bv_cvxbook_extra_exercises

Consider a matrix x x t rnn partitioned as x a bt b c

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Unformatted text preview: = b, x 0, with variable x ∈ Rn , where A ∈ Rm×n , with m < n. Throughout these exercises we will assume that A is full rank, and the sublevel sets {x | Ax = b, x 0, cT x ≤ γ } are all bounded. (If this is not the case, the centering problem is unbounded below.) (a) Centering step. Implement Newton’s method for solving the centering problem minimize cT x − subject to Ax = b, n i=1 log xi with variable x, given a strictly feasible starting point x0 . Your code should accept A, b, c, and x0 , and return x⋆ , the primal optimal point, ν ⋆ , a dual optimal point, and the number of Newton steps executed. 81 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 xi is very small, you might get a warning about the condition number of the KKT matrix.) Plot λ2 /2 versus iteration k , for various problem data and initial poin...
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