### Abstract

This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.

Original language | English (US) |
---|---|

Pages (from-to) | 309-328 |

Number of pages | 20 |

Journal | Transportation Research Part B: Methodological |

Volume | 126 |

DOIs | |

State | Published - Aug 2019 |

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### All Science Journal Classification (ASJC) codes

- Civil and Structural Engineering
- Transportation

### Cite this

}

*Transportation Research Part B: Methodological*, vol. 126, pp. 309-328. https://doi.org/10.1016/j.trb.2018.08.015

**The mathematical foundations of dynamic user equilibrium.** / Friesz, Terry L.; Han, Ke.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The mathematical foundations of dynamic user equilibrium

AU - Friesz, Terry L.

AU - Han, Ke

PY - 2019/8

Y1 - 2019/8

N2 - This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.

AB - This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE; then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples.

UR - http://www.scopus.com/inward/record.url?scp=85052965680&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052965680&partnerID=8YFLogxK

U2 - 10.1016/j.trb.2018.08.015

DO - 10.1016/j.trb.2018.08.015

M3 - Article

AN - SCOPUS:85052965680

VL - 126

SP - 309

EP - 328

JO - Transportation Research, Series B: Methodological

JF - Transportation Research, Series B: Methodological

SN - 0191-2615

ER -