Tighter Bound Estimation for Efficient Biquadratic Optimization Over Unit Spheres

TL;DR

This paper introduces a tighter and more efficiently computable upper bound for the M-spectral radius of nonnegative fourth-order PS-tensors, improving the convergence of heuristic algorithms for bi-quadratic optimization over unit spheres.

Contribution

It provides a novel tight upper bound for the M-spectral radius, extends it to exact solutions for certain tensor classes, and demonstrates improved algorithmic convergence.

Findings

The proposed bound is tight and computationally efficient.

The bound enhances the convergence speed of the BIM algorithm.

Numerical experiments confirm the practical utility of the method.

Abstract

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.