Optimality conditions under relaxed quasiconvexity assumptions using star and adjusted subdifferentials

A set-constrained optimization problem and a mathematical programming problem are considered. We assume that the sublevel sets of the involving functions are convex only at the point under question and hence these functions are not assumed quasiconvex. Using the two star subdifferentials and the adjusted subdifferential, we establish optimality conditions for usual minima and strict minima. Our results contain and improve some recent ones in the literature. Examples are provided to explain the advantages of each of our results.