A Note on a Boundedness Property of Normal Barriers of Convex Cones

نویسنده

  • Bahman Kalantari
چکیده

Let K be a closed convex pointed cone in a finite dimensional 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 forK◦. We prove that for each d ∈ K◦, the operator D ≡ ∇2F (d) 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.

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تاریخ انتشار 2007