Mathematical Problems in Engineering
Volume 2020 (2020), Article ID 7986351, 10 pages
https://doi.org/10.1155/2020/7986351

Research Article

A Smoothing Newton Method with a Mixed Line Search for Monotone Weighted Complementarity Problems

Xiaoqin Jiang¹ and He Huang²

¹Department of Public Basic, Wuhan Technology and Business University, Wuhan 430065, China
²College of Management and Economics, Tianjin University, Weijin South Road 192, 300372 Tianjin, China

Correspondence should be addressed to He Huang; hhuangtju@163.com

Received 25 February 2020; Accepted 7 May 2020; Published 30 June 2020

Academic Editor: Quanxin Zhu

Copyright © 2020 Xiaoqin Jiang and He Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we present a smoothing Newton method for solving the monotone weighted complementarity problem (WCP). In each iteration of our method, the iterative direction is achieved by solving a system of linear equations and the iterative step length is achieved by adopting a line search. A feature of the line search criteria used in this paper is that monotone and nonmonotone line search are mixed used. The proposed method is new even when the WCP reduces to the standard complementarity problem. Particularly, the proposed method is proved to possess the global convergence under a weak assumption. The preliminary experimental results show the effectiveness and robustness of the proposed method for solving the concerned WCP.

1. Introduction

The weighted linear complementarity problem, introduced by Potra [1], is to find a vector such that

where , , and are the given matrices, is a given vector,

and is a given nonnegative weighted vector. We denote this problem by the WMLCP. When , WMLCP (1) reduces to a mixed linear complementarity problem which has extensively been studied in the literature [2].

In [1], Potra showed that the problem of Fisher market equilibrium may be modeled as a WMLCP, and particularly, it could be solved more efficiently than a corresponding complementarity problem model; Anstreicher [3] proved that the problem of the weighted centering may also be reformulated as a WMLCP and proposed an interior point method to solve it. Since then, this kind of problem has been studied a lot. For example, Potra [4] studied some theories and proposed an interior point method to solve the WMLCP with the involved matrix being sufficient; Zhang [5] presented a smoothing Newton method to solve WMLCPs; Tang [6] proposed a variant nonmonotone smoothing algorithm to solve WMLCPs; Chi et al. [7] investigated the existence and uniqueness of the solution for a class of weighted horizontal linear complementarity problems.

Recall that a standard finite-dimensional complementarity problem (denoted by the CP) is to find a vector pair such that

where is a given mapping and the subscript denotes transpose. Problem (3) has many applications in fields such as engineering and economics (see, for example, [8]), and it has received great attention [2, 9–11]. Recently, a subclass of CPs, tensor complementarity problems, was also studied extensively (see, for example, survey papers [12–14]). Two extensions of tensor complementarity problems and an application to the problem of traffic equilibrium problems were given in [15].

Inspired by the above papers, we investigate an extension of the CP, called the weighted complementarity problem (denoted by WCP), which is to find a vector pair such that

where is a given mapping. Obviously, when , WCP (4) becomes a CP. When is a linear mapping, say with being an real matrix and , the corresponding WCP is named as a weighted linear complementarity problem (WLCP); otherwise, it is named as a weighted nonlinear complementarity problem. Recall that a mapping is called to be monotone if and only if holds for all . If is a monotone mapping, then the corresponding WCP (4) is called a monotone WCP. We will investigate the numerical method for solving the WCP.

It is well known that CP (3) may be reformulated as a system of parameterized smoothing equations in terms of some complementarity function [16–18]. Thus, in order to achieve a solution of CP (3), one may use some Newton-type method to iteratively solve the obtained system of equations and make the smoothing parameter tend to zero. This is the so-called smoothing Newton method. This class of methods has been successfully applied to solving lots of optimization and related problems, including linear complementarity problems [19–24], linear programs [25], nonlinear complementarity problems [26–30], variational inequalities [27, 31], semidefinite complementarity problems [32–37], system of inequalities [19, 34], symmetric cone complementarity problems [35, 36], mathematical programs with complementarity constraints [38, 39], and absolute value equations [40]. In order to achieve the global convergence, most of the known smoothing Newton methods require the assumption that the solution set of the concerned problem is nonempty and compact. In [29], the author proposed a smoothing Newton method for solving CP (3) with a monotone mapping and showed that the smoothing Newton method is globally convergent when the problem has at least a solution. This assumption is weaker than the ones required in the global convergence of smoothing Newton methods (see, for example, [41]). Moreover, since the nonmonotone line search technique can improve the likelihood of finding a global optimal solution and convergence speed in cases where the involving function is highly nonconvex and has a valley in a small neighbourhood of some point, the nonmonotone line search criterion has been used in some smoothing Newton methods for solving CPs (see, for example, [42–45]). More nonmonotone techniques can be found in [46–50]. In addition, some related Newton-type methods can be found in [51–55].

Inspired by the above methods, we investigate the nonmonotone smoothing Newton method for solving WCP (4). We use the following assumption.

Assumption 1. is a continuously differentiable monotone mapping and WCP (4) has at least a solution.
In terms of the algorithmic framework of the one studied in [29] for CP (3) and some nonmonotone line search criteria used in [42, 43], by using a symmetric perturbed smoothing function, we propose a smoothing Newton method with a mixed line search criterion to solve WCP (4), where, in each iteration, a system of linear equations is solved to find the iterative direction and a line search is performed to achieve the iterative step length. Particularly, we show that the proposed smoothing Newton method is globally convergent if Assumption 1 is satisfied. This assumption is weaker than most of them used in the analysis on the global convergence of smoothing Newton methods for CP (3). When WCP (4) reduces to CP (3), our proof of main results given in this article is simpler than that of the corresponding ones in [29].
The article is organized as follows. In Section 2, we describe a reformulation of WCP (4) and propose a smoothing Newton method for solving WCP (4). In Section 3, we prove the global convergence of the proposed smoothing Newton method. The preliminary numerical experiments and conclusions are given in Sections 4 and 5, respectively.
In this article, we use to represent the nonnegative orthant in and to represent the positive orthant in . Denote

For any , means the th component of and stands for simplicity. For any , we denote and . Moreover, the solution set of WCP (4) is denoted by .

2. A Smoothing Newton Method

For any given , we define a mapping by

for any . In fact, when , the above mapping (i.e., ) is just the symmetric perturbed smoothing function introduced in [28].

Lemma 1. Given . Then, we have the following two results:

(a) holds if and only if

(b) holds if and only if

Proof. These two results can be easily proved by using the definition of . We omit the proof here.
Furthermore, for any given weighted vector , we define a mapping for the WCP, , by

for any , where

Then, by using (9) and Lemma 1 (a), it follows that

The following results can be easily obtained.

Proposition 1. Suppose that is a continuously differentiable monotone mapping. Then,

(a)For each , the mapping is continuously differentiable at , and

 where is the identity matrix, , , and with

 and and for all with .(b) is nonsingular at each point .

Therefore, by (11) and the continuity of the mapping , we may find a solution of WCP (4) in the following way: use some Newton-type method to iteratively solve and make .

Algorithm 1. (a smoothing Newton method).

 Step 0. Given . Take , , , and a positive integer . Set . Take satisfying . Denote . Set and . Choose and a sufficiently small positive number . Set and . Step 1. If , stop. Step 2. Let solve

 where is defined by (4). Step 3. Let be the maximum in , satisfying

 Step 4. Set . If , then we set ; otherwise, we choose . Set

 Step 5. Let and . Go to Step 1.

In Algorithm 1, we set a selection condition: . If it is satisfied for some , then ; hence, by (16), we have that . In this case, the line search in the iteration is a monotone line search. Moreover, if the above selection condition is not satisfied for some , then the line search in the iteration is a nonmonotone line search. Thus, the line search criterion designed in Algorithm 1 is a mixed line search that switches between monotone and nonmonotone line search.
Algorithm 1 is new even in the case of the weighted vector involved being the zero vector (in this case, the WCP reduces to the CP). Moreover, Algorithm 1 is simpler than many smoothing Newton methods for solving the CP in the sense that it requires only to solve a system of linear equations and to do a line search in each iteration of the algorithm. In fact, a similar algorithmic framework (without the nonmonotone line search) was proposed in [29] for solving (3).
In the following, we show that Algorithm 1 is well defined.

Lemma 2. Suppose that is a continuously differentiable monotone mapping and the sequence is generated by Algorithm 1. Then, we have the following results:

(a) for all (b) for all (c) for all (d) for all (e)Algorithm 1 is well defined

Proof. (a) For any , if , then we have that

where the first inequality holds due to (16) and the second inequality holds due to (15); otherwise, we have that , and hence, by (16) and (15), we can obtain that

Thus, for all .
(b) From Step 0 in Algorithm 1, it is easy to see that . Suppose that for some , then

where the first equality follows from the first equation in (14), the first inequality from the inductive assumption, and the last inequality from the above (a). Thus, for all .
(c) By the first equation in (14), we have for all . Moreover, for any , we have that

where the first equality follows from the first equation in (14) and the inequality in (b). Thus, for all .
(d) By (16), we have that for any ,

where the inequality follows from the above (a). Thus, for all .
(e) On the one hand, since for all , it follows from Proposition 1 that (see (12)) is invertible, and hence, equation (14) is solvable for all . On the other hand, for any , if we let , then, by (14), we have that

where the second inequality holds due to the above (d). Since the function is continuously differentiable at any , it follows from (a) that for all . This means that the line search (15) is well defined. Thus, Algorithm 1 is well defined.

3. Global Convergence of Algorithm 1

We now show that Algorithm 1 is globally convergent under Assumption 1.

Theorem 1. Let Assumption 1 be satisfied. Then, the following results hold:

(a)The sequence generated by Algorithm 1 is bounded(b)Every accumulation point of solves WCP (4)

Proof. (a) We verify the boundedness of by the following three parts.
Part 1. We verify the boundedness of . Such a result holds directly from Lemma 2 (c).
Part 2. We verify the boundedness of . Suppose, on the contrary, that is unbounded. For our purposes, we need to construct sequences and by

For any , by Lemma 2 (b), we have , and by Lemma 2 (d), we have . Thus, for all . This, together with (23) and (24) as well as the definition of the mapping in (9), implies that

These mean that both sequences and are uniformly bounded.
Furthermore, we construct sequences and by

Then, by using Lemma 1(b) and the definition of in (24), we have

which leads to

Moreover, by Assumption 1 it follows that , and hence, there exists satisfying

Thus, by using (28) and inequalities given in (27) and (29), we further obtain that

Substituting (26) into (30), we obtain that

holds for any . In what follows, we investigate the right-hand side of (31). Since Assumption 1 holds, we have that the mapping is monotone, and hence,

Hence, by using (23) and (24), we obtain that for any ,

where the first equality holds due to (23) and (29) and

Denote

Then, it holds that either is bounded or as . Moreover, we have that

holds for any . Thus, by combining (31), (33), (34), and (36), we obtain that

By the assumption that is unbounded, it holds that the right-hand side of (37) tends to as , which is a contradiction to the fact that the left-hand side of (37) is a constant. Therefore, the assumption that is unbounded does not hold, i.e., is bounded.
Part 3. We verify the boundedness of . On the one hand, we have that is bounded since is bounded and the mapping is continuous. On the other hand, by (23), we have that holds for all . Thus, by the boundedness of sequences , and , we can obtain that is bounded. Therefore, by combining Part 1 with Part 2 and Part 3, we obtain that is bounded.
(b) We verify that every accumulation point of solves WCP (3). From Lemma 2 (a)–(d), we have that for all ,

Thus, by using the boundedness of the iterative sequence in (a) and taking the subsequence if necessary, we can assume that is the limiting point of the sequence and denote that and . Since the mapping is continuous, it is obvious that . Moreover, it is not difficult to verify from (16) that . If , then we can derive the desired result by a simple continuity discussion.
In the following, assume . Then, and . We consider the following two cases.

Case 1. Suppose that for all , where is an s constant. Then, by (15), we have that for any ,

Let , so we can obtain that , which leads to a contradiction.

Case 2. Suppose that . Then, for any sufficiently large , the stepsize does not satisfy the line search criteria (15), i.e.,

where the second inequality holds from Lemma 2 (d). Thus, for any sufficiently large ,

Since , it is easy to show from (14) that the sequence is convergent. Denote . Let , and it follows from (41) that

which leads to . This is a contradiction.
Therefore, by combining Case 1 with Case 2, we obtain that every accumulation point of solves WCP (4).
Combining (a) with (b), we complete the proof of the theorem.
The basic idea in showing the boundedness of the iterative sequence (i.e., the proof of Theorem 1 (a)) is similar to the ones in [29], Lemma 3.4 and Theorem 3.1. However, the proof of Theorem 1 is more simpler than those of Lemma 3.4 and Theorem 3.1 in [29].

4. Numerical Experiments

In this section, we implement Algorithm 1 for solving weighted linear complementarity problems. All experiments were performed on a Thinkpad notebook computer with 2.5 GHz CPU and 8 GB memory. The codes are run in MATLAB R2016a under Win 10. Our experiments are divided into two parts.

Part 1. In this part, we implement Algorithm 1 for solving WCP (4). Specifically, we test the following problem:

Problem 1. Consider the following WLCP:

which is randomly generated in the following way: for any given positive integer , take with and . Let , and take in the following way: for all and for all . Let .
Obviously, this problem is a monotone WLCP, which is solvable. In this problem, for all . In the experiments, we take

Suppose is taken as those listed in Table 1. Take the starting point . Set and . Take . We take , and for any , if the condition is satisfied, we set ; otherwise, we set . We terminate Algorithm 1 when .
For every choice of , the corresponding problem is tested five times and the experimental results are reported in Table 1, where It denotes the number of iterations, Itm denotes the number of iterations when the monotone line search is used (i.e., the case of ), NF denotes the number of evaluations for the mapping , Val denotes the value of , and Cpu denotes the CPU time in seconds, respectively.
From Table 1, we have the following observations:

(i)From columns “It” and “Itm” in Table 1, we see that the monotone line search and the nonmonotone line search both work in Algorithm 1. So, this algorithm is really an algorithm with a mixed line search.(ii)It can be seen that Algorithm 1 is effective for solving WLCPs since each tested problem was successfully solved in few iterations and short CPU time.(iii)Algorithm 1 is robust since different tested problems having the same size were successfully solved with small differences among the numbers of iterations.

Part 2. In fact, Algorithm 1 can be applied to solve WMLCP (1). For this purpose, instead of the mapping defined by (9), we define a mapping for the WMLCP, , by

for any , where the mapping is the same as the one defined in Section 2 and is used to denote the Jacobian matrix of at any .
In this part, we implement Algorithm 1 for solving this type of weighted complementarity problem and compare it with two algorithms studied in the literature. The test problem is constructed in a similar way as those in [5, 6]. Specifically, we test the following problem.

Table 1: The numerical results of Problem 1.

It
Itm
NF
Val
Cpu

1000
7
5
13
9.07e − 07
1.94e + 00

8
5
20
1.00e − 07
3.08e + 00

7
5
15
1.17e − 07
2.09e + 00

6
5
8
4.23e − 07
1.69e + 00

8
5
13
3.55e − 07
2.22e + 00

3000
10
5
29
5.69e − 07
3.56e + 01

9
8
22
2.75e − 09
3.30e + 01

9
5
24
7.07e − 07
3.28e + 01

10
5
26
8.09e − 08
3.60e + 01

11
5
28
4.99e − 07
4.57e + 01

5000
10
9
26
1.12e − 07
1.50e + 02

10
5
26
1.21e − 07
1.50e + 02

11
5
27
1.11e − 08
1.63e + 02

10
5
27
1.49e − 07
1.66e + 02

10
5
25
9.83e − 08
158

Problem 2. Consider WMLCP (1), where matrices , , and and vector are generated as follows:

where with and being an identity matrix, , is an matrix with all entries being zero, with , and . Moreover, with .
In order to generate a strictly feasible starting point, we first take with and , and then, we set the starting point by

We implement our algorithm (i.e., Algorithm 1), Algorithm 1 in [5], and Algorithm 1 in [1] for solving Problem 2, respectively. For our algorithm, all parameters are selected as the same as those in experiments of Part 1; and for Algorithm 1 in [5], all parameters are selected as the same as those in experiments in [5]. Denote

For the first two algorithms, the termination criterion is ; while for the last algorithm, the termination criterion is .
In the experiments of each algorithm, for every choice of and , the corresponding problem is tested five times. The experimental results are reported in Table 2, where It and Cpu denote the number of iterations and the CPU time in seconds, respectively, and Val denotes for the first two algorithms and for the last algorithm.
From the numerical results in Table 2, it is easy to see that our algorithm is comparable to Algorithm 1 in [5], and these two algorithms are more efficient than Algorithm 1 in [1].

Table 2: The numerical results of Problem 2.

Algorithm 1
Algorithm 1 in [5]
Algorithm 1 in [1]

It/Cpu/Val
It/Cpu/Val
It/Cpu/Val

1000
1500
8/5.41e + 00/5.30e − 07
8/4.56e + 00/7.04e − 07
19/1.85e + 01/1.64e − 07

9/5.64e + 00/5.32e − 08
9/4.78e + 00/7.03e − 09
20/2.00e + 01/1.25e − 07

8/4.58e + 00/7.84e − 07
8/4.66e + 00/2.68e − 08
15/1.41e + 01/1.10e − 07

8/3.84e + 00/6.81e − 08
8/4.67e + 00/6.36e − 10
18/1.75e + 01/2.19e − 07

8/4.22e + 00/2.96e − 08
9/4.89e + 00/8.51e − 09
17/18/5.41e − 07

1500
2000
9/1.19e + 01/6.63e − 08
9/1.13e + 01/3.42e − 10
21/5.45e + 01/2.53e − 07

10/1.35e + 01/2.51e − 07
12/1.60e + 01/6.32e − 09
23/5.53e + 01/5.90e − 07

8/1.07e + 01/1.22e − 07
9/1.23e + 01/1.61e − 08
22/5.48e + 01/8.99e − 07

10/1.31e + 01/2.76e − 08
10/1.26e + 01/4.22e − 09
22/5.32e + 01/2.85e − 07

9/1.23e + 01/6.89e − 08
10/1.28e + 01/7.38e − 08
20/4.80e + 01/9.96e − 07

2500
3000
10/5.18e + 01/1.65e − 08
12/5.68e + 01/5.01e − 07
33/2.92e + 02/8.33e − 07

8/4.13e + 01/7.81e − 07
10/4.96e + 01/1.50e − 10
26/2.34e + 02/2.16e − 07

10/4.80e + 01/1.03e − 09
12/5.99e + 01/3.24e − 08
39/3.63e + 02/6.98e − 07

9/4.29e + 01/1.82e − 08
12/5.68e + 01/1.78e − 07
35/3.11e + 02/1.07e − 07

9/4.53e + 01/7.03e − 07
10/4.86e + 01/9.47e − 08
30/2.71e + 02/2.00e − 07

5. Conclusions

We proposed a smoothing Newton method with a mixed line search for solving monotone WCPs and showed the global convergence of the method under a weak assumption. The preliminary experimental results demonstrate the effectiveness and robustness of the method for solving the monotone WCP. We believe that many other smoothing Newton methods can also be modified to solve the WCP. Two future issues which have to be investigated are the theoretical properties of WCPs and solution methods for large-scale WCPs.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partially supported by the Science Fund of Educational Commission of Hubei Province in China (no. B2015335) and the Science Fund of Wuhan Technology and Business University (no. A2014024).

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