Multi-block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates Under Kurdyka–?ojasiewicz Property

We study the convergence and rates of a multi-block proximal alternating direction method multipliers (PADMM) for solving linearly constrained separable nonconvex nonsmooth optimization problems. This algorithm is an important variant (ADMM) which includes term in each subproblem, to cancel out complicated terms applications where subproblems are not easy solve or do admit simple closed form solution. consider over-relaxation step size dual update provide detailed proof any \(\beta \in (0,2)\). prove sequence generated by PADMM showing that it has finite length Cauchy. Under powerful Kurdyka–?ojasiewicz (K?) property, we establish values iterates, show various K?-exponent associated with objective function can raise three different rates. More precisely, if exponent \(\theta =0\), converges numbers iterations. If (0,1/2]\), then sequential rate \(cQ^{k}\) \(c>0\), \(Q\in (0,1)\), \(k\in {\mathbb {N}}\) iteration number. (1/2,1]\), \({\mathcal {O}}(1/k^{r})\) \(r=(1-\theta )/(2\theta -1)\) achieved.