Extended and strongly extended well-posedness of set-valued optimization problems.

*(English)*Zbl 1018.49019Summary: In this paper, we deal with the extended well-posedness and strongly extended well-posedness of set-valued optimization problems. These two concepts are generalizations of the extended well-posedness of real-valued optimization problems defined by Zolezzi. We obtain some criteria and characterizations of these two types of extended well-posedness, further generalizing most results obtained by Zolezzi for the extended well-posedness of scalar optimization problems. In the mean time, many results obtained by us for the extended well-posedness of vector optimization problems have been generalized to set-valued optimization. Finally, we present an approximate variational principle for set-valued maps, derive a necessary approximate optimality condition for set-valued optimization, based on which we introduce a condition, which is somewhat analogous to the Palais-Smale condition (C), and provide sufficient conditions for the extended and strongly extended well-posedness of set-valued optimization problems.

##### MSC:

49K40 | Sensitivity, stability, well-posedness |

90C29 | Multi-objective and goal programming |

90C31 | Sensitivity, stability, parametric optimization |