# zbMATH — the first resource for mathematics

Surrogate duality for vector optimization. (English) Zbl 0609.49012
Using our theorems on separation of convex sets by linear operators in the sense of the lexicographical order on $$R^ n$$, we prove some theorems of surrogate duality for vector optimization problems with convex constraints (but no regularity assumption), where the surrogate constraint sets are generalized half-spaces and the surrogate multipliers are linear operators, or isomorphisms, or isometries. For some of these results, we define surrogate subdifferentials of extended vector-valued functions, as well as surrogate Lagrangians, surrogate dual problems and surrogate Lagrange multipliers for vector optimization problems. When the level sets of the objective function are convex, we obtain a saddle-point characterization of the Pareto-minima. In the case of inequality constraints, we prove that the surrogate multipliers can be taken lexicographically as non-negative isometries or non-negative (in the usual order) linear isomorphisms.

##### MSC:
 49N15 Duality theory (optimization) 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 90C31 Sensitivity, stability, parametric optimization 49J45 Methods involving semicontinuity and convergence; relaxation 58E17 Multiobjective variational problems, Pareto optimality, applications to economics, etc.
Full Text:
##### References:
 [1] Akilov G. P., Ordered vector spaces (1978) [2] Crouzeix J.-P., Contributions d l’étude des fonctions quasi-convexes (1977) [3] DOI: 10.1287/opre.13.6.879 · Zbl 0163.41301 · doi:10.1287/opre.13.6.879 [4] DOI: 10.1287/opre.18.5.924 · Zbl 0232.90059 · doi:10.1287/opre.18.5.924 [5] DOI: 10.1215/S0012-7094-55-02209-2 · Zbl 0064.16601 · doi:10.1215/S0012-7094-55-02209-2 [6] DOI: 10.1007/BF02594784 · Zbl 0497.90067 · doi:10.1007/BF02594784 [7] DOI: 10.1137/0116088 · Zbl 0212.23905 · doi:10.1137/0116088 [8] Martinez-Legaz J.-E., Lecture Notes in Pure and Appl. Math. 86 pp 45– (1983) [9] DOI: 10.1007/BF01916921 · Zbl 0522.90069 · doi:10.1007/BF01916921 [10] Martinez-Legaz J.-E., Lecture Notes in Control and Information Sciences 59 pp 203– (1984) [11] MartinerLegaz J.-E., Lexicographical order and duality in multiobjective proqramming · Zbl 0646.90076 · doi:10.1016/0377-2217(88)90178-6 [12] Martinez-Legaz J.-E., Lexicographical separation in R n · Zbl 0901.52008 [13] Podinovskii V.V., Pareto-optimal solutions of multicriterial problems (1982) [14] Singer I., In: Optimization: Theory and algorithms Lecture Notes in Pure and Appl.Math. 86 pp 13– (1983) [15] DOI: 10.1080/01630568208816110 · Zbl 0496.49014 · doi:10.1080/01630568208816110 [16] DOI: 10.1080/01630568208816118 · Zbl 0497.49022 · doi:10.1080/01630568208816118 [17] DOI: 10.1016/0022-247X(84)90277-4 · Zbl 0584.49006 · doi:10.1016/0022-247X(84)90277-4 [18] Singer I., Lecture Notes in Econ. and Math. Sys- tems 226 pp 49– (1984) [19] DOI: 10.1016/0022-247X(84)90002-7 · Zbl 0607.90089 · doi:10.1016/0022-247X(84)90002-7 [20] DOI: 10.1080/00207728208926352 · Zbl 0487.49021 · doi:10.1080/00207728208926352 [21] DOI: 10.1016/0022-247X(68)90206-0 · Zbl 0157.16004 · doi:10.1016/0022-247X(68)90206-0
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.