On stability and well-posedness in the vector optimization
Darinka Dentcheva, Siegfried Helbig · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1994
We deal with vector optimization problems (f, Z), where f : X {yields} Y is a mapping, Z {improper_subset} X is a non-empty set. X and Y are metric spaces and Y is partially ordered by a non-trivial cone C. The problem (f, Z) is to find efficient points of f with respect to Z and C. We introduce a notion of level sets L(q, {epsilon}) for the problem, where q {element_of} Y and {epsilon} {>=} 0 is a real number. The level sets consist of {epsilon}-efficient points for certain values of q and of efficient points, if {epsilon} = 0. Therefore the continuity properties of the set valued mapping L are of interest and have the meaning of stability of the underlying problem. We give some sufficient conditions for Kuratowski-convergency of level sets to a certain set L(q, 0). Further, we suggest an approach towards well-posedness in the vector optimization. Notions of well-posedness in Tykhonov sense and strong well-posedness are introduced. Some results are presented, which establish similar characterizations of the well-posed vector optimization problems are they are known for scalar problems. Variational principles with well-posed perturbed functions are formulated.