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Unformatted text preview: the original LP is strictly feasible if and
only if t⋆ < 1, where t⋆ is the optimal value of the phase I problem.
We can initialize x and t for the phase I problem with any x0 satisfying Ax0 = b, and
t0 = 2 − mini x0 . (Here we can assume that min x0 ≤ 0; otherwise x0 is already a strictly
feasible point, and we are done.) You can use a change of variable z = x +(t − 1)1 to transform
the phase I problem into the form in part (b).
Check your LP solver against cvx on several numerical examples, including both feasible and
9.6 Barrier method for LP. Consider a standard form LP and its dual
minimize cT x
subject to Ax = b
x0 maximize bT y
subject to AT y c, with A ∈ Rm×n and rank(A) = m. In the barrier method the (feasible) Newton method is applied
to the equality constrained problem
minimize tcT x + φ(x)
subject to Ax = b,
where t > 0 and φ(x) = − n
i=1 log xi . The Newton equation at a strictly feasible x is given by
ˆ ∇2 φ(ˆ) AT
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- Fall '13
- The Aeneid