{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

l26_bw_bandbound

l26_bw_bandbound - Integer Programming and Branch and Bound...

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

View Full Document Right Arrow Icon
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
Background image of page 1

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

View Full Document Right Arrow Icon
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
Background image of page 2
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
Background image of page 3

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

View Full Document Right Arrow Icon
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.
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.

{[ snackBarMessage ]}

Page1 / 21

l26_bw_bandbound - Integer Programming and Branch and Bound...

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

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