Now suppose that x is strictly feasible but not

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Unformatted text preview: mize (1/2)xT x + log m T i=1 exp(ai x + bi ) . Give an efficient method for solving the Newton system, assuming the matrix A ∈ Rm×n (with rows aT ) is dense with m ≪ n. Give an approximate flop count of your method. i 8.8 We consider the equality constrained problem minimize tr(CX ) − log det X subject to diag(X ) = 1. The variable is the matrix X ∈ Sn . The domain of the objective function is Sn . The matrix ++ C ∈ Sn is a problem parameter. This problem is similar to the analytic centering problem discussed in lecture 11 (p.18–19) and pages 553-555 of the textbook. The differences are the extra linear term tr(CX ) in the objective, and the special form of the equality constraints. (Note that the equality constraints can be written as tr(Ai X ) = 1 with Ai = ei eT , a matrix of zeros except for the i, i i element, which is equal to one.) (a) Show that X is optimal if and only if X ≻ 0, X −1 − C is diagonal, diag(X ) = 1. (b) The Newton step ∆X at a feasible X is defined as the solution of the Newton equations X −1 ∆XX −1 + diag(w) = −C + X −1 , diag(∆X ) = 0, with variables ∆X ∈ Sn , w ∈ Rn . (Note the...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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