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

The local values of κ and ν are given byκ(x) = maxiWe need to find values of κ and ν such thatAnalysis of the algorithm (2)K(v i ), ν(x) = 2n∑i=1N (v i ) 2 .κ(x) ≤ κ,ν(x) ≤ ν<strong>for</strong> each v that occurs during the course of the algorithm. This certainly holds ifφ µ (x, s) = 2n∑i=1The left-hand side of this implication impliesψ (v i ) ≤ 126κ 2 ⇒ max K(v i ) ≤ κ, 2iψ (v i ) ≤ 152κ2, i = 1, . . . , n.According to Lemma 8 this implies( ) ( )1 1χ52κ 2 ≤ v i ≤ ̺52κ 2 , i = 1, . . . , n.n∑i=1N (v i ) 2 ≤ ν.37