Abstract
In this paper, we study the convergence rate of the proximal difference of the convex algorithm for the problem with a strong convex function and two convex functions. By making full use of the special structure of the difference of convex decomposition, we prove that the convergence rate of the proximal difference of the convex algorithm is linear, which is measured by the objective function value.
1. Introduction
Difference of convex programming (DCP) is a kind of important optimization problem that the objective function can be written as the difference of convex (DC) functions. The DCP problem has found many applications in assignment and power allocation [1], digital communication system [2], compressed sensing [3], and so on [4–6].
Up to now, one of the classical algorithms for DCP is the DC algorithm (DCA) [7] in which the nonconvex part of the objective function is replaced by a linear approximation. By DCA, only a convex optimization subproblem needs to be solved at each iteration. After that, the DCA has been attracted by a lot of researchers. Le Thi et al. [8] proved the linear convergence rate of DCA by employing the Kurdyka–Lojasiewicz inequality. Assuming that the subproblem of DCA can be easily solved [6], Gotoh et al. [4] proposed the proximal DC algorithm (PDCA) for solving the DCP, in which not only the nonconvex part in the objective function is replaced by the same technique as in DCA but also the convex part is replaced by a quadratic approximal. The PDCA reduces to the classical proximal gradient algorithm for convex programming if the nonconvex part of the objective function is void [9]. To accelerate the PDCA, Wen et al. [10] introduced a new type of proximal algorithm () with the help of an extrapolation technique. Since the convergence rate of heavily depends on the Kurdyka–Lojasiewicz inequality, converges linearly in general [10].
In this paper, we study the linear convergence rate of PDCA by the structure, which is different from the techniques in [8, 10]. Under conditions that the objection function can be divided into difference of a strong convex function and two convex functions with Lipschitz continuous gradient, we prove the linear convergence rate of PDCA, which is measured by the objective function value.
The remainder of the paper is organized as follows. In Section 2, several useful preliminaries are recalled. In Section 3, more details about the DC optimization problem are given, and the PDCA proposed in [4] is listed for the sake of simplicity. The linear convergence rate of the PDCA is established in Section 4. Final remarks are given in Section 5.
2. Preliminaries
In this section, we recall some useful definitions and properties.
Let be an extended real function. The domain of is denoted by
If never equals for all and dom , we say that is a proper function. If the proper function is lower semicontinuous, then it is called a closed function. A proper closed function is said to be level bounded if the lower level set of (i.e., ) is bounded.
Let be a proper closed function. Then, the limit subdifferential of at is defined as follows:where denote and . Note that dom It is well known that the limit subdifferential reduces to the classical subdifferential in convex analysis when is a convex function, that is,
Furthermore, if is continuously differentiable, then the limit subdifferential reduces to the gradient of denoted by .
3. DC Programming and PDCA
In this section, we begin to consider the DC programming problem:where is a strong convex function with constant and are convex functions, and their gradients are Lipschitz continuous with constants and , respectively. Throughout the paper, we assumed that is level bounded and . Apparently, (4) is a DC optimization problem and can be solved by the following DCA Algorithm 1.
| ||||||||
Although the subproblem (Algorithm 2) is convex, it may not have closed solutions. To solve this drawback, Gotoh et al. proposed the following PDCA.
| ||||||||
4. The Convergence Rate of PDCA
In this section, we give the linear convergence rate of PDCA. To continue, the following lemma is useful.
Lemma 1. Let be a continuous differentiable function with Lipschitz continuous gradient with Lipschitz constant . Then, for any , it holds that
By Lemma 1, we have the following result.
Lemma 2. Let be generated in Algorithm 2. Then,
Proof. Since is strongly convex with parameter , it holds thatwhere .
Since is Lipschitz continuous with constant , by (5), there exists such thatthat is,Since is a convex function, we haveSumming (7), (9), and (10), we getOn the contrary, since is a convex function, we havewhich is equivalent to the following form:Since is Lipschitz continuous with constant , by (5), there exists such thatSumming (13) and (14), we haveAdding on both sides of (15), we getTaking , it follows thatBy optimality conditions of Algorithm 2, we know thatwhere , which means thatBy (11) and (17), it holds thatwhere the first equality follows from (19) and the last inequality follows from . The desired result follows.
Now, we are at a position to prove the main theorem as follows.
Theorem 1. Let be generated in Algorithm 2. Then,where is the stationary point of (4).
Proof. By Lemma 2, let , and we have thatThen, it follows from that , which means that the sequence is nonincreasing. Then, for any , it follows thatBy Lemma 2 again, let , and we have thatwhich implies thatBy (23) and (25), it yields thatand the desired result follows.
5. Conclusions
In this paper, we give the linear convergence rate of PDCA for the case that the objective function is divided into a strong convex function and two convex functions. Different from the method in [8, 10], which depends heavily on the Kurdyka–Lojasiewicz inequality, we give a simple proof by the special structure of the optimization problem. Actually, there may be some other potential applications about the proposed PDCA. We leave this work in the future. For example, we will study further applications of the PDCA algorithm to some nonconvex problems [11, 12], tensor optimization problems [13, 14], and so on [15–18].
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
Each author contributed equally to this paper and read and approved the final manuscript.
Acknowledgments
This project was supported by the National Natural Science Foundation of China (Grant nos. 11801309 and 12071249) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant no. KJ1713334).