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

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Effect of a µ-update and the barrier parameter νLet λ = λ µ (x) and µ + = (1 − θ)µ. Our aim is to estimate λ µ +(x). We haveg µ +(x) = cµ + ∇φ(x) = c+ (1 − θ)µ + ∇φ(x)( )cµ + ∇φ(x) − θ∇φ(x)= 11−θHence, denoting H(x) shortly as H,λ µ +(x) = 11−θ ‖g µ(x) − θ∇φ(x)‖ H −1 ≤ 11−θ⎛⎜= 11−θ (g µ(x) − θ∇φ(x)) .⎝ ‖g µ(x)‖ H −1} {{ }λ µ (x)+θ ‖g(x)‖ H −1Definition 1 Let ν ≥ 0. The self-concordant barrier function φ is called a ν-barrier ifλ(x) 2 = ‖g(x)‖ 2 H −1 ≤ ν, ∀x ∈ D.⎞⎟⎠ .An immediate consequence of this definition isLemma 2 If φ is a self-concordant ν-barrier then λ µ +(x) ≤ λ µ(x)+θ √ ν1−θ.10
Input:Algorithm with full <strong>Newton</strong> <strong>step</strong>sAn accuracy parameter ǫ > 0;proximity parameter τ > 0;update parameter θ, 0 < θ < 1;x = x 0 ∈ D and µ = µ 0 > 0 such that λ µ (x) ≤ τ < 1 κ .(while µ ν + τ (τ+ √ ν)1−κτµ := (1 − θ)µ;x := x + ∆x;endwhile)≥ ǫ doTheorem 2 If τ =9κ 1 and θ = 5requires not more than⌈9+36κ √ ν2 ( 1 + 4κ √ ν ) ln 2µ0 νǫ, then the algorithm with full <strong>Newton</strong> <strong>step</strong>siterations. The output is a point x ∈ D such that c T x ≤ c T x ∗ + ǫ, where x ∗ denotes anoptimal solution of (P).⌉11
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: Newton ste
Page 13 and 14: Relevant part of the analysis of th
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

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

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