Associative Algebras, Symmetric Cones and Polynomial Time Interior Point Algorithms

We present a generalization of Monteiro's polynomiality proof of a large class of primal-dual interior point algorithms for semideenite programs. We show that this proof, essentially verbatim, applies to all optimization problems over almost all symmetric cones, that is those cones that can be derived from classes of normed associative algebras and certain Jordan algebras obtained from them. As a consequence, we show that Monteiro's proof can be extended to convex quadrat-ically constrained quadratic optimization problems. We also extend the notion of Zhang family of algorithms, and show that it can be applied to all symmetric cones, in particular the quadratic cone.