\(E\)-Benson proper efficiency in vector optimization. (English) Zbl 1310.90094
Summary: Starting from the innovative ideas of M. Chicco et al. [J. Optim. Theory Appl. 150, No. 3, 516–529 (2011; Zbl 1231.90329)], in this paper, the concepts of improvement set and \(E\)-efficiency are introduced in a real locally convex Hausdorff topological vector space. Furthermore, some properties of the improvement sets are given and a kind of proper efficiency, named as \(E\)-Benson proper efficiency, which unifies some proper efficiency and approximate proper efficiency, is proposed via the improvement sets in vector optimization. Moreover, the concept of \(E\)-subconvexlikeness of set-valued maps is introduced via the improvement sets and an alternative theorem is proved. In the end, some scalarization theorems and Lagrange multiplier theorems of \(E\)-Benson proper efficiency are established for a vector optimization problem with set-valued maps.
MSC:
| 90C26 | Nonconvex programming, global optimization |
| 90C29 | Multi-objective and goal programming |
| 90C30 | Nonlinear programming |
Keywords:
vector optimization; \(E\)-Benson proper efficiency; \(E\)-subconvexlikeness; scalarization; Lagrange multipliersCitations:
Zbl 1231.90329References:
| [1] | DOI: 10.1007/978-3-540-24828-6 · doi:10.1007/978-3-540-24828-6 |
| [2] | Ehrgott M, Multicriteria optimization (2005) |
| [3] | DOI: 10.1016/0022-247X(68)90201-1 · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1 |
| [4] | DOI: 10.1016/0022-247X(79)90226-9 · Zbl 0418.90081 · doi:10.1016/0022-247X(79)90226-9 |
| [5] | DOI: 10.1137/0315004 · Zbl 0369.90096 · doi:10.1137/0315004 |
| [6] | DOI: 10.1007/BF00934353 · Zbl 0452.90073 · doi:10.1007/BF00934353 |
| [7] | DOI: 10.2307/2154446 · Zbl 0796.90045 · doi:10.2307/2154446 |
| [8] | DOI: 10.1023/A:1022676013609 · Zbl 0913.90235 · doi:10.1023/A:1022676013609 |
| [9] | DOI: 10.1023/A:1022689517921 · Zbl 0908.90225 · doi:10.1023/A:1022689517921 |
| [10] | DOI: 10.1023/A:1026407517917 · Zbl 0966.90071 · doi:10.1023/A:1026407517917 |
| [11] | DOI: 10.1023/A:1017535631418 · Zbl 1012.90061 · doi:10.1023/A:1017535631418 |
| [12] | Kutateladze S, Soviet Math. Dokl 20 pp 391– (1979) |
| [13] | DOI: 10.1007/BF00940762 · Zbl 0573.90085 · doi:10.1007/BF00940762 |
| [14] | DOI: 10.1081/NFA-100108312 · Zbl 1052.90075 · doi:10.1081/NFA-100108312 |
| [15] | DOI: 10.1137/05062648X · Zbl 1119.49020 · doi:10.1137/05062648X |
| [16] | DOI: 10.1023/A:1004657412928 · Zbl 1028.90057 · doi:10.1023/A:1004657412928 |
| [17] | DOI: 10.1007/s10957-011-9891-6 · Zbl 1246.90134 · doi:10.1007/s10957-011-9891-6 |
| [18] | Ghaznavi-ghosoni BA, Khorram E, Soleimani-damaneh M. Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization. Optimization. 2012. doi: 10.1080/02331934.2012.668190. · Zbl 1273.90185 · doi:10.1080/02331934.2012.668190 |
| [19] | DOI: 10.1007/s10957-011-9851-1 · Zbl 1231.90329 · doi:10.1007/s10957-011-9851-1 |
| [20] | DOI: 10.1016/S0893-9659(99)00087-7 · Zbl 0944.90081 · doi:10.1016/S0893-9659(99)00087-7 |
| [21] | Rong WD, OR Trans 4 pp 21– (2000) |
| [22] | DOI: 10.3934/jimo.2011.7.483 · Zbl 1246.90135 · doi:10.3934/jimo.2011.7.483 |
| [23] | Zhao KQ, Yang XM. A unified stability result with perturbations in vector optimization. Optim. Lett. 2012. doi: 10.1007/s11590-012-0533-1. · Zbl 1287.90062 · doi:10.1007/s11590-012-0533-1 |
| [24] | DOI: 10.1080/02331934.2011.641018 · Zbl 1266.90162 · doi:10.1080/02331934.2011.641018 |
| [25] | DOI: 10.1006/jmaa.1997.5568 · Zbl 0893.90150 · doi:10.1006/jmaa.1997.5568 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
