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# a2 - CO350 Assignment 2 Question 1(LPa Consider the...

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CO350 Assignment 2 Due: May 18 at 2pm. Question 1: Consider the following linear programs. ( LP a ) min x 1 + 2 x 3 Subject to 2 x 1 + x 2 - 3 x 3 4 3 x 1 + 2 x 2 - 4 x 3 = 3 - 3 x 1 - x 2 + x 3 3 x 1 ,x 2 0 ( LP b ) min x 1 - x 3 Subject to 2 x 1 + x 2 + 3 x 3 4 x 1 + 2 x 2 - 2 x 3 1 - 3 x 1 + x 3 = 2 x 1 0 ,x 2 0 (a) Write ( LP a ) in standard inequality form. (b) Write ( LP b ) in standard equality form. Question 2: Consider the linear program max( c T x : Ax = 0 , x 0), where A R m × n and c R n . Show that if x = 0 is not an optimal solution, then the linear program is unbounded. Question 3: Consider the linear program max( c T x : Ax = b, x 0), where A R m × n , b R m , and c R n . Show that, if the linear program has two distinct optimal solutions, then it has infinitely many optimal solutions. Question 4: Let A R m × n and b R m . (a) Show that, if there exists y R m such that y T A 0 and y T b > 0, then the system ( Ax = b, x 0) is infeasible. (b) Show that the following linear program is infeasible by using the result proved in part (a)
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