The traffic assignment problem aims to calculate an equilibrium route flow vector, generally by seeking a zero of an appropriate objective function. If a continuous dynamical system follows a descent direction for this objective function at each nonequilibrium route flow vector, the system converges to equilibrium. It is shown that when this dynamical system is discretized with a fixed step length, the system eventually approaches close to equilibrium provided that the objective function is continuously differentiable and that the rate of descent is bounded below. The method of successive averages is widely used in traffic assignment; it has a decreasing step size at each iteration. With the same conditions as above, it is shown that the resulting dynamical system converges to equilibrium. In the steady-state model, the necessary conditions are shown to be satisfied, provided that the route cost vector is a continuously differentiable monotone function of the route flow vector. However, continuous differentiability of the cost function is shown not to hold in the dynamic queueing model.
Bibliographical noteReference text: Bar-Gera H, Boyce D (2006) Solving a non-convex combined travel forecasting model by the method of successive averages with constant step sizes. Transportation Res. Part B 40:351–367.
Bernstein DH (1990) Programmability of continuous and discrete network equilibria. Unpublished doctoral thesis, University of Illinois at Chicago, Chicago.
Dial RB (2006) A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration. Transportation Res. Part B 40:917–936.
Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer. Math. 1:269–271.
Du J, Wong SC, Shu C, Xiong T, Zhang M, Choi K (2013) Revisiting Jiang’s dynamic continuum model for urban cities. Transportation Res. Part B 56:96–119.
Dunn JC (1976) Convexity, monotonicity, and gradient processes in Hilbert space. J. Math. Anal. Appl. 53:145–158.
Dunn JC, Harshbarger S (1978) Conditional gradient algorithms with open loop step size rules. J. Math. Anal. Appl. 62:432–444.
Kreyszig E (1978) Introductory Functional Analysis with Applications (John Wiley & Sons, New York).
LeBlanc LJ, Morlok EK, Pierskalla W (1975) An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Res. Part B 9:309–318.
Liu HX, He X, He B (2009) Method of successive weighted averages (MSWA) and self-regulated averaging schemes for solving stochastic user equilibrium problem. Networks and Spatial Econom. 9:485–503.
Lyapunov AM (1907) Problème général de la stabilité du mouvement. Ann. Fac. Sci. Univ. Toulouse 9(2):203–474. Reprint (1949) Ann. Math. Studies, no. 17 (Princeton University Press, Princeton, NJ). [Original paper published in 1892 in Comm. Soc. Math. Kharkov (Russian).]
Magnanti TL, Perakis G (1997) Averaging schemes for variational inequalities and systems of equations. Math. Oper. Res. 22:568–587.
Mounce R (2006) Convergence in a continuous dynamic queueing model for traffic networks. Transportation Res. Part B 40:779–791.
Mounce R, Carey M (2010) Route swap processes and convergence measures in dynamic traffic assignment. Tampére CMJ, Viti F, Immers LH, eds. New Developments in Transport Planning: Advances in Dynamic Traffic Assignment (Edward Elgar, Cheltenham, UK), 107–130.
Mounce R, Carey M (2011) Route swapping in dynamic traffic networks. Transportation Res. Part B 45:102–111.
Mounce R, Smith M (2007) Uniqueness of equilibrium in steady state and dynamic traffic networks. Allsop RE, Bell MGH, Heydecker BG, eds. Transportation and Traffic Theory (Emerald Group Publishing, Bingley, UK), 281–299.
Nie Y (2010) A class of bush-based algorithms for the traffic assignment problem. Transportation Res. Part B 44:73–89.
Powell WB, Sheffi Y (1982) The convergence of equilibrium algorithms with predetermined step sizes. Transportation Sci. 16:45–55.
Schauder J (1930) Der Fixpunktsatz in Funktionalräumen. Studia Mathematica 2:171–180.
Sheffi Y (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods (Prentice-Hall, Englewood Cliffs, NJ).
Smith MJ (1984a) A descent algorithm for solving monotone variational inequalities and monotone complementarity problems. J. Optim. Theory Appl. 44:485–498.
Smith MJ (1984b) The stability of a dynamic model of traffic assignment—An application of a method of Lyapunov. Transportation Sci. 18:245–252.
Smith MJ, Wisten MB (1995) A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium. Ann. Oper. Res. 60:59–79.
Williams JWJ (1964) Algorithm 232: Heapsort. Comm. ACM 7:347–348.
- method of successive averages
- traffic assignment
- step length