An inertial Douglas–Rachford splitting algorithm for nonconvex and nonsmooth problems

Abstract

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-Łojasiewicz 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.

An Inertial Parametric Douglas–Rachford Splitting Method for Nonconvex Problems

“…The parameters of various methods are listed in Table 1, and for the definition of the algorithmic parameters in Table 1, please refer to [11,16,23,30].…”

“…where P C is the projector onto the set C and P D is the projector onto the set D; α is the parameterized coefficient; β is the inertial parameter; and γ > 0 is the step-size. We compare our method with that of several other algorithms, including DR [11], PR (the Peaceman-Rachford splitting method [30]), PDR [16], Alt (the alternating projection method [11]), and IDRS [23].…”

“…In Feng et al [23], they study the IDRS method for the signal recovery problem. We give the IDRS method for the feasibility problem and make the same substitution as the PDR method for the functions f , g. The IDRS method is given below:…”

“…Meanwhile, Han et al [19] study the randomized r-sets-Douglas-Rachford (RrDR) method with the inertial scheme and show that it converges at an accelerated linear rate. Additionally, Feng et al [23] developed an inertial Douglas-Rachford splitting (IDRS) method and demonstrated its validity in terms of signal recovery. The IDRS method is given as follows:…”

Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems

“…Recently, there are increasing interests in studying inertial type algorithms, such as Nicolas et al [18] proposed inertial versions of block coordinate descent methods for solving nonconvex nonsmooth optimization problems. Feng et al [19] focused on a minimization optimization model that is nonconvex and nonsmooth and established an inertial Douglas-Rachford splitting (IDRS) algorithm, which incorporate the inertial technique into the framework of the Douglas-Rachford splitting algorithm. Specially, for solving problem (1.1), Zhang and He [20] introduced an inertial version of the proximal alternating minimization method, Pock and Sabach [21] proposed the following inertial proximal alternating linearized minimization (iPALM) algorithm:…”