Image regularity conditions based on nonconvex separation with applications

This paper focuses on the investigation of regularity conditions by virtue of the image space analysis, based on nonconvex separation. We consider a scalar constrained optimization problem and employ the Gerstewitz separation function, which is well known from scalarization of vector optimization problems to present and investigate a collection of nonlinear weak separation functions in connection with methods of image space analysis. With the separation theorems associated with the Gerstewitz function, some image regularity conditions which guarantee the existence of a weak separation function, conducting a regular separation, are studied. In addition, a Lagrange type function and a penalty function are constructed, while the existence of saddle points and exact penalty functions are established by means of the image regularity conditions.