  Bi-quadratic programming over unit spheres is a fundamental problem in
quantum mechanics introduced by pioneer work of Einstein, Schr\"odinger, and
others. It has been shown to be NP-hard; so it must be solve by efficient
heuristic algorithms such as the block improvement method (BIM). This paper
focuses on the maximization of bi-quadratic forms, which leads to a rank-one
approximation problem that is equivalent to computing the M-spectral radius and
its corresponding eigenvectors. Specifically, we provide a tight upper bound of
the M-spectral radius for nonnegative fourth-order partially symmetric (PS)
tensors, which can be considered as an approximation of the M-spectral radius.
Furthermore, we showed that the proposed upper bound can be obtained more
efficiently, if the nonnegative fourth-order PS-tensors is a member of certain
monoid semigroups. Furthermore, as an extension of the proposed upper bound, we
derive the exact solutions of the M-spectral radius and its corresponding
M-eigenvectors for certain classes of fourth-order PS-tensors. Lastly, as an
application of the proposed bound, we obtain a practically testable sufficient
condition for nonsingular elasticity M-tensors with strong ellipticity
condition.
  We conduct several numerical experiments to demonstrate the utility of the
proposed results. The results show that: (a) our proposed method can attain a
tight upper bound of the M-spectral radius with little computational burden,
and (b) such tight and efficient upper bounds greatly enhance the convergence
speed of the BIM-algorithm, allowing it to be applicable for large-scale
problems in applications.
