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

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χAnalysis of the algorithm (3)( 152κ 2 )≤ v i ≤ ̺( 152κ 2 ), i = 1, . . . , n.If we choose κ such that( ( )) 1K ̺52κ 2 ≤ κ (19)then the barrier function is locally κ-SC. The above inequality certainly has a solution, becauseif κ goes to infinity then the left-hand side approaches K(1), which is finite, whereas theright-hand side goes to ∞. Let ¯κ denote the smallest solution of (19).Finally, if we take ¯ν such that( ( ( ))) 1 2¯ν = 2n N χ52¯κ 2then the barrier function is a locally (¯κ,¯ν)-SC barrier function.38
Analysis of the algorithm (4)Substitution of the chosen values of κ and ν yields (also using that µ 0 = 1) the followingiteration bound <strong>for</strong> the algorithm:⌈2 ( 1 + 4¯κ √¯ν ) ln 2¯νǫ⌉=⎡⎢2(1 + 4¯κN( ( 1)) √)χ 2n52¯κ 2ln 2n( N ( χ ( 1ǫ))) 2⎤52¯κ 2 ⎥Note that apart from n the coefficients occurring in this expression depend only on the kernelfunction ψ, and not on n. Thus we may safely state that <strong>for</strong> every kernel function satisfyingour conditions the iteration bound is( √nlog)nO .ǫ.39
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 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: The local values of κ and ν are g
Page 41: Some references• Y.Q. Bai, M. El

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