Branch and Bound Algorithm For Integer Program

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Branch and Bound AlgorithmFor Integer Program
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Integrality Conditions
MAX 350X1 300X2 S.T. 1X1 1X2 lt
200 9X1 6X2 lt 1566 12X1 16X2
lt 2880 X1, X2gt 0 X1, X2 must
be integers
Integrality conditions are easy to state but make
the problem much more difficult (and sometimes
impossible) to solve.
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LP Relaxation

• Original IP
• MAX 2X1 3X2
• S.T. X1 3X2 lt 8.25
• 2.5X1 X2 lt 8.75
• X1, X2 gt 0 and integers
• LP Relaxation
• MAX 2X1 3X2
• S.T. X1 3X2 lt 8.25
• 2.5X1 X2 lt 8.75
• X1, X2 gt 0

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IP vs. LP Feasible Region
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Relation Between LP and IP

• 1) LP has a large feasible region, so will have
  better objectives
• For minimization, ZLP lt ZIP an upper bound.
• For maximization, ZLP gt ZIP a lower bound
• The optimal sol. to an LP relaxation of an IP
  problem gives us a bound on the optimal IP.
• 2) Solution Wise, Can we Use LP Solution
• Can we solve an LP instead of an IP?
• Can we approximate IP solution with LP?

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The Concept of Bound

• The optimal solution to an LP relaxation of an
  ILP problem gives us a bound on the optimal
  objective function value.
• For maximization problems, the optimal relaxed
  objective function values is an upper bound on
  the optimal integer value.
• For minimization problems, the optimal relaxed
  objective function values is a lower bound on the
  optimal integer value.

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Solving IP Problems

• 1) When solving an LP relaxation, sometimes you
  get lucky and obtain an integer feasible
  solution.
• There are a set of problems where you can always
  get lucky.
• Network Flow problem
• Problem with Totally Unimodular (TU) Property

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Solve IP Problems

• 2) But often, you get fractional solutions
• Round LP solution
• Derive Solutions closes to IP
• In general, this does not work reliably
• The rounded solution may be infeasible.
• The rounded solution may be suboptimal

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Rounding ?Infeasible Solution
X2
3
Integer Infeasible obtained by rounding up
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optimal relaxed solution
infeasible solution obtained by rounding down
1
0
X1
0
1
2
3
4
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Round ? Suboptimal Solution
X2
3
One of them is optimal
LP optimal
X1 2.769, X21.826
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1
Suboptimal
0
0
3
4
X1
1
2
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There is no Obvious way to Round
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Branch-and-Bound

• Branch and Bound
• The Branch and Bound (BB) algorithm can be used
  to solve IP problems.
• The Basic Technique Divide and Conquer
• The Traditional LP Based BB Algorithm
• Divide the problem into a series of LP
  sub-problems and solve/conquer each one
• If all the sub-problems are conquered, I got the
  answer -- the best of all them is the solution to
  the original problem.

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Branch and Bound Algorithm
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Branch and Bound Technique
LP (x1,x2) (2.44,3.26) obj 14.66
Solve the LP relaxation, we have faction sol.
(2.44, 3.26) with obj14.66 Not IP Feasible Let
us round it, (2,3) obj 13.
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Branch and Bound Technique
UB14.66 LB13.00
x12.44 x23.26 obj14.66
LP (x1,x2) (2.44,3.26) obj 14.66
x1 can not be fraction, so there are two
possibilities x1 lt 2 and x1gt 3 Let us divide
the orig. problem into two subproblems. One
requires that X1 lt 2 and the other requires x1gt
3
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Branch and Bound Technique
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Branch and Bound Technique
UB14.66 LB13.00
x12.44 x23.26 ov14.66
UB13.90 LB13.00
x12 x23.3 obj13.90
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Branch and Bound Technique
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Branch and Bound Technique
UB14.66, 14.58 LB13.00, 13.00
x12.44 x23.26 ov14.66
UB13.90 LB13.00
UB14.58 LB13.00
x13 x22.8 obj14.58
x12 x23.3 obj13.90
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Branch and Bound Technique
Two branches can be created. x2 gt3 and x2 lt2
x2 lt2
x2 lt2
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Branch and Bound Technique
UB14.66, 14.58 LB13.00, 13.00
x12.44 x23.26 ov14.66
UB13.90 LB13.00
UB14.58 LB13.00
x12 x23.3 ov13.90
x13 x22.8 ov14.58
infeasible
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Branch and Bound Technique
UB14.66, 14.58, 14.00 LB13.00, 13.00, 14.00
x12.44 x23.26 ov14.66
UB13.90 LB13.00
UB14.58 LB13.00
x1lt 2
x1gt 3
x12 x23.3 ov13.90
x13 x22.8 ov14.58
x2lt 2
x2gt 3
x14 x22 ov14
UB14.00 LB14.00
infeasible
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Branch and Bound

• General idea Divide-and-conquer
• Consider the IP zIP max cx x ? S
• Divide the solution space S in to K subsets S1,
  , SK
• Instead of solving IP directly, solve IPi
  max cx x ? Si
• Then, zIP maxi
• We break a big problem into manageable size
  problems and solve them

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Branch and Bound Tree
SS0
S1
S2
S12
S22
S13
S11
S21
S122
S121
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Pruning

• At each node, we decide whether we should divide
  the problem further or not
• If we can establish that no further division of a
  node is necessary, we say that the tree can be
  pruned at the node
• How do we determine if the tree can be pruned at
  a node?

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Pruning Criteria

• At node Si of a BB tree, if any one of the
  following conditions hold
• Infeasibility Si ?
• Optimality An optimal sol of IPi is found -- all
  sol are integer, nothing to branch
• Value Dominance We can establish that ZIPi ?
  zIP. There would be any better solution in this
  region.

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Generic Algorithm

• (Initialization) ? IP, S0 S, ,
  and
• (Termination test) If ? ?, then solution
  x0 is optimal
• (Problem Selectionrelaxation) Select and remove
  a problem IPi from ? and solve its relaxation
  RPi
• (Pruning) Test if the tree can be pruned at Ipi
• ? then go to Termination test
• go to next step
• , set . Delete from
  ? all problems with . If
  goto step 2 otherwise step 5

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End of BB

• Homework

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Other Approaches for IP

• Exact Algorithms
• General Algorithms Branch and bound, cutting
  planes, branch and cut
• Special Purpose Dynamic programming,
  decomposition techniques
• Heuristic Techniques
• No guarantees, but they work well in practice

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Geometric Representation
Optimal Relaxed Solution X1 2.769, X21.826
Obj 11.019