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

Huang, X. X.

Math. Methods Oper. Res. 53, No. 1, 101-116 (2001).

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

Keywords:

set-valued optimization; asymptotically minimizing sequence; well-posedness; set-valued variational principle; stability; necessary approximate optimality condition