论文标题
通过数值范围来界定实际张量优化
Bounding Real Tensor Optimizations via the Numerical Range
论文作者
论文摘要
我们展示了如何使用矩阵的数值范围来限制某些优化问题的最佳值，而不是实际张量的产品向量。我们的界限比基于特征值的微不足道的界限强，并且可以比半决赛编程松弛所提供的界限要快得多。我们讨论了对其他硬线性代数问题的许多应用，例如表明矩阵的真实子空间不包含一个级别矩阵，并且表明作用于矩阵的线性图是正面的。
We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed significantly faster than bounds provided by semidefinite programming relaxations. We discuss numerous applications to other hard linear algebra problems, such as showing that a real subspace of matrices contains no rank-one matrix, and showing that a linear map acting on matrices is positive.
