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Unformatted text preview: OR 3300 Optimization I Summer 2008 HW 2 Geometry of LP; Sensitivity Analysis Due date: Friday, June 6, by 3:20pm in the drop box on 2nd floor Rhodes. References: Lecture notes; Bradley, Hax and Magnanti chapter 3. 1. Show that the intersection of two convex sets is convex. Is it true that the union of two convex sets is always convex? Either prove this or give a counterexample. 2. Can (1 , 1 , , 3 2 , 0) be an extreme point of the polyhedron specified by the following linear system? Explain your answer. 3 x 1 + 4 x 2 + x 3 + 4 x 4 = 13 x 2 + 2 x 4 + x 5 = 2 x 1 , x 2 , x 3 , x 4 , x 5 ≥ 3. Consider the standard linear programming problem: max cx s.t. Ax ≤ b ( LP ) x ≥ Show that if ¯ x and x * are optimal solutions for ( LP ) then any vector of the form αx * + (1 α )¯ x, ≤ α ≤ 1 , is also an optimal solution for ( LP ) . 4. Recall that a single linear inequality defines a halfspace , e.g., H = { x ∈ < n : α 1 x 1 + ··· + α n x n ≤ β } , where α 1 , . . . , α, ....
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 Spring '08
 TODD
 Optimization, basic feasible solution, extreme point

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