A practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor

In this paper, we consider a bi-quadratic homogeneous polynomial optimization problem over two unit spheres arising in nonlinear elastic material analysis and in entanglement studies in quantum physics. The problem is equivalent to computing the largest M-eigenvalue of a fourth-order tensor. To solve the problem, we propose a practical method whose validity is guaranteed theoretically. To make the sequence generated by the method converge to a good solution of the problem, we also develop an initialization scheme. The given numerical experiments show the effectiveness of the proposed method.