论文标题
一种分析中心切割平面方法，以确定矩阵的完全阳性
An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix
论文作者
论文摘要
我们提出了一种分析中心切割平面方法，以确定矩阵是否完全正面，并返回将其与完全正锥体分开的切割。这是Berman，Dür和Shaked-Monderer的开放（计算）问题[线性代数电子杂志，2015年]。我们的方法优化了球与共同锥体的交集，其中会员资格是通过求解Xia，Vera和Zuluaga建议的混合智能线性程序来确定的[Indors on Computing on Computing，2018年]。因此，只要知道一个包含最佳溶液的球的半径，我们的算法就可以用来解决任何共同优化问题。数值实验表明，我们实现量表的Oracle调用数量（矩阵共阳性检查）与矩阵大小相当，大致像$ O（D^2）$（对于$ d \ times d $ d $矩阵）。该方法在Julia中实现，可在https://github.com/rileybadenbroek/copositiveanalyticcenter.jl上找到。
We propose an analytic center cutting plane method to determine if a matrix is completely positive, and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman, Dür, and Shaked-Monderer [Electronic Journal of Linear Algebra, 2015]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like $O(d^2)$ for $d\times d$ matrices. The method is implemented in Julia, and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl.
