A Primal Barrier Function Phase I Algorithm for Nonsymmetric Conic Optimization Problems

We call a positive semidefinite matrix whose elements are nonnegative a doubly nonnegative matrix, and the set of those matrices the doubly nonnegative cone (DNN cone). The DNN cone is not symmetric but can be represented as the projection of a symmetric cone embedded in a higher dimension. In \cite{aYOSHISE10}, the authors demonstrated the efficiency of the DNN relaxation using the symmetric cone representation of the DNN cone. They showed that the DNN relaxation gives significantly tight bounds for a class of quadratic assignment problems, but the computational time is not affordable as long as we employ the symmetric cone representation. They then suggested a primal barrier function approach for solving the DNN optimization problem directly, instead of using the symmetric cone representation. However, most of existing studies on the primal barrier function approach assume the availability of a feasible interior point. This fact means that those studies are not inextricably tied to the practical usage. Motivated by these observations, we propose a primal barrier function Phase I algorithm for solving conic optimization problems over the closed convex cone $K$ having the following properties: (a) $K$ is not necessarily symmetric, (b) a self-concordant function $f$ is defined over the interior of $K$, and (c) its dual cone $K^*$ is not explicit or is intractable, all of which are observed when $K$ is the DNN cone. We analyze the algorithm and provide a sufficient condition for finite termination.