Cutting Plane Method: Theory, Applications, and Insights
Telechargé par
Yasmine Cherfaoui
The Integer Programming Challenge
Consider the problem:
max 3x1+ 4x2
s.t. x1+ 2x2≤8
3x1+x2≤10
x1,x2≥0 and integer
The Question: How do we solve this efficiently?
LP Relaxation: Remove integrality constraints solve easily
But: LP solution is fractional (2.67,2.67) with value ≈16
Gap: Integer solution is (2,2) with value 14 — we lose optimality!
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Why Cutting Planes Matter
Naive Approach: Branch and
Bound
Enumerates tree of possibilities
Can be exponential in worst case
Slow for large instances
Smart Approach: Cutting Planes
Add constraints (cuts)
progressively
Tighten LP relaxation
Converge to integer optimum
Core Idea: Iteratively add constraints that:
1Cut off fractional solutions
2Preserve all integer feasible solutions
Optimization Course The Cutting Plane Method November 23, 2025 4 / 1
What is a Valid Cut?
Definition: A valid inequality (valid cut) for integer program (IP) is a
linear constraint αTx≤βsuch that:
1Is satisfied by ALL feasible integer solutions: αTx≤βfor all x∈X
(integer feasible set)
2Does not eliminate the integer optimum
Mathematically: If X={x:Ax ≤b,x∈Zn}, then αTx≤βis valid iff:
max{αTx:x∈X} ≤ β
Key Property: Valid inequalities define the integer hull PI= conv(X)
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