| Title:
|
Locally Lipschitz vector optimization with inequality and equality constraints (English) |
| Author:
|
Ginchev, Ivan |
| Author:
|
Guerraggio, Angelo |
| Author:
|
Rocca, Matteo |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
55 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
77-88 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The present paper studies the following constrained vector optimization problem: $\min _Cf(x)$, $g(x)\in -K$, $h(x)=0$, where $f\colon\Bbb R^n\to \Bbb R^m$, $g\colon\Bbb R^n\to \Bbb R^p$ are locally Lipschitz functions, $h\colon\Bbb R^n\to \Bbb R^q$ is $C^1$ function, and $C\subset \Bbb R^m$ and $K\subset \Bbb R^p$ are closed convex cones. Two types of solutions are important for the consideration, namely $w$-minimizers (weakly efficient points) and $i$-minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point $x^0$ to be a $w$-minimizer and first-order sufficient conditions for $x^0$ to be an $i$-minimizer are obtained. Their effectiveness is illustrated on an example. A comparison with some known results is done. (English) |
| Keyword:
|
vector optimization |
| Keyword:
|
locally Lipschitz optimization |
| Keyword:
|
Dini derivatives |
| Keyword:
|
optimality conditions |
| MSC:
|
49J52 |
| MSC:
|
90C29 |
| MSC:
|
90C30 |
| MSC:
|
90C46 |
| idZBL:
|
Zbl 1224.90154 |
| idMR:
|
MR2585562 |
| DOI:
|
10.1007/s10492-010-0003-y |
| . |
| Date available:
|
2010-07-20T13:32:36Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140388 |
| . |
| Reference:
|
[1] Aghezzaf, B., Hachimi, M.: Second-order optimality conditions in multiobjective optimization problems.J. Optim. Theory Appl. 102 (1999), 37-50. Zbl 1039.90062, MR 1702841, 10.1023/A:1021834210437 |
| Reference:
|
[2] Amahroq, T., Taa, A.: On Lagrange-Kuhn-Tucker multipliers for multiobjective optimization problems.Optimization 41 (1997), 159-172. Zbl 0882.90114, MR 1459915, 10.1080/02331939708844332 |
| Reference:
|
[3] Antczak, T., Kisiel, K.: Strict minimizers of order $m$ in nonsmooth optimization problems.Commentat. Math. Univ. Carol. 47 (2006), 213-232. Zbl 1150.90007, MR 2241528 |
| Reference:
|
[4] Auslender, A.: Stability in mathematical programming with nondifferentiable data.SIAM J. Control Optim. 22 (1984), 239-254. Zbl 0538.49020, MR 0732426, 10.1137/0322017 |
| Reference:
|
[5] Bednařík, D., Pastor, K.: On second-order conditions in unconstrained optimization.Math. Program. 113 (2008), 283-298. MR 2375484, 10.1007/s10107-007-0094-8 |
| Reference:
|
[6] Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces.Math. Program. Study 18 (1982), 39-76. Zbl 0494.49020, MR 0669725, 10.1007/BFb0120982 |
| Reference:
|
[7] Clarke, F. H.: Optimization and Nonsmooth Analysis.John Wiley & Sons New York (1983). Zbl 0582.49001, MR 0709590 |
| Reference:
|
[8] Craven, B. D.: Nonsmooth multiobjective programming.Numer. Funct. Anal. Optim. 10 (1989), 49-64. Zbl 0645.90076, MR 0978802, 10.1080/01630568908816290 |
| Reference:
|
[9] Ginchev, I., Guerraggio, A., Rocca, M.: First-order conditions for $C^{0,1}$ constrained vector optimization.In: Variational Analysis and Applications. Proc. 38th Conference of the School of Mathematics ``G. Stampacchia'' in Memory of G. Stampacchia and J.-L. Lions, Erice, Italy, June 20--July 1, 2003 F. Giannessi, A. Maugeri Springer New York (2005), 427-450. MR 2159985 |
| Reference:
|
[10] Ginchev, I., Guerraggio, A., Rocca, M.: Second-order conditions in $C\sp {1,1}$ constrained vector optimization.Math. Program., Ser. B 104 (2005), 389-405. MR 2179243, 10.1007/s10107-005-0621-4 |
| Reference:
|
[11] Ginchev, I., Guerraggio, A., Rocca, M.: From scalar to vector optimization.Appl. Math. 51 (2006), 5-36. Zbl 1164.90399, MR 2197320, 10.1007/s10492-006-0002-1 |
| Reference:
|
[12] Ginchev, I., Guerraggio, A., Rocca, M.: Second-order conditions in $C^{1,1}$ vector optimization with inequality and equality constraints.In: Recent Advances in Optimization. Proc. 12th French-German-Spanish Conference on Optimization, Avignon, France, September 20-24, 2004. Lecture Notes in Econom. and Math. Systems, Vol. 563 A. Seeger Springer Berlin (2006), 29-44. MR 2191149 |
| Reference:
|
[13] Li, Z.: The optimality conditions of differentiable vector optimization problems.J. Math. Anal. Appl. 201 (1996), 35-43. Zbl 0851.90105, MR 1397884, 10.1006/jmaa.1996.0239 |
| Reference:
|
[14] Liu, L., Neittaanmäki, P., Křížek, M.: Second-order optimality conditions for nondominated solutions of multiobjective programming with $C^{1,1}$ data.Appl. Math. 45 (2000), 381-397. MR 1777017, 10.1023/A:1022272728208 |
| Reference:
|
[15] Malivert, C.: First and second order optimality conditions in vector optimization.Ann. Sci. Math. Qué. 14 (1990), 65-79. Zbl 0722.90065, MR 1070607 |
| Reference:
|
[16] Maruşciac, I.: On Fritz John type optimality criterion in multi-objective optimization.Anal. Numér. Théor. Approximation 11 (1982), 109-114. MR 0692476 |
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