Target-following framework for symmetric cone programming

We extend the target map, together with the weighted barriers and the notions of weighted analytic centers, from linear programming to general convex conic programming. This extension is obtained from a novel geometrical perspective of the weighted barriers, that views a weighted barrier as a weighted sum of barriers for a strictly decreasing sequence of faces. Using the Euclidean Jordan-algebraic structure of symmetric cones, we give an algebraic characterization of a strictly decreasing sequence of its faces, and specialize this target map to produce a computationally-tractable target-following algorithm for symmetric cone programming. The analysis is made possible with the use of triangular automorphisms of the cone, a new tool in the study of symmetric cone programming. As an application of this algorithm, we demonstrate that starting from any given any pair of primal-dual strictly feasible solutions, the primal-dual central path of a symmetric cone program can be efficiently approximated. 2000 Mathematics Subject Classification. 90C25; 90C51; 52A41.