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 -stability for vector functions.
Keywords: $C^{1,1}$ function, ${\ell }$-stable function, generalized second-order directional derivative, Dini derivative, weakly efficient minimizer, isolated minimizer of second-order
@article{RO_2009__43_4_359_0,
author = {Bedna\v{r}{\'\i}k, Du\v{s}an and Pastor, Karel},
title = {Decrease of $C^{1,1}$ property in vector optimization},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {359--372},
year = {2009},
publisher = {EDP Sciences},
volume = {43},
number = {4},
doi = {10.1051/ro/2009023},
mrnumber = {2573992},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2009023/}
}
TY - JOUR
AU - Bednařík, Dušan
AU - Pastor, Karel
TI - Decrease of $C^{1,1}$ property in vector optimization
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2009
SP - 359
EP - 372
VL - 43
IS - 4
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/ro/2009023/
DO - 10.1051/ro/2009023
LA - en
ID - RO_2009__43_4_359_0
ER -
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%A Pastor, Karel
%T Decrease of $C^{1,1}$ property in vector optimization
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2009
%P 359-372
%V 43
%N 4
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ro/2009023/
%R 10.1051/ro/2009023
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%F RO_2009__43_4_359_0
Bednařík, Dušan; Pastor, Karel. Decrease of $C^{1,1}$ property in vector optimization. RAIRO - Operations Research - Recherche Opérationnelle, Tome 43 (2009) no. 4, pp. 359-372. doi: 10.1051/ro/2009023
[1] and , Differential inclusions, Springer Verlag, Berlin (1984). | Zbl | MR
[2] and , On characterization of convexity for regularly locally Lipschitz functions. Nonlinear Anal. 57 (2004) 85-97. | Zbl | MR
[3] and , Elimination of strict convergence in optimization. SIAM J. Control Optim. 43 (2004) 1063-1077. | Zbl | MR
[4] and , Using the Peano derivative in unconstrained optimization. Math. Program. 113 (2008) 283-298. | Zbl
[5] and , Differentiability properties of functions that are -stable at a point. Nonlinear Anal. 69 (2008) 3128-3135. | Zbl | MR
[6] and , -stable functions are continuous. Nonlinear Anal. 70 (2009) 2317-2324. | Zbl | MR
[7] and , Directional derivatives in nonsmooth optimization. J. Optim. Theory Appl. 47 (1985) 483-490. | Zbl | MR
[8] , and , On generalized second-order derivatives and Taylor expansions in nonsmooth optimization. SIAM J. Control Optim. 32 (1994) 591-611. | Zbl | MR
[9] and , A generalized second-order derivative in nonsmooth optimization. SIAM J. Control Optim. 28 (1990) 789-809. | Zbl | MR
[10] and , Second-order Subdifferentials of Functions and Optimality Conditions. Set-Valued Anal. 4 (1996) 101-117. | Zbl | MR
[11] , Higher order optimality conditions in nonsmooth optimization. Optimization 51 (2002) 47-72. | Zbl | MR
[12] , and , Second order conditions for constrained vector optimization. Math. Program. Ser. B 104 (2005) 389-405. | Zbl | MR
[13] , and , From scalar to vector optimization. Applications of Mathematics 51 (2006) 5-36. | Zbl | MR
[14] and , Optimality conditions for vector optimization problems. J. Optim. Theory Appl. 109 (2001) 615-629. | Zbl | MR
[15] , and , Generalized Hessian matrix and second order optimality conditions for problems with data. Appl. Math. Optim. 11 (1984) 169-180. | Zbl | MR
[16] and , 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] , Vector optimization, Springer Verlag, New York (2004). | Zbl | MR
[18] and , First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl. 284 (2003) 496-510. | Zbl | MR
[19] and , First order optimality conditions in vector optimization involving stable functions. Optimization 57 (2008) 449-471. | MR
[20] and , Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133 (2007) 341-357. | Zbl | MR
[21] , Upper Lipschitz behavior of solutions to perturbed programs. Math. Program. (Ser B) 88 (2000) 285-311. | Zbl | MR
[22] , The second-order conditions of nondominated solutions for generalized multiobjective mathematical programming. J. Systems Sci. Math. Sci. 4 (1991) 128-138. | Zbl | MR
[23] and , The second-order optimality conditions for nonlinear mathematical programming with data. Appl. Math. 42 (1997) 311-320. | Zbl | MR
[24] , and , Second-order optimality conditions for nondominated solutions of multiobjective programming with data. Appl. Math. 45 (2000) 381-397. | Zbl | MR
[25] , 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] , Convexity and generalized second-order derivatives for locally Lipschitz functions. Nonlinear Anal. 60 (2005) 547-555. | Zbl | MR
[27] , Fréchet approach to generalized second-order differentiability. to appear in Studia Scientiarum Mathematicarum Hungarica 45 (2008) 333-352.
[28] , Convex analysis, Princeton University Press, Princeton (1970). | Zbl | MR
[29] , , Variational Analysis, Springer Verlag, New York (1998). | Zbl | MR
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