A new characterization of symmetric $H^+$-tensors

In this work, we present a new characterization of symmetric $H^+$-tensors. It is known that a symmetric tensor is an $H^+$-tensor if and only if it is a generalized diagonally dominant tensor with nonnegative diagonal elements. By exploring the diagonal dominance property, we derive new necessary and sufficient conditions for a symmetric tensor to be an $H^+$-tensor. Based on these conditions, we propose a new method that allows to check if a tensor is a symmetric $H^+$-tensor in polynomial time. In particular, this allows to efficiently compute the minimum $H$-eigenvalue of tensors in the related and important class of $M$-tensors. Furthermore, we show how this result can be used to approximately solve polynomial optimization problems.