MIT1_204S10_lec19

MIT1_204S10_lec19 - 1.204 Lecture 19 Continuous constrained...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
2 Network equilibrium problem formulation x a min z ( x ) = t a ( ω ) d 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 ) = y a for all feasible y Gradient of z(x) is just the arc travel times: The linear program becomes: arcs a x 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.
Background image of page 2
3 Convex combinations algorithm 2 The linear program minimizes travel times over a The linear program minimizes travel times over a network with fixed, not flow-dependent times. Total time is minimized by assigning each traveler to shortest O-D path Thus, a shortest path algorithm, plus loading flow on the links used by each O-D pair, solves the linear program Line search step uses bisection method which, for a minimization problem, requires a derivative It happens to be easy to compute. After a lot of algebra: z [ x n + α ( y n x n )] = ( y n x n ) t ( x n + ( y n a )) a x n a a a a a Convex combinations steps Step 0: Initialization. Find shortest paths based on t a =t a (0). Assign flows to obtain {x 1 a } Step 1 Update times Step 1: Update times. Set t n a =t a (x n a ) for all a Step 2: Direction finding. Find shortest paths based on {t n a } Assign flows to obtain auxiliary flows {y n a } Step 3: Line search. Find α n that solves x n + y n x n a ( a a ) Step 4: Move. Set Step 5: Convergence test. If not converged, go to step 1 0 min 1 t a ( ω ) d a 0 x n + 1 = x n + n ( y n x n )
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example network Example program output Iter Step 0 Update Move t: x: Link: 1 10.0 10.00 2 20.0 0.00 3 25.0 0.00 Objective fn 0.00 Alpha 1 Update Direction Move t: t: y: x: 947 5 20 0 947.5 20.0 0.00 10.00 4.03 5.97 25 0 25.0 0.00 0.00 1975 00 1975.00 0.597 2 Update Direction Move t: y: x: 34.8 0.00 3.38 34.8 25.0 0.00 10.00 5.00 1.61 197.40 0.161 3 Update Direction Move t: y: x: 22.3 10.00 3.62 27.3 0.00 4.83 25.3 0.00 1.55 189.99 0.036 4 Update Direction Move t: y: x: 26.1 0.00 3.55 26.4 25.3 0.00 10.00 4.73 1.73 189.45 0.020 5 Update Direction Move t: y: x: 24.8 10.00 3.59 25.9 0.00 4.69 25.4 0.00 1.71 189.36 0.007 4 Figure by MIT OpenCourseWare.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/04/2011 for the course ESD 1.204 taught by Professor Georgekocur during the Spring '10 term at MIT.

Page1 / 13

MIT1_204S10_lec19 - 1.204 Lecture 19 Continuous constrained...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online