An adaptive descent BFGS method with fast Fourier transform for generalized Hankel tensor equations

For solving generalized Hankel tensor equations, we propose an adaptive descent BFGS method with Fast Fourier Transform. First, by fully exploiting the structural characteristics of Hankel tensors, an efficient Hankel tensor-vector product computation method with Fast Fourier Transform is designed, which effectively reduces the computational complexity. Second, by using a finite difference to approximate gradient information, the search direction is determined without explicitly calculating the gradient. Simultaneously, an adaptive factor is introduced to modify the BFGS update formula, improving the condition number of the approximate Hessian matrix and thereby enhancing the numerical stability. Global convergence of the method is established under standard assumptions. Numerical experiments demonstrate that the Fast Fourier Transform significantly reduces CPU time compared to Tensor Toolbox, and the adaptive factor effectively improves the computational efficiency and robustness of the method.