164(Summer10)_PracticeMidterm

# 164(Summer10)_PracticeMidterm - minimize z = 3 x 1-x 2 5 x...

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MATH 164 - Lecture 1 - Summer 2010 Practice Midterm - July 9, 2010 NAME: STUDENT ID #: This is a closed-book and closed-note examination. Calculators are not allowed. Please show all your work. Use only the paper provided. You may write on the back if you need more space, but clearly indicate this on the front. There are 6 problems for a total of 100 points. POINTS: 1. 2. 3. 4. 5. 6. TOTAL: 1

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2 1. (20 points) Suppose x R n is an extreme point of the set { x : Ax = b,x 0 } where A is an m × n matrix and b R m . Show that x is a basic feasible solution.
3 2. (15 points) Suppose ¯ x is a feasible point for the program maximize f ( x ) subject to x 1 + 2 x 2 - 4 x 3 = 8 2 x 1 - x 2 2 x 1 + x 3 ≥ - 1 x 1 ,x 2 0 (a) Formulate the criterium for p to be a feasible direction at ¯ x (b) Find all feasible directions for ¯ x = (0 , 2 , - 1) T (c) Determine whether p = (4 , - 2 , 0) T is a feasible direction at ¯ x = (1 , 1 . 5 , - 1). If yes, ﬁnd the maximal step α

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4 3. (15 points) Convert the following linear program to standard form

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Unformatted text preview: minimize z = 3 x 1-x 2 + 5 x 3 subject to 2 x 1 + 2 x 2-x 3 ≤ 8 x 1-3 x 2 + x 3 = 4 x 1 + x 2 + x 3 ≥ -1-1 ≤ x 1 ≤ 4 x 2 ≥ -2 in a standard form. 5 4. (15 points) Let f be a convex function on a convex set S and r is a real number. Prove that the set T = { x ∈ S : f ( x ) ≤ r } is also convex. 6 5. (20 points) Consider a system of constraints Ax = b, x ≥ 0 with A = 1 0 0 2 1 0 1-1 3 0-1 and b = 1 3 2 (a) Determine all basic solutions. (b) Determine all basic feasible solutions. (c) Give an example of the direction of unboundedness or prove that there are none. 7 6. (15 points) Consider a linear program with the constraints Ax = b, x ≥ . Prove that a nonzero vector d is a direction of unboundedness if and only if Ad = 0 and d ≥ 0....
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164(Summer10)_PracticeMidterm - minimize z = 3 x 1-x 2 5 x...

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