On the Central Paths in Symmetric Cone Programming

Abstract

This paper is devoted to the study of optimal solutions of symmetric coneprograms by means of the asymptotic behavior of central paths with respect to a broadclass of barrier functions. This class is, for instance, larger than that typically foundin the literature for semidefinite positive programming. In this general framework,we prove the existence and the convergence of primal, dual and primal–dual centralpaths. We are then able to establish concrete characterizations of the limit points ofthese central paths for specific subclasses. Indeed, for the class of barrier functionsdefined at the origin, we prove that the limit point of a primal central path minimizesthe corresponding barrier function over the solution set of the studied symmetric coneprogram. In addition, we show that the limit points of the primal and dual centralpaths lie in the relative interior of the primal and dual solution sets for the case of thelogarithm and modified logarithm barriers.