Appl Math Optim 53:67–77 (2006)
2005 Springer Science+Business Media, Inc.
Jordan-Algebraic Aspects of Nonconvex Optimization
over Symmetric Cones
Leonid Faybusovich and Ye Lu
Department of Mathematics, University of Notre Dame,
255 Hurley Hall, Notre Dame, IN 46556, USA
Communicated by M. Kojima
Abstract. We illustrate the usefulness of Jordan-algebraic techniques for non-
convex optimization by considering a potential-reduction algorithm for a non-
convex quadratic function over the domain obtained as the intersection of a sym-
metric cone with an afﬁne subspace.
Key Words. Jordan algebras, Symmetric cone, Nonconvex optimization.
AMS Classiﬁcation. 17C20, 90C26, 90C51.
Jordan-algebraic techniques proved to be very useful for the analysis of convex opti-
mization problems over symmetric cones. See, e.g., –,  and . Since the
class of symmetric cones contains the positive orthant in R
, the second-order cone and
cone of positive-deﬁnite symmetric matrices, this technique is applicable to a broad class
of optimization problems. However, it is also quite useful for nonconvex optimization
problems over symmetric cones. To illustrate this point we have chosen a potential-
reduction algorithm for the minimization of a nonconvex quadratic function developed
and analyzed by Ye  for the case of polyhedral constraints. Many other algorithms
(e.g. path-following algorithms based on trust-region ideas) can be analyzed using the
same technique. An excellent introduction to the theory of Euclidean Jordan algebras
This paper is partially based on the work supported by NSF Grants DMS0402740 and DMS0102628.