Exactness of penalty functions for solving MPEC model of the transportation network optimization problems with user equilibrium constraints

Now, both bilevel programming problem (BLPP) and mathematical program with equilibrium constraints (MPEC) can be used to describe a: class of transportation network optimization problems with deterministic user-equilibrium constraints (TNO-UEC). However, these models are very hard to solve due to their complicate structure. By investigating the natures of the marginal function for the lower level problem in these models, a partial augmented Lagrange penalty algorithm has been used to solve these problems successfully. By studying the special structure of TNO-UEC we find that the partial critical direction plays a very important role in the behavior of penalty functions. In this paper, we first give definitions of two penalty functions-simple penalty function (SPF) and partial augmented Lagrange penalty function (ALPF), hence, studying the MPEC models for solving TNO-UEC problem is equivalent to discussing the penalty problems correspondingly. Then, studying the exactness of penalty functions is very important for us to discuss the optimal solutions of MPEC problem. So by defining a partial critical direction, we mainly discuss the sufficient conditions and necessary conditions under which the simple penalty functions and the partial augmented Lagrange penalty function are exact for the MPEC model of TNO-UEC Problems in this paper, at the same time, we also give a sufficient condition for a direction to be a partial critical direction.