t3key - MAT419 Linear Optimization Apr 10 2008 TEST 3 NAME...

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MAT419 Linear Optimization Apr. 10, 2008 TEST 3 NAME KEY 1. Let γ 1 = (2 , 9 , - 5) T , γ 2 = (6 , 4 , 1) T , γ 3 = (0 , - 4 , 4) T , γ 4 = (3 , - 7 , - 4) T , and γ 5 = (4 , 6 , - 9) T . Define P = vhull( { γ 1 , γ 2 , γ 3 , γ 4 , γ 5 } ). Write the inequality that corresponds to one of the facets of P that contains γ 2 and γ 4 but not γ 5 . Describe your method of solution. First we find the equations of the planes defined by X = { γ 2 , γ 4 , γ 1 } and Y = { γ 2 , γ 4 , γ 3 } . WebSim computes these as 91 x 1 + 2 x 2 - 59 x 3 = 495 and 73 x 1 - 39 x 2 + 42 x 3 = 324 , respectively. Next we test each plane to see if the remaining points defining P are on the same side of the plane. For X we test γ 3 and γ 5 , while for Y we test γ 1 and γ 5 . In the former case, γ 3 yields 91(0) + 2( - 4) - 59(4) < 495 , while γ 5 yields 91(4) + 2(6) - 59( - 9) > 495 , which means they are on opposite sides of the plane. In the latter case, γ 1 yields 73(2) - 39(9) + 42( - 25) < 324 , while γ 5 yields 73(4) - 39(6) + 42( - 9) < 324 , which means they are on the same side of the plane. Hence the appropriate constraint is 73 x 1 - 39 x 2 + 42 x 3 324 .
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2. Let H be the polyhedron in R 2 defined by the following system of constraints.
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