Yuan’s lemma is a basic proposition on homogeneous quadratic function system. In this paper, we extend Yuan’s lemma to 4th-order tensor system. We first give two gen- eralized definitions of positive semidefinite of 4th-order tensor, and based on them, two extensions of Yuan’s lemma are proposed. We illustrate the difference between our ex- tensions and existing another extension of Yuan’s lemma. We also put forward several 4th-order tensor optimization problems and show extended Yuan’s lemma how to be applied.
School of Mathematical Sciences and LPMC, Nankai University, 1/2018