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l26_bw_bandbound

# l26_bw_bandbound - Integer Programming and Branch and Bound...

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Integer Programming and Branch and Bound Brian C. Williams 16.410-13 Session 26 Adapted from slides by Eric Feron , 16.410, 2002. Cooperative Vehicle Path Planning Vehicle Waypoint Obstacle Cooperative Vehicle Path Planning Vehicle Waypoint Obstacle 1

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2 Cooperative Vehicle Path Planning Objective: Find most fuel-efficient 2-D paths for all vehicles. Constraints: – Operate within vehicle dynamics – Avoid static and moving obstacles – Avoid other vehicles – Visit waypoints in specified order – Satisfy timing constraints Cooperative Path Planning MILP Encoding: Constraints • Min J T Receding Horizon Fuel Cost Fn • s ij = w ij , etc. State Space Constraints s i +1 = As i + Bu i State Evolution Equation x i = x min + My i1 -x i = -x max + My i2 y i = y min + My i3 Obstacle Avoidance -y i = -y max + My i4 At least one enabled S y ik = 3 At least one enabled • Similar constraints for Collision Avoidance (for all pairs of vehicles) Integer Programs LP: Maximize 3x 1 + 4x 2 Subject to: x 1 = 4 2x 2 = 12 IP: Maximize 3x 1 + 4x 2 Subject to: x 1 = 4 2x 2 = 12 3x 1 + 2x 2 = 18 x 1 , x 2 = 0 x 1 , x 2 integers x 1 3x 1 + 2x 2 x 1 , x 2 e) x 2 = 18 = 0
Integer Programming Integer programs are LPs where some variables are integers Why Integer programs? 1. Some variables are not real-valued: Boeing only sells complete planes, not fractions. 2. Fractional LP solutions poorly approximate integer solutions: For Boeing Aircraft Co., producing 4 versus 4.5 airplanes results in radically different profits. Often a mix is desired of integer and non-integer variables Mixed Integer Linear Programs (MILP). Outline • Review of Integer Programming (IP) • How do we solve using Branch and Bound? – Solving Binary IPs Appendices: • How do we encode decisions using IP? – Exclusion between choices – Exclusion between constraints • Solving Mixed IPs and LPs Solving Integer Programs: Characteristics • Fewer feasible solutions than LPs. • Worst-case exponential in # of variables. • Solution time tends to: – Increase with increased # of variables. – Decrease with increased # of constraints. • Commercial software: – Cplex 3

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Methods To Solve Integer Programs • Branch and Bound – Binary Integer Programs – Integer Programs – Mixed Integer (Real) Programs • Cutting Planes Branch and Bound Problem: Optimize f(x) subject to A(x) = 0, x ˛ D B & B - an instance of Divide & Conquer: I. Bound D’s solution and compare to best alternative. 1) Bound solution to D quickly. Perform quick check by relaxing hard part of problem and solve. Relax integer constraints. Relaxation is LP. 2) Use bound to “ fathom” (finish) D if possible. a. If relaxed solution is integer , Then keep soln if best found to date (“incumbent”), delete D i b. If relaxed solution is worse than incumbent, Then delete D.
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l26_bw_bandbound - Integer Programming and Branch and Bound...

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