Computing the Largest C-Eigenvalue of a Tensor Using Convex Relaxation

A piezoelectric-type tensor is of order three that is symmetric with respect to the last two indices. Its largest C-eigenvalue determines the highest piezoelectric coupling constant. However, computing the largest C-eigenvalue is NP-hard. This paper addresses this problem using convex relaxation. To this end, we first establish an equivalence property that helps to rewrite the problem as a matrix optimization over rank-1 together with the partially symmetric tensor constraints. Then, via Lagrangian dual, convex relaxations are derived, yielding the problems of maximization of a linear function over one or two matrix nuclear norm constraint(s), together with a partially symmetric tensor constraint. Such relaxations define new norms, which are smaller than the spectral norms of the corresponding unfolding matrices. Several insights are provided for the tightness issues of the convex relaxations, including a certification, a sufficient criterion, and an equivalence condition. The spectral property of the dual variable, in particular, determines the tightness. When the convex relaxations are not tight, an approximation algorithm is proposed to extract a feasible approximation solution, with a theoretical lower bound provided. We provide several types of tensors to justify the tightness of the convex relaxations. In case that the relaxations are not tight, their optimal values, serving as upper bounds, are still tighter than those in the literature.