### Abstract

Original language | English |
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Journal | INFORMS Journal on Computing |

DOIs | |

Publication status | E-pub ahead of print - Nov 2019 |

### Fingerprint

### Keywords

- quadratic shortest path problem
- semidefinite programming
- alternating direction method of multipliers
- branch and bound

### Cite this

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**On solving the quadratic shortest path problem.** / Hu, Hao; Sotirov, Renata.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - On solving the quadratic shortest path problem

AU - Hu, Hao

AU - Sotirov, Renata

PY - 2019/11

Y1 - 2019/11

N2 - The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order m+1, where m is the number of arcs in the graph. We use the alternating direction method of multipliers to solve the semidefinite programming relaxations. Numerical results show that our bounds are currently the strongest bounds for the quadratic shortest path problem. We also present computational results on solving the quadratic shortest path problem using a branch and bound algorithm. Our algorithm computes a semidefinite programming bound in each node of the search tree, and solves instances with up to 1300 arcs in less than an hour.

AB - The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order m+1, where m is the number of arcs in the graph. We use the alternating direction method of multipliers to solve the semidefinite programming relaxations. Numerical results show that our bounds are currently the strongest bounds for the quadratic shortest path problem. We also present computational results on solving the quadratic shortest path problem using a branch and bound algorithm. Our algorithm computes a semidefinite programming bound in each node of the search tree, and solves instances with up to 1300 arcs in less than an hour.

KW - quadratic shortest path problem

KW - semidefinite programming

KW - alternating direction method of multipliers

KW - branch and bound

U2 - 10.1287/ijoc.2018.0861

DO - 10.1287/ijoc.2018.0861

M3 - Article

JO - INFORMS Journal on Computing

JF - INFORMS Journal on Computing

SN - 1091-9856

ER -