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Extended and strongly extended well-posedness of set-valued optimization problems. (English) Zbl 1018.49019
Summary: 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
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