Hands-on Tutorial on Optimization F. Eberle, R. Hoeksma, and N. Megow September 26, Branch & Bound

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1 Hands-on Tutorial on Optimization F. Eberle, R. Hoeksma, and N. Megow September 6, 8 Branh & Bound

2 Branh & Bound: A General Framework for ILPs Introdued in the 96 s by Land and Doig Based on two priniple ideas. Branhing. Bounding Complete enumeration might be performed

3 A First Glane Branhing: min T x s.t. Ax b x x Z n Compute a solution x of the urrent subproblem If x Z n, ompare to the urrent upper bound. If better, store it as the new urrent best solution. If worse, prune the urrent branh. If x / Z n, hoose x i / Z and reate two subproblems Add x i x i. Add x i x i.

4 A First Glane Branhing: min T x s.t. Ax b x x Z n Compute a solution x of the urrent subproblem If x Z n, ompare to the urrent upper bound. If better, store it as the new urrent best solution. If worse, prune the urrent branh. If x / Z n, hoose x i / Z and reate two subproblems Add x i x i. Add x i x i. Bounding: If T x U, where x is the LP solution of the urrent subproblem and U the urrent upper bound, the urrent branh an be pruned.

5 Example: Integer LP P max x s.t. x + x + x

6 Example: Integer LP max x s.t. x + x + x Z P P I x

7 Example: Branhing. solve LP relaxation x () P I x

8 Example: Branhing. solve LP relaxation. x () i / Z branhing x i x () i x i x () i x () P I x x x

9 Example: Branhing. solve LP relaxation. x () i / Z branhing x i x () i x i x () i. two subproblems P () P () x

10 Example: Branh & Bound Tree x x x () x

11 Example: Branh & Bound Tree x x P () P () x

12 Example: Branh & Bound Tree x x P () x

13 Example: Branh & Bound Tree x x x () P () x

14 Example: Branh & Bound Tree x x x 4 x () P () x

15 Example: Branh & Bound Tree x x x 4 P () x

16 Example: Branh & Bound Tree x x x 4 P () x

17 Example: Branh & Bound Tree x x x 4 P () x

18 Example: Branh & Bound Tree x x x 4 x () P () x

19 Example: Branh & Bound Tree x x 4 x () 5x x 6 P () x

20 Example: Branh & Bound Tree x x 4 5x x 6 P (5) x

21 Example: Branh & Bound Tree x x 4 5x x 6 P (5) x

22 Example: Branh & Bound Tree x x 4 5x x 6 P (5) x

23 Example: Branh & Bound Tree x x 4 x () 5x x 6 P (5) x

24 Example: Branh & Bound Tree x x x () = (, ) T, T x () = 4 x () integral 5x x 6 P (5) x

25 Bounding and Pruning problem: large branh & bound tree

26 Bounding and Pruning problem: large branh & bound tree idea: prune nodes

27 Bounding and Pruning problem: large branh & bound tree idea: prune nodes for eah node: determine upper bound S S T x for best known solution x prune node

28 Bounding and Pruning problem: large branh & bound tree idea: prune nodes for eah node: determine upper bound S S T x for best known solution x prune node for max-ilp: upper bound = LP relaxation

29 Example: Bounding and Pruning x x 4 T x = 5x x 6 x

30 Example: Bounding and Pruning x x T x () =.5 4 T x = 5x x 6 x

31 Example: Bounding and Pruning x x T x () =.5 4 T x = 5x x 6 x

32 Example: Bounding and Pruning x x T x () =.5 4 Optimum! 5x x 6 x

33 Interative: Knapsak Problem: Knapsak Given: Task: n N items with values v i N and weights w i N knapsak apaity W N Maximize the total value of the knapsak while not exeeding the apaity

34 Interative: Knapsak Problem: Knapsak Given: Task: n N items with values v i N and weights w i N knapsak apaity W N Maximize the total value of the knapsak while not exeeding the apaity max s.t. ni= v i x i ni= w i x i W x {, } n

35 Interative: Knapsak (Multiple Copies) Problem: Knapsak (Multiple Copies) Given: Task: n N items with values v i N, weights w i N, and number of opies p i N knapsak apaity W N Maximize the total value of the knapsak while not exeeding the apaity

36 Interative: Knapsak (Multiple Copies) Problem: Knapsak (Multiple Copies) Given: Task: n N items with values v i N, weights w i N, and number of opies p i N knapsak apaity W N Maximize the total value of the knapsak while not exeeding the apaity max s.t. ni= v i y i ni= w i y i W y i p i for all i [n] y N n

37 Interative: TSP Problem: TSP Given: n N ities with distanes d ij between i, j [n] Task: Minimize the length of a tour that visits all ities exatly one

38 Interative: TSP Problem: TSP Given: n N ities with distanes d ij between i, j [n] Task: Minimize the length of a tour that visits all ities exatly one max s.t. i,j d ijx ij j x ij = for all i [n] j x ji = for all i [n] x ij {, } for all i, j [n]

39 Interative: TSP Problem: TSP Given: n N ities with distanes d ij between i, j [n] Task: Minimize the length of a tour that visits all ities exatly one max s.t. i,j d ijx ij j x ij = for all i [n] j x ji = for all i [n] u i u j + n x ij n for all i j {,..., n} x ij {, } for all i, j [n]

to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting

$to work with) can be solved by solving their LP relaxations with the Simplex method I Cutting plane algorithms, e.g., Gomory s fractional cutting$ Summary so far z =max{c T x : Ax apple b, x 2 Z n +} I Modeling with IP (and MIP, and BIP) problems I Formulation for a discrete set that is a feasible region of an IP I Alternative formulations for the