Journal of Global Optimization

, Volume 55, Issue 3, pp 611–626 | Cite as

Characterizations of the nonemptiness and compactness for solution sets of convex set-valued optimization problems

Article

Abstract

In this paper, we first derive several characterizations of the nonemptiness and compactness for the solution set of a convex scalar set-valued optimization problem (with or without cone constraints) in which the decision space is finite-dimensional. The characterizations are expressed in terms of the coercivity of some scalar set-valued maps and the well-posedness of the set-valued optimization problem, respectively. Then we investigate characterizations of the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) in which the objective space is a normed space ordered by a nontrivial, closed and convex cone with nonempty interior and the decision space is finite-dimensional. We establish that the nonemptiness and compactness for the weakly efficient solution set of a convex vector set-valued optimization problem (with or without cone constraints) can be exactly characterized as those of a family of linearly scalarized convex set-valued optimization problems and the well-posedness of the original problem.

Keywords

Cone convex set-valued map Set-valued optimization Solution set Weakly efficient solution Well-posedness of optimization problem 

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Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.School of Economics and Business AdministrationChongqing UniversityChongqingChina
  2. 2.Center for General EducationKaohsiung Medical UniversityKaohsiungTaiwan

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