Linear systems in Jordan algebras and primal-dual interior-point algorithms

We discuss a possibility of the extension of a primal-dual interior-point algorithm suggested recently by Alizadeh et al. (1994). We consider optimization problems defined on the intersection of a symmetric cone and an affine subspace. The question of solvability of a linear system arising in the implementation of the primal-dual algorithm is analyzed. A nondegeneracy theory for the considered class of problems is developed. The Jordan algebra technique suggested by Faybusovich (1995) plays major role in the present paper.