Abstract:

Wang et al proposed the alternating direction method of multipliers with Bregman distance (Bregman ADMM) for solving multi-block separable nonconvex optimization problems with linear constraints, and proved its convergence.In this paper, we will further study the convergence rate of Bregman ADMM for solving multi-block separable nonconvex optimization problems with linear constraints, and the sufficient conditions for the boundedness of the iterative point sequence generated by the algorithm.Under the Kurdyka-Łojasiewicz property of benefit function, this paper establish the convergence rates for the values and iterates, and we show that various values of KŁ-exponent associated with the objective function can obtain Bregman ADMM with three different convergence rates. More precisely, this paper proves the following results:if the(KŁ) exponent of the benefit function$\theta=0$, then the sequence generated by Bregman ADMM converges in a finite numbers of iterations; if$\theta \in \left (0, \frac{1}{2}\right ]$, then Bregman ADMM is linearly convergent; if$\theta \in \left ( \frac{1}{2}, 1\right )$, then Bregman ADMM is sublinear convergent.

Key words: Nonconvex optimization problem, The alternating direction method of multipliers, Kurdyka-Łojasiewicz property, Bregman distance, Convergence rates, Boundedness