一类自适应广义交替方向乘子法

A SELF-ADAPTIVE GENERALIZED ALTERNATING DIRECTION METHOD OF MULTIPLIERS

广义交替方向乘子法是求解凸优化问题的有效算法．当实际问题中子问题难以求解时，可以采用在子问题中添加邻近项的方法处理，邻近矩阵正定时，算法收敛，然而这也会使迭代步长较小．最新研究表明，邻近矩阵可以有一定的不正定性．本文在基于不定邻近项的广义交替方向乘子法框架下，提出一种自适应的广义交替方向乘子法，动态地选择邻近矩阵，增大迭代步长．在一些较弱的假设下，证明了算法的全局收敛性．我们进行一些初等数值实验，验证了算法的有效性．

Generalized alternating direction method of multipliers （G-ADMM） is effective in solving the convex optimization problem. When the subproblem is difficult to solve in practical problem, we can add the proximal term in the subproblem. The positive definiteness of the proximal matrix guarantees the convergence while resulting in the tiny step size. A new study indicates that the proximal matrix can be indefinite. In this paper, based on the frame of G-ADMM with indefinite proximal term, we propose a self-adaptive G-ADMM while the proximal matrix is dynamically selected to increase the step size. Under mild assumptions, we prove the global convergence of the proposed method. The preliminary numerical results indicate that the new algorithm is efficient.