- 著者
- Matsukawa Yasuaki Yoshise Akiko
- 出版者
- Springer
- 雑誌
- Japan journal of industrial and applied mathematics (ISSN:09167005)
- 巻号頁・発行日
- vol.29, no.3, pp.499-517, 2012-10
- 被引用文献数
- 10 4
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 int 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.
- 著者
- MATSUKAWA Yasuaki YOSHISE Akiko
- 出版者
- University of Tsukuba. Graduate School of Systems and Information Engineering. Doctoral Program in Social Systems & Management
- 巻号頁・発行日
- 2011-11
We call a positive semidefinite matrix whose elements are nonnegative a doubly nonnegativematrix, and the set of those matrices the doubly nonnegative cone (DNN cone). The DNN coneis not symmetric but can be represented as the projection of a symmetric cone embedded in ahigher dimension. In [16], the authors demonstrated the efficiency of the DNN relaxation usingthe symmetric cone representation of the DNN cone. They showed that the DNN relaxation givessignificantly tight bounds for a class of quadratic assignment problems, but the computational timeis not affordable as long as we employ the symmetric cone representation. They then suggested aprimal barrier function approach for solving the DNN optimization problem directly, instead of usingthe symmetric cone representation. However, most of existing studies on the primal barrier functionapproach assume the availability of a feasible interior point. This fact means that those studies arenot inextricably tied to the practical usage. Motivated by these observations, we propose a primalbarrier function Phase I algorithm for solving conic optimization problems over the closed convexcone K having the following properties: (a) K is not necessarily symmetric, (b) a self-concordantfunction f is defined over intK, and (c) its dual cone K∗ is not explicit or is intractable, all of whichare observed when K is the DNN cone. We analyze the algorithm and provide a sufficient conditionfor finite termination.