We develop a practical approach to semidefinite programming (SDP) that includes the von Neumann entropy, or an appropriate variant, as a regularization term. In particular we solve the dual of the regularized program, demonstrating how a carefully chosen randomized trace estimator can be used to estimate dual gradients effectively. We also introduce specialized optimization approaches for common SDP, specifically SDP with diagonal constraint and the problem of the determining the spectral projector onto the span of extremal eigenvectors. We validate our approach on such problems with applications to combinatorial optimization and spectral embedding.

翻译：我们提出了一种半定规划（SDP）的实用方法，该方法将冯·诺依曼熵或其适当变体作为正则化项。具体而言，我们求解正则化问题的对偶形式，并展示了如何通过精心选择的随机化迹估计器有效估计对偶梯度。我们还针对常见的SDP问题引入了专门的优化方法，特别是带有对角约束的半定规划以及确定延伸到极端特征向量张成的子空间的谱投影算子问题。我们通过组合优化和谱嵌入等应用中的问题验证了该方法的有效性。