Linear Programming in Low Dimensions

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Linear Programming in Low Dimensions

• Presented by Nati Srebro

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Linear Programming
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Linear Programming in 2D
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Linear Programming in 2D
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An Infeasible Linear Program
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An Unbounded LP
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An Unbounded LP
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Types of LPs

• Unique optimum
• Optimal edge
• Unbounded
• Infeasible

• Find the optimum
• Find an optimum
• Find unbounded ray
• Declare as unfeasible

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Adding a New Constraint
If its not broken dont fix it !
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Adding a New Constraint
????
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint
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Adding a New Constraint

• Check is optimum is feasible
• Optimum is feasible
• Were fine, dont do anything
• Optimum isnt feasible
• Find optimum on new constraint (line)
• No feasible points on new constraint
• LP isnt feasible

O(1)
O(n)
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Incremental Algorithm

• Add new constraints one by one,keeping track of
  current optimum

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Initialization
Find two constraints that together bound the LP
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Initialization

• Constraint h closest to top half-plane

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Initialization

• Constraint h closest to top half-plane
• For every other constraint, check if it bounds
  the LP with h
• If no constraint is good--- LP is unbounded

or unfeasible because of parallel constraints
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Incremental Algorithm

• Find two constraints that bound LP
• If none exist, LP is unbounded
• Add all other constraints one by one, keeping
  track of current optimum

O(n)
O(n2)
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Adding a New Constraint

• Check is optimum is feasible
• Optimum is feasible
• Were fine, dont do anything
• Optimum isnt feasible
• Find optimum on new constraint (line)
• No feasible points on new constraint
• LP isnt feasible

O(1)
O(n)
Maybe we only rarely have to update optimum ?
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O(n) updates
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But
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Use a random permutation !
Expected time spent updating
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Probability of update at round i
First i constraints
Update only if ith constraint is one of two
defining constraints
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Expected Run-Time Analysis
Expectation is over algorithm randomness, not
over input
Expected time spent updating
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Cool man, but what about higher dimensions ?

• Incrementally add new constraints
• Probability of update d/(i-d)
• On update solve d-1 dimensional LP