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Bounding Real Tensor Optimizations via the Numerical Range

Nathaniel JohnstonLogan Pipes
Dec 2022
摘要
We show how the numerical range of a matrix can be used to bound the optimalvalue of certain optimization problems over real tensor product vectors. Ourbound is stronger than the trivial bounds based on eigenvalues, and can becomputed significantly faster than bounds provided by semidefinite programmingrelaxations. We discuss numerous applications to other hard linear algebraproblems, such as showing that a real subspace of matrices contains no rank-onematrix, and showing that a linear map acting on matrices is positive.
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