We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

我们开发技术来构造一系列半定锥的稀疏多面体近似。受Tanaka和Yoshise提出的半定（SD）基的启发（Ann Oper Res 265：155–182，2018），我们提出了SD基的简单扩展，以保持组成它的矩阵的稀疏性。我们证明使用扩展的SD基的多面体逼近包含所有对角优势矩阵的集合，并且包含在所有缩放后的对角优势矩阵的集合中。我们还证明，可以使用无限数量的扩展SD基数来表示所有比例对角占优矩阵的集合。我们使用近似值作为切割平面方法中的初始近似值，以解决最大稳定集问题的半确定松弛。