Enclosing ellipsoids and elliptic cylinders of semialgebraic sets and their application to error bounds in polynomial optimization

• 成果类型: Article
• 署名作者: Kojima, Masakazu; Yamashita, Makoto
• 署名单位: Institute of Science Tokyo; Tokyo Institute of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-012-0515-1
• 发表日期: 2013
• 页码: 333-364
• 关键词: semidefinite programming relaxation; determinant maximization; exploiting sparsity; sdp-relaxations; squares; sums; algorithms; extension

摘要

This paper is concerned with a class of ellipsoidal sets (ellipsoids and elliptic cylinders) in which are determined by a freely chosen m x m positive semidefinite matrix. All ellipsoidal sets in this class are similar to each other through a parallel transformation and a scaling around their centers by a constant factor. Based on the basic idea of lifting, we first present a conceptual min-max problem to determine an ellipsoidal set with the smallest size in this class which encloses a given subset of . Then we derive a numerically tractable enclosing ellipsoidal set of a given semialgebraic subset of as a convex relaxation of the min-max problem in the lifting space. A main feature of the proposed method is that it is designed to incorporate into existing SDP relaxations with exploiting sparsity for various optimization problems to compute error bounds of their optimal solutions. We discuss how we adapt the method to a standard SDP relaxation for quadratic optimization problems and a sparse variant of Lasserre's hierarchy SDP relaxation for polynomial optimization problems. Some numerical results on the sensor network localization problem and polynomial optimization problems are also presented.