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方法提出了一种新型步长，用于高效求解部分具有挑战性的优化问题，并将其命名为正则化BB（RBB）步长。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步长包含一类有效步长（如BB步长的另一种变体）。我们证明了相应RBB算法在求解凸二次优化问题时的全局收敛性，并提出了一种自适应生成正则化参数的方案（即自适应两步参数）。通过增强型RBB步长，可更高效地求解二次优化及一般优化问题。RBB步长能克服BB步长在病态优化问题中的不稳定性，且数值实验表明其鲁棒性优于BB步长。数值实例生动展示了所提步长在解决部分具有挑战性优化问题中的优势。