非凸多分块优化部分对称正则化交替方向乘子法

A Partially Symmetric Regularized Alternating Direction Method of Multipliers for Nonconvex Multi-block Optimization

交替方向乘子法求解两分块优化的研究已逐渐成熟和完善,但对于非凸多分块优化的研究相对较少.本文提出带线性约束的非凸多分块优化的部分对称正则化交替方向乘子法.首先,在适当的假设条件下,包括部分对称乘子修正中参数的估值区域,证明了算法的全局收敛性.其次,当增广拉格朗日函数满足Kurdyka-Lojasiewicz(KL)性质时,证明了算法的强收敛性.当KL性质关联函数具有特殊结构时,保证了算法的次线性和线性收敛率.最后,对算法进行了初步数值试验,结果表明算法的数值有效性.

The researches on the alternating direction method of multiplier method(ADMM)for solving two-block optimization have been gradually mature and perfect.However,the studies on ADMM for solving nonconvex multi-block optimization are relatively few.In this paper,we first propose a partially symmetric regularized ADMM for nonconvex multi-block optimization with linear constraints.Second,under appropriate assumptions including the region of the two parameters in the updating formulas for the multiplier,the global convergence of the proposed method is proved.Third,when the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz(KL)property,the strong convergence of the method is proved.Furthermore,when the associated KL property function has a special structure,the sublinear and linear convergence rate of the method are obtained.Finally,some preliminary numerical experiments are carried out,and this shows that the proposed method is numerically effective.