Optimization and Engineering

, Volume 8, Issue 3, pp 241–257 | Cite as

A globally convergent algorithm for transportation continuous network design problem



The continuous network design problem (CNDP) is characterized by a bilevel programming model, in which the upper level problem is generally to minimize the total system cost under limited expenditure, while at the lower level the network users make choices with regard to route conditions following the user equilibrium principle. In this paper, the bilevel programming model for CNDP is transformed into a single level convex programming problem by virtue of an optimal-value function tool and the relationship between System Optimum (SO) and User Equilibrium (UE). By exploring the inherent nature of the CNDP, the optimal-value function for the lower level user equilibrium problem is proved to be continuously differentiable and its derivative in link capacity enhancement can be obtained efficiently by implementing user equilibrium assignment subroutine. However, the reaction (or response) function between the upper and lower level problem is implicit and its gradient is difficult to obtain. Although, here we approximately express the gradient with the difference concept at each iteration, based on the method of successive averages (MSA), we propose a globally convergent algorithm to solve the single level convex programming problem. Comparing with widely used heuristic algorithms, such as sensitivity analysis based (SAB) method, the proposed algorithm needs not strong hypothesis conditions and complex computation for the inverse matrix. Finally, a numerical example is presented to compare the proposed method with some existing algorithms.


Continuous network design problem B-level programming Method of successive averages Global convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abdulaal M, LeBlanc LJ (1979) Continuous equilibrium network design models. Transp Res B 13:19–32 CrossRefGoogle Scholar
  2. Boyce DE (1984) Urban transportation network equilibrium and design models: recent achievements and future prospectives. Environ Plan A 16:1445–1474 CrossRefGoogle Scholar
  3. Chiou SW (1999) Optimization of area traffic control for equilibrium network flows. Transp Sci 33:279–289 MATHGoogle Scholar
  4. Cho HJ (1988) Sensitivity analysis of equilibrium network flows and its application to the development of solution methods for equilibrium network design problems. PhD dissertation. University of Pennsylvania, Philadelphia Google Scholar
  5. Cree ND, Maher MJ (1998) The continuous equilibrium optimal network design problem: a genetic approach. In: Transportation networks: recent methodological advances. Elsevier, Netherlands, pp 163–174 Google Scholar
  6. Dafermos S (1980) Traffic equilibria and variational inequalities. Transp Sci 14:42–54 MathSciNetGoogle Scholar
  7. Dafermos S, Nagurney A (1984) Sensitivity analysis for the asymmetric network equilibrium problem. Math Program 28:174–184 MATHCrossRefMathSciNetGoogle Scholar
  8. Friesz TL (1981) An equivalent optimization problem with combined multiclass distribution assignment and modal split which obviates symmetry restriction. Transp Res B 15:361–369 CrossRefMathSciNetGoogle Scholar
  9. Friesz TL (1985) Transportation network equilibrium, design and aggregation: key developments and research opportunities. Transp Res A 19:413–427 CrossRefGoogle Scholar
  10. Friesz TL, Harker PT (1985) Properties of the iterative optimization equilibrium algorithm. Civ Eng Syst 2:142–154 Google Scholar
  11. Friesz TL et al (1990) Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints. Math Program 48:265–284 Google Scholar
  12. Friesz TL et al (1993) The multiobjective equilibrium network design problem revisited: a simulated annealing approach. Eur J Oper Res 65:44–57 MATHCrossRefGoogle Scholar
  13. Gao ZY, Song YF (2002) A reserve capacity model of optimal signal control with user-equilibrium route choice. Transp Res B 36:313–323 CrossRefGoogle Scholar
  14. Gao ZY, Song YF Si BF (2000) Urban transportation continuous equilibrium network design problem: theory and method. China Railway Press, Beijing Google Scholar
  15. Gao ZY, Wu JJ, Sun HJ (2005) Solution algorithm for the bi-level discrete network design problem. Transport Res B 39:479–495 CrossRefGoogle Scholar
  16. Kim TJ (1990) Advanced transport and spatial systems models: applications to Korea. Springer, New York Google Scholar
  17. Kim TJ, Suh S (1988) Toward developing a national transportation planning model: a bilevel programming approach for Korea. Ann Reg Sci XXSPED:65–80 CrossRefGoogle Scholar
  18. Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge Google Scholar
  19. Magnanti TL, Wong RT (1984) Network design and transportation planning: models and algorithms. Transp Sci 18:1–55 Google Scholar
  20. Mangasarian OL, Rosen JB (1964) Inequalities for stochastic nonlinear programming problems. Oper Res 12:143–154 MATHMathSciNetGoogle Scholar
  21. Marcotte P (1983) Network optimization with continuous control parameters. Transp Sci 17:181–197 Google Scholar
  22. Marcotte P (1986) Network design problem with congestion effects: a case of bi-level programming. Math Program 34:142–162 MATHCrossRefMathSciNetGoogle Scholar
  23. Marcotte P, Marquis G (1992) Efficient implementation of heuristics for the continuous network design problem. Ann Oper Res 34:163–176 MATHCrossRefGoogle Scholar
  24. Marcotte P, Zhu DL (1996) Exact and inexact penalty methods for the generalized bilevel programming problems. Math Program 74:141–157 MathSciNetGoogle Scholar
  25. Meng Q, Yang H, Bell MGH (2001) An equivalent continuously differentiable model and a locally convergent algorithm for the continuous networks design problem. Transp Res B 35:83–105 CrossRefGoogle Scholar
  26. Patriksson M (1994) The traffic assignment problem models and methods. VSB BV, Netherlands Google Scholar
  27. Powell WB, Sheffi Y (1982) The convergence of equilibrium algorithms with predetermined step size. Transp Sci 6:5–55 MathSciNetGoogle Scholar
  28. Rockafellar RT (1970) Convex analysis. Princeton University Press, Princeton MATHGoogle Scholar
  29. Sheffi Y (1985) Urban transportation networks: equilibrium analysis with mathematical programming methods. Prentice–Hall, Englewood Cliffs Google Scholar
  30. Shimizu K, Ishizuka Y, Bard JF (1997) Nondifferentiable and two-level mathematical programming. Kluwer Academic, Massachusetts MATHGoogle Scholar
  31. Suwansirikul C, Friesz TL, Tobin RL (1987) Equilibrium decomposed optimization: a heuristic for the continuous equilibrium network design problem. Transp Sci 21:254–263 MATHGoogle Scholar
  32. Tan HN, Gershwin SB, Athans M (1979) Hybrid optimization in urban traffic networks. Report No. DOT-TSC-RSPA-79-7. Laboratory for Information and Decision System, MIT, Cambridge, MA Google Scholar
  33. Tobin RL, Friesz TL (1988) Sensitivity analysis for equilibrium network flows. Transp Sci 22:242–250 MATHCrossRefMathSciNetGoogle Scholar
  34. Wong SC, Yang H (1997) Reserve capacity of a signal-controlled road network. Transp Res Part B 31:397–402 CrossRefGoogle Scholar
  35. Yang H (1995) Sensitivity analysis for queuing equilibrium network flow and its application to traffic control. Math Comput Model 22:247–258 MATHGoogle Scholar
  36. Yang H (1997) Sensitivity analysis for the elastic demand network equilibrium problem with applications. Transp Res B 31:55–70 CrossRefGoogle Scholar
  37. Yang H, Bell MGH (1998) Models and algorithm for road network design: a review and some new developments. Transp Rev 18(3):257–278 CrossRefGoogle Scholar
  38. Yang H, Yagar S (1994) Traffic assignment and traffic control in general freeway-arterial corridor systems. Transp Res B 28:463–486 CrossRefGoogle Scholar
  39. Yang H, Meng Q, Liu GS (2004) The generalized transportation network optimization problem: models and algorithms. Working Paper, The Hong Kong University of Science and Technology Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.State Key Laboratory of Rail Traffic Control and Safety, School of Traffic and TransportationBeijing Jiaotong UniversityBeijingPeoples Republic of China

Personalised recommendations