This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result in [Usevich et al. SIOPT 2020], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multi-block Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.

翻译：本文主要研究基于梯度的雅可比型算法，用于最大化两类带正交约束的齐次多项式，并建立其收敛性质。针对第一类受Stiefel流形约束的齐次多项式，我们将其重新表述为酉群上的优化问题，这使得应用基于梯度的雅可比型（Jacobi-G）算法成为可能。接着，若子问题总能表示为二次型，我们在三种条件中的任一种下建立Jacobi-G的全局收敛性。第一种条件的收敛结果是[Usevich等, SIOPT 2020]中结果的简单推广，而另两种条件是新的。该算法及收敛性质适用于著名的联合近似对称张量对角化问题。针对第二类受Stiefel流形乘积约束的齐次多项式，我们将其重新表述为酉群乘积上的优化问题，进而提出一种新的基于梯度的多块雅可比型（Jacobi-MG）算法进行求解。若子问题总能表示为二次型，我们在上述三种条件中的任一种下建立Jacobi-MG的全局收敛性。该算法及收敛性质适用于著名的联合近似张量对角化问题。作为Jacobi-G和Jacobi-MG的近端变体，我们还提出了Jacobi-GP和Jacobi-MGP算法，并在无需额外条件的情况下建立其全局收敛性。数值结果表明所提算法的有效性。