We focus on the convergence analysis of the extended linearized alternating direction method of multipliers (L-ADMM) for solving convex minimization problems with three or more separable blocks in the objective functions. Previous convergence analysis of the L-ADMM needs to reduce the multi-block convex minimization problems to two blocks by grouping the variables. Moreover, there has been no rate of convergence analysis for the L-ADMM. In this paper, we construct a counter example to show the failure of convergence of the extended L-ADMM. We prove the convergence and establish the sublinear convergence rate of the extended L-ADMM under the assumptions that the proximal gradient step sizes are smaller than certain values, and any two coefficient matrices in linear constraints are orthogonal.
In the fields of wireless communication and data processing, there are varieties of mathematical optimization problems, especially nonconvex and nonsmooth problems.For these problems, one of the biggest difficulties is how to improve the speed of solution. To this end, here we mainly focused on a minimization optimization model that is nonconvex and nonsmooth. Firstly, an inertial Douglas-Rachford splitting (IDRS) algorithm was established, which incorporate the inertial technology into the framework of the Douglas-Rachford splitting algorithm. Then, we illustrated the iteration sequence generated by the proposed IDRS algorithm converges to a stationary point of the nonconvex nonsmooth optimization problem with the aid of the Kurdyka-艁ojasiewicz property. Finally, a series of numerical experiments were carried out to prove the effectiveness of our proposed algorithm from the perspective of signal recovery. The results are implicit that the proposed IDRS algorithm outperforms another algorithm.
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