Optimality Conditions Under a New Constraint Qualification for Nonconvex Optimization Problems

The aim of this paper is to give necessary and sufficient optimality conditions for nonconvex optimization problems such that the constraint inequalities are both nonnegative and nonpositive, where the objective function and the constraint functions are tangentially convex, but are not necessarily convex. We do this first by introducing a novel constraint qualification, call “tangential nearly convexity” ((TNC), in short). Next, by using the cone of tangential subdifferentials together with the novel constraint qualification, we show that Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient for the optimality. Several examples are presented to clarify and compare the novel constraint qualification with the other well known constraint qualifications.