Completely positive factorization by a Riemannian smoothing method
Completely positive factorization by a Riemannian smoothing method
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DOI：
10.1007/s10589-022-00417-4
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发表时间：
2022-10-09
影响因子：
2.2
通讯作者：
Yoshise，Akiko
中科院分区：
文献类型：
--
作者：
Lai，Zhijian;Yoshise，Akiko
余正优化是凸锥规划的一种特殊情况，它包括在线性约束下优化所有完全正矩阵的锥上的线性函数。余正优化为NP难二次问题或组合问题提供了强有力的松弛，但关于余正或完全正矩阵仍有许多未解决的问题。在本文中，我们专注于这样一个问题：找到一个完全正（CP）分解为一个给定的完全正矩阵。我们将其视为非光滑黎曼优化问题，即，黎曼流形上非光滑函数的极小化问题。为了解决这个问题，我们提出了一个一般的光滑框架，解决非光滑黎曼优化问题，并显示收敛到一个稳定点的原始问题。一个优点是，我们可以通过直接使用现有的标准光滑黎曼解算器（如Manopt）来快速实现它。数值实验表明，我们的方法的效率，特别是对大规模的CP分解。
Copositive optimization is a special case of convex conic programming, and it consists of optimizing a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful relaxations of NP-hard quadratic problems or combinatorial problems, but there are still many open problems regarding copositive or completely positive matrices. In this paper, we focus on one such problem; finding a completely positive (CP) factorization for a given completely positive matrix. We treat it as a nonsmooth Riemannian optimization problem, i.e., a minimization problem of a nonsmooth function over a Riemannian manifold. To solve this problem, we present a general smoothing framework for solving nonsmooth Riemannian optimization problems and show convergence to a stationary point of the original problem. An advantage is that we can implement it quickly with minimal effort by directly using the existing standard smooth Riemannian solvers, such as Manopt. Numerical experiments show the efficiency of our method especially for large-scale CP factorizations.