Question: Q4. (20 pt) Consider a generic LP in standard form minimize cu subject to Ar=b (LP) => 0 where c, I ER, A Rmxn, and

Q4. (20 pt) Consider a generic LP in standard

Q4. (20 pt) Consider a generic LP in standard form minimize cu subject to Ar=b (LP) => 0 where c, I ER", A Rmxn, and be A In addition, A is assumed to have full row rank. As an alternative to the two-phase approach, consider using the big-M method to solve (LP): m minimize c+x+Mx i=1 subject to Ar + r = b (bigM) 2,">0, where r = (21,..., is a vector of artificial variables, and M is a large constant. If (bigM) is solved by the simplex method, prove the following: (4.a) If an optimal solution is found with r = 0 (a zero vector), then the corresponding 2 is an optimal basic feasible solution to the original problem. (5 pt) (4.b) If for every M > 0, the problem (bigM) is unbounded, then the original problem is either unbounded or infeasible. (7 pt) Q4. (20 pt) Consider a generic LP in standard form minimize cu subject to Ar=b (LP) => 0 where c, I ER", A Rmxn, and be A In addition, A is assumed to have full row rank. As an alternative to the two-phase approach, consider using the big-M method to solve (LP): m minimize c+x+Mx i=1 subject to Ar + r = b (bigM) 2,">0, where r = (21,..., is a vector of artificial variables, and M is a large constant. If (bigM) is solved by the simplex method, prove the following: (4.a) If an optimal solution is found with r = 0 (a zero vector), then the corresponding 2 is an optimal basic feasible solution to the original problem. (5 pt) (4.b) If for every M > 0, the problem (bigM) is unbounded, then the original problem is either unbounded or infeasible. (7 pt)

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