A primal barrier function Phase I algorithm for nonsymmetric conic optimization problems

作者：Matsukawa Yasuaki; Yoshise Akiko^*

来源：Japan Journal of Industrial and Applied Mathematics, 2012, 29(3): 499-517.

DOI：10.1007/s13160-012-0081-1

• Doubly nonnegative cone
• Nonsymmetric conic optimization
• Primal barrier function
• Self-concordant function
• Phase I algorithm

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 Yoshise and Matsukawa (Proceedings of 2010 IEEE Multi-conference on Systems and Control, 2010), 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 have assumed 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) its interior int K is not necessarily symmetric, (b) a self-concordant function f is defined over

• 出版日期2012