Set Relations and Weak Minimal Solutions for Nonconvex Set Optimization Problems with Applications

In this paper, we give some properties concerned with weak minimal solutions of nonconvex set optimization problems. We also give some properties of the nonconvex separation functional and apply them to characterize weak minimal solutions of nonconvex set optimization problems. Moreover, we derive some new existence results for weak minimal solutions of nonconvex set optimization problems whose image spaces have no topology. Finally, we establish a set-valued version of Ekeland's variational principle via set relations and present a weak minimization for a nonconvex set optimization problem. As applications, we obtain the existence of weak minimal solutions for nonconvex vector optimization problems.