Abstract
Let K be a closed convex pointed cone in a finite imensional Hilbert space E. Assume K has a nonempty interior K◦. Let r = inf{(x, y) : x, y ∈ K, ||x|| = ||y|| = 1}, where (·, ·) denotes the inner product and || · || the corresponding induced norm. The quantity r gives a measure of obtuseness of K and it necessarily lies in (−1, 1]. Let F be any θ-normal barrier for K◦. We prove that for each d ∈ K◦, the operator D ≡ ∇2F(d)−1/2 satisfies the inequality ||D|| ≤ ρ||d||, where ρ ≤ 1 + √2/1+r . In particular, if K is acute, i.e., (x, y) ≥ 0, ∀x, y ∈ K, then ρ ≤ 1 + √2. We prove the above using a property from the theory of self-concordance of Nesterov and Nemirovskii [3], and a result in Kalantari [2]. The existence of this bound implies that the polynomial-time potential-reduction and path-following algorithms which were described in [2], for self-concordant homogeneous programming and for a generalizations of the diagonal matrix scaling problem, are applicable over any arbitrary cone K. The effect of ρ in the total complexity of these algorithms is the addition of a number of iterations, only O(ln ln ρ). Since any symmetric cone, and more generally any self-dual cone, is necessarily acute, the bound proved here implies that the corresponding iteration complexity of the above mentioned algorithms is independent of ρ.