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MIT1_204S10_lec19

# MIT1_204S10_lec19 - 1.204 Lecture 19 Continuous constrained...

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1.204 Lecture 19 Continuous constrained nonlinear optimization: Convex combinations 2: Network equilibrium Constrained optimization 1 6 5 5 4 4 3 3 2 2 1 1 Z = -22.5 Z = -20 Z = -15 Z = -10 Z = -5 x 1 x 2 Unconstrained solution Z(5.0, 5.0) = -25.0 Solution Z(4.0, 4.5) = -24.5 Min Z(x) = x 1 2 + 2x 2 2 + 2x 1 x 2 - 10x 2 s.t. 0 < x 1 < 4 _ _ 0 < x 2 < 6 _ _ Figure by MIT OpenCourseWare.

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2 Network equilibrium problem formulation x a min z ( x ) = t a ( ω ) d ω arcs a 0 subject to f rs k = q rs OD pairs r , s paths k f rs k 0 k , r , s x ∑ ∑ rs a = f k r , s , k ∑ ∑ ∑ , , i j k if a on path from r to s Figures, examples from Sheffi Convex combinations algorithm 1 Applying the convex combinations algorithm requires solution of a linear program at each step z n z ( x n ) min ( y ) = x y a for all feasible y Gradient of z(x) is just the arc travel times: The linear program becomes: arcs a a z ( x n ) = t n x a a min z n ( y ) = t n y a a a s . t . g rs k = q rs r , s k g rs k 0 k , r , s y a = ∑ ∑ g rs k a in path rs k t n = t ( x n ) a a a Link Travel Time [min] Link Flow [veh/hr] Free flow travel time X3 Capacity ω t Figure by MIT OpenCourseWare.