Duality
In this chapter we present duality assertions for set-valued optimization problems in infinite dimensional spaces where the solution concept is based on vector approach, on set approach as well as on lattice approach. For set-valued optimization problems
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Duality
In this chapter we present duality assertions for set-valued optimization problems in infinite dimensional spaces where the solution concept is based on vector approach, on set approach as well as on lattice approach. For set-valued optimization problems where the solution concept is based on vector approach we present conjugate duality statements. The notions of conjugate maps, subdifferential and a perturbation approach used for deriving these duality assertions are given. Furthermore, Lagrange duality for set-valued problems based on vector approach is shown. Moreover, we consider set-valued optimization problems where the solution concept is given by a set order relation introduced by Kuroiwa and derive corresponding saddle point assertions. For set-valued problems where the solution concept is based on lattice structure, we present duality theorems that are based on an consequent usage of infimum and supremum (in the sense greatest lower and least upper bounds with respect to a partial ordering). We derive conjugate duality assertions as well as Lagrange duality statements. A comparison of different approaches to duality in set-valued optimization is given at the end of this chapter. It is an old idea to study additionally to a given optimization problem (p.x/ ! inf with infimal value I) a corresponding dual problem (d.u/ ! sup with supremal value S; S I ), remember the dual variational principles of Dirichlet and Thompson or simply the pair of dual programs in linear optimization. The reasons for the introduction of a useful dual problem are the following: • The dual problem has (under additional conditions) the same optimal value as the given “primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics. • An approximate solution of the given minimization problem gives an estimation of the infimal value I from above, whereas an approximate solution of the dual problem is an estimation of I from below, so that one gets intervals containing I.
© Springer-Verlag Berlin Heidelberg 2015 A.A. Khan et al., Set-valued Optimization, Vector Optimization, DOI 10.1007/978-3-642-54265-7__8
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8 Duality
• Recalling Lagrange method, saddle points, equilibrium points of two person games, shadow prices in economics, perturbation methods or dual variational principles, it becomes clear, that optimal dual variables often have a special meaning for the given problem. Of course, the just listed advantages require a skilfully chosen dual program. Nevertheless, the mentioned points are motivation enough, to look for dual problems in set-valued optimization with corresponding properties too. There are a lot of papers, which are dedicated to that aim, also a lot of survey papers (see the books in [73, 292, 293, 402] and references therein).
8.1 Duality Assertions for Set-Valued Problems Based on Vector Approach 8.1.1 Conjugate Duality for Set-Valued Problems Based on Vector Approach A comprehensive and detailed theory for set-valued conjugate
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