# a5 - Hint Modify the algorithm that constructs a basic...

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CO350 Assignment 5 Due: June 8 at 2pm. Question 1: Consider the linear program max( c T x : Ax = b, x 0) where A = 1 0 0 - 1 - 2 - 3 0 1 0 3 1 1 0 0 1 1 1 2 , b = 2 10 4 , and c = [0 , 0 , 0 , - 1 , 1 , 1] T . Solve this linear program using the simplex method; start with the basis { 1 , 2 , 3 } . Question 2: Consider the linear program max( c T x : Ax = b, x 0) where A = 1 0 1 - 1 0 - 2 1 0 1 0 - 2 0 0 1 1 , b = 3 4 6 , and c = [ - 1 , 0 , 0 , 1 , 0] T . Solve this linear program using the simplex method; start with the basis { 2 , 3 , 5 } . Question 3: Let A R m × n , b R m , and c R n , where rank( A ) = m . Consider the linear program. ( P ) max( c T x : Ax = b, x 0). Prove that, if x is a feasible solution for ( P ), then either (1) there is a basic feasible solution x to ( P ) with c T x c T x , or (2) ( P ) is unbounded.
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Unformatted text preview: Hint: Modify the algorithm that constructs a basic feasible solution from a feasible solution. Remark: Question 3 can be used to prove the Fundamental Theorem of Linear Program-ming (that is, if ( P ) is feasible and bounded, then ( P ) has an optimal solution). Note that there are only ±nitely many basic feasible solutions to ( P ). So, if ( P ) is feasible, we can choose a basic feasible solution x ∗ maximizing c T x ∗ . If ( P ) is bounded, then, by Question 3, x ∗ is an optimal solution to (P)....
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