We can initialize x and t for the phase i problem

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Unformatted text preview: sume that the matrix P is large, positive definite, and sparse, and that P −1 is dense. ‘Efficient’ means that the complexity of the method should be much less than O(n3 ). 9.4 Dual feasible point from incomplete centering. Consider the SDP minimize 1T x subject to W + diag(x) 0, with variable x ∈ Rn , and its dual maximize − tr W Z subject to Zii = 1, i = 1, . . . , n Z 0, 80 with variable X ∈ Sn . (These problems arise in a relaxation of the two-way partitioning problem, described on page 219; see also exercises 5.39 and 11.23.) Standard results for the barrier method tell us that when x is on the central path, i.e., minimizes the function φ(x) = t1T x + log det(W + diag(x))−1 for some parameter t > 0, the matrix 1 Z = (W + diag(x))−1 t is dual feasible, with objective value − tr W Z = 1T x − n/t. Now suppose that x is strictly feasible, but not necessarily on the central path. (For example, x might be the result of using Newton’s method to minimize φ, but with early termination....
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