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sol7 - C O350 L INEAR O PTIMIZATION S OLUTION HW7 Exercise...

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CO350 L INEAR O PTIMIZATION - S OLUTION HW7 Exercise 4.8.9 Claim. The LP (P): min e T x subject to Ax = 0 x 0 is unbounded, if and only if, the system Ax = 0 x 0 x 6 = 0 (i) has a solution. Proof. Suppose (P) is unbounded. Then there exists a non-trivial solution to { Ax = 0 , x 0 } which is a solution to (i). Suppose (i) has a non-zero solution x * , then for any λ 0 , λx * is a feasible solution to (P) of value λe T x * where e T x * > 0 . Hence (P) is unbounded. / The dual of (D) is given by min 0 T y subject to A T y e Suppose (i) has a solution. Then by the Claim, (P) is unbounded. It follows from weak duality that (D) is infeasible, so (ii) has no solution. Suppose (ii) has a solution. Clearly, (D) cannot be unbounded, since its objective function is 0 . It follows from strong duality that (P) has an optimal solution. By the claim, (i) has no solution. Exercise 1 (a) Suppose that C is not convex. Then there exist two elements x 1 , x 2 C and a real number λ [0 , 1] such that λx 1 + (1 - λ ) x 2 6∈ C. (1) Thus, for points x 1 , x 2 , x 1 and λ 1 = λ, λ 2 = 1 - λ 1 , λ 3 = 0 , ( λ 1 +
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