Efficiency and approachability of nonconvex bicriteria programs

In this paper we investigate equivalences between an efficient solution of a bicriteria program and a lower envelope point of a certain image set of the bicriteria program. We also employ various kinds of approachabilities to characterize efficiency and proper efficiency for nonconvex bicriteria programs. In particular, nonlinear Lagranian functions are applied to construct dual problems and to study stability for the corresponding constrained scalar optimization problems. Under certain conditions we show that the finite approachability, proper efficiency, stability, and exact penalization of the relevant constrained scalar optimization problems are equivalent.