Complete Classification of Self-scaled Barrier Functions

Stefan H. Schmieta

Self-scaled barrier functions are an important tool in the study of conic linear programming problems over symmetric cones. They are also closely related to Euclidean Jordan algebras. In the first part of this article we show that every self-scaled barrier function defines a Euclidean Jordan algebra. In the second part we use this algebra to obtain a complete classification of self-scaled barrier functions. In particular we show that the barrier function with the smallest possible self-concordance parameter is unique up to an additive constant and can be written in terms of the determinant of the associated Jordan algebra.

TR-2000-01 (July 2000), CORC, Columbia University, New York, NY