Donald W. Hearn

Solving Congestion Toll Pricing Models

Congestion toll pricing addresses the classic traffic assignment problem for which Wardrop enunciated two principles of traffic flow: user-optimal behavioral hypothesis and the notion and system-optimality. (See Florian and Hearn, 1995, for a recent review of the traffic assignment problem and Johnson and Mattson, 1992 for a recent volume of papers on road pricing.) The traditional objective of congestion pricing has been to determine link tolls which will cause the solution of the tolled user-optimal problem to be optimal for the untolled system problem (Arnott and Small, 1994). In most of the literature, the one choice given has been the vector of marginal social cost pricing tolls.

An MPEC approach to second-best toll pricing

Abstract.This paper addresses two second-best toll pricing problems, one with fixed and the other with elastic travel demands, as mathematical programs with equilibrium constraints. Several equivalent nonlinear programming formulations for the two problems are discussed. One formulation leads to properties that are of interest to transportation economists. Another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems. Numerical results using transportation networks from the literature are also presented.

Continuous Characterizations of the Maximum Clique Problem

Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program max{xTAx ∣ xTe = 1, x ≥ 0}. It is well known that the global optimum value of this QP is 1-1/ωG, where ωG is the clique number of G. Here, we characterize the following properties pertaining to the above QP: 1 first order optimality, 2 second order optimality, 3 local optimality, 4 strict local optimality. These characterizations reveal interesting underlying discrete structures, and are polynomial time verifiable. A parametrization of the Motzkin-Strauss QP is then introduced and its properties are investigated. Finally, an extension of the Motzkin-Strauss formulation is provided for the weighted clique number of a graph.

Simplical decomposition of the asymmetric traffic assignment problem

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 gap function of a convex program

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.

Efficient Algorithms for the Weighted Minimum Circle Problem

The weighted minimum covering circle problem is a well-known single facility location problem used in emergency facility models. This paper introduces a classification scheme, based on fundamental mathematical programming concepts, for algorithms which solve both weighted and unweighted versions. One result of this classification is proof that a recently developed method is identical to one developed in the nineteenth century. Also, within the classification scheme, efficient new algorithms are given for the weighted problem. The results of some extensive computational tests identify the empirically fastest methods.

Congestion Toll Pricing of Traffic Networks

This paper concerns tolling methodologies for traffic networks which ensure that the resultant equilibrium flows are system optimal. A nonnegative vector β is defined to be a valid toll vector, if the set of tolled user equilibrium solutions is a subset of the set of untolled system optimal solutions. The problem of characterizing the toll set τ, which is the set of all valid toll vectors, is studied. Descriptions and characterizations of τ are given for the cases when either the cost map is strictly monotonic or is affine monotonic. In the latter case, the cost map is of the form Qυ + c, where Q is a not necessarily symmetric matrix and Q + Q T is positive semidefinite. The results are illustrated with several examples.

A Toll Pricing Framework for Traffic Assignment Problems with Elastic Demand

This paper extends the notion of toll pricing and the toll pricing framework previously developed for fixed demand traffic assignment (Bergendorff, Hearn and Ramana, 1997; Hearn and Ramana, 1998) to the problem with elastic demand. The system problem maximizes net benefit to the network users (Gartner, 1980; Yang and Huang, 1998) and the user problem is the usual one of finding equilibrium with elastic demand. We define and characterize T, the set of all tolls for the user problem that achieve the system optimal solution. When solutions to the two problems are unique, T is a polyhedron defined by the optimal solution of the system problem, similar to the case in (Bergendorff, Hearn and Ramana, 1997; Hearn and Ramana, 1998). The Toll Pricing Framework in (Hearn and Ramana, 1998) is also extended to allow optimization of secondary criteria over T. Examples include minimizing the number of toll booths and minimizing the maximum toll on any link. A numerical example illustrates the results.

A new dynamic programming algorithm for the single item capacitated dynamic lot size model

We develop a new dynamic programming method for the single item capacitated dynamic lot size model with non-negative demands and no backlogging. This approach builds the Optimal value function in piecewise linear segments. It works very well on the test problems, requiring less than 0.3 seconds to solve problems with 48 periods on a VAX 8600. Problems with the time horizon up to 768 periods are solved. Empirically, the computing effort increases only at a quadratic rate relative to the number of periods in the time horizon.

Finiteness in restricted simplicial decomposition

Simplicial decomposition is an important form of decomposition for large non-linear programming problems with linear constraints. Von Hohenbalken has shown that if the number of retained extreme points is n + 1, where n is the number of variables in the problem, the method will reach an optimal simplex after a finite number of master problems have been solved (i.e., after a finite number of major cycles). However, on many practical problems it is infeasible to allocate computer memory for n + 1 extreme points. In this paper, we present a version of simplicial decomposition where the number of retained extreme points is restricted to r, 1 =< r =< n + 1, and prove that if r is sufficiently large, an optimal simplex will be reached in a finite number of major cycles. This result insures rapid convergence when r is properly chosen and the decomposition is implemented using a second order method to solve the master problem.