Combinatorial Optimization

1
Combinatorial Optimization

• Chapter 14
• Solving ILPs by Cutting Planes

2
ILP in Standard form

• min c x
• s.t. A x b
• x 0
• x integer
• LP Relaxation
• min c x
• s.t. A x b
• x 0

3
Cutting planes...
4
Cutting planes (2)

• Definition A cutting plane, or simply a cut, is a
  constraint which reduces the feasible set of the
  LP relexaxation of an ILP, without excluding
  integer feasible points.
• Definition 14.1 Given a real number y, y
  (called the integer part of y) is defined to be
  the largest integer q such that q y.

5
Simplex tableau (cf. Chapter 2)
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Optimal LP solution

• Typical equation
• xB(i) Sj not in B yij xj yi0. (1)
• The variables xi are constrained to be
  nonnegative, thus
• xB(i) Sj not in B yij xj yi0.
• Since xj is constrained to be integer, the right
  hand side can be assumed to be integer
• xB(i) Sj not in B yij xj yi0 . (2)

7
Deducing Gomory Cuts

• Subtracting (2) from (1) yields
• Sj not in B (yij -yij )xj yi0 - yi0 .
• Defining the fractional part fij of yij as
• fij yij -yij
• we have derived the constraint
• Sj not in B fij xj fi0.
• which is called the Gomory cut corresponding to
  row i.

8
Adding Gomory cuts

• We then add the constraint
• - Sj not in B fij xj s - fi0. (3)
• To the simplex tableau, where s has entered the
  basis. We have derived the correctness of the
  following Lemma....

9
Formal statement on Gomory cuts

• Lemma 14.1 If the cut defined in (3) is added to
  an optimal tableau of an LP, no integer feasible
  points are excluded, and the new tableau is
  basic, primal infeasible if yi0 is not integer,
  and dual feasible.

10
Fractional dual algorithm

• Solve the ILP relaxation, giving solution x.
• Feasible ? true
• While ( x is not integer and feasible ) do
• begin
• choose a row i
• add the Gomory cut corresponding to i
• and update the tableau (with s)
• apply dual simplex
• if dual is unbounded
• then feasible ? false
• else x ? new optimal solution
• end

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Properties of Fractional Dual

• You can skip 14.2 and 14.3, but be aware of the
  following theorem that goes without proof here
• Theorem 14.1 It is possible to implement the
  fractional dual method such that it terminates
  with an integer solution in a finite number of
  steps if one exists, (and else find that there is
  no feasible integer solution.)

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Example 14.2

• Max x2
• s.t. 3 x1 2 x2 6
• - 3 x1 2 x2 0
• x1, x2 0 and integer

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Example 14.2 (2) graphics
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Example 14.2 (3) initial tableau
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Example 14.2 (3) reference slide

• x3 6 3 x1 2 x2
• x4 0 3 x1 2 x2
• For future reference
• ¼ x3 3/2 3/4 x1 2/4 x2
• ¼ x4 0 3/4 x1 2/4 x2

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Example 14.2 (4) optimal tableau
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Example 14.2 (5) generating a cut

• Take the constraint
• x2 ¼ x3 ¼ x4 3/2
• ¼ x3 ¼ x4 1/2
• (the fractional variables take care of the
  fractional part of the right hand side)
• x2 1
• (substitute from the future reference before)
• (or ¼ x3 ¼ x4 s 1/2 ) ? s ½ - ¼ x3 - ¼ x4

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Example 14.2 (6) graphics update
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Example 14.2 (7) new tableau
20
Example 14.2 (8) pivot step
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Example 14.2 (9) generating a cut

• First row
• x1 - 1/3 x4 2/3 s 2/3
• ?
• 2/3 x4 2/3 s 2/3
• ?
• Together with s - ½ ¼ x3 ¼ x4
• 2/3 x4 2/3 (- ½ ¼ x3 ¼ x4 ) 2/3
• ? 5/6 x4 1/6 x3 1
• Substituting back with x3 6 3 x1 2 x2, x4
  0 3 x1 2 x2)
• ? 5/2 x1 5/3 x2 1 - 1/2 x1 - 1/3 x2 1
• ? 2 x1 - 2 x2 0 ? 2 x1 2 x2
• which leads to x1 x2.

22
Example 14.2 (10) graphic update
23
Example 14.2 (11)

• Next iteration gives optimal solution

24
A method to solve the ILP of TSP?

• 0. Consider the relaxed LP, and relax also the
  exponentially many subtour elimination STE)
  constraints.
• Solve this (transportation) problem (it has an
  integral optimal solution (TUM ?)
• Check all (STE) constraints, by solving max flow
  problems.
• If all STE are satisfied, stop. Current solution
  is optimal, otherwise go to step 4.
• Add violated STE and solve again LP relaxation
  (not TUM ? )
• Make solution of 4. integral by adding Gomory
  cuts. Go to step 2.

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Exercises

• 1. Consider the algorithm on the previous slide
• Why is the problem under 1. TUM.
• Work out the details of step 2.
• Prove the validity of step 3.
• Why is the relaxation under 4 not TUM
• 2. Complete the algorithm as on slide 23.