Xiangdong Xu

A self-adaptive gradient projection algorithm for the nonadditive traffic equilibrium problem

Gradient projection (GP) algorithm has been shown as an efficient algorithm for solving the traditional traffic equilibrium problem with additive route costs. Recently, GP has been extended to solve the nonadditive traffic equilibrium problem (NaTEP), in which the cost incurred on each route is not just a simple sum of the link costs on that route. However, choosing an appropriate stepsize, which is not known a priori, is a critical issue in GP for solving the NaTEP. Inappropriate selection of the stepsize can significantly increase the computational burden, or even deteriorate the convergence. In this paper, a self-adaptive gradient projection (SAGP) algorithm is proposed. The self-adaptive scheme has the ability to automatically adjust the stepsize according to the information derived from previous iterations. Furthermore, the SAGP algorithm still retains the efficient flow update strategy that only requires a simple projection onto the nonnegative orthant. Numerical results are also provided to illustrate the efficiency and robustness of the proposed algorithm.

Computation and application of the paired combinatorial logit stochastic user equilibrium problem

The paired combinatorial logit (PCL) model is one of the recent extended logit models adapted to resolve the overlapping problem in the route choice problem, while keeping the analytical tractability of the logit choice probability function. However, the development of efficient algorithms for solving the PCL model under congested and realistic networks is quite challenging, since it has large-dimensional solution variables as well as a complex objective function. In this paper, we examine the computation and application of the PCL stochastic user equilibrium (SUE) problem under congested and realistic networks. Specifically, we develop an improved path-based partial linearization algorithm for solving the PCL SUE problem by incorporating recent advances in line search strategies to enhance the computational efficiency required to determine a suitable stepsize that guarantees convergence. A real network in the city of Winnipeg is applied to examine the computational efficiency of the proposed algorithm and the robustness of various line search strategies. In addition, in order to acquire the practical implications of the PCL SUE model, we investigate the effectiveness of how the PCL model handles the effects of congestion, stochasticity, and similarity in comparison with the multinomial logit stochastic traffic equilibrium problem and the deterministic traffic equilibrium problem.

A self-adaptive Armijo stepsize strategy with application to traffic assignment models and algorithms

Stepsize determination is an important component of algorithms for solving several mathematical formulations. In this article, a self-adaptive Armijo strategy is proposed to determine an acceptable stepsize in a more efficient manner. Instead of using a fixed initial stepsize in the original Armijo strategy, the proposed strategy allows the starting stepsize per iteration to be self-adaptive. Both the starting stepsize and the acceptable stepsize are thus allowed to decrease as well as increase by making use of the information derived from previous iterations. This strategy is then applied to three well-known algorithms for solving three traffic equilibrium assignment problems with different complexity. Specifically, we implement this self-adaptive strategy in the link-based Frank–Wolfe algorithm, the route-based disaggregate simplicial decomposition algorithm and the route-based gradient projection algorithm for solving the classical user equilibrium problem, the multinomial logit-based stochastic user equilibrium (MNL SUE) and the congestion-based C-logit SUE problem, respectively. Some numerical results are also provided to demonstrate the efficiency and applicability of the proposed self-adaptive Armijo stepsize strategy implemented in traffic assignment algorithms.

Improved Partial Linearization Algorithm for Solving the Combined Travel-Destination-Mode-Route Choice Problem

Combined travel demand models (CTDM) that integrate trip generation, trip distribution, modal split, and traffic assignment have been developed to resolve the inconsistency problem between the level-of-service and flow values of the sequential four-step travel demand forecasting procedure. In this paper, an improved partial linearization algorithm for solving the logit-based combined travel-destination-mode-route choice model formulated as a convex mathematical programming is developed. The improvements mainly focus on exploring recent advances in line search strategies to minimize the computational efforts required to determine a suitable stepsize that guarantees convergence. Specifically, the quadratic interpolation and the self-regulated averaging schemes are examined. Numerical results show that the self-regulated averaging line search scheme is more effective and efficient for solving the convex mathematical programming with a complex objective function in terms of solution quality and computational effort.

Reformulating Environmentally Constrained Traffic Equilibrium via a Smooth Gap Function

Various government laws have recently been enacted to alleviate the environmental deterioration of transportation systems. Environmental constraint is a valid means to explicitly reflect various environmental protection requirements imposed by the government. In this paper, we examine the environmentally constrained traffic equilibrium problem (EC-TEP), which is a fundamental tool for modeling and evaluating environmental protection requirements. Specifically, we provide an equivalent reformulation for the EC-TEP. The proposed reformulation adapts the concept of gap function to simultaneously reformulate the nonlinear complementarity conditions associated with the generalized user equilibrium conditions, environmental constraints, and conservation constraints as an equivalent unconstrained optimization problem. This gap function reformulation has two desirable features: (1) it can handle a general environmental constraint structure (linear or nonlinear; link-based or area-based) and a general link and route cost structure, enhancing the modeling adaptability and flexibility; (2) it is smooth and unconstrained, permitting a number of existing efficient algorithms for its solution. A gradient-based solution algorithm with a self-regulated averaging stepsize scheme is customized to solve the reformulated unconstrained optimization problem. Numerical examples are also provided to demonstrate the modeling flexibility of the proposed EC-TEP reformulation.

C-logit stochastic user equilibrium model with elastic demand

Abstract Modeling the elasticity of travel demand in network equilibrium analysis has several important transportation applications. In this paper, we provide a mathematical programming formulation for the C-logit stochastic user equilibrium problem with elastic demand (CL-SUE-ED) in the route domain. The proposed model is capable of explicitly modeling the elasticity of travel demand and the effect of route overlapping on travel choice and route choice simultaneously. Some qualitative properties of the model, including the equivalency and uniqueness of the solution, are also rigorously proved. To solve the CL-SUE-ED model, a partial linearization method is developed to handle the elastic demand and route overlapping considerations. In addition, a self-regulated averaging stepsize scheme is adopted to smartly determine the stepsize while avoiding evaluating the complex objective function. Numerical examples are also provided to demonstrate the features of the proposed model and solution algorithm.

Stochastic Network Design Problem with Fuzzy Goals

The transportation network design problem (NDP) is a high capital investment decision-making problem that inherently involves both subjective and objective uncertainties as well as multiple objectives. Goal programming is a practically useful approach with an explicit consideration of planners’ goal setting and priority structure among the multiple objectives. This paper describes the development of a hybrid goal programming (HGP) approach for modeling both subjective and objective uncertainties simultaneously in the NDP decision-making process. Planners’ subjective uncertainty regarding the linguistic setting of goals and priority structure is characterized as a set of fuzzy variables with nonlinear achievement and satisfaction functions, and the objective travel demand uncertainty is characterized as a set of random variables with predefined probability distributions. The HGP-NDP is formulated as a chance-constrained model in a bi-level programming framework and solved by a genetic algorithm procedure based on random simulation and fuzzy evaluation. The paper provides numerical examples and a real case study to demonstrate the features and applicability of the proposed HGP approach in solving the NDP under an uncertain environment.

Goal Programming Approach to Solve the Stochastic Multi-Objective Network Design Problem

The network design problem (NDP) is one of the optimizing improvements of a transportation network with respect to a set of system-wide objectives while considering the route choice behavior of network users. It involves making the optimal decisions at the strategic, tactical, and operational levels as to how to choose improvements for the network in such a way as to make efficient use of limited resources to achieve the stated objectives (e.g., minimizing total travel time, minimizing pollution, and minimizing inequity).

Constrained newton methods for transport network equilibrium analysis

Abstract A set of constrained Newton methods were developed for static traffic assignment problems. The Newton formula uses the gradient of the objective function to determine an improved feasible direction scaled by the second-order derivatives of the objective function. The column generation produces the active paths necessary for each origin-destination pair. These methods then select an optimal step size or make an orthogonal projection to achieve fast, accurate convergence. These Newton methods based on the constrained Newton formula utilize path information to explicitly implement Wardrops principle in the transport network modelling and complement the traffic assignment algorithms. Numerical examples are presented to compare the performance with all possible Newton methods. The computational results show that the optimal-step Newton methods have much better convergence than the fixed-step ones, while the Newton method with the unit step size is not always efficient for traffic assignment problems. Furthermore, the optimal-step Newton methods are relatively robust for all three of the tested benchmark networks of traffic assignment problems.

Considering On-Time and Late Arrivals in Multi-Class Risk-Averse Traffic Equilibrium Model with Elastic Demand

Recent empirical studies have revealed that travel time variability plays an important role in travelers’ route choice decision processes (Abdel-Aty et al. 1995; Brownstone et al. 2003; Liu et al. 2004; de Palma and Picard 2005; Fosgerau and Karlstrom 2010). Travelers treat the travel time variability as a risk in their travel choices, because it introduces uncertainty for an on-time arrival at the destination. Due to its importance, modeling route choice under uncertainty is receiving more attention. Some of the recent models can be classified as the travel time budget (TTB)-based, schedule delay-based, and mean-excess travel time (METT)-based models according to the studied aspects of the travel time variability.