In unconstrained optimization problems, gradient descent method is the most basic algorithm, and its performance is directly related to the step size. In this paper, we develop a family of gradient step sizes based on Barzilai-Borwein method, named regularized Barzilai-Borwein (RBB) step sizes. We indicate that the reciprocal of the RBB step size is the close solution to an \( \varvec{\ell }_{\varvec{2}}^{\varvec{2}} \) -regularized least squares problem. We propose an adaptive regularization parameter scheme based on the principle of the alternate Barzilai-Borwein (ABB) method and the local mean curvature of the objective function. We introduce a new alternate step size criterion into the ABB method, forming a three-term alternate step size, thereby establishing an enhanced RBB method for solving quadratic and general unconstrained optimization problems efficiently. We apply the proposed algorithms to solve typical quadratic and non-quadratic optimization problems, and further employ them to address spherical t-design, which is a nonlinear nonconvex optimization problem on an Oblique manifold.
