Q. Meng, BLK E1, 07-10, and D.-H. Lee, BLK E1A, 07-16, Department of Civil Engineering, National University of Singapore, 1 Engineering Drive 2, Kent Ridge Crescent, Singapore 117576. H. Yang, Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. H.-J. Huang, School of Management, Beijing University of Aeronautics and Astronautics, Beijing 100083, People’s Republic of China. acterize the aforementioned transportation network optimization problems. In these resulting bilevel programming models, the upper- level subproblems are constrained optimization problems that plan- ners require to make decisions about control, management, design, and investment improvement to optimize systemwide objective func- tions. The lower-level subproblems are network equilibrium prob- lems for given upper-level decisions. For presentation purposes, the transportation network optimization problems are classified into two major categories in this paper: problems of transportation net- work optimization with DUE constraints (TNO-DUEC) and prob- lems of transportation network optimization with SUE constraints (TNO-SUEC). The past two decades have witnessed the development of a vast and growing body of research focused on the modeling and devel- opment of algorithms for various TNO-DUEC problems. This paper will not review these achievements in detail; and interested readers can refer to a comprehensive survey paper (5), a Ph.D. dissertation (6 ), and a special issue of the journal Transportation Research, Part B (Vol. 35, No. 1, 2001). The establishment of the bilevel program- ming model for a TNO-DUEC problem is an easy task, but the design of efficient algorithms to solve such a model is a difficult, yet challenging task, since DUE link flows in general are nondifferen- tiable implicit functions with respect to decision variables. Further- more, mathematical models for the DUE problem undoubtedly involve path flows or origin-based link flows that define the set of feasible link flows in which the number of paths or origin-based links may be huge for a large-scale problem. In particular, neither DUE path flows nor DUE origin-based link flows are unique, even if the DUE link flow pattern is unique. These features do not allow the numerous existing solution procedures for the bilevel programming or MPEC problems in operations research (7–11) to be applied directly to solve TNO-DUEC problems. On the other hand, such problems also provide opportunities for transportation researchers to develop different algorithms for a TNO-DUEC problem. Over the years, most of the algorithms developed for the TNO-DUE prob- lems have been heuristics, such as sensitivity analysis-based algo- rithms or stochastic search methods like the genetic algorithm or the simulated annealing method (12–16 ). Nevertheless, the first cate- gory lacks theoretical guarantees, whereas the second may be in- efficient for large-scale problems. More recently, the construction of equivalent single-level optimization models for TNO-DUEC problems and their solution by some existing convergent algorithms (17–19) have reactivated the earlier gap function approach (20) and resulted in another research methodology that seems promising. Compared with the TNO-DUEC problems, the TNO-SUEC prob- lems have received only limited attention in the literature (21–25). It might appear that a TNO-SUEC problem is a straightforward A comprehensive study of static transportation network optimization problems with stochastic user equilibrium constraints is presented. It is explicitly demonstrated that the formulation of the fixed-point problem— in terms of link flows for the general stochastic user equilibrium problem in which the Jacobian matrix of link travel cost functions may not be symmetric—possesses a unique solution with mild conditions. By devel- oping a sensitivity analysis method for the stochastic user equilibrium problem, the study proves that the perturbed equilibrium link flows are continuously differentiable implicit functions with respect to perturbation parameters. Accordingly, it can be concluded that the proposed unified bilevel programming model, which can characterize transportation network optimization problems subject to stochastic user equilibrium constraints, is a smooth optimization problem. In addition, the study presents a single-level continuously differentiable optimization formu- lation that is equivalent to the unified bilevel programming model. Furthermore, as a unified solution method, a successive quadratic pro- gramming algorithm based on the sensitivity analysis method is used to solve the transportation network optimization problems with stochastic user equilibrium constraints. Finally, two examples are used to demon- strate the proposed models and algorithm. Transportation network optimization problems that take into account the behaviors of network users’ and their route choices have been the focus of considerable research in transportation network modeling for many years. Typical examples include network design problems, optimal toll-pricing problems, optimal signal-setting problems, origin–destination (O-D) matrix estimation problems, and so on. From the viewpoint of game theory, these problems can be described by the standard Stackelberg game, in which the leader is a planner tak- ing charge of a transportation network optimization project and the follower is the network user complying with either deterministic user equilibrium (DUE) or stochastic user equilibrium (SUE) principles. It is known that the DUE and SUE conditions can be modeled by optimization problems or more general variational inequalities and fixed-point problems (1–4). Accordingly, the bilevel programming approach or its extension, which is known as mathematical programs with equilibrium constraints (MPECs), can fully and perfectly char- Transportation Network Optimization Problems with Stochastic User Equilibrium Constraints Qiang Meng, Der-Horng Lee, Hai Yang, and Hai-Jun Huang Transportation Research Record: Journal of the Transportation Research Board, No. 1882, TRB, National Research Council, Washington, D.C., 2004, pp. 113–119. 113