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On the irreducibility, Lyapunov rank, and automorphisms of speical Bishop-Phelps cones

M. Seetharama Gowda(gowda***at***math.umbc.edu)
David Trott(dtrott1***at***umbc.edu)

Abstract: Motivated by optimization considerations, we consider special Bishop-Phelps cones in R^n which are of the form {(t,x): t \geq ||x||} for some norm on R^(n-1). We show that for n bigger than 2, such cones are always irreducible. De fining the Lyapunov rank of a proper cone K as the dimension of the Lie algebra of the automorphism group of K, we show that the Lyapunov rank of any special Bishop-Phelps polyhedral cone is one. Extending an earlier known result for the l_1 cone (which is a special Bishop-Phelps cone with 1-norm), we show that any l_p cone, for p different from 2, has Lyapunov rank one. We also study automorphisms of special Bishop-Phelps cones, in particular giving a complete description of the automorphisms of the l_1 cone.

Keywords: Complementarity set, Lyapunov rank, Bishop-Phelps cone, Irreducible cone

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Complementarity and Variational Inequalities

Citation: Research Report TRGOW13-01, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA. August 2013

Download: [PDF]

Entry Submitted: 10/16/2013
Entry Accepted: 10/16/2013
Entry Last Modified: 10/16/2013

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