Abstract:In this paper, we consider a set optimization problem with a partial order relation, which is defined by Minkowski difference. By using the image space analysis, we establish the relationships among the set optimization problem, a vector optimization problem and a set-valued optimization with vector criterion related to the image of the set optimization problem. In addition, two nonlinear regular weak separation functions are proposed for the set optimization problem. Based on the two nonlinear regular weak se… Show more
“…However, the converses are not necessarily valid (see Example 2.2 of [30]). Now, we recall an example given in [47] to show the difference between ⪯ m1 C and…”
The aim of this paper is to investigate the Painlevé-Kuratowski convergence and the extended well-posedness of a set optimization problem with respect to the both perturbations of the order cone and the feasible set. Firstly, we introduce the concepts of the approximate minimal solutions for the set optimization problem with set order relations involving the Minkowski difference. Some properties, in particular, the density and the nonlinear scalarization result of approximate minimal solutions and approximate weak minimal solutions, are presented. In addition, based on the density theorem and some appropriate assumptions, we establish the Painlevé-Kuratowski convergence of the approximate solution sets of the set optimization problem with respect to the both perturbations of the order cone and the feasible set. We also discuss the extended well-posedness and weak extended well-posedness for the set optimization problem with respect to the both perturbations of the order cone and the feasible set. Finally, we put forward the equivalence between the weak extended well-posedness for the set optimization problem and the generalized well-posendess of a scalar optimization problem.
“…However, the converses are not necessarily valid (see Example 2.2 of [30]). Now, we recall an example given in [47] to show the difference between ⪯ m1 C and…”
The aim of this paper is to investigate the Painlevé-Kuratowski convergence and the extended well-posedness of a set optimization problem with respect to the both perturbations of the order cone and the feasible set. Firstly, we introduce the concepts of the approximate minimal solutions for the set optimization problem with set order relations involving the Minkowski difference. Some properties, in particular, the density and the nonlinear scalarization result of approximate minimal solutions and approximate weak minimal solutions, are presented. In addition, based on the density theorem and some appropriate assumptions, we establish the Painlevé-Kuratowski convergence of the approximate solution sets of the set optimization problem with respect to the both perturbations of the order cone and the feasible set. We also discuss the extended well-posedness and weak extended well-posedness for the set optimization problem with respect to the both perturbations of the order cone and the feasible set. Finally, we put forward the equivalence between the weak extended well-posedness for the set optimization problem and the generalized well-posendess of a scalar optimization problem.
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