Vol. 12, No. 1, 2019

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On the complexity of detecting positive eigenvectors of nonlinear cone maps

Bas Lemmens and Lewis White

Vol. 12 (2019), No. 1, 141–150

In recent work with Lins and Nussbaum, the first author gave an algorithm that can detect the existence of a positive eigenvector for order-preserving homogeneous maps on the standard positive cone. The main goal of this paper is to determine the minimum number of iterations this algorithm requires. It is known that this number is equal to the illumination number of the unit ball Bv of the variation norm, xv := maxixi minixi on V 0 := {x n: xn = 0}. In this paper we show that the illumination number of Bv is equal to ( n n2), and hence provide a sharp lower bound for the running time of the algorithm.

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nonlinear maps on cones, positive eigenvectors, illumination problem, Hilbert's metric
Mathematical Subject Classification 2010
Primary: 47H07, 47H09
Secondary: 37C25
Received: 29 August 2017
Revised: 21 November 2017
Accepted: 14 December 2017
Published: 31 May 2018

Communicated by Kenneth S. Berenhaut
Bas Lemmens
School of Mathematics, Statistics & Actuarial Science
University of Kent
United Kingdom
Lewis White
School of Mathematics, Statistics & Actuarial Science
University of Kent
United Kingdom