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 Güler and Tunçel (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.
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Communicated by F.A. Potra.
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Shevchenko, O. Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones. J Optim Theory Appl 140, 339–354 (2009). https://doi.org/10.1007/s10957-008-9451-x
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DOI: https://doi.org/10.1007/s10957-008-9451-x