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

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Central pathThe basic idea of IPMs is to replace the so-called complementarity condition xs = 0 <strong>for</strong>(SP), by the parameterized equation xs = µ1, with µ > 0. Thus we consider the systems = Mx + q, x ≥ 0, s ≥ 0,xs = µ1.Clearly, any solution (x, s) will satisfy x > 0 and s > 0. Note that x = s = 1 and µ = 1satisfy this system.Surprisingly enough, a solution exists <strong>for</strong> each µ > 0, and this solution is unique. It isdenoted as (x(µ), s(µ)) and we call x(µ) the µ-center of (SP); s(µ) is the correspondingslack vector. The set of µ-centers (with µ running through all positive real numbers) givesa homotopy path, which is called the central path of (SP). If µ → 0 then the limit ofthe central path exists and since the limit point satisfies the complementarity condition, thelimit yields an optimal solution <strong>for</strong> (SP). Moreover, this solution can be shown to be strictlycomplementary.We will start our method at x = s = 1 and µ = 1. The method uses nonnegative barrierfunctions φ µ (x, s), <strong>for</strong> each µ > 0, such that φ µ (x(µ), s(µ)) = 0. If s = Mx + q thenwe denote φ µ (x, s) as Φ µ (x).16
Kernel-function-<strong>based</strong> barrier functionsFirst we choose a kernel function ψ : (0, ∞) → [0, ∞). We require that ψ(t) is three<strong>time</strong>s differentiable and strictly convex, and moreover that ψ(t) is minimal at t = 1, whereasψ(1) = 0. Then we define√ n∑xsΦ µ (x) := φ µ (x, s) = 2 ψ (v i ) where v := , s = Mx + q.µi=1The barrier function Φ µ (x) <strong>based</strong> on the kernel function ψ(t) is defined on the interior ofthe domain of (SP).φ µ (x, s) is strictly convex and minimal when v = 1, and then x = x(µ) (and s = s(µ)).Provided that θ is small enough, after a full <strong>Newton</strong> <strong>step</strong> we get a good enough approximationof x = x(µ). Then we repeat the above process: reduce µ by the factor 1 − θ, do a full<strong>Newton</strong> <strong>step</strong>, etc., until µ is close enough to zero. At the end this yields an ǫ-solution of theproblem (SP).In earlier papers we used a search direction determined by the systemM∆x = ∆s,s∆x + x∆s = −µ v ψ ′ (v).17
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: Linear optimization via self-dual e
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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