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Unformatted text preview: 6.079/6.975, Fall 200910 S. Boyd & P. Parrilo Homework 9 1. Eﬃcient numerical method for a regularized leastsquares problem. We consider a reg ularized least squares problem with smoothing, k n − 1 n minimize ( a T i x − b i ) 2 + δ ( x i − x i +1 ) 2 + η x 2 i , i =1 i =1 i =1 where x ∈ R n is the variable, and δ,η > are parameters. (a) Express the optimality conditions for this problem as a set of linear equations involving x . (These are called the normal equations.) (b) Now assume that k n . Describe an eﬃcient method to solve the normal equations found in (1a). Give an approximate ﬂop count for a general method that does not exploit structure, and also for your eﬃcient method. (c) A numerical instance. In this part you will try out your eﬃcient method. We’ll choose k = 100 and n = 2000, and δ = η = 1. First, randomly generate A and b with these dimensions. Form the normal equations as in (1a), and solve them using a generic method. Next, write (short) code implementing your eﬃcient method, and run it on your problem instance. Verify that the solutions found by the two methods are nearly the same, and also that your eﬃcient method is much faster than the generic one. Note: You’ll need to know some things about Matlab to be sure you get the speedup from the eﬃcient method. Your method should involve solving linear equations with tridiagonal coeﬃcient matrix. In this case, both the factorization and the back sub stitution can be carried out very eﬃciently. The Matlab documentation says that banded matrices are recognized and exploited, when solving equations, but we found this wasn’t always the case. To be sure Matlab...
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 Fall '06
 boyde
 Least Squares, Potential Energy, Trigraph, Maximum likelihood, Linear least squares, Convex Optimization

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