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Unformatted text preview: minimize z = 3 x 1x 2 + 5 x 3 subject to 2 x 1 + 2 x 2x 3 ≤ 8 x 13 x 2 + x 3 = 4 x 1 + x 2 + x 3 ≥ 11 ≤ 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 11 3 01 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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 Summer '10
 Brown
 Vector Space, Optimization, Convex set, Convex function

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