# An Active-set Algorithm for Discrete Network Design Problems

## Abstract

In this paper, we formulate a discrete network design problem as a mathematical program with complementarity constraints and propose an active set algorithm to solve the problem. Each complementarity constraint requires the product of a pair of nonnegative variables to be zero. Instead of dealing with this type of constraints directly, the proposed algorithm assigns one of the nonnegative variables in each pair a value of zero. Doing so reduces the design problem to a regular nonlinear program. Using the multipliers associated with the constraints forcing nonnegative variables to be zero, the algorithm then constructs and solves binary knapsack problems to make changes to the zero-value assignments in order to improve the system delay. Numerical experiments with data from networks in the literature indicate that the algorithm is effective and has the potential for solving larger network design problems.

## Keywords

System Delay Network Design Problem User Equilibrium Complementarity Constraint Transportation Research Part## Preview

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## Notes

### Acknowledgments

This research is supported in part by grants from National Science Foundation (CMMI-0653804) and the Center for Multimodal Solutions for Congestion Mitigation at the University of Florida.

## References

- Ahuja, R.K., Magnanti, T.L. and Orlin, J.B. (1998).
*Network Flows: Theory, Algorithms, and Applications*. Prentice Hall, New York.Google Scholar - Beckmann, M., McGuire, C.B. and Winsten, C.B. (1955).
*Studies in the Economics of Transportation*. Yale University Press, New Haven, Connecticut.Google Scholar - Bell, M.G.H. and Iida, Y. (1997).
*Transportation Network Analysis*. John Wiley and Sons, West Sussex, England.Google Scholar - Boyce, D.E. (1984). Urban transportation network equilibrium and design models: recent achievements and future prospective.
*Environment and Planning A*, 16, 1445–1474.CrossRefGoogle Scholar - Boyce, D.E., Farhi, A. and Weishedel, R. (1973). Optimal network problem: a branch-and-bound algorithm.
*Environment and Planning*, 5, 519–533.CrossRefGoogle Scholar - Brooke, A., Kendirck, D. and Meeraus, A. (1992).
*GAMS: A User’s Guide*. The Scientific Press, South San Francisco, California.Google Scholar - Chen, A. and Alfa, A.S. (1991). A network design algorithm using a stochastic incremental traffic assignment approach.
*Transportation Science*, 25, 215–224.CrossRefGoogle Scholar - Chen, Y., Shen, T.S. and Mahaba, N.M. (1989). Transportation-network design problem: application of a hierarchical search algorithm.
*Transportation Research Record*, 1251, 24–34.Google Scholar - Chiou, S. (2005). Bi-level programming for the continuous transport network design problem.
*Transportation Research Part B*, 39, 362–383.Google Scholar - CPLEX (2008).
*ILOG Inc*. Sunnyvale, California.Google Scholar - Dantzig, G.B., Harvey, R.P., Lansdowne, Z.F., Robinson, D.W. and Maier, S.F. (1979). Formulating and solving the network design problem by decomposition.
*Transportation Research Part B*,13, 5–17.CrossRefGoogle Scholar - Davis, G.A. (1994). Exact local solution of the continuous network design problem via stochastic user equilibrium assignment.
*Transportation Research Part B*,28, 61–75.CrossRefGoogle Scholar - Drud, A. (1994). CONOPT – a large scale GRG code.
*ORSA Journal on Computing*, 6, 207-216.Google Scholar - Facchinei, F. and Pang, J.S. (2003).
*Finite-dimensional Variational Inequalities and Complementarity Problem*. Springer, New York.Google Scholar - Fletcher, R., Leyffer, S., Ralph, D. and Scholtes, S. (2006). Local convergence of SQP methods for mathematical programs with equilibrium constraints.
*SIAM Journal of Optimization*, 17, 259–286.CrossRefGoogle Scholar - Florian M., Guelat J. and Spiess H. (1987). An efficient implementation of the PARTAN variant of the linear approximation method for the network equilibrium problem.
*Networks*, 17, 319-339.CrossRefGoogle Scholar - Friesz, T.L (1985). Transportation network equilibrium, design and aggregation: key developments and research opportunities.
*Transportation Research Part A*, 19, 413–427.CrossRefGoogle Scholar - Gao, Z., Wu, J. and Sun, H. (2005). Solution algorithm for the bi-level discrete network design problem.
*Transportation Research Part B*,39, 479-495.CrossRefGoogle Scholar - Hearn, D.W., Lawphongpanich, S., and Ventura, J. (1987). Restricted simplicial decomposition: computation and extensions.
*Math Program Study*, 31, 99-118.Google Scholar - Heydecker, B. (2002). Dynamic equilibrium network design.
*Proceedings of the 15th International Symposium on Transportation and Traffic Theory*.Google Scholar - Hong, H.H. (1982). Topological optimization of networks: a nonlinear mixed integer model employing generalized Benders decomposition.
*IEEE Transactions on Automatic Control*, 27(1), 164-169.CrossRefGoogle Scholar - Jeon, K., Lee, J., Ukkusuri, S. and Waller, S.T. (2006). Selectorecombinative genetic algorithm to relax computational complexity of discrete network design problem.
*Transportation Research Record*,1964, 91-103.CrossRefGoogle Scholar - Lawphongpanich, S. and Hearn, D.W. (2004). An MPEC approach to second-best toll pricing.
*Mathematical Programming Series B*, 7, 33–55.Google Scholar - LeBlanc L.J., Morlok E.K. and Pierskalla W.P. (1975). An efficient approach to solving the road network equilibrium traffic assignment problem.
*Transport Research Part B*, 9, 309-318.Google Scholar - LeBlanc, L.J. (1975). An algorithm for the discrete network design problem.
*Transportation Science*, 9, 183-199.CrossRefGoogle Scholar - Luo, Z.Q., Pang, J.S. and Ralph, D. (1996).
*Mathematical Programs with Equilibrium Constraints*. Cambridge University Press, New York.Google Scholar - Magnanti, T.L. and Wong, R.T. (1984). Network design and transportation planning: models and algorithms.
*Transportation Science*, 18, 1-55.CrossRefGoogle Scholar - Mangasarian, O.L. and Fromovitz, S. (1967). The Fritz-John optimal necessary optimality conditions in the presence of equality and inequality constraints.
*Journal of Mathematical Analysis and Applications*, 17, 37-47.CrossRefGoogle Scholar - Martello, S. and Toth, P. (1990).
*Knapsack problems: algorithms and computer implementations*. John Wiley & Sons, New York.Google Scholar - Meng, Q., Yang, H. and Bell, M.G.H. (2001). An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem.
*Transportation Research Part B*, 35, 83-105.CrossRefGoogle Scholar - Nocedal, J. and Wright, S.J. (2006).
*Numerical Optimization*. Springer, New York.Google Scholar - Poorzahedy, H. and Turnquist, M.A. (1982). Approximate algorithms for the discrete network design problem.
*Transportation Research Part B*, 16, 45-55.CrossRefGoogle Scholar - Poorzahedy, H. and Abulghasemi, F. (2005). Application of ant system to network design problem.
*Transportation*, 32, 251-273.CrossRefGoogle Scholar - Scheel, H. and Scholtes, S. (2000). Mathematical programs with complementarity constraints: stationarity, optimality and sensitivity.
*Mathematics of Operations Research*, 25, 1-22.CrossRefGoogle Scholar - Steenbrink, P.A. (1974).
*Optimization of Transportation Networks*. John Wiley and Sons, London, England.Google Scholar - Ukkusuri, S. and Waller, S.T. (2007). Linear programming models for the user and system optimal dynamic network design problem: formulations, comparisons and extensions.
*Network and Spatial Economics*, DOI 10.1007/s11067-007-9019-6.Google Scholar - Ukkusuri, S., Mathew, T.V. and Waller, S.T. (2007). Robust transportation network design under demand uncertainty.
*Computer-Aided Civil and Infrastructure Engineering*, 22(1), 6-18.CrossRefGoogle Scholar - Waller, S.T. and Ziliaskopoulous, A.K. (2001). A dynamic and stochastic approach to network design.
*Transportation Research Record*, 1771, 106-114.CrossRefGoogle Scholar - Xiong, Y. and Schneider, J.B. (1992). Transportation network design using a cumulative genetic algorithm and neural network.
*Transportation Research Record*, 1364, 37-44.Google Scholar - Yang, H. and Bell, M.G.H. (1998). Models and algorithms for road network design: a review and some new development.
*Transportation Review*, 18, 257-278.CrossRefGoogle Scholar - Yin, Y. and Lawphongpanich, S. (2007). A robust approach to continuous network designs with demand uncertainty. In
*Transportation and Traffic Theory 2007*, R.E. Allsop, M.G.H. Bell, B.G. Heydecker (Eds.), Elsevier, Amsterdam, The Netherlands, 110-126.Google Scholar