Stochastic Equilibrium Bilevel Programming Based On Transit Charging-flow Allocation Problem

Considering the transit charging-flow allocation problem as the background, taking the stochastic equilibrium bi-level programming as the theory basis, this paper presents us an ap-plication model for the transit problem and discusses the optimum condition and convergence properties of the SAA method.The first part firstly introduces the origin of the random user equilibrium problem and its current research situation, and explains why we introduce the oligopolistic traffic assignment model as the application.The second part mainly introduces the related mathematical background which we may need next, such as convex programming problem, variational inequality followed their proving, this has prepared for the next model derivation and theorem proving theoretical process.The third part is the mathematical description under the condition of demand elasticity with random user equilibrium condition of the urban transportation charging-flow allocation model, proved that the elastic assignment equilibrium condition can be expressed as a variational inequality problem, and based on this gives the definition of the General Nash equilibrium state of traffic assignment problem.The fourth part is a mathematical modeling under the fixed flow condition, specifically de-picts a random equilibrium bi-level programming model, and proved that this bi-level problem can degrade to a fee decision making problem. Finally, according to the fee decision mak-ing problem, when the objective function is not smooth, we use the sample average estimation method to discuss the convergence properties of the problems’solutions.The last part is the example analysis under fixed flow condition problems using smooth-SAA method, and gives corresponding numerical result which shows that the smooth-SAA method is feasible to the non-smooth stochastic programming problem.