A Class Of Bilevel Programming Problem, Optimality Conditions And Applications

Bilevel Programming problems have a wide range of applications in real world,but they are difficult to solve.Even if the objective functions and constraint functions in both levels are linear, the whole system would be a nonconvex problem.Since the optimality conditions play an important role in the algorithm design,researchers have to pay great attentions on the optimality conditions of the bilevel programming problems.Meanwhile,principal agent problems as the application of bilevel programming problems have been a central topic in modern economics.The principal agent theory is to study how to design effective incentives.In this paper,we consider some special bilevel programming problem,where the constraint of the upper level problem is linear,the objective of the lower level problem is quadratic and the constraint is linear.We provide the optimality conditions for this bilevel programming problem when both of its decision variables are real numbers as well as both of them are multi-dimensional. We use the KKT condition instead of the lower level problem and get a single level optimization problem with an equilibrium constraint.Then we use the Fritz-John condition for this single level problem and get its optimality conditions.At the same time,we get the optimality conditions for the bilevel programming problem.And assumptions of the theorem provided in this paper do not satisfy the conditions provided by Flegel in[18].Then we study the application of the bilevel programming problem on principal agent problems. The usual method for principal agent problems is the first order approach.However,this method is not always valid.Mirrlees gave the monotone likelihood ratio condition and the convexity of the distribution function which guarantee the validity of the first order approach.But few distributions satisfy both of the two conditions at the same time.Therefore we provide some distributions satisfy both of the two conditions and extend the applications of the first order approach to principal agent problems.And the first distribution contains some of the results provided by LiCalzi in[26].