| Title:
|
Necessary conditions for vector optimization in infinite dimension (English) |
| Author:
|
Dvorská, Marie |
| Author:
|
Pastor, Karel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
51 |
| Issue:
|
6 |
| Year:
|
2015 |
| Pages:
|
1023-1034 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper we present second-order necessary conditions for constrained vector optimization problems in infinite-dimensional spaces. In this way we generalize some corresponding results obtained earlier. (English) |
| Keyword:
|
$C^{1{},1}$–function |
| Keyword:
|
${\ell }$–stable function |
| Keyword:
|
generalized second-order directional derivative |
| Keyword:
|
Dini derivative |
| Keyword:
|
vector optimization |
| MSC:
|
49J50 |
| MSC:
|
49J52 |
| MSC:
|
49K10 |
| MSC:
|
90C29 |
| MSC:
|
90C30 |
| idZBL:
|
Zbl 06537793 |
| idMR:
|
MR3453683 |
| DOI:
|
10.14736/kyb-2015-6-1023 |
| . |
| Date available:
|
2016-01-21T18:31:57Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144822 |
| . |
| Reference:
|
[1] Bednařík, D, Berková, A: On some properties of l-stable functions..In: Proc.14th WSEAS International Conference on Mathematical Methods, Comput. Techniques and Intelligent Systems (A. J. Viamonte, ed.), Porto 2012, pp. 52-55. |
| Reference:
|
[2] Bednařík, D., Pastor, K.: Elimination of strict convergence in optimization..SIAM J. Control Optim. 43 (2004), 1063-1077. Zbl 1089.49023, MR 2114389, 10.1137/s0363012903424174 |
| Reference:
|
[3] Bednařík, D., Pastor, K.: On second-order conditions in unconstrained optimization..Math. Programming, Ser. A 113 (2008), 283-298. Zbl 1211.90276, MR 2375484, 10.1007/s10107-007-0094-8 |
| Reference:
|
[4] Bednařík, D., Pastor, K.: Differentiability properties of functions that are ${\ell}$-stable at a point..Nonlinear Analysis 69 (2008), 3128-3135. Zbl 1146.49017, MR 2452120, 10.1016/j.na.2007.09.006 |
| Reference:
|
[5] Bednařík, D., Pastor, K.: $\ell-$stable functions are continuous..Nonlinear Analysis 70 (2009), 2317-2324. Zbl 1158.49022, MR 2498316, 10.1016/j.na.2008.03.011 |
| Reference:
|
[6] Bednařík, D., Pastor, K.: A note on second-order optimality conditions..Nonlinear Analysis 71 (2009), 1964-1969. MR 2524410, 10.1016/j.na.2009.01.035 |
| Reference:
|
[7] Bednařík, D., Pastor, K.: On l-stable mappings with values in infinite dimensional Banach spaces..Nonlinear Analysis 72 (2010), 1198-1209. MR 2577520, 10.1016/j.na.2009.08.004 |
| Reference:
|
[8] Bednařík, D., Pastor, K.: On second-order condition in constrained vector optimization..Nonlinear Analysis 74 (2011), 1372-1382. MR 2746815, 10.1016/j.na.2010.10.009 |
| Reference:
|
[9] Cominetti, R., Correa, R.: A generalized second-order derivative in nonsmooth optimization..SIAM J. Control Optim. 28 (1990), 789-809. Zbl 0714.49020, MR 1051624, 10.1137/0328045 |
| Reference:
|
[10] Chan, W. L., Huang, L. R., Ng, K. F.: On generalized second-order derivatives and Taylor expansions in nonsmooth optimization..SIAM J. Control Optim. 32 (1994), 591-611. Zbl 0801.49016, MR 1269984, 10.1137/s0363012992227423 |
| Reference:
|
[11] Dvorská, M., Pastor, K.: On comparison of $\ell-$stable vector optimization results..Mathematica Slovaca 64 (2014), 1-18. MR 3255866, 10.2478/s12175-014-0252-4 |
| Reference:
|
[12] Dvorská, M., Pastor, K.: Generalization of $C^{1,1}$ property in infinite dimension..Acta Mathematica Vietnamica, accepted. 10.1007/s40306-015-0129-9 |
| Reference:
|
[13] Fabian, M, Hájek, P., Habala, P., Santalucía, V. M., Pelant, J., Zizler, V.: Functional Analysis and Infinite-Dimensional Geometry..Springer-Verlag, New York 2001. Zbl 0981.46001, MR 1831176, 10.1007/978-1-4757-3480-5 |
| Reference:
|
[14] Ginchev, I.: On scalar and vector $\ell-$stable functions..Nonlinear Analysis 74 (2011), 182-194. Zbl 1205.26007, MR 2734988, 10.1016/j.na.2010.08.032 |
| Reference:
|
[15] Ginchev, I., Guerraggio, A.: Second-order conditions for constrained vector optimization problems with ${\ell}$-stable data..Optimization 60 (2011), 179-199. Zbl 1237.90221, MR 2772143, 10.1080/02331930903578718 |
| Reference:
|
[16] Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization..Applications of Mathematics 51 (2006), 5-36. Zbl 1164.90399, MR 2197320, 10.1007/s10492-006-0002-1 |
| Reference:
|
[17] Ginchev, I., Guerraggio, A., Rocca, M.: Second order conditions for $C^{1,1}$ constrained vector optimization..Math. Programming, Ser. B 104 (2005), 389-405. MR 2179243, 10.1007/s10107-005-0621-4 |
| Reference:
|
[18] Ginchev, I., Guerraggio, A., Rocca, M.: First order conditions for $C^{0,1}$ constrained vector optimization..In: Variational Analysis and Applications (F. Gianessi, A. Maugeri, ed.), Kluwer Academic, London 2005, pp. 427-450. Zbl 1148.90011, MR 2159985, 10.1007/0-387-24276-7_27 |
| Reference:
|
[19] Guerraggio, A., Luc, D. T.: Optimality conditions for $C^{1,1}$ vector optimization problems..J. Optim. Theory Appl. 109 (2001), 615-629. Zbl 1038.49027, MR 1835076, 10.1023/a:1017519922669 |
| Reference:
|
[20] Guerraggio, A., Luc, D. T.: Optimality conditions for $C^{1,1}$ constrained multiobjective problems..J. Optim. Theory Appl. 116 (2003), 117-129. Zbl 1030.90115, MR 1961031, 10.1023/a:1022114319999 |
| Reference:
|
[21] Georgiev, P. G., Zlateva, N. P.: Second-order subdifferentials of $C^{1,1}$ functions and optimality conditions..Set-Valued Analysis 4 (1996), 101-117. MR 1394408, 10.1007/bf00425960 |
| Reference:
|
[22] 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. Comput. 11 (1984), 43-56. MR 0726975, 10.1007/bf01442169 |
| Reference:
|
[23] Jahn, J.: Vector Optimization..Springer-Verlag, Berlin 2004. Zbl 1298.90092, MR 2058695, 10.1007/978-3-540-24828-6 |
| Reference:
|
[24] Jameson, G.: Ordered Linear Spaces..Springer-Verlag, Berlin 1970. Zbl 0234.46007, MR 0438077, 10.1007/bfb0059130 |
| Reference:
|
[25] Jeyakumar, V., Luc, D. T.: Nonsmooth Vector Functions and Continuous Optimization..Springer-Verlag, Berlin 2008. Zbl 1138.90002, MR 2354918 |
| Reference:
|
[26] Khanh, P. Q., Tuan, N. D.: First and second order optimality conditions using approximations for nonsmooth vector optimization in Banach spaces..J. Optim. Theory Appl. 130 (2006), 289-308. Zbl 1143.90386, MR 2282352, 10.1007/s10957-006-9103-y |
| Reference:
|
[27] Khanh, P. Q., Tuan, N. D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives..J. Optim. Theory Appl. 133 (2007), 341-357. Zbl 1149.90134, MR 2333819, 10.1007/s10957-007-9169-1 |
| Reference:
|
[28] Khanh, P. Q., Tuan, N. D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming I: ${\ell}$-stability and set-valued directional derivatives..J. Math. Anal. Appl. 403 (2013), 695-702. MR 3037499, 10.1016/j.jmaa.2012.12.076 |
| Reference:
|
[29] Khanh, P. Q., Tuan, N. D.: Second-order optimality conditions with the envelope-like effect in nonsmooth multiobjective mathematical programming II: Optimality conditions..J. Math. Anal. Appl. 403 (2013), 703-714. Zbl 1291.90225, MR 3037500, 10.1016/j.jmaa.2012.12.075 |
| Reference:
|
[30] Li, S. J., Xu, S.: Sufficient conditions of isolated minimizers for constrained programming problems..Numer. Funct. Anal. Optim. 31 (2010), 715-727. Zbl 1218.90214, MR 2682838, 10.1080/01630563.2010.490970 |
| Reference:
|
[31] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications..Springer-Verlag, Berlin 2006. Zbl 1100.49002, MR 2191744 |
| Reference:
|
[32] Mordukhovich, B. S., Treiman, J. S., Zhu, Q. J.: An extended extremal principal with applications to multiobjective optimization..SIAM J. Optim. 14 (2003), 359-379. MR 2048166, 10.1137/s1052623402414701 |
| Reference:
|
[33] Pastor, K.: Differentiability properties of ${\ell}$-stable vector functions in infinite-dimensional normed spaces..Taiwan. J. Math. 18 (2014), 187-197. MR 3162119, 10.11650/tjm.18.2014.2605 |
| Reference:
|
[34] Rockafellar, R. T.: Convex Analysis..Princeton University Press, Princeton 1970. Zbl 1011.49013, MR 0274683, 10.1515/9781400873173 |
| Reference:
|
[35] Rockafellar, R. T., Wets, J. B.: Variational Analysis..Springer-Verlag, New York 1998. Zbl 0888.49001, MR 1491362, 10.1007/978-3-642-02431-3 |
| . |
