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

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.

[1]  Yosef Sheffi,et al.  Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods , 1985 .

[2]  Robert B. Dial,et al.  A path-based user-equilibrium traffic assignment algorithm that obviates path storage and enumeration , 2006 .

[3]  M. Smith,et al.  A descent algorithm for solving monotone variational inequalities and monotone complementarity problems , 1984 .

[4]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[5]  Richard Mounce,et al.  Route swapping in dynamic traffic networks , 2011 .

[6]  A. M. Li︠a︡punov.  Problème général de la stabilité du mouvement , 1949 .

[7]  J. Dunn,et al.  Convexity, monotonicity, and gradient processes in Hilbert space , 1976 .

[8]  Henry X. Liu,et al.  Method of Successive Weighted Averages (MSWA) and Self-Regulated Averaging Schemes for Solving Stochastic User Equilibrium Problem , 2009 .

[9]  M. J. Smith,et al.  A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium , 1995, Ann. Oper. Res..

[10]  Yu Nie,et al.  A class of bush-based algorithms for the traffic assignment problem , 2010 .

[11]  R. Mounce.  Convergence in a continuous dynamic queueing model for traffic networks , 2006 .

[12]  D. Boyce,et al.  Solving a Non-Convex Combined Travel Forecasting Model by the Method of Successive Averages with Constant Step Sizes , 2006 .

[13]  J. Dunn,et al.  Conditional gradient algorithms with open loop step size rules , 1978 .

[14]  Richard Mounce,et al.  Route swap processes and convergence measures in dynamic traffic assignment , 2010 .

[15]  Chi-Wang Shu,et al.  Revisiting Jiang’s dynamic continuum model for urban cities , 2013 .

[16]  Larry J. LeBlanc,et al.  AN EFFICIENT APPROACH TO SOLVING THE ROAD NETWORK EQUILIBRIUM TRAFFIC ASSIGNMENT PROBLEM. IN: THE AUTOMOBILE , 1975 .

[17]  Georgia Perakis,et al.  Averaging Schemes for Variational Inequalities and Systems of Equations , 1997, Math. Oper. Res..

[18]  Michael J. Smith,et al.  The Stability of a Dynamic Model of Traffic Assignment - An Application of a Method of Lyapunov , 1984, Transp. Sci..

[19]  Warrren B Powell,et al.  The Convergence of Equilibrium Algorithms with Predetermined Step Sizes , 1982 .

[20]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[21]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[22]  David Howard Bernstein,et al.  Programmability of continuous and discrete network equilibria , 1990 .

[23]  Richard Mounce,et al.  Uniqueness of Equilibrium in Steady State and Dynamic Traffic Networks , 2007 .

[24]  J. Schauder,et al.  Der Fixpunktsatz in Funktionalraümen , 1930 .

[25]  Yu Nie,et al.  A class of bush-based algorithms for the traffic assignment problem , 2009 .