Application of jordan algebras to the design and analysis of interior-point algorithms for linear, quadratically constrained quadratic, and semidefinite programming

We present two methods of generalizing algorithms for semi-definite programming to linear optimization problems over all symmetric cones. Both use the Euclidean Jordan algebra associated with a symmetric cone. In the first approach, the algorithm and its analysis are extended by replacing all statements about symmetric matrices by their counterparts for elements of a Jordan algebra. The second approach uses the notion of a Euclidean Associative Jordan (EAJ) system. Here an analogue of general matrices is provided to enable a “word-by-word” translation of algorithms and proofs. We show that such an EAJ system exists for almost all symmetric cones, with the exception of those that include a special 27-dimensional cone. We first develop the necessary technical results for Euclidean Jordan algebras and EAJ systems and then investigate optimality conditions, (strict) complementarity, the central path, and primal/dual degeneracy. The notion of similarly scaled search directions is extended from semi-definite programming to all symmetric cones, and scaling invariant neighborhoods of the central path are defined. Then, as an example of the first approach, we apply this technique to Tsuchiya's recent analysis of short-, semi-long, and long-step path-following methods for quadratic cone programming which is based on Monteiro and Zhang's work for semi-definite programming. As an example of the second approach, we apply it to Monteiro's proof of polynomial iteration complexity of two path-following methods for semi-definite programming using the whole Monteiro-Zhang family of search directions, obtaining in particular polynomial iteration complexity of these algorithms for optimization over quadratic cones.