An extension of Chubanov's algorithm to symmetric cones

• 成果类型: Article
• 署名作者: Lourenco, Bruno F.; Kitahara, Tomonari; Muramatsu, Masakazu; Tsuchiya, Takashi
• 署名单位: Seikei University; Institute of Science Tokyo; Tokyo Institute of Technology; University of Electro-Communications - Japan; National Graduate Institute for Policy Studies
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1207-7
• 发表日期: 2019
• 页码: 117-149
• 关键词: cutting plane method; convex feasibility problem; CONVERGENCE

摘要

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone K. As in Chubanov's method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and K. Following an earlier work by Kitahara and Tsuchiya on second order cone feasibility problems, progress is measured through the volumes of those intersections: when they become sufficiently small, we know it is time to stop. We never have to explicitly compute the volumes, it is only necessary to keep track of the reductions between iterations. We show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem. Another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that K is decomposed as simple cones. In several key cases, including semidefinite programming and second order cone programming, these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by Pena and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.