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Jordan algebras, Symmetric cones and Interior-point methods

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Leonid Faybusovich

In two recent remarkable papers Y.Nesterov and M.Todd developed a
comprehensive theory of long-step interior-point algorithms for
optimization problems involving symmetric (i.e. self-dual,
homogeneous) cones. As is well understood by now many interior-point
linear programming algorithms can be carried over to the semidefinite
programming case (along with complexity estimates, proofs etc )
without significant changes. In the present paper we explain that
Euclidean Jordan algebras play essentially the same role for symmetric
cones as the (Jordan) algebra of real symmetric matrices plays for the
cone of positive definite symmetric matrices. In particular, we s how
how results of Nesterov and Todd can be obtained by simple
computations in Jordan alge bras. We describe the optimal barrier
function of arbitrary symmetric cone in terms of the attached Jordan
algebra (this includes for the first time quaternionic and octonion co
nes). As a by-product we prove Guler's conjecture:optimal self-scaling
barrier for an irre ducible symmetric cone is proportional to the
characteristic function.
Necessary results from the theory of Euclidean Jordan algebras are
briefly described.

Research Report,October, 1995,
University of Notre Dame, Notre Dame, USA.

Contact leonid.faybusovich.1@nd.edu for copy of paper.