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

  • Authors:

  • X. X. Huang;J. C. Yao

  • Affiliations:

  • School of Economics and Business Administration, Chongqing University, Chongqing, China 400030;Center for General Education, Kaohsiung Medical University, Kaohsiung, Taiwan 807

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2013

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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.