On the Convergence of the Method of Successive Averages for Calculating Equilibrium in Traffic Networks

References

• 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.Crossref, Google Scholar
• Bernstein DH (1990) Programmability of continuous and discrete network equilibria. Unpublished doctoral thesis, University of Illinois at Chicago, Chicago.Google Scholar
• Dial RB (2006) A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration. Transportation Res. Part B 40:917–936.Crossref, Google Scholar
• Dijkstra EW (1959) A note on two problems in connexion with graphs. Numer. Math. 1:269–271.Crossref, Google Scholar
• 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.Crossref, Google Scholar
• Dunn JC (1976) Convexity, monotonicity, and gradient processes in Hilbert space. J. Math. Anal. Appl. 53:145–158.Crossref, Google Scholar
• Dunn JC, Harshbarger S (1978) Conditional gradient algorithms with open loop step size rules. J. Math. Anal. Appl. 62:432–444.Crossref, Google Scholar
• Kreyszig E (1978) Introductory Functional Analysis with Applications (John Wiley & Sons, New York).Google Scholar
• 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.Crossref, Google Scholar
• 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 Spatial Econom. 9:485–503.Crossref, Google Scholar
• 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).]Google Scholar
• Magnanti TL, Perakis G (1997) Averaging schemes for variational inequalities and systems of equations. Math. Oper. Res. 22:568–587.Link, Google Scholar
• Mounce R (2006) Convergence in a continuous dynamic queueing model for traffic networks. Transportation Res. Part B 40:779–791.Crossref, Google Scholar
• 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.Crossref, Google Scholar
• Mounce R, Carey M (2011) Route swapping in dynamic traffic networks. Transportation Res. Part B 45:102–111.Crossref, Google Scholar
• 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.Google Scholar
• Nie Y (2010) A class of bush-based algorithms for the traffic assignment problem. Transportation Res. Part B 44:73–89.Crossref, Google Scholar
• Powell WB, Sheffi Y (1982) The convergence of equilibrium algorithms with predetermined step sizes. Transportation Sci. 16:45–55.Link, Google Scholar
• Schauder J (1930) Der Fixpunktsatz in Funktionalräumen. Studia Mathematica 2:171–180.Google Scholar
• Sheffi Y (1985) Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods (Prentice-Hall, Englewood Cliffs, NJ).Google Scholar
• Smith MJ (1984a) A descent algorithm for solving monotone variational inequalities and monotone complementarity problems. J. Optim. Theory Appl. 44:485–498.Crossref, Google Scholar
• Smith MJ (1984b) The stability of a dynamic model of traffic assignment—An application of a method of Lyapunov. Transportation Sci. 18:245–252.Link, Google Scholar
• 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.Crossref, Google Scholar
• Williams JWJ (1964) Algorithm 232: Heapsort. Comm. ACM 7:347–348.Google Scholar