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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.
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