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Unformatted text preview: mize (1/2)xT x + log m
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.
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
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.
- Fall '13
- The Aeneid