Outlook and Further Application Areas
In this final chapter we give a short outlook on the appearance of variable ordering structures in vector variational inequalities, vector complementarity and equilibrium problems. We show that the theory of consumer demand in economics is also related to
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Outlook and Further Application Areas
In this final chapter we discuss some additional applications and related mathematical areas in which variable ordering structures appear. While the relation between these other areas and the results presented in this book are not yet fully explored, the application in intensity-modulated radiation therapy motivates to study also more general concepts of variable ordering structures. As we have seen in Sect. 1.3, variable ordering structures appear in vector optimization problems, for instance, when modeling problems from medical image registration or portfolio optimization. Closely related to the study of vector optimization problems are vector variational inequalities and there, variable ordering structures are also studied. In Sect. 10.1 we shortly present in which form variable ordering structures appear in that context. In Sect. 10.2 we recall the concept of local preferences as discussed in the theory of consumer demand and point out their relation to variable ordering structures. Finally, we end this chapter by another application problem: in intensity-modulated radiation therapy the problem of finding an optimal treatment plan is modeled so far as a multiobjective optimization problem with the objective space partially ordered by the componentwise (natural) partial ordering. We discuss the limitations of this model and show that variable ordering structures have to be considered. We also point out that variable ordering structures based on a cone-valued ordering map may only locally be the right concept.
10.1 Vector Variational Inequalities and Related Problems Vector variational inequalities and their relation to vector optimization problems in partially ordered spaces have been intensively studied in the last decades since their introduction by Giannessi in 1980 [67]. Thus it is a natural consequence that also vector variational inequalities with variable ordering cones are studied. In [28], Chen examines the following vector variational inequality (VVI): Let X , Y be real Banach spaces, S X a nonempty closed convex set and T W S ! L.X; Y / a map with L.X; Y / the space of continuous linear maps from X into Y . Additionally, we G. Eichfelder, Variable Ordering Structures in Vector Optimization, Vector Optimization, DOI 10.1007/978-3-642-54283-1__10, © Springer-Verlag Berlin Heidelberg 2014
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10 Outlook and Further Application Areas
assume a set-valued map CW S ! 2Y to be given with C.x/ a closed pointed convex cone with nonempty interior for all x 2 S . The task, which is related to optimality conditions in mathematical optimization, is now to find some xN 2 S such that T .x/.x N x/ N 62 int.C.x// N for all x 2 S:
(10.1)
In [107], Lee, Kim and Kuk use in this context the notion of a generalized efficient solution of a vector optimization problem minx2S f .x/, which is similar to the notion of a weakly minimal solution given in Definition 2.33. They consider a vector-valued objective map f and assume the map C to be defined as above. An element xN 2
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