Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones

Abstract

We give a recursive formula for optimal dual barrier functions on homogeneous cones. This is done in a way similar to the primal construction of Guler and Tuncel (Math. Program. 81(1):55---76, 1998) by means of the dual Siegel cone construction of Rothaus (Bull. Am. Math. Soc. 64:85---86, 1958). We use invariance of the primal barrier function with respect to a transitive subgroup of automorphisms and the properties of the duality mapping, which is a bijection between the primal and the dual cones. We give simple direct proofs of self-concordance of the primal optimal barrier and provide an alternative expression for the dual universal barrier function.