Branch & Bound Branch & Bound: A General Framework for ILPs - - PowerPoint PPT Presentation

Hands-on Tutorial on Optimization

• F. Eberle, R. Hoeksma, and N. Megow

September 25, 2019

Branch & Bound

Branch & Bound: A General Framework for ILPs

◮ Introduced in the 1960’s by Land and Doig ◮ Based on two principle ideas

• 1. Branching
• 2. Bounding

◮ Complete enumeration might be performed

A First Glance

max cTx s.t. Ax ≤ b x ≥ 0 x ∈ Zn Branching:

◮ Compute a solution x′ of the current subproblem ◮ If x′ ∈ Zn, compare to the current lower bound.

◮ If better, store it as the new current best solution. ◮ If worse, prune the current branch.

◮ If x′ /

∈ Zn, choose x′

i /

∈ Z and create two subproblems

◮ Add xi ≤ ⌊x′

i ⌋.

◮ Add xi ≥ ⌈x′

i ⌉.

A First Glance

max cTx s.t. Ax ≤ b x ≥ 0 x ∈ Zn Branching:

◮ Compute a solution x′ of the current subproblem ◮ If x′ ∈ Zn, compare to the current lower bound.

◮ If better, store it as the new current best solution. ◮ If worse, prune the current branch.

◮ If x′ /

∈ Zn, choose x′

i /

∈ Z and create two subproblems

◮ Add xi ≤ ⌊x′

i ⌋.

◮ Add xi ≥ ⌈x′

i ⌉.

Bounding: If cTx′ ≤ L, where x′ is the LP solution of the current subproblem and L the current lower bound, the current branch can be pruned.

Example: Integer LP

max x1 + x2 s.t. x2 ≥

1 2

2x1 + x2 ≤

11 2

−2x1 + x2 ≤ −1

2

x1 x2

1 1 2 2 3 3

P c

Example: Integer LP

max x1 + x2 s.t. x2 ≥

1 2

2x1 + x2 ≤

11 2

−2x1 + x2 ≤ −1

2

x ∈ Z2 x1 x2

1 1 2 2 3 3

P c PI

Example: Branching

• 1. solve LP relaxation

x1 x2

1 1 2 2 3 3

c PI x(0)

Example: Branching

• 1. solve LP relaxation
• 2. x(0)

i

/ ∈ Z → branching xi ≤ ⌊x(0)

i

⌋ xi ≥ ⌈x(0)

i

⌉ x1 x2

1 1 2 2 3 3

c PI x(0)

x1 ≥ 2 x1 ≤ 1

Example: Branching

• 1. solve LP relaxation
• 2. x(0)

i

/ ∈ Z → branching xi ≤ ⌊x(0)

i

⌋ xi ≥ ⌈x(0)

i

⌉

• 3. two subproblems

x1 x2

1 1 2 2 3 3

P(1) P(2)

Example: Branch & Bound Tree

1

x1 ≤ 1

2

x1 ≥ 2

x1 x2

1 1 2 2 3 3

c x(0)

Example: Branch & Bound Tree

1

x1 ≤ 1

2

x1 ≥ 2

x1 x2

1 1 2 2 3 3

c P(1) P(2)

Example: Branch & Bound Tree

1

x1 ≤ 1

2

x1 ≥ 2

x1 x2

1 1 2 2 3 3

c P(2)

Example: Branch & Bound Tree

1

x1 ≤ 1

2

x1 ≥ 2

x1 x2

1 1 2 2 3 3

c P(2) x(1)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3

x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

x1 x2

1 1 2 2 3 3

c P(2) x(1)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3

x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

x1 x2

1 1 2 2 3 3

c P(3)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3

x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 x1 x2

1 1 2 2 3 3

c P(3)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3

x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 x1 x2

1 1 2 2 3 3

c P(3)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3

x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 x1 x2

1 1 2 2 3 3

c P(3) x(2)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 x1 x2

1 1 2 2 3 3

c P(3) x(2)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 x1 x2

1 1 2 2 3 3

c P(5)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 6 x1 x2

1 1 2 2 3 3

c P(5)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 6 x1 x2

1 1 2 2 3 3

c P(5)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 6 x1 x2

1 1 2 2 3 3

c P(5) x(3)

Example: Branch & Bound Tree

1

x1 ≤ 1

2 3 5

x1 ≤ 2

6

x1 ≥ 3 x2 ≤ 1

4

x2 ≥ 2 x1 ≥ 2

4 6 integral x1 x2

1 1 2 2 3 3

c P(5) x(3) x(3) = (2, 1)T, cTx(3) = 3

Bounding and Pruning

◮ problem: large branch & bound tree

Bounding and Pruning

◮ problem: large branch & bound tree ◮ idea: prune nodes

Bounding and Pruning

◮ problem: large branch & bound tree ◮ idea: prune nodes

◮ for each node: determine upper bound S ◮ S ≤ cTx∗ for best known solution x∗

→ prune node

Bounding and Pruning

◮ problem: large branch & bound tree ◮ idea: prune nodes

◮ for each node: determine upper bound S ◮ S ≤ cTx∗ for best known solution x∗

→ prune node

◮ for min-ILP:

◮ upper bound U for the problem (current best solution) ◮ lower bound S for each node ◮ if S ≥ U → prune node

Example: Bounding and Pruning

1 x1 ≤ 1 2 3 5 x1 ≤ 2 6 x

1

≥ 3 x2 ≤ 1 4 x

2

≥ 2 x1 ≥ 2 cTx∗ = 3

x1 x2

1 1 2 2 3 3

c

Example: Bounding and Pruning

1 x1 ≤ 1 2 3 5 x1 ≤ 2 6 x

1

≥ 3 x2 ≤ 1 4 x

2

≥ 2 x1 ≥ 2 cTx∗ = 3

x1 x2

1 1 2 2 3 3

c cTx(1) = 2.5

Example: Bounding and Pruning

1 x1 ≤ 1 2 3 5 x1 ≤ 2 6 x

1

≥ 3 x2 ≤ 1 4 x

2

≥ 2 x1 ≥ 2 cTx∗ = 3 1

x1 x2

1 1 2 2 3 3

c cTx(1) = 2.5

Example: Bounding and Pruning

1 x1 ≤ 1 2 3 5 x1 ≤ 2 6 x

1

≥ 3 x2 ≤ 1 4 x

2

≥ 2 x1 ≥ 2 Optimum! 1

x1 x2

1 1 2 2 3 3

c cTx(1) = 2.5

Interactive: Knapsack

Problem: Knapsack

Given: n ∈ N items with values vi ∈ N and weights wi ∈ N knapsack capacity W ∈ N Task: Maximize the total value of the knapsack while not exceeding the capacity

Interactive: Knapsack

Problem: Knapsack

Given: n ∈ N items with values vi ∈ N and weights wi ∈ N knapsack capacity W ∈ N Task: Maximize the total value of the knapsack while not exceeding the capacity max

n

i=1 vixi

s.t.

n

i=1 wixi

≤ W x ∈ {0, 1}n

Interactive: Knapsack (Multiple Copies)

Problem: Knapsack (Multiple Copies)

Given: n ∈ N items with values vi ∈ N, weights wi ∈ N, and number of copies pi ∈ N knapsack capacity W ∈ N Task: Maximize the total value of the knapsack while not exceeding the capacity

Interactive: Knapsack (Multiple Copies)

Problem: Knapsack (Multiple Copies)

Given: n ∈ N items with values vi ∈ N, weights wi ∈ N, and number of copies pi ∈ N knapsack capacity W ∈ N Task: Maximize the total value of the knapsack while not exceeding the capacity max

n

i=1 viyi

s.t.

n

i=1 wiyi

≤ W yi ≤ pi for all i ∈ [n] y ∈ Nn