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practice_prelim1-solution

practice_prelim1-solution - 1(a[2 marks Dene the term...

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1. (a) [2 marks] Define the term extreme point . A point x is an extreme point of a convex set C if it does not lie on any open line-segment in C . Algebraically: there do not exist distinct points y, z C and a number λ (0 , 1) satisfying x = λy + (1 - λ ) z . Now consider a system of constraints Ax = b x 0 . Suppose that the vector [2 , 3 , 0 , 4 , 0] T is a basic feasible solution, and that the vector [0 , 1 , 2 , 2 , 4] T is a feasible solution that may or may not be basic. Answer the following questions, justifying your answers carefully . (b) [2 marks] Exactly one of the following statements is correct. State which one, and explain why it is correct. (i) A must have no more than 3 rows. (ii) A must have at least 3 rows. Three of the components of the basic feasible solution [2 , 3 , 0 , 4 , 0] T are nonzero, so the corresponding basis must contain the three indices 1,2, and 4. Hence the basis matrix has at least three columns. Since the basis matrix is square, it must contain at least three rows, and hence so does the matrix A . So statement (ii) is correct. (c) [2 marks] Find an extreme point of the feasible region. Basic feasible solutions are the same as extreme points. Hence [2 , 3 , 0 , 4 , 0] T is an extreme point. (d) [2 marks] Find a feasible solution that is not an extreme point of the feasible region. Consider the point halfway between the two given points: [1 , 2 , 1 , 3 , 2] T = 1 2 [2 , 3 , 0 , 4 , 0] T + 1 2 [0 , 1 , 2 , 2 , 4] T . According to the definition, this point cannot be an extreme point. 1
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2. [6 marks] Use the simplex method to solve the linear program maximize x 1 - x 2 - x 3 subject to 2 x 1 + x 2 - x 3 2 3 x 1 + 2 x 2 4 - x 1 + x 3 0 x 1 , x 2 , x 3 0 .
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