一类非凸优化问题广义交替方向法的收敛性

Convergence of the Generalized Alternating Direction Method of Multipliers for a Class of Nonconvex Optimization Problems

考虑利用广义交替方向法(GADMM)求解线性约束两个函数和的最小值问题,其中一个函数为凸函数,另一个函数可以表示为两个凸函数的差.对GADMM的每一个子问题,采用两个凸函数之差算法中的线性化技术来处理.通过假定相应函数满足Kurdyka-Lojasiewicz不等式,当增广Lagrange(拉格朗日)函数的罚参数充分大时,证明了GADMM所产生的迭代序列收敛到增广Lagrange函数的稳定点.最后,给出了该算法的收敛速度分析.

The generalized alternating direction method of multipliers (GADMM)for the minimization problems of the sum of 2 functions with linear constraints was considered,where one function was convex and the other can be expressed as the difference of 2 convex functions. For each subproblem in the GADMM,the linearization technique in the convex function difference algorithm was employed.Under the assumptions that the associated functions satisfy the Kurdyka-Lojasiewicz inequality,the sequence generated with the GADMM converges to a critical point of the augmented Lagrangian function,while the penalty parameter in the augmented Lagrangian function is sufficiently large.At last,the convergence rate of the algorithm was established.