A Feasible Conjugate Gradient Method for Calculating $\mathcal{B}$-Eigenpairs of Symmetric Tensors

Abstract

In this paper, we propose a feasible conjugate gradient (FCG) method for calculating $\mathcal{B}$-eigenpairs of a symmetric tensor $\mathcal{A}$. The method is an extension of the well-known conjugate gradient method for unconstrained optimization problems to some curve constrained optimization problems. The proposed FCG method can find a $\mathcal{B}$-eigenpair of a symmetric tensor $\mathcal{A}$ without the requirement that the orders of $\mathcal{A}$ and $\mathcal{B}$ are equal. We pay particular attention to the Polak-Ribíre-Polyak (PRP) type conjugate gradient method. We show that the FCG method with some Armijo-type line search is globally convergent. Our numerical experiments indicate the promising performance of the proposed method.