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432pre2 - Solution one x = A T AA T-1 b Solution two Let x...

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Math 432/532 Spring 2012 Announcement of March 14 test Our second test will be on Wednesday, March 14. It will cover the material from our last three homework assignments. The main topics are these: The definitions of convex and concave functions and the basic facts about these func- tions. The AGM inequality and its use in solving certain types of optimization problems. Least squares solution of a system of equations: min { b - Ax | x R n } , where A has independent columns. Solution: x * = A b , where A = ( A T A ) - 1 A T . Nearest point in a subspace min { x - y | y M } , where M is a subspace of R m . Solution: y * = AA x , where M = R ( A ). (Any A whose columns form a basis of M may be used.) Minimum norm solution of a system min { x | Ax = b } , where A has linearly independent rows.
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Unformatted text preview: Solution one: x * = A T ( AA T )-1 b . Solution two: Let x be any point such that Ax = b , and let C be a matrix such that M = R ( C ), where M is the subspace M = { y | Ay = 0 } . Then the solution is x * = x-CC † x . Test questions could include • functions to classify as convex, strictly convex, concave, strictly concave, or none of these. • optimization problems to solve by the AGM inequality. • least squares optimization problems, including curve fitting, least squares solution of a linear system, nearest point and minimum norm problems. • true-false questions. • some statement to prove....
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