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Unformatted text preview: program. Prove that x * is a global minimizer. 5 4. (15 points) Consider the linear program maximize z = 7 x 15 x 2 subject to x 14 x 2 ≤ 13 x 1 + 4 x 2 ≤ 12 x 1 + 2 x 2 ≤ 11 x 1 ,x 2 ≥ Represent point (3 , 2) as a convex combination of extreme points, plus, if applicable, a direction of unboundedness. 6 5.(15 points) Solve the following linear program using the general formulas for the simplex method. maximize z = 4 x 1x 2 subject to x 1 + x 2 ≤ 6 x 1x 2 ≤ 3 x 1 + 2 x 2 ≥ 2 x 1 ,x 2 ≥ 7 6. (15 points) Let f be a realvalued function of n variables and assume that ∇ 2 f is Lipschitz continuous. Suppose ∇ 2 f ( y ) is a positive deﬁnite for some point y . Prove that there exist constants ² > 0 and β > 0 such that k∇ f ( x ) ∇ f ( y ) k ≥ β k xy k for all x satisfying k xy k < ² ....
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 Summer '10
 Brown
 Optimization, #, linear program, Lipschitz, unboundedness

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