Transport Bilevel Programming Problems: Unified Models and Algorithms

Abstract

This paper is concerned with a class of transport bilevel programming problems investigated by analytical analysis approaches. It begins with a state-of-art review on transport bilevel programming problems taking into account behavior of network users' routing choice. These problems in reality can be classifies into two major categories: transport bilevel programming problems with deterministic user equilibrium constraints and transport bilevel programming problems with stochastic user equilibrium constraints. It is well recognized that the bilevel programming model or mathematical program with equilibrium constraints as unified modeling approach can perfectly characterize these two categories of problems. Nevertheless, induced bilevel programming models for the former category usually belong to a subject of nondifferentiable optimization problems, whereas that for the latter category becomes the continuously differentiable optimization problems. It should be pointed out that designing an efficient solution method for a nondifferentiable optimization problem is not an easy task. This study thus introduces the recent unified modeling approach for the problems in the preceding category, which aims at transforming a bilevel programming model into a single level continuously differentiable optimization problem. As unified algorithms, it can be seen that the augmented Lagrangian method and sequential quadratic programming method based on the sensitivity analysis for the stochastic user equilibrium problem are capable of solving any problem in the first and second categories. Finally, three examples are provided to demonstrate the unified continuously differentiable optimization approach.

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