A Nonmonotone Projection Method for Constrained System of Nonlinear Equations

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.


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][2][3][4][5][6][7][8][9][10][11].For instance, there are some variants of the Levenberg-Marquardt type methods [1][2][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.

The Algorithm and Its Convergence Analysis
The basic idea of our algorithm is as follows.Taking a welldetermined 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.
Step 0. Choose  > 2 and  >  which are positive integer numbers.Take  ∈ (0, 1). is an arithmetic number which is as large as possible.
where   is the smallest nonnegative integer that satisfies (8).
Compared with the existing algorithms, we attempt to interrupt the monotone behavior of the iterative sequence {  } ∞ =0 by taking a big step at different moments and introducing  +1 at every appropriate step.Therefore, for some starting points, the nonmonotone technique may avoid the tunneling effect and improve the algorithms convergence.
Lemma 2 (see [15,Lemma 3.2]).Suppose that the underlying mapping  is monotone.Then Lemma 3 (see [12,Theorem 3.1]).When  ∉ { + } +∞ =0 , take  1   in (9) and use iteration (10).Then From the inequality above we have the conclusion that ‖  − * ‖ is monotonically decreasing and converges.Namely, {  } is bounded.Proof.First, we claim that {  } +∞ =0 is bounded.From the demonstration in [12], we have (  )      √ +1 exists and is equal to zero.Combined with Theorem 4, the conclusion above is proved.Remark 6.We know that the value of  +1 generated by algorithm in [12] may be very small if the algorithm encounters "tunneling effect" during the process of iteration from the ( + 1 − )th step to the ( + 1)th step.In order to make the current iterate point   as close as possible to the optimum point,  +1 needs to be more maximized.In Algorithm 1, we use the nonmonotone technique so that  +1 may be very large which is the superior place of the Algorithm 1.
Next, we accomplish the demonstration in the following three steps.

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  (..,  0 ≈ 0),  = 0.95,  = 0.6, and  1 =  2 = 1.We put  = 6,  = 12, and  = 0.9.The stop criterion is ‖(  )‖ ≤ 10 −6 .
, where  is an integer from 1 to 120; M is a 120 * 120 asymmetrical positive definite matrix;  is the vector and  0 is a constant vector.In addition, elements of  are produced in (−5, 5) randomly and  is produced by an interval range from −10 to 10.
There are two cases below to consider: Table 1 gives the numbers of iterations that are required, in order to get the approximate solutions for the above two cases with  = 100 and  = 200 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."

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.