求解不可分离非凸非光滑问题的线性惯性ADMM算法

Linear Inertial ADMM for Nonseparable Nonconvex and Nonsmooth Problems

针对目标函数中包含耦合函数H(x,y)的非凸非光滑极小化问题,提出了一种线性惯性交替乘子方向法(Linear Inertial Alternating Direction Method of Multipliers,LIADMM)。为了方便子问题的求解,对目标函数中的耦合函数H(x,y)进行线性化处理,并在x-子问题中引入惯性效应。在适当的假设条件下,建立了算法的全局收敛性;同时引入满足Kurdyka-Lojasiewicz不等式的辅助函数,验证了算法的强收敛性。通过两个数值实验表明,引入惯性效应的算法比没有惯性效应的算法收敛性能更好。

In this paper,a linear inertial alternating direction multiplier method(LIADMM)is proposed for the nonconvex non-smooth miniaturisation problem of the objective function containing the coupling function H(x,y),and to facilitate the solution of the subproblems,the objective function is linearised and the inertial effect is introduced into the x-subproblem.To facilitate the solution of the subproblem,the coupling function H(x,y)is linearised in the objective function and the inertial effect is introduced into the x-subproblem.The global convergence of the algorithm is established under appropriate assumptions,and the strong convergence of the algorithm is proved by introducing auxiliary functions satisfying the Kurdyka-Lojasiewicz inequality.Two numerical experiments show that the algorithm with the introduction of inertial effect converges has better convergence performance than the algorithm without inertial effect.