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Unformatted text preview: two meanings of the diag function: diag(w) is
the diagonal matrix with the vector w on its diagonal; diag(∆X ) is the vector of the diagonal
elements of ∆X .) Eliminating X from the ﬁrst equation gives an equation
diag(X diag(w)X ) = 1 − diag(XCX ).
This is a set of n linear equations in n variables, so it can be written as Hw = g . Give a
simple expression for the coeﬃcients of the matrix H . 77 (c) Implement the feasible Newton method in Matlab. You can use X = I as starting point. The
code should terminate when λ(X )2 /2 ≤ 10−6 , where λ(X ) is the Newton decrement.
You can use the Cholesky factorization to evaluate the cost function: if X = LLT where L is
triangular with positive diagonal then log det X = 2 i log Lii .
To ensure that the iterates remain feasible, the line search has to consist of two phases. Starting
at t = 1, you ﬁrst need to backtrack until X + t∆X ≻ 0. Then you continue the backtracking
until the condition of suﬃcient decrease
f0 (X + t∆X ) ≤ f0 (X ) +...
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 Fall '13
 F.Borrelli
 The Aeneid

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