mathematics, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic; [email protected]
Karel Pastor
Affiliation:
Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tř. Svobody 26, 771 46 Olomouc,Czech Republic; [email protected].
Get access

Abstract

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 ℓ-stability for vector functions.

Keywords

C 1,1 functionℓ-stable functiongeneralized second-order directional derivativeDini derivativeweakly efficient minimizerisolated minimizer of second-order.
Type
Research Article
Information
RAIRO - Operations Research , Volume 43 , Issue 4: ROADEF 07 , October 2009 , pp. 359 - 372
Copyright
© EDP Sciences, ROADEF, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.P. Aubin and A. Cellina, Differential inclusions, Springer Verlag, Berlin (1984).
Bednařík, D. and Pastor, K., On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57 (2004) 85–97.
Bednařík, D. and Pastor, K., Elimination of strict convergence in optimization. SIAM J. Control Optim. 43 (2004) 1063–1077.
Bednařík, D. and Pastor, K., Using the Peano derivative in unconstrained optimization. Math. Program. 113 (2008) 283–298. CrossRef
Bednařík, D. and Pastor, K., Differentiability properties of functions that are ℓ-stable at a point. Nonlinear Anal. 69 (2008) 3128–3135. CrossRef
Bednařík, D. and Pastor, K., ℓ-stable functions are continuous. Nonlinear Anal. 70 (2009) 2317–2324. CrossRef
Ben-tal, A. and Zowe, J., Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47 (1985) 483–490. CrossRef
Chan, W.L., Huang, L.R. and On, K.F. Ng generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32 (1994) 591–611. CrossRef
Cominetti, R. and Correa, R., A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28 (1990) 789–809. CrossRef
Georgiev, P.G. and Zlateva, N.P., Second-order Subdifferentials of C 1,1 Functions and Optimality Conditions. Set-Valued Anal. 4 (1996) 101–117. CrossRef
Ginchev, I., Higher order optimality conditions in nonsmooth optimization. Optimization 51 (2002) 47–72. CrossRef
Ginchev, I., Guerraggio, A. and Rocca, M., Second order conditions for C 1,1 constrained vector optimization. Math. Program. Ser. B 104 (2005) 389–405. CrossRef
Ginchev, I., Guerraggio, A. and Rocca, M., From scalar to vector optimization. Applications of Mathematics 51 (2006) 5–36. CrossRef
Guerraggio, A. and Luc, D.T., Optimality conditions for C 1,1 vector optimization problems. J. Optim. Theory Appl. 109 (2001) 615–629. CrossRef
Hiriart–Urruty, J.J., Strodiot, J.J. and Nguyen, V.H., Generalized Hessian matrix and second order optimality conditions for problems with C 1,1 data. Appl. Math. Optim. 11 (1984) 169–180.
Huang, L.R. and On, K.F. Ng lower bounds of the second-order directional derivatives of Ben-Tal-Zowe and Chaney. Math. Oper. Res. 22 (1997) 747–753. CrossRef
J. Jahn, Vector optimization, Springer Verlag, New York (2004).
Jiménez, B. and Novo, V., First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284 (2003) 496–510. CrossRef
Jiménez, B. and Novo, V., First order optimality conditions in vector optimization involving stable functions. Optimization 57 (2008) 449–471. CrossRef
Khanh, P.Q. and Tuan, N.D., Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133 (2007) 341–357. CrossRef
Klatte, D., Upper Lipschitz behavior of solutions to perturbed C 1,1 programs. Math. Program. (Ser B) 88 (2000) 285–311. CrossRef
Liu, L., The second-order conditions of nondominated solutions for C 1,1 generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 4 (1991) 128–138.
Liu, L. and Křížek, M., The second-order optimality conditions for nonlinear mathematical programming with C 1,1 data. Appl. Math. 42 (1997) 311–320. CrossRef
Liu, L., NeittaanmÄki, P. and Křížek, M., Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data. Appl. Math. 45 (2000) 381–397. CrossRef
Maruyama, Y., 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. CrossRef
Pastor, K., Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal. 60 (2005) 547–555. CrossRef
K. Pastor, Fréchet approach to generalized second-order differentiability. to appear in Studia Scientiarum Mathematicarum Hungarica 45 (2008) 333–352.
R.T. Rockafellar, Convex analysis, Princeton University Press, Princeton (1970).
R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, Springer Verlag, New York (1998).