EMIS 8373: Integer Programming

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EMIS 8373 Integer
Programming

• Column Generation
• updated 12 April 2005

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Example 1 Adapted from Optimal Placement of
ADD/DROP Multiplexers Heuristic and Exact
Algorithms by Alain Sutter, Francis
Vanderbeck, Laurence Wolsey Operations Research,
Vol. 46 (5), pp. 719-728, 1998
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Formulation

• Sets
• N is the set of nodes in the demand graph
• E ? N ? N is the set of edges in the demand
  graph
• M is a set of candidate SONET rings to may be
  used to construct the network
• Parameters
• de is weight of edge e ? E
• b is the ring capacity
• aej 1 if demand e can be routed on ring j
• cj is cost of ring j (the number of ADMs
  required)

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Formulation Continued

• Decision Variables
• zj 1 if ring j is select, and 0, otherwise.
• BIP Formulation
• Constraint set (1) ensures that each demand is
  assigned to exactly one of the candidate rings.
• Views the problem as edge partitioning

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Example Demand Graph and Candidate Set with b 18
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• (1, 2) c1 2
• (1, 3) c2 2
• (1, 4) c3 2
• (2, 3) c4 2
• (2, 5) c5 2
• (3, 4) c6 2
• (3, 5) c7 2
• (4, 5) c8 2
• (1, 2), (1, 3), (2, 3) c9 3
• (1, 2), (1, 3), (1, 4), (2, 3), (3, 5),
  (4, 5) c10 5

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2
3
4
8
2
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Optimal Solution z5 z6 z10 1 cost 9 ADMs
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Comments

• The best solution for the given demand graph uses
  two rings and eight ADMs
• (1, 2), (2, 3), (2, 5), (3, 5)
• (1, 3), (1, 4), (3, 4), (4, 5)
• The edge-partitioning formulation cannot find
  this solution unless these candidate rings are
  given as part of the input.
• Let BIP(M) be the edge-partitioning formulation
  for a given candidate set M.
• The exact formulation is BIP(M) where M is the
  set of all feasible candidate rings.
• For a given demand graph M O(2E)

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Matrix Representation of BIP(M)
BIP(M) could have up to 28 1 255 columns
depending on b.
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LP Relaxation of BIP(M)
Dual Problem
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Additional Notation

• For a given basic feasible solution (BFS) for
  LP(M)
• Let B denote the basis matrix
• Let A?j denote column j of the constraint matrix
  A
• Let cB denote the vector of objective
  coefficients of the basic variables
• Let ? denote the vector of corresponding dual
  variables
• Recall that the reduced cost for variable zj is
  given by the formula cj- cB B-1 A?j cj- ? A?j.
• - B is optimal if all variables have a
  non-negative reduced cost.

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Reduced Cost Example 1
Suppose z1,z2, , z8 are the basic variables.
The current BFS uses 16 ADMs. Bringing ring 9
into the basis reduces this to 16 3 13
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Reduced Cost Example 1

• Suppose z1,z2, , z8 are the basic variables
• Each demand assigned to its own ring
• Using ring 9 to cover three demands saves 3 ADMs.

The current BFS uses 16 ADMs. Bringing ring 9
into the basis reduces this to 16 3 13
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Reduced Cost Example 2
Suppose z1,z2, , z8 are the basic variables.
The current BFS uses 16 ADMs. Bringing ring 10
into the basis reduces this to 16 5 11
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A Column-Generation Heuristic

• Solve restricted LP master problem LP(M) and let
  B be the optimal basis matrix
• LP(M) is referred to as the linear programming
  master problem.
• Look for a ring (column) j that is not in M, but
  would have a negative reduced cost if it were
  added to M
• This is referred as as solving the pricing
  problem.
• If j is found then add j to M and goto step 1.
• Solve BIP(M)
• At this point an optimal solution to LP(M) is
  also an optimal solution to LP(M).

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Feasible Rings with Negative Reduced Costs
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Generating Feasible Rings

• Let yij 1 if the new ring contains edge (i,j),
  and 0, otherwise
• Let xi 1 if the new ring requires an ADM at
  node i
• For the xs and ys to represent a feasible ring
  we need

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Ring Generation BIP

• This problem is NP-hard, but in practical terms
  is much easier to solve than BIP(M).
• If the optimal value of the objective function
  is zero then the optimal basis for LP(M) is
  optimal for L(M).

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Column Generation Flow Chart
Solve LP(M)
Add column to M
Solve the Pricing Problem
Yes
No
Restore Integrality Constraints and Solve BIP(M)
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Optimal Solution for LP(M) Iteration 1
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Pricing Problem for Iteration 1
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Optimal Solution for LP(M) Iteration 1
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Pricing Problem for Iteration 2
Optimal Solution
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Optimal Solution for LP(M) Iteration 3
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Pricing Problem for Iteration 3
Optimal Solution
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Optimal Solution for LP(M) Iteration 4
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Comments

• Repeat until optimal solution to the pricing
  problem has objective function value zero.
• In many cases, the pricing problem is NP-hard.
• There is no guarantee that the column-generation
  procedure will generate all of the columns that
  are selected in the optimal solution to BIP(M)
• A Branch-and-Price procedure does column
  generation at each node of the branch-and-bound
  tree
• The extra constraints added by branching usually
  complicate the pricing the problem.