Intrinsic volumes of symmetric cones and applications in convex programming

• 成果类型: Article
• 署名作者: Amelunxen, Dennis; Buergisser, Peter
• 署名单位: University of Manchester; Technical University of Berlin
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-013-0740-2
• 发表日期: 2015
• 页码: 105-130
• 关键词: interior-point methods; Barrier functions

摘要

We express the probability distribution of the solution of a random (standard Gaussian) instance of a convex cone program in terms of the intrinsic volumes and curvature measures of the reference cone. We then compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions in terms of integrals related to Mehta's integral. In particular, we obtain a closed formula for the probability that the solution of a random (standard Gaussian) semidefinite program has a certain rank.