Abstract
This paper deals with the nonmonotone projection algorithm for constrained nonlinear equations. For some starting points, the previous projection algorithms for the problem may encounter slow convergence which is related to the monotone behavior of the iterative sequence as well as the iterative direction. To circumvent this situation, we adopt the nonmonotone technique introduced by Dang to develop a nonmonotone projection algorithm. After constructing the nonmonotone projection algorithm, we show its convergence under some suitable condition. Preliminary numerical experiment is reported at the end of this paper, from which we can see that the algorithm we propose converges more quickly than that of the usual projection algorithm for some starting points.
1. Introduction
Recall that is a nonlinear mapping with continuity and is a nonempty closed set in with convexity; then the constrained nonlinear equations are defined as seeking a point so that the following equation is established:
Many iteration methods and algorithms for solving such problem have been proposed in [1–11]. For instance, there are some variants of the Levenberg-Marquardt type methods [1–3] which have strong convergence property. In addition, Wang et al. presented a projection algorithm [5] for solving problem (1) in 2007 and a superlinearly convergent projection method [12] in 2009. From the numerical performances given in [12], we can see that the algorithm in [12] is more efficient than the method in [5] for solving such problem. Recently, a hybrid conjugate gradient projection algorithm has been established which is on the basis of the Dai-Yuan and Hestenes-Stiefel conjugate gradient method, seen in [11].
However, the projection algorithms may encounter “tunneling effect” [13] which will result in slow convergence. That is to say, during the iteration, the projection onto two or more convex sets may encounter a narrow channel, and the projection iterative sequences will become very slow. Applying the nonmonotone technique to the projection algorithm is an effective way to avoid this effect, which is based on the idea of taking a big step to interrupt the monotone behavior. The “tunneling effect” is associated with the monotone iterative sequence. Inspired by the work of Dang and Gao [14] for convex feasibility problem, we propose a nonmonotone projection algorithm, which has already been confirmed to converge faster than average in the “tunneling.” From the numerical experiment, it can be verified that, comparing with the projection method in [12], this method is more effective.
The remaining part of this article is distributed as follows. In the next section, some fundamental properties will been given which is useful in the following demonstration. In Section 3, the nonmonotonic projection method will be shown and the algorithm convergence is proved theoretically. At the end of this article, an example will be given which elucidates the algorithm we propose, which converges more quickly than the existing algorithms. Based on the above understanding, we come to the conclusion.
2. Preliminaries
Let be a nonlinear continuous mapping; then is said to be monotone if, for , it holds that In addition, is convex if is monotone.
Let be a convex set where and . Define : as a projection; it can be expressed as
It is well known that the projection has some fundamental properties. It holds that According to (4) we know that is nonexpansive. In this paper, we mainly use formula (5).
3. The Algorithm and Its Convergence Analysis
The basic idea of our algorithm is as follows. Taking a well-determined big step at each of the a priori fixed moments, we try to interrupt the monotone behavior of the iteration sequence by introducing an appropriate parameter at suitable steps so as to ensure that both of the nonmonotone sequence and the iteration within the interval are monotonically decreasing. In this way, the whole sequence may converge to a point in the solution set.
Algorithm 1 (the nonmonotone projection algorithm).
Step 0. Choose and which are positive integer numbers. Take . is an arithmetic number which is as large as possible.
Step 1. Pick an initial point ; set the parameters such that , , , , and .
Step 2. If , stop. Or else let ; solve the linear equation below: Find the solution to (6) so that can satisfy Step 3. Get by such thatwhere is the smallest nonnegative integer that satisfies (8).
Step 4. Set
(1) When ( is nonnegative integer), put Construct by
(2) When , determine as in (1). Put Construct by where Then replace by and turn to Step 2.
Compared with the existing algorithms, we attempt to interrupt the monotone behavior of the iterative sequence by taking a big step at different moments and introducing at every appropriate step. Therefore, for some starting points, the nonmonotone technique may avoid the tunneling effect and improve the algorithms convergence.
Next, we analyse the convergence of our algorithm. To this end, we need the assumptions below:() is a monotone mapping.() is nonempty and convex.()Algorithm 1 always generates an infinite sequence.
Lemma 2 (see [15, Lemma ]). Suppose that the underlying mapping is monotone. Then
Lemma 3 (see [12, Theorem ]). When , take in (9) and use iteration (10). Then
From the inequality above we have the conclusion that is monotonically decreasing and converges. Namely, is bounded.
Theorem 4. Suppose , when , where and , under Assumptions and , for the sequence produced by Algorithm 1; then
Proof. First, we claim that is bounded.
From the demonstration in [12], we have Substituting it into the Cauchy-Schwartz inequality with , we obtain Due to the boundedness of and the continuity of , is bounded.
Second, we show that .
For one thing, when , according to (16), we can see that and are limited to the same number. So we can draw a conclusion that For another,and by the substitution of and , we get and, combined with the equality of , it leads to Finally, we show that , and In association with Step 2, we obtain the desired conclusion.
Theorem 5. Suppose , when , where and , for the sequence produced by Algorithm 1; then
Proof. First, according to (12), the following equality exists, for any : Combining this with (4), we obtain SoSecond, when was chosen as in (12), hence From (28), is not incremental; thus and tend to the same number. It leads to Thus, exists and is equal to zero. Combined with Theorem 4, the conclusion above is proved.
Remark 6. We know that the value of generated by algorithm in [12] may be very small if the algorithm encounters “tunneling effect” during the process of iteration from the ()th step to the ()th step. In order to make the current iterate point as close as possible to the optimum point, needs to be more maximized. In Algorithm 1, we use the nonmonotone technique so that may be very large which is the superior place of the Algorithm 1.
Theorem 7. Under Assumptions , combined with Theorems 4 and 5, we have the conclusion that constructed by our method globally converges to .
Proof. In our algorithm, contains the subsequence which was generated by the nonmonotone technique. For brevity, we denote as .
Next, we accomplish the demonstration in the following three steps.
Firstly, there is a subsequence in converging to a point . For one thing, from Theorem 5, is convergent; thus there is a subsequence converging to a point . For another, we denote the last subsequence as . It also converges to . The following iterative yieldsFrom (11), we have By converging to , also converges to .
Secondly, each convergent sequence in converges to the same point . From Theorem 5, we know that ; from the above, when . Hence, we get the above proposition.
Last but not least, converges to . From the above analysis, we see that the sequence converges to . Let be an arbitrary index. Then there are successive indices and of , where and . When , , and when . Thus when . When , the results were significant.
4. Numerical Examples
Here we utilize our algorithm to solve a constrained system of nonlinear equations. To test the algorithm, we compare the results with the ones of the projection algorithm in [12]. For convenience, we denote our algorithm as NMPA and the projection algorithm in [12] as PA. We take the example in [12]. Set the parameters used in this example as , , , and . We put , and . The stop criterion is .
Example 1. Let the domain set be taken as , where and . Let the nonlinear equations be taken as , among which is a constant; , where is an integer from 1 to 120; M is a asymmetrical positive definite matrix; is the vector and is a constant vector. In addition, elements of are produced in randomly and is produced by an interval range from to 10.
There are two cases below to consider: Case 1: Case 2: Table 1 gives the numbers of iterations that are required, in order to get the approximate solutions for the above two cases with and of Example 1 by Alg-PA and Alg-NMPA, respectively.
From Table 1, by choosing the proper initial point, we show that the sequences are generated by the nonmonotone convergent projection algorithm. Comparing with the PA, the most prominent advantage is that our algorithm can avoid the “tunnel effect.”
5. Conclusion
This paper presented a nonmonotone projection method for constrained nonlinear equations. With the introduction of monotone technology, the monotone behavior of the iterative sequence has been disorganized. Based on some assumption, algorithm global convergence is guaranteed. In comparison with the extant projection methods, the most prominent characteristics in this paper are that, for some starting points, the nonmonotone projection algorithm can circumvent the “tunneling effect,” which leads to slow convergence.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported in part by Natural Science Foundation of Shanghai (no. 14ZR1429200) and Innovation Program of Shanghai Municipal Education Commission (no. 15ZZ073). The authors would like to express their gratitude to all those who have helped them during the writing of this paper.