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

Summary of resultsFrom now on we assume that s = Mx + q. Our ingredients areTheorem 3 ν(x) = 2 ‖N(v)‖ 2 , where N(t) = √−ψ′ b (t) .ψb ′′(t)Theorem 4 κ(x) = ‖K(v)‖ ∞ , where K(t) =.1¯ρ(t) √ √ 2 3−¯ρ(t)−ψ ′′′b (t)(ψ′′b (t))3 2, withρ(t) = ψ′ b(t) ψ′′′b (t)ξ b (t)ψb ′′(t), ¯ρ(t) = min[2, ρ(t)] , ξ b(t) = ψ b ′′ (t) − ψ′ b (t) .tLemma 7 During the course of the algorithm we have λ(x) ≤ 1 4κ .This lemma impliesφ µ (x, s) = 2n∑i=1ψ (v i ) ≤ ω(−1 4 )κ 2 = 0.0376821κ 2 ≤ 126κ 2 33