notes_on_optimization - 6.079/6.975, Fall 2009-10 S. Boyd...

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6.079/6.975, Fall 2009-10 S. Boyd & P. Parrilo (Part of) Homework 10: Standard form LP barrier method In the following three exercises, you will implement a barrier method for solving the standard form LP minimize c T x subject to Ax = b, x 0 , with variable x R n , where A R m × n , with m < n . Throughout this exercise we will assume that A is full rank, and the sublevel sets { x | Ax = b, x 0 , c T x γ } are all bounded. (If this is not the case, the centering problem is unbounded below.) 1. Centering step. Implement Newton’s method for solving the centering problem minimize c T x i n =1 log x i subject to Ax = b, with variable x , given a strictly feasible starting point x 0 . Your code should accept A , b , c , and x 0 , and return x , the primal optimal point, ν , a dual optimal point, and the number of Newton steps executed. Use the block elimination method to compute the Newton step. (You can also compute the Newton step via the KKT system, and compare the result to the Newton step computed via block elimination. The two steps should be close, but if any x i is very small, you might get a warning about the condition number of the KKT matrix.) Plot λ 2 / 2 versus iteration k , for various problem data and initial points, to verify that your implementation gives asymptotic quadratic convergence. As stopping criterion, you can use λ 2 / 2 10 6 . Experiment with varying the algorithm parameters α and β , observing the effect on the total number of Newton steps required, for a ±xed problem instance. Check that your computed
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notes_on_optimization - 6.079/6.975, Fall 2009-10 S. Boyd...

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