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

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Graphical illustration of full-<strong>Newton</strong>-<strong>step</strong> path-following methodz 0µecentral pathλ(x) ≤ τz k = x k s k(1 − θ)µez 1One iteration.12
Relevant part of the analysis of the algorithmAt the start of the first iteration we have x ∈ D and µ = µ 0 such that λ µ (x) ≤ τ. When the barrier parameteris updated to µ + = (1 − θ)µ, Lemma 2 givesλ µ +(x) ≤ λ µ(x) + θ √ ν1 − θ≤ τ + θ√ ν1 − θ . (4)Then after the <strong>Newton</strong> <strong>step</strong>, the new iterate is x + = x + ∆x and( ) 2λ µ +(x + λµ +(x)) ≤ κ. (5)1 − κλ µ +(x)The algorithm is well defined if we choose τ and θ such that λ µ +(x + ) ≤ τ. To get the lowest iteration bound,we need at the same <strong>time</strong> to maximize θ. From (5) we deduce that λ µ +(x + ) ≤ τ certainly holds ifλ µ +(x)1 − κλ µ +(x) ≤ √ τ√ κ,which is equivalent to λ µ +(x) ≤the following condition on θ:√ τκ √ τ+ √ κ . According to (4) this will hold if τ+θ√ ν1−θθ ≤ √ τ1 − κτ − √ κτ√ τ +√ νκ( 1 +√ κτ)≤√ τκ √ τ+ √ . This leads toκWe choose τ = 1 9κ . The upper bound <strong>for</strong> θ gets the value 59+36κ √ ν ≤ 12+8κ √ ν , and then λ µ+(x) ≤ 1 4κ .This justifies the choice of the value of τ and θ in the theorem. For the rest of the proof we refer to the relevantreferences.13
Page 1 and 2: FullNewt
Page 3 and 4: Self-concordant univariate function
Page 5 and 6: Newton ste
Page 7 and 8: Algorithm with full Newton<
Page 9 and 10: Newton ste
Page 11: Input:Algorithm with full N
Page 15 and 16: Linear optimization via self-dual e
Page 17 and 18: Kernel-function-based</stro
Page 19 and 20: Local self-concordancy of the barri
Page 21 and 22: Consequences of the assumptionFor x
Page 23 and 24: Proof of Theorem 3We apply the comp
Page 25 and 26: The case n = 1 (2)v(α) 2 = v 2 (1
Page 27 and 28: Computation of νTo compute the bar
Page 29 and 30: Proof of Theorem 4 (1)We first cons
Page 31 and 32: Proof of Theorem 4 (3)Now assuming
Page 33 and 34: Summary of resultsFrom now on we as
Page 35 and 36: Analysis of the algorithm (1)Note t
Page 37 and 38: The local values of κ and ν are g
Page 39 and 40: Analysis of the algorithm (4)Substi
Page 41: Some references• Y.Q. Bai, M. El

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