Jordan algebras, Symmetric cones and Interior-point methods

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.

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