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Linear Programming and Combinatorial Optimization Tutorial 2

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Linear Programming and Combinatorial Optimization Tutorial 2

January 18, 2019

1. Show how to compute the inverse of ann×nmatrix. What is the time complexity?

2. Given a non-zero vectorc, find a vectorxsuch thatcTx >0. A point inS={x:Ax≤b} is said to be on theboundaryif at least one of the inequalities is an equality i.e. ∃i Aix=bi. Prove that in such a case, there is a direction vector y such that moving along the direction of y takes you outsideS. That is,x+y /∈S for any >0.

3. Letx0be on the boundary. LetA0x0=b0be the equalities andA00x0< b00be the strict inequalities.

Let rank ofA0ber. Then the set{y|A0y=b0, A00y < b00}is called ann−rfaceorn−rdimensional face ofS.

What is the maximum number ofkdimensional faces possible as a function ofm, n, k?

4. During the execution of simplex algorithm, show how to detect if there is a degeneracy at a given extreme point.

5. In continuation to the above, show how a degeneracy can be eliminated without affecting the feasible region.

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