Solving Integer Programs

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Solving Integer Programs
2
Natural solution ideas that dont work well

• Solution idea 1
• Explicit enumeration Try all possible solutions
  and pick the best one.
• Example max 3x1 2x2 4x3
• s. t. x1 x2 x3 1
• x1 2x2 x3 ? 2
• x1 x2 - 2x3 0
• x1 , x2 , x3 binary
• Checking for all possible solutions
• x1 1 1 1 1 0 0 0 0
• x2 1 1 0 0 1 1 0 0
• x3 1 0 1 0 1 0 1 0
• feasible no no no yes no yes no no
• value 3 2
• Select the best solution among the feasible ones
• x1 1 x2 0 x3 0 with value 3

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Natural solution ideas that dont work well

• Number of possible solutions in the example is 23
  8 .
• Generally, if we have n binary variables,
• then the number of possible solutions is 2n .
• We have exponential growth of number of
  solutions
• which makes the algorithm very slow for large
  n.
• This method might work only when
• the number of binary variables is small in a
  mixed integer program.
• In general, more sophisticated algorithms are
  needed
• Structure the enumeration procedure so that
• only a small fraction of the possible
  solutions need to be examined.

n10 n30 n50 n100
2n 103 109 1015 1030
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Natural solution ideas that dont work well

• Solution idea 2 Solve the corresponding LP and
  round the solution.
• For every IP (or MIP) there is a corresponding
  linear program which is obtained by dropping the
  integrality constraints on the variables.
• That linear program is called LP-relaxation of
  the original IP (MIP).
• E.g., for the following IP
• max x2 (IP)
• s.t. -2x1 2x2 ? 1
• 2x1 2x2 ? 7
• x1 , x2 0 integer
• the LP-relaxation is
• max x2 (LP)
• s.t. -2x1 2x2 ? 1
• 2x1 2x2 ? 7
• x1 , x2 0

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Natural solution ideas that dont work well

• There are efficient algorithms (e.g., Simplex
  method) to solve LPs.
• So a natural algorithm for solving an IP is
• (i) solve the LP-relaxation
• (ii) round its solution to get an integral
  solution for the original IP.
• Lets see how it works on our example.
• max x2
• s.t. -2x1 2x2 ? 1
• 2x1 2x2 ? 7
• x1 , x2 0
• Optimal solution is (1.5, 2)
• Rounded solutions are
• (1, 2) and (2, 2)
• but both are infeasible

-2x1 2x2 1
2x1 2x2 7
Z2
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Natural solution ideas that dont work well

• Consider another example.
• Integer Program LP relaxation
• max x1 5x2 max x1 5x2
• s.t. x1 10x2 ? 20 s.t. x1 10x2 ? 20
• x1 ? 2 x1 ? 2
• x1 , x2 0 integer x1 , x2 0
• Solution to LP-relaxation (2, 1.8)
• Rounded IP solution
• (2, 1) with value 7
• IP optimal solution
• (0, 2) with value 10
• Conclusion Rounded solution too far
• from optimal solution

x1 2
x1 10x2 20
Z11
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Natural solution ideas that dont work well

• Summarizing,
• Solution idea 2 has the following two pitfalls
• The rounded solution might be infeasible.
• The rounded solution might be too far from
  optimal solution.
• Solution idea 2 works well for a class of
  problems
• which have the following property
• All CPF solutions of LP-relaxation are
  integral.
• For this class of problems,
• the optimal solution of LP-relaxation
• is also optimal solution for the original IP.
• Generally, we need more sophisticated algorithms
  to solve IPs
• (still using the idea of LP-relaxation).

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LP-relaxation-based solution methods for Integer
Programs

• Branch-and-Bound Technique
• (next handout)
• Cutting Plane Algorithms