Bounding real tensor optimizations via the numerical range

A new method of using the numerical range of a matrix to bound the optimal value of certain optimization problems over real tensor product vectors is presented. This bound is stronger than the trivial bounds based on eigenvalues and can be computed significantly faster than bounds provided by semidefinite programming relaxations. Numerous applications to other hard linear algebra problems are discussed, 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.