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Complexity of LP and running time of the Simplex Method

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• The worst case complexity of the Linear Program
  • Fact:

    You can solve an LP in polynomial time using the Ellipsoid Method Wikipedia: click here

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  • However:

    The most frequently used method to solve an LP is using the Simplex Method.... Reference: click here

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• Running time of the Simplex Method
  • Fact:

    The worst-case running time of the Simplex method is exponential !!! Because the number of constraints that you can formulate using n variables is exponential large (one constraint can be formulated using one subset of the n variables --- you can have exponentially large number of "corner" point that the Simplex Method must visit...)

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  • Real life problems:

    The number of constraints used to solve real life problems are not expenentially large The Simplex Method will therefore run pretty fast