## Mathematical programming algorithms for large scale network equilibrium and network design problems |

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arc i,j assignment algorithm assumption average travel budget Chapter column congestion conservation of flow cost function Dantzig-Wolfe decomposition decision variables denote developed DIST driver efficient equal equilibrium flows feasible solution Figure find the equilibrium FIRST(K flow along arc flow equations flow on arc Frank-Wolfe algorithm Golden Section search incidence matrix incremental assignment technique integer programming lemma level of flow linear programming problem linking constraints lower bound matrix multiplication mC mC minimize network design problem NLP problem node network non-zero entry Northwestern University number of arcs number of iterations objective function optimal solution optimal value origin and destination origin-destination pair partial solution procedure shortest path shortest route problems simplex method solution technique solve subproblem system optimal flows traffic assignment problem travel time function tree unit of flow unit vector urban transportation volume-delay zero