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

Analysis of the algorithm (1)Note that ψ(t) is monotonically decreasing <strong>for</strong> t ≤ 1 and monotonically increasing <strong>for</strong> t ≥ 1.In the sequel we denote by ̺ : [0, ∞) → [1, ∞) the inverse function of ψ(t) <strong>for</strong> t ≥ 1 andby χ : [0, ∞) → (0,1] the inverse function of ψ(t) <strong>for</strong> 0 < t ≤ 1. So we haveand̺(s) = t ⇔ s = ψ(t), s ≥ 0, t ≥ 1. (17)χ(s) = t ⇔ s = ψ(t), s ≥ 0, 0 < t ≤ 1. (18)Note that χ(s) is monotonically decreasing and ̺(s) is monotonically increasing in s ≥ 0.Lemma 8 Let t > 0 and ψ(t) ≤ s. Then χ(s) ≤ t ≤ ̺(s) .Proof: This is almost obvious. Since ψ(t) is strictly convex and minimal at t = 1, withψ(1) = 0, ψ(t) ≤ s implies that t belongs to a closed interval whose extremal points areχ(s) and ̺(s).•35