Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems

This paper concentrates on solving bilevel programming problems where the lower level programs are max-min optimization problems and the upper level programs have max-max or max-min objective functions. Because these bilevel programming problems include nonconvex and nonsmooth lower level program problems, it is a challenging undone work. Giving some assumptions, we translate these problems into general single level optimization problems or min-max optimization problems. To deal with these equivalent min-max optimization problems, we propose a class of regularization methods which approximate the maximum function by using a family of maximum entropy functions. In addition, we examine the limit situations of the proposed regularization methods and show that any limit points of the global optimal solutions obtained by the approximation methods are the same as the ones of the original problems. Finally, we apply the proposed methods to newsvendor problems and use a numerical example to show their effectiveness.