# Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones

@article{Schmieta2001AssociativeAJ, title={Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones}, author={Stefan Schmieta and Farid Alizadeh}, journal={Math. Oper. Res.}, year={2001}, volume={26}, pages={543-564} }

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We prove the case of the Lorentz cone by using the embedding of its associated Jordan algebra in the…

## 126 Citations

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A generalization of interior-point methods for linear optimization based on kernel functions to symmetric optimization covering the three standard cases of conic optimization is presented.

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Let G be a primitive strongly regular graph G such that the regularity is less than half of the order of G and A its matrix of adjacency, and let 𝒜 be the real Euclidean Jordan algebra of real…

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The primal-dual Dikin-type affine scaling method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras and is shown to be viable and robust when compared to SeDuMi, MOSEK and SDPT3.

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The Q Method for Symmetric Cone Programming

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It is proved that in the presence of certain non-degeneracies the Jacobian of the Newton system is nonsingular at the optimum and can be used to “warm-start” a slightly perturbed symmetric cone program.

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