bv_cvxbook_extra_exercises

# Now suppose that x is strictly feasible but not

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online