Lagrange duality in set optimization
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SubjectSet optimization; Lagrangian; Set relation; Convex duality; Complete lattice; Saddle points; Risk measures
Based on the complete-lattice approach, a new Lagrangian type duality theory for set-valued optimization problems is presented. In contrast to previous approaches, set-valued 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 set-valued risk measures are indicated.
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Hamel, AH (TAYLOR & FRANCIS LTD, 2011)A duality theorem of the Fenchel-Rockafellar type for set-valued optimization problems is presented along with a result for the conjugate of the sum of two set-valued functions and a chain rule. The underlying solution ...
Hamel, AH; Heyde, F; Löhne, A; Tammer, C; Winkler, K (2004)Using a set-valued dual cost function we give a new approach to duality theory for linear vector optimization problems. We develop the theory very close to the scalar case. Especially, in contrast to known results, we avoid ...
Hamel, AH (SPRINGER, 2009)It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre-Fenchel ...