Positivity 1: 331–357, 1997.
© 1997 Kluwer Academic Publishers. Printed in the Netherlands.
Euclidean Jordan Algebras and Interior-point
Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556-5683 USA
(Received: 30 October 1996; Accepted: 9 October 1997)
Abstract. We provide an introduction to the theory of interior-point algorithms of optimization
based on the theory of Euclidean Jordan algebras. A short-step path-following algorithm for the
convex quadratic problem on the domain , obtained as the intersection of a symmetric cone with
an afﬁne subspace, is considered. Connections with the Linear monotone complementarity problem
are discussed. Complexity estimates in terms of the rank of the corresponding Jordan algebra are
obtained. Necessary results from the theory of Euclidean Jordan algebras are presented.
Mathematics Subject Classiﬁcations (1991): 17A10, 90C05, 90C25
Key words: Jordan algebras, Interior-point methods, Complexity estimates
The goal of this paper is to provide a reasonably self-contained introduction to so-
called interior-point algorithms based on the theory of Euclidean Jordan algebras.
The theory of Euclidean Jordan algebras in the form we use in the present paper
has been developed by Max Koecher  and since then played an important role in
various branches of analysis and differential geometry (see e.g. , ,, ,
). A connection with interior-point algorithms has been established in  and
is based on the fact that an arbitrary symmetric (i.e. self-dual, homogeneous ) cone
can be realized as a cone of squares in the attached Euclidean Jordan algebra. This
provides a one-to-one correspondence between isomorphism classes of Euclidean
Jordan algebras and isomorphism classes of symmetric cones (see e.g. ). This
connection enables us to describe the optimal barrier function (in the sense of the
theory of interior- point algorithms ) for an arbitrary symmetric cone in terms
of the attached Jordan algebra. Since the class of symmetric cones contains the
positive orthant in R
, the second order cone and the cone of nonnegative deﬁnite
symmetric matrices, our approach enables one to analyze major classes of opti-
mization problems (e.g. linear programming, semideﬁnite programming, problems
with convex quadratic constraints and various combinations of problems of these
types) with the help of a simple and unifying Jordan-algebraic technique. There are
clear indications that one can go much further in this direction.
The plan of the paper is as follows. In section 2 we consider a convex quadratic
programming problem on the domain obtained as the intersection of a symmetric
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