We develop a novel stepsize based on \BB method for solving some challenging optimization problems efficiently, named regularized \BB (RBB) stepsize. We indicate that RBB stepsize is the close solution to a $\ell_{2}^{2}$-regularized least squares problem. When the regularized item vanishes, the RBB stepsize reduces to the original \BB stepsize. RBB stepsize includes a class of valid stepsizes, such as another version of \BB stepsize. The global convergence of the corresponding RBB algorithm is proved in solving convex quadratic optimization problems. One scheme for adaptively generating regularization parameters was proposed, named adaptive two-step parameter. An enhanced RBB stepsize is used for solving quadratic and general optimization problems more efficiently. RBB stepsize could overcome the instability of BB stepsize in many ill-conditioned optimization problems. Moreover, RBB stepsize is more robust than BB stepsize in numerical experiments. Numerical examples show the advantage of using the proposed stepsize to solve some challenging optimization problems vividly.

翻译：我们基于BB方法提出了一种新颖的步长，称为正则化BB（RBB）步长，用以高效求解若干具有挑战性的优化问题。研究表明，RBB步长是$\ell_{2}^{2}$正则化最小二乘问题的近似解。当正则化项消失时，RBB步长退化为原始BB步长。RBB步长包含一类有效的步长，例如BB步长的另一变体。证明了相应RBB算法在求解凸二次优化问题时的全局收敛性。提出了一种自适应生成正则化参数的方案，称为自适应两步参数。增强型RBB步长被用于更高效地求解二次优化问题和一般优化问题。RBB步长能够克服BB步长在许多病态优化问题中的不稳定性。此外，数值实验表明RBB步长比BB步长具有更强的鲁棒性。数值算例生动展示了所提步长在求解若干具有挑战性优化问题时的优势。