Full-Newton step polynomial-time methods for LO based on locally ...

Algorithm with full <strong>Newton</strong> <strong>step</strong>sAssuming that we know a point x ∈ D with λ(x) ≤3κ 1 we can easily obtain a point x ∈ Dsuch that λ(x) ≤ ǫ, <strong>for</strong> prescribed ǫ > 0, with the following algorithm.Input:An accuracy parameter ǫ ∈ (0,1);x ∈ D such that λ(x) ≤ 1 3κ .while λ(x) ≤ ǫ dox := x + ∆xendwhileTheorem 1 Let x ∈ D and λ(x) ≤3κ 1 . Then the algorithm with full <strong>Newton</strong> <strong>step</strong>s requiresat most⎡⎢ log 2⎛⎝ log ǫlog 3 4⎞⎤⌈⎠≤ ⎥(log 2 3.5log 1 ǫiterations. The output is a point x ∈ D such that λ(x) ≤ ǫ.)⌉7