Perfect Copositive Matrices

在这个项目中，我们发展了一个完美的共正矩阵的理论。例如，这些矩阵可以用于以系统的方式离散共正矩阵的凸锥。这可能会导致应用在共正优化，如算法计算的理性证书的完全正矩阵。我们的项目的一个中心组成部分-和重要的算法应用的新理论-是调查的邻域图的完美正矩阵。该图表现为多面体表面的边图，其刻面由共正矩阵的最小向量确定。因此，对于邻域图的算法处理，计算共正矩阵的最小向量是必要的。为了完成这项重要任务，开发和测试了新的方法。这些新的算法，确定最小向量本身有潜在的有趣的未来应用程序。作为该项目的一部分，我们将开发各种算法的实现，并与已知的方法进行广泛的比较。最后，我们的目标是发布软件，易于使用，连同相关的数据集的测试实例，为未来的研究。

In this project we develop a theory of perfect copositive matrices. These matrices can be used, for example, to discretize the convex cone of copositive matrices in a systematic way. This results in possible applications in copositive optimization, such as the algorithmic calculation of rational certificates for completely positive matrices. A central component of our project — and important for algorithmic applications of the new theory — is the investigation of the neighborhood graph of perfect copositive matrices. This graph appears as an edge graph of a polyhedral surface, whose facets are determined by minimal vectors of copositive matrices. For the algorithmic treatment of the neighborhood graph it is therefore essential to compute minimal vectors of copositive matrices. For this important task new approaches are developed and tested. These new algorithms for determining minimal vectors themselves have the potential for interesting future applications.As part of the project, we will develop various implementations of algorithms and compare them extensively with known approaches. Towards the end we aim to publish software that is easy to use, together with an associated data set of test instances for future research.