164(Summer10)_PracticeFinal

164(Summer10)_PracticeFinal - program. Prove that x * is a...

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MATH 164 - Lecture 1 - Summer 2010 Practice Final - July 23, 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. (15 points) Let f ( x 1 ,x 2 ) = x 2 1 + 2 x 2 2 - 2 x 1 x 2 + 2 x 3 2 + x 4 2 Determine the minimizers/maximizers of f and indicate what kind of min- ima or maxima (local, global, strict, etc.) they are.
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3 2. (25 points) (a) Find the dual to the problem minimize z = c t x subject to Ax = b x 0 by converting the problem to canonical form, finding its dual and then simplifying the result. (b) Consider linear problem minimize z = x 1 - 2 x 2 - 38 x 3 subject to x 1 + 2 x 2 - 12 x 3 = 1 x 1 - x 2 + x 3 = 2 x 1 ,x 2 ,x 3 0 Write a dual and solve it graphically. Then use complimentary slack- ness to obtain solution to the primal.
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4 3. (15 points) Suppose x * is a local minimizer for a convex linear
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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 1-5 x 2 subject to x 1-4 x 2 ≤ -1-3 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 for-mulas for the simplex method. maximize z = 4 x 1-x 2 subject to x 1 + x 2 ≤ 6 x 1-x 2 ≤ 3 x 1 + 2 x 2 ≥ 2 x 1 ,x 2 ≥ 7 6. (15 points) Let f be a real-valued 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 x-y k for all x satisfying k x-y k < ² ....
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This note was uploaded on 01/11/2011 for the course MATH Math 164 taught by Professor Brown during the Summer '10 term at UCLA.

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164(Summer10)_PracticeFinal - program. Prove that x * is a...

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