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Linear Programming in Low Dimensions. Presented by Nati Srebro. Linear Programming. Linear Programming in 2D. Linear Programming in 2D. An Infeasible Linear Program. An Unbounded LP. An Unbounded LP. Unique optimum Optimal edge Unbounded Infeasible. Types of LPs. Find the optimum

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linear programming in low dimensions

Linear Programming in Low Dimensions

Presented by Nati Srebro

types of lps
Unique optimum

Optimal edge

Unbounded

Infeasible

Types of LPs
  • Find the optimum
  • Find an optimum
  • Find unbounded ray
  • Declare as unfeasible
adding a new constraint
Adding a New Constraint

If its not broken– don’t fix it !

adding a new constraint25
Adding a New Constraint
  • Check is optimum is feasible
  • Optimum is feasible:

We’re fine, don’t do anything

  • Optimum isn’t feasible:

Find optimum on new constraint (line)

No feasible points on new constraint:

LP isn’t feasible

O(1)

O(n)

incremental algorithm
Incremental Algorithm
  • Add new constraints one by one,keeping track of current optimum
initialization
Initialization

Find two constraints that together bound the LP

initialization28
Initialization
  • Constraint h closest to top half-plane
initialization29
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

incremental algorithm30
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)

adding a new constraint31
Adding a New Constraint
  • Check is optimum is feasible
  • Optimum is feasible:

We’re fine, don’t do anything

  • Optimum isn’t feasible:

Find optimum on new constraint (line)

No feasible points on new constraint:

LP isn’t feasible

O(1)

O(n)

Maybe we only rarely have to update optimum ?

slide40

Update time duringround i

Update time duringround i

Update at roundi

Use a random permutation !

Expected time spent updating:

probability of update at round i

non-initial constraints

Probability of update at round i

First i constraints:

Update only if ith constraint is one of two defining constraints

expected run time analysis
Expected Run-Time Analysis

Update time duringround i

Update time duringround i

Update at roundi

Expectation is over algorithm randomness, not over input

Expected time spent updating:

slide43

Cool man, but what about higher dimensions ?

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