A vector variational inequality and optimization over an efficient set. (English) Zbl 0693.90091
Some relations are obtained between weak vector minimization, a vector variational inequality, and the optimization of a utility function over a set of efficient points. In order to prove the existence of subgradients a generalized Hahn-Banach extension theorem is introduced. The paper also contains Kuhn-Tucker conditions for optimization over an efficient set.
Reviewer: J.W.Nieuwenhuis
MSC:
| 90C31 | Sensitivity, stability, parametric optimization |
| 49J40 | Variational inequalities |
| 49J52 | Nonsmooth analysis |
| 91B16 | Utility theory |
Keywords:
multiobjective optimization; duality; weak vector minimization; vector variational inequality; utility function; efficient points; existence of subgradients; generalized Hahn-Banach extension theorem; Kuhn-Tucker conditionsReferences:
| [1] | Benson HP (1984) Optimization over the efficient set. J Math Anal Appl 98:562-580 · Zbl 0534.90077 · doi:10.1016/0022-247X(84)90269-5 |
| [2] | Benson HP (1986) An algorithm for optimizing over the weakly efficient set. European J Operational Research 25(2):192-199 · Zbl 0594.90082 · doi:10.1016/0377-2217(86)90085-8 |
| [3] | Chen Guang-Ya, Cheng Ging-Min (1987) Vector variational inequality and vector optimization problems. Proceedings of 7th MCDM Conference, Kyoto, Japan. Springer-Verlag, Berlin, pp 408-416 |
| [4] | Craven BD (1978) Mathematical programming and control theory. Chapman & Hall, London · Zbl 0431.90039 |
| [5] | Craven BD (1977) Lagrangean conditions and quasiduality. Bull Austral Math Soc 16:325-339 · Zbl 0362.90106 · doi:10.1017/S0004972700023431 |
| [6] | Giles JR (1982) Convex analysis with application in differentiation of convex functions. Pitman Advanced Publishing Program, Boston · Zbl 0486.46001 |
| [7] | Jameson G (1970) Ordered linear spaces. Lectture Notes in Mathematics 141. Springer-Verlag, Berlin · Zbl 0196.13401 |
| [8] | Philip J (1972) Algorithms for one vector maximization problem. Math Programming 2:207-229 · Zbl 0288.90052 · doi:10.1007/BF01584543 |
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