Lagrange duality in set optimization
Subject
Set optimization; Lagrangian; Set relation; Convex duality; Complete lattice; Saddle points; Risk measuresAbstract
Based on the completelattice approach, a new Lagrangian type duality theory for setvalued optimization problems is presented. In contrast to previous approaches, setvalued versions for the known scalar formulas involving infimum and supremum are obtained. In particular, a strong duality theorem, which includes the existence of the dual solution, is given under very weak assumptions: The ordering cone may have an empty interior or may not be pointed. “Saddle sets” replace the usual notion of saddle points for the Lagrangian, and this concept is proven to be sufficient to show the equivalence between the existence of primal/dual solutions and strong duality on the one hand, and the existence of a saddle set for the Lagrangian on the other hand. Applications to setvalued risk measures are indicated.
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