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

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• Self-concordant (barrier) functionsOutline– Definitions– <strong>Newton</strong> <strong>step</strong> and proximity measure– Algorithm with full <strong>Newton</strong> <strong>step</strong>s– Complexity analysis– Minimization of a linear function over a convex domain– Algorithm with full <strong>Newton</strong> <strong>step</strong>s– Complexity analysis• Kernel-function-<strong>based</strong> approach– Linear optimization via self-dual embedding– Central path of self-dual problem– Kernel-function-<strong>based</strong> barrier functions– Complexity results• Local self-concordancy of kernel-function-<strong>based</strong> barrier functions• Analysis of the full <strong>Newton</strong> <strong>step</strong> method• Concluding remarks• Some references2
Self-concordant univariate functionsWe start by considering a univariate function φ : D → R. The domain D of the function φmust be an open interval in R. One calls φ a κ-self-concordant (SC) function if there existsa nonnegative number κ such that∣∣∣φ ′′′ (x) ∣ ≤ 2κ ( φ ′′ (x) )32 , ∀x ∈ D. (1)Note that this definition assumes that φ ′′ (x) is nonnegative, whence φ is convex, and moreoverthat φ is three <strong>time</strong>s differentiable.Moreover, if φ ′′ (x) > 0 <strong>for</strong> all x ∈ D, then φ is SC if and only ifis bounded above (by 4κ 2 ).φ ′′′ (x) 2φ ′′ (x) 33
Page 1: FullNewt
Page 5 and 6: Newton ste
Page 7 and 8: Algorithm with full Newton<
Page 9 and 10: Newton ste
Page 11 and 12: Input:Algorithm with full N
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

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