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Adaptation of a Modified Newton Method for Solving the Asymmetric Traffic Equilibrium Problem
Abstract
In this paper a restriction (simplicial decomposition) strategy is employed to efficiently implement a modified Newton method to solve the general asymmetric network equilibrium problem. Implementation details are discussed. Numerical results on one small-scale and one medium-scale problem, with varying asymmetry levels, are presented. Performance of the algorithms with respect to the asymmetry level of the cost mapping is assessed.
... The third variant also represents the feasible set by (23) and the optimality conditions of its analytic center are (24)(25)(26)(27). However, unlike the second variant, the starting point is heuristically obtained as ...
... Equations (29) are approximately the optimality conditions (24)(25)(26)(27), but for the primal feasibility. Indeed, it can be easily proved that (29)-setting ǫ = 0-are the optimality conditions of the perturbed analytic center problem max λ,s ...
... For γ > 1 a nondiagonal dominant matrix is obtained and thus monotonicity of F is not guaranteed. In general, the pattern of interactions used in the computational tests of this work led to sparse Jacobian matrices whose asymmetric level, as defined by some authors [27], can be very high [29]. The w ab weights for flows on links b interacting with the current link a are equal and proportionally computed in order to satisfy a preselected γ value. ...
En la UE se ha estimado que los costes de la congesti�n representan el 2% de su PIB y que el coste de la poluci�n del aire y ruido supera el 0,6% del PIB, siendo alrededor del 90% de los mismos ocasionados por el transporte terrestre. Ante este hecho y el continuo aumento de la demanda del transporte privado frente al p�blico para los desplazamientos, muchos abogan por una conjunci�n de medidas tanto restrictivas como alternativas al uso del coche. Dentro de las primeras se encuentra el establecimiento de un peaje o una tarifa por el uso de las carreteras, medida que aunque desde el punto de vista de la Teor�a Econ�mica es la manera m�s eficiente para corregir el fallo de mercado que supone la congesti�n, desde la visi�n de pol�ticos y del p�blico no goza de gran aceptaci�n. En este trabajo se pretende hacer una simulaci�n de los efectos que tendr�a sobre el bienestar social de la implantaci�n de una medida de este tipo en la Bah�a de C�diz. In the European Union it has been estimated that the congestion cost are the 2% of the gross domestic product and the cost of pollution and noise is over 0,6%, olso it is known that the 90% of this cost are caused by overland transport. For this reason and for the always increasing demand of private transport, there are professionals who thinks that the solution have to be restrictive measures added to alternatives to the car. road pricing is a restrictive measures that for the economic theory is the most efficient way to solve congestion cost but for politicians and user of transport is not always accepted. In this study we are going to simulate road pricing for commuters in the Bah�a of C�diz and then it will be estimated welfare effects.
... (An example of such an algorithm is Newton's algorithm, for which line search rules have been proposed in e.g. [62,63,64,65,66,77,91].) ...
... [1,3,53,46]. A gap-decreasing algorithm (for Φ ≡ 0, see Section 5.3) based on Newton's method is given in [66]. ...
... The line searches can also ensure global convergence where else only local convergence can be ensured. (An example is Newton's method, which is not globally convergent in its successive approximation version, but globally convergent under an additional line search with respect to a gap function [66].) The possibility of solving subproblems in gap-decreasing algorithms only approximately [52,98] and still ensuring global convergence makes the algorithms even more interesting for practical use. ...
The purpose of this paper is to provide a unified description of iterative algorithms for the solution of traffic equilibrium problems. We demonstrate that a large number of well known solution techniques can be described in a unified manner through the concept of partial linearization, and establish close relationships with other algorithmic classes for nonlinear programming and variational inequalities. In the case of nonseparable travel costs, the class of partial linearization algorithms are shown to yield new results in the theory of finite-dimensional variational inequalities. The possibility of applying truncated algorithms within the framework is also discussed. Keywords: Transportation, Networks, Mathematical Programming 1 Introduction Traffic planning and transportation analysis problems have motivated a large amount of academic research into mathematical modelling and the design of algorithms during the last 25 years. The analysis of mathematical models of traffic planning ...
... Then, because T 2 like T 1 is maximal monotone, it follows that T 1 + T 2 is maximal monotone, cf. Rockafellar [40,Thm. 2]. ...
... Furthermore the latter mapping, by virtue of linearity and positive definiteness, is maximal monotone, even strongly monotone. It follows through [40,Thm. 2] that H + λT 2 is maximal monotone. ...
Forward-backward splitting methods,provide a range of approaches to solving large-scale optimization problems and variational inequalities in which structure conducive to de- composition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out, eorts at evaluating the convergence potential of such methods have so far relied on Lipschitz properties and strong monotonicity, or inverse strong mono- tonicity, of the mapping involved in the forward step, the perspective mainly being that of projection algorithms. Here convergence is analyzed by a technique that allows properties of the mapping,in the backward step to be brought in as well. For the first time in such a general setting, global and local contraction rates are derived, moreover in a form making it possible to determine the optimal step size relative to certain constants associated with the given problem. Insights are thereby gained into the eects,of shifting strong monotonicity between the forward and backward mappings when a splitting is selected. Key words. Forward-backward splitting, numerical optimization, variational inequalities, pro-
... Formalmente, el esquema iterativo anterior se define como sigue: (a) Decomposición simplicial (SD). La primera conduce a la clásica formulación del algoritmo SD (ver Lawphongpanich y Hearn [144], Pang y Yu [192], Marcotte y Guélat [161], Montero y Barceló [174]) y ésta es <psD{h'-\s) = C{h'-^)-A{h'-'). ...
... El objetivo de esta sección es especializar los dos algoritmos CG/SD, discutidos en la sección anterior, y mostrar que en este caso son equivalentes a aplicar un algoritmo CG/SD al problema general de asignación de tráfico. Las consideraciones efectuadas para este modelo (ver por ejemplo: Lawphongpanich y Hearn [144], Pang y Yu [192], Marcotte y Guélat [161], Montero y Barceló [174]) también pueden ser aplicadas al TAP-MVIP(c -A, üf). ...
La tesis doctoral desarrolla métodos y modelos de la investigación de operaciones aplicados a la planificación del transporte urbano, haciendo especial énfasis en los viajes con modos combinados (en más de un modo de transporte). La metodología empleada son los modelos de asignación en equilibrio entre la oferta y la demanda, formulados mediante programación matemática y desigualdades variacionales, los modelos de diseño de redes y los de estimación de matrices origen-destino, formulados mediante programación matemática binivel y programación matemática con restricciones de equilibrio. Se ha introducido una clase de algoritmos de generación de columnas / descomposición simplicial para la resolución de los problemas de optimización convexa diferenciable, estableciendo su convergencia asintótica y dando condiciones suficientes para su convergencia finita. Se ha realizado un estudio computacional sobre dos tipos de problemas de flujos en redes no lineales, mostrando que forman parte del Estado-del-Arte para problemas estructurados de grandes dimensiones. Se han elaborado modelos binivel para el diseño de intercambiadores multimodales urbanos (problema mixto de diseño de redes), diseño de aparcamientos disuasorios (problema continuo de diseño de redes) y para la estimación de matrices origen-destino. Se han elaborado algoritmos heurísticos para su resolución, debido a su no-convexidad, no-diferenciabilidad y grandes dimensiones. Se han realizado pruebas numéricas, mostrando su eficacia y su utilización.
... Other classic works that analyze the formulation of the equilibrium problem, the uniqueness of the solution and the solution algorithm are Dafermos [9,15], Florian and Spiess [16], Fisk and Nguyen [17], Fisk and Boyce [18], Nagurney [19], Hammond [20], Nguyen and Dupuis [21], Marcotte and Guelatt [22], and Auchmuty [23]. More recent analyses, focusing for the most part on the design, implementation, and comparison of solution algorithms for the trafc assignment problem, including cases with asymmetric interactions, are found in Chen et al. [24], Panicucci et al. [25], and Sancho et al. [26]. ...
An optimization model is developed to solve the deterministic traffic assignment problem under congested transport networks with cost functions that have an asymmetric Jacobian. The proposed formulation is a generalization of Beckmann’s transformation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions between the flows and costs of the different links. The objective function is built around a line integral that generalizes the simple definite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satisfies Wardrop’s first principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach. Our new approach could be extended to supply-demand equilibria models in other markets than transportation, with complementary or substitute goods/services in which there are asymmetric interactions between prices.
... Meneguzzer (1995) provided an overview of the advances in the field of diagonalization for A-TAP and convergence for explicit modeling of intersections. Marcotte and Guélat (1988) applied the modified Newton method to A-TAP, comparing its performance with cutting plane methods and diagonalization, observing their method performing better than other methods for complex asymmetric interactions. Dupuis and Darveau (1986) assessed the convergence conditions for A-TAP solutions using projection and diagonalization methods. ...
Extensions of the static traffic assignment problem with link interactions were studied extensively in the past. Much of the network modeling community has since shifted to dynamic traffic assignment incorporating these interactions. We believe there are several reasons to re-examine static assignment with link interactions. First, if link interactions can be captured in a symmetric, monotone manner, equilibrium always exists and is unique, and provably-correct algorithms exist. We show that several of the most efficient algorithms for the separable traffic assignment problem can be readily applied with symmetric interactions. We discuss how the (asymmetric) Daganzo merge model can be approximated by symmetric linear cost functions. Second, we present computational evidence suggesting that convergence to equilibrium is faster when symmetric, monotone link interactions are present. This is true even when interactions are asymmetric, despite the lack of a provable convergence result. Lastly, we present convergence behavior analysis for commonly used network and link metrics. For these reasons, we think static assignment with link interactions deserves additional attention in research and practice.
... However, traffic assignment problems based on link flows have generally been addressed using variational inequality, to include asymmetric cost functions, or multivariable cost functions. Notable among works that treat the formulation of the equilibrium problem, the existence of a solution and the solution algorithm are the following: Dafermos and Sparrow [4], Smith [5], Dafermos [6,7], Florian and Spiess [8], Fisk and Nguyen [9], Fisk and Boyce [10], Nagurney [11], Hammond [12], Marcotte and Guelatt [13], Auchmuty [14], Gabriel and Bernstein [15], and Patriksson [16]. All these works represent mainly theoretical contributions and algorithmic implementation. ...
A new model is presented that determines the traffic equilibrium on congested networks using link densities as well as costs in a manner consistent with the fundamental traffic equation. The solution so derived satisfies Wardrop’s first principle. This density-based approach recognizes traffic flow reductions that may occur when network traffic congestion is high; also, it estimates queue lengths (i.e., the number of vehicles on saturated links), and it explicitly takes into account the maximum flow a link can handle, which is defined by the fundamental traffic equation. The model is validated using traffic microsimulations and implemented on a typical Nguyen-Dupuis network to compare it with a flow-based approach. System optimal assignment model based on link densities is also presented.
... In the literature, relaxation or diagonal methods have been used to compute UE of asymmetric traffic assignment problems (Dafermos, 1982b;Fisk and Nguyen, 1982;Smith, 1984a;Nguyen and Dupius, 1984;Marcotte and Guélat, 1988;Patriksson, 1993;Marcotte and Wynter, 2004;Nagurney andZhang, 1996, 1997). Due to their internal connection with the mathematical programming formulation, these methods cannot guarantee convergence to unstable UE. ...
The stability of a user equilibrium determines whether it could be realized or sustained in a transportation network, when subject to perturbations in traffic conditions and drivers' route choice behaviors. In this paper, with a simple deterministic dynamical system model of the traffic assignment problem, we first develop a systematic approach to analyzing asymptotic stability and instability of user equilibria in static transportation networks with fixed demand. We then discuss two examples of non-monotone traffic assignment problem. Compared with existing approaches, the new approach is simpler to use and leads to results consistent with the former. This study could be helpful for analysis and design of real transportation networks.
... Finally let us mention that Marcotte and Gu61at [20] have successfully implemented Algorithm N to solve large-scale network equilibrium problems when the mapping F, i.e., its Jacobian matrix, is highly asymmetric. ...
Applied to strongly monotone variational inequalities, Newton's algorithm achieves local quadratic convergence. It is shown how the basic Newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the 'gap function' associated with the variational inequality. Each iteration consists in the solution of a linear program in the space of primal-dual variables and of a linesearch. Convergence does not depend on strong monotonicity. However, under strong monotonicity and geometric stability assumptions, the set of active constraints at the solution is implicitly identified, and quadratic convergence is achieved.
... Hammond (1984) provides insightful surveys of numerous convergence results in this literature. In Marcotte and Gélat (1988) a restriction strategy is used to apply a Newton method to resolve the general network equilibrium problem with asymmetric link cost functions. Auchmuty (1989) introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. ...
In this paper, we present a general formulation for the deterministic traffic assignment problem, using an equivalent optimization problem applicable to the case of asymmetric linear cost functions. We present a resolution approach for this problem in such a way that in equilibrium Wardrop's first principle or Nash equilibrium is satisfied. We conclude that many deterministic traffic assignment problems with asymmetric linear costs can be formulated as an optimization problem whose objective is defined by a line integral, and whose constraints correspond to non-negativity and flows conservation. By adequately defining the integration path, it is feasible to resolve the problem, obtaining Wardrop's equilibrium. This approach can be applied in other economic contexts, including microeconomic theory and consumer surplus analysis.
... The column generation/simplicial decomposition strategy for the (asymmetric) traffic assignment problem has been utilized in different forms by Bertsekas and Gafni (1982), Pang and Yu (1984), Lawphongpanich and Hearn (1984), Marcotte and Gu elat (1988), Larsson and Patriksson (1992), Montero and Barcel o (1996), Larsson et al. (1997). In Wu and Florian (1993) an adaptation of the simplicial decomposition algorithm for the transit equilibrium assignment problem was proposed. ...
In this paper we propose a new model for the equilibrium multi-modal assignment problem with combined modes (MAPCM) for the case of asymmetric costs. MAPCM is stated on a generic passenger assignment equilibrium model, on a generalized traffic assignment model, and on a nested logit distribution as demand model which explicitly takes into account the choice of mode of transport and transfer node among modal networks. This model is formulated as a variational inequality problem in the space of the hyperpath flows and then solved by the disaggregate simplicial decomposition (DSD) algorithm.Illustrations of the model and of the numerical approach are reported on two test networks with asymmetric cost functions.
... It is well known that when rs(v) is symmetric and positive deenite on v , V IP v corresponds to the optimality condition of a convex program; if rs(v) does not satisfy the condition, then there is usually no such convex program. Both V IP v and V IP h have been widely used to model network equilibrium problems arising in transportation system analysis (Smith, 1979, Dafermos, 1980, Bertsekas and Gafni, 1982, Florian and Spiess, 1984, Florian, 1986, Marcotte and Gu elat, 1988) and regional science (Florian and Los, 1982, Nagurney, 1987, Dafermos and Nagurney, 1987). The study of variational inequalities draws on nonlinear optimization methodology, solution of nonlinear equations and xed point theory. ...
We present a framework for descent algorithms that solve the monotone variational inequality problem VIPv
which consists in finding a solutionv
*∈Ω
v satisfyings(v
*)T(v−v
*)⩾0, for allv∈Ω
v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.
An optimization model is developed to solve the deterministic traffic assignment problem under congested transport networks with cost functions that have an asymmetric Jacobian. The proposed formulation is a generalization of Beckmann’s transformation that can incorporate network links with multivariate vector cost functions to capture the asymmetric interactions between the flows and costs of the different links. The objective function is built around a line integral that generalizes the simple definite integral in Beckmann’s transformation and is parameterised to ensure the solution of the new problem satisfies Wardrop’s first principle of network equilibrium. It is shown that this method is equivalent to the variational inequality approach.
Forecasting Urban Travel presents in a non-mathematical way the evolution of methods, models and theories underpinning travel forecasts and policy analysis, from the early urban transportation studies of the 1950s to current applications throughout the urbanized world. From original documents, correspondence and interviews, especially from the United States and the United Kingdom, the authors seek to capture the spirit and problems faced in different eras, as changing information requirements, computing technology and planning objectives conditioned the nature of forecasts.
This article examines the application of a path-based algorithm to the static and fixed demand asymmetric traffic assignment problem. The algorithm is of the simplicial decomposition type and it solves the equilibration or master problem step by means of five existing projection methods for variational inequality problems to evaluate their performance on real traffic networks. The projection methods evaluated are: (1) a cost approximation-based method for minimizing the Fukushima's gap function, (2) the modified descent method of Zhu and Marcotte (), (3) the double projection method of Khobotov () and three of its recently developed variants (Nadezhkina and Takahashi, ; Wang et al., ; and He et al., 2012); (4) the method of Solodov and Svaiter (); and (5) the method of Solodov and Tseng (). These projection methods do not require evaluation of the Jacobians of the path cost functions. The source for asymmetries are link costs with interactions, as in the case of priority ruled junctions. The path-based algorithm has been computationally tested using the previous projection methods on three medium to large networks under different levels of congestion and the computational results are presented and discussed. Comparisons are also made with the basic projection algorithm for the fixed demand asymmetric traffic assignment problem. Despite the lack of monotonicity properties of the test problems, the only method that failed to converge under heavy congestion levels was the basic projection algorithm. The fastest convergence was obtained in all cases solving the master problem step using the method of He et al. (2012), which is a variant of Khobotov's method.
This paper considers user equilibrium as a nonlinear complementarity problem when the Jacobian of arc travel cost function is symmetric and asymmetric. It shows solutions of user equilibrium exist and corresponding arc flow is unique. It also presents a novel modified FBLSA algorithm which makes full use of the advantage of dealing with large-size road network of column generation method.Finally, numerical results show this algorithm successfully decreases the number of variables and enjoys quick rate of convergence.
This chapter presents the main theoretical and algorithmical results pertaining to the traffic equilibrium problem (TEP), along the way improving theoretical results that were established some 20 years ago, and describes the relationship between the Nash and Wardrop concepts. The subject of traffic equilibrium is the description, through analytical tools, of the stationary distribution of vehicles in a transportation network. Assuming that travelers seek to minimize their individual travel cost, equilibrium is reached when no traveler has an incentive to modify its travel decision. Traffic equilibrium is the cornerstone, or the inner loop, of any modern network analysis. Much of the analysis in this chapter can be directly transposed from traffic networks to computer communication networks, where origin–destination pairs are connected computer communication centers and where vehicles correspond to message packets being routed between them.
The Traffic Equilibrium (TE) problem is to predict traffic flows in a transportation network given the travel demand between every pair of nodes. The existing TE models assume deterministic networks. However, real world transportation networks are stochastic in the sense that arc attributes are random variables. This paper addresses the TE problem in a stochastic network. The general problem is difficult to solve but special cases are solvable when: arc attributes are statistically independent; and disutility (or cost) functions for route evaluation are linear, exponential or quadratic. The proposed models incorporate travellers' risk taking behaviour under stochastic conditions whereas the existing models assume risk neutral travellers.
We propose a model for the transit equilibrium assignment problem (TEAP) and develop two algorithms for its solution. The behavior of the transit users is modeled by using the concept of hyperpaths (strategies) on an appropriate network (general network) which is obtained from the road network and the transit lines by a transformation which makes explicit the walk, wait, in-vehicle, transfer and alight arcs. The waiting (generalized) cost is a function of both frequency of the transit lines and congestion effects due to queues at stops. The TEAP is stated and formulated as a variational inequality problem, in the space of hyperpath flows, and then solved by the linearized Jacobi method and the projection method. We prove the global convergence of these two algorithms for strongly monotone arc cost mappings. The implementation of the algorithms and computational experiments are presented as well.
This chapter introduces the formulation, analysis and solution of the network equilibrium model (NEM). In the sequel, the terminology used will be that associated with the movement of vehicles over a network of streets, such as in an urban traffic network, since this context is a very familiar one. Nevertheless, the use of the model for problems arising in different physical contexts will be outlined, once the model formulation, analysis of its properties, and solution algorithms has been presented. The general version of the network equilibrium problem is formulated and its mathematical structure is analyzed, in order to characterize the existence and uniqueness of solutions. Then, solution algorithms for versions of the network equilibrium problem which may be formulated as nonlinear cost network optimization problems are presented. This is followed by solution algorithms for network equilibrium problems which are formulated as variational inequality problems with an embedded network structure. Then, a relatively short presentation of stochastic network equilibrium models is given. The applicability of the network equilibrium problem in other physical contexts is outlined.
The aim of this article is to give a semi-technical and somewhat journalistic account of the contributions to the methods used for quantitative transportation planning by professors, researchers and graduate students who have been active at the Centre for Research on Transportation (CRT) of the University of Montreal since its inception.
In this paper, we present an efficient implementation of heuristic procedures for solving the continuous network design problem where network users behave according to Wardrop's first principle of traffic equilibrium. Numerical results involving a standard benchmark problem are given. Also, it is shown that the cost mapping arising in the Iterative-Optimization-Assignment algorithm is integrable if and only if the volume-delay function is of either the BPR or some logarithmic form.
Recently, Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functio...
In this paper we present a version of the (static) traffic equilibrium problem in which the cost incurred on a path is not simply the sum of the costs on the arcs that constitute that path. We motivate this nonadditive version of the problem by describing several situations in which the classical additivity assumption fails. We also present an algorithm for solving nonadditive problems that is based on the recent NE/SQP algorithm, a fast and robust method for the nonlinear complementarity problem. Finally, we present a small example that illustrates both the importance of using nonadditive costs and the effectiveness of the NE/SQP method. 1 The research was conducted while this author was a member of the Mathematical Sciences Department of The Johns Hopkins University and the Mathematics and Computer Science Division of Argonne National Laboratory (ANL). At ANL, this work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Co...
In this paper we propose a new iterative method for solving the asymmetric traffic equilibrium problem when formulated as a variational inequality whose variables are the path flows. The path formulation leads to a decomposable structure of the constraints set and allows us to obtain highly accurate solutions. The proposed method is a column generation scheme based on a variant of the Khobotov’s extragradient method for solving variational inequalities. Computational experiments have been carried out on several networks of a medium-large scale. The results obtained are promising and show the applicability of the method for solving large-scale equilibrium problems.
The class of simplicial decomposition methods has been shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. This paper presents a particular implementation of such an algorithm, with emphasis on its ability to solve large scale problems efficiently.
The convergence of the algorithm is monitored by the primal gap function, which arises naturally in simplicial decomposition schemes. The gap function also serves as an instrument for maintaining a reasonable subproblem size, through its use in column dropping criteria. The small dimension and special structure of the subproblems also allows for the use of very efficient algorithms; several algorithms in the class of linearization methods are presented.
When restricting the number of retained extremal flows in a simplicial decomposition scheme, the number of major iterations tends to increase. For large networks the shortest path calculations, leading to new extremal flow generation, require a large amount of the total computation time. A special study is therefore made in order to choose the most efficient extremal flow generation technique.
Computational results on symmetric problems are presented for networks of some large cities, and on asymmetric problems for some of the networks used in the literature. Computational results for bimodal models of some large cities leading to asymmetric problems are also discussed.
This paper describes the implementation of a framework for incorporating detailed models of intersection operation into a user-optimal route choice model. It is assumed that signal settings are flow-responsive; this yields a combined route choice-intersection control problem, which is formulated as a non-separable network equilibrium problem with asymmetric cost functions. Such cost functions represent delays incurred by vehicles at intersections under various types of control (signalized, major/minor priority and all-way stop), and are based on an approach to capacity analysis widely used in traffic engineering practice, namely the methodologies of the 1985 Highway Capacity Manual and related amendments. The combined route choice-control problem is solved using a diagonalization algorithm; computational tests carried out on a real network indicate that the algorithm converges to a user equilibrium consistent with the control policy, despite violation of the sufficient conditions for convergence. The application also provides empirical evidence that the values of signal timing parameters, which are usually chosen solely on the basis of traffic engineering considerations, may have a substantial impact on the equilibration pattern. Finally, numerical experiments carried out to test the model for solution uniqueness suggest that 'reasonably' close equilibrium flow patterns arise starting from greatly different initial solutions. The modelling framework described in this paper could be typically applied in urban road networks for the assessment of alternative intersection control strategies, or, more generally, to carry out detailed traffic analyses and forecasts for the evaluation of TSM options.
This paper considers the problem of estimating traffic performance characteristics at traffic roundabouts. One and two lane roundabouts are considered with the results consisting of analytical expressions for capacity, queue length, and delay. Where drivers have a choice of lane for a manoeuvre, a user optimal model is used to allocate flow to alternative lanes. The analysis is suitable for inclusion in traffic assignment models which explicitly represent traffic conflicts at intersections, or in stand-alone analytically based computer programs which analyse a single roundabout.
This paper investigates the lane choice problem as it affects intersection delay analysis and traffic assignment. A model framework is proposed and solution properties of the lane choice and assignment problems are examined.
. In this paper we develop an algorithm for solving a version of the (static) traffic equilibrium problem in which the cost incurred on each path is not simply the sum of the costs on the arcs that constitute that path. The method we describe is based on the recent NE/SQP algorithm, a fast and robust technique for solving nonlinear complementarity problems. Finally, we present an example that illustrates both the importance of using nonadditive costs and the effectiveness of the NE/SQP method. 1 Introduction In modeling (static) traffic equilibria, researchers have generally made use of what is known as the additive model. In this approach, the path costs faced by users of the traffic network are simply the sum of the arc costs for all the arcs on the path in question. While this modeling assumption is computationally attractive, it is not appropriate in a variety of realistic and important situations. Gabriel and Bernstein [8] provide several examples in which nonadditivity is more a...
Bibliography: leaves 26-29. Supported in part by the Transportation Advanced Research Program of the U.S. Department of Transportation under contract. DOT-TSC-1058 Supported in part by the National Science Foundation under grant. 79-26225-ECS by Hedayat Z. Aashtiani and Thomas L. Magnanti.
The gap function expresses the duality gap of a convex program as a function of the primal variables only. Differentiability and convexity properties are derived, and a convergent minimization algorithm is given. An example gives a simple one-variable interpretation of weak and strong duality. Application to user-equilibrium traffic assignment yields an appealing alternative optimization problem.
We consider the general traffic equilibrium network model where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network. The model has been designed in order to handle situations where there is interaction between traffic on different links e.g., two-way streets, intersections or between different modes of transportation on the same link. For this model, we use the techniques of the theory of variational inequalities to establish existence of a traffic equilibrium pattern, to design an algorithm for the construction of this pattern and to derive estimates on the speed of convergence of the algorithm.
In the presence of several user categories or transportation modes, or when transportation costs on each arc of a network depend on the flows on adjacent arcs, the traffic equilibrium problem may be expressed as a variational problem. Methods for determining traffic equilibira are then adaptations of techniques for solving variational inequalities. In this paper, a new convex optimization formulation for the general traffic equilibrium problem is presented, and a simple iterative method is proposed for calculating traffic equilibria, which essentially involves postoptimizing a linear subproblem at each iteration. Preliminary computational results are reported.
This paper reviews the concept of a diagonalization algorithm for use in solving traffic network equilibrium problems for which the arc cost and/or the origin-destination travel demand functions are asymmetric. Such functions are known to occur in realistic settings involving multiple modes or users. The computational performance of this algorithm for different degrees of travel demand asymmetry is then explored by a detailed numerical experiment since no previous results of this type have been reported. It is found that, through the use of progressive stopping tolerances, the impact of high degrees of travel demand function asymmetry on the computational burden associated with finding a traffic network equilibrium may be mitigated; in effect equilibrium problems with high degrees of demand asymmetry are little more difficult to solve than perfectly symmetric problems.
This article discusses the solution of the fixed-demand traffic equilibrium problem by certain linearized simplicial decomposition methods. These methods are derived from the family of linear approximation methods for solving a general variational inequality problem. The central idea of a linearized simplicial decomposition method is that instead of solving linear variational inequality subproblems over the entire set of feasible flows as in a typical linear approximation method, one solves the same subproblems over subsets of feasible flows where each such subset is defined explicitly by certain extreme points of the (polyhedral) set of feasible flows. A global convergence result of the linearized decomposition methods will be established under suitable assumptions on the change of the set of “working” extreme points in each iteration plus some standard conditions on the linear approximating mappings used. Extensive computational results with the use of such methods are reported. Sizes of problems solved range from relatively small to reasonably large.
The variational inequality problem in Euclidian space is formulated as a nonconvex, nondifferentiable optimization problem. We show that any stationary point is optimal, and we propose a solution algorithm that decreases the nondifferential objective monotonically. Application to the asymmetric traffic assignment problem is considered.
This paper presents a convergent simplicial decomposition algorithm for the variational inequality formulation of the asymmetric traffic assignment problem. It alternates between generating minimum path trees based on the cost function evaluated at the current iterate and the approximate solving of a master variational inequality subject to simple convexity constraints. Thus it generalizes the popular Frank-Wolfe method (where the master problem is a line search) to the asymmetric problem. Rules are given for dropping flow patterns which are not needed to express the current iterate as a convex combination of previous patterns. The results of some computational testing are reported.
The paper gives two user objective functions for the asymmetric assignment problem, and an algorithm of descent type. The algorithm produces a sequence of flows which converges to the set of equilibria if the cost-flow function is continuous.
We provide a sufficient condition for the convergence of diagonalization algorithms for equilibrium traffic assignment problems with asymmetric Jacobian matrix B(v) of the link user cost mapping s(v) of the flow v. When , where D(v*) > 0 is the diagonal of B(v*) and v* is the equilibrium flow, we demonstrate a local convergence theorem for nonlinear cost functions. The implication of this result for practical applications of the model are outlined.
The paper formulates link-flow definitions of equilibrium and stability, and gives conditions which guarantee the existence, uniqueness and stability of traffic equilibria. Junction delays in towns usually depend on the traffic flow along intersecting links; the theory presented here is designed to be applicable when there are such junction interactions.