Research Article | Open Access

Volume 2020 |Article ID 7986351 | 10 pages | https://doi.org/10.1155/2020/7986351

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

Accepted07 May 2020
Published30 Jun 2020

#### 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 , is to find a vector such thatwhere , , 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 .

In , 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  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  studied some theories and proposed an interior point method to solve the WMLCP with the involved matrix being sufficient; Zhang  presented a smoothing Newton method to solve WMLCPs; Tang  proposed a variant nonmonotone smoothing algorithm to solve WMLCPs; Chi et al.  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 thatwhere is a given mapping and the subscript denotes transpose. Problem (3) has many applications in fields such as engineering and economics (see, for example, ), and it has received great attention [2, 911]. Recently, a subclass of CPs, tensor complementarity problems, was also studied extensively (see, for example, survey papers ). Two extensions of tensor complementarity problems and an application to the problem of traffic equilibrium problems were given in .

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 thatwhere 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 . 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 , linear programs , nonlinear complementarity problems , variational inequalities [27, 31], semidefinite complementarity problems , system of inequalities [19, 34], symmetric cone complementarity problems [35, 36], mathematical programs with complementarity constraints [38, 39], and absolute value equations . 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 , 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, ). 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, ). More nonmonotone techniques can be found in . In addition, some related Newton-type methods can be found in .

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  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 .
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 . DenoteFor 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 byfor any . In fact, when , the above mapping (i.e., ) is just the symmetric perturbed smoothing function introduced in .

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, , byfor any , whereThen, by using (9) and Lemma 1 (a), it follows thatThe 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 , andwhere is the identity matrix, , , and withand 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 solvewhere is defined by (4).Step 3. Let be the maximum in , satisfyingStep 4. Set . If , then we set ; otherwise, we choose . SetStep 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 1is 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  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 thatwhere 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 thatThus, for all .
(b) From Step 0 in Algorithm 1, it is easy to see that . Suppose that for some , thenwhere 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 thatwhere 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 thatwhere 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 byFor 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 thatThese mean that both sequences and are uniformly bounded.
Furthermore, we construct sequences and byThen, by using Lemma 1(b) and the definition of in (24), we havewhich leads toMoreover, by Assumption 1 it follows that , and hence, there exists satisfyingThus, by using (28) and inequalities given in (27) and (29), we further obtain thatSubstituting (26) into (30), we obtain thatholds 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) andDenoteThen, it holds that either is bounded or as . Moreover, we have thatholds for any . Thus, by combining (31), (33), (34), and (36), we obtain thatBy 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) thatwhich 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 , 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 .

#### 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 takeSuppose 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, , byfor 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.

 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 byWe implement our algorithm (i.e., Algorithm 1), Algorithm 1 in , and Algorithm 1 in  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 , all parameters are selected as the same as those in experiments in . DenoteFor 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 , and these two algorithms are more efficient than Algorithm 1 in .

 Algorithm 1 Algorithm 1 in  Algorithm 1 in  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).

1. F. A. Potra, “Weighted complementarity problems—a new paradigm for computing equilibria,” SIAM Journal on Optimization, vol. 22, no. 4, pp. 1634–1654, 2012. View at: Publisher Site | Google Scholar
2. R. W. Cottle, J. S. Pang, and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, MA, USA, 1992.
3. K. M. Anstreicher, “Interior-point algorithms for a generalization of linear programming and weighted centering,” Optimization Methods and Software, vol. 27, no. 4-5, pp. 605–612, 2012. View at: Publisher Site | Google Scholar
4. F. A. Potra, “Sufficient weighted complementarity problems,” Computational Optimization and Applications, vol. 64, no. 2, pp. 467–488, 2016. View at: Publisher Site | Google Scholar
5. J. Zhang, “A smoothing Newton algorithm for weighted linear complementarity problem,” Optimization Letters, vol. 10, no. 3, pp. 499–509, 2016. View at: Publisher Site | Google Scholar
6. J. Tang, “A variant nonmonotone smoothing algorithm with improved numerical results for large-scale LWCPs,” Computational and Applied Mathematics, vol. 37, no. 3, pp. 3927–3936, 2018. View at: Publisher Site | Google Scholar
7. X. Chi, M. S. Gowda, and J. Tao, “The weighted horizontal linear complementarity problem on a Euclidean Jordan algebra,” Journal of Global Optimization, vol. 73, no. 1, pp. 153–169, 2019. View at: Publisher Site | Google Scholar
8. M. C. Ferris and J. S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Review, vol. 39, no. 4, pp. 669–713, 1997. View at: Publisher Site | Google Scholar
9. F. Facchinei and J. S. Pang, “Finite-dimemsional variational inequalities and complementarity problems,” Operations Research, vol. 1, Springer-Verlag, New York, NY, USA, 2003. View at: Google Scholar
10. F. Facchinei and J. S. Pang, “Finite-dimemsional variational inequalities and complementarity problems,” Operations Research, vol. 2, Springer-Verlag, New York, NY, USA, 2003. View at: Google Scholar
11. J. Han, N. Xiu, and H. D. Qi, Nonlinear Complementarity Theory and Algorithms, Shanghai Science and Technology Press, Shanghai, China, 2006, in Chinese.
12. Z.-H. Huang and L. Qi, “Tensor complementarity problems—part I: basic theory,” Journal of Optimization Theory and Applications, vol. 183, no. 1, pp. 1–23, 2019. View at: Publisher Site | Google Scholar
13. L. Qi and Z.-H. Huang, “Tensor complementarity problems—part II: solution methods,” Journal of Optimization Theory and Applications, vol. 183, no. 2, pp. 365–385, 2019. View at: Publisher Site | Google Scholar
14. Z.-H. Huang and L. Qi, “Tensor complementarity problems—part III: applications,” Journal of Optimization Theory and Applications, vol. 183, no. 3, pp. 771–791, 2019. View at: Publisher Site | Google Scholar
15. Y. Xu, W. Gu, and H. Huang, “Solvability of two classes of tensor complementarity problems,” Mathematical Problems in Engineering, vol. 2019, Article ID 6107517, 8 pages, 2019. View at: Publisher Site | Google Scholar
16. D. Sun and L. Qi, “On NCP-functions,” Computational Optimization and Applications, vol. 13, no. 1–3, pp. 201–220, 1999. View at: Google Scholar
17. S.-L. Hu, Z.-H. Huang, and J.-S. Chen, “Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 69–82, 2009. View at: Publisher Site | Google Scholar
18. J. Sun, X.-R. Wu, B. Saheya, J.-S. Chen, and C.-H. Ko, “Neural network for solving SOCQP and SOCCVI based on two discrete-type classes of SOC complementarity functions,” Mathematical Problems in Engineering, vol. 2019, Article ID 4545064, 18 pages, 2019. View at: Publisher Site | Google Scholar
19. C. Chen and O. L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Mathematical Programming, vol. 71, no. 1, pp. 51–69, 1995. View at: Publisher Site | Google Scholar
20. C. Kanzow, “Some noninterior continuation methods for linear complementarity problems,” SIAM Journal on Matrix Analysis and Applications, vol. 17, no. 4, pp. 851–868, 1996. View at: Publisher Site | Google Scholar
21. J. Burke and S. Xu, “A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem,” Mathematical Programming, vol. 87, no. 1, pp. 113–130, 2000. View at: Publisher Site | Google Scholar
22. X. Chen and Y. Ye, “On smoothing methods for the P0 matrix linear complementarity problem,” SIAM Journal on Optimization, vol. 11, no. 2, pp. 341–363, 2000. View at: Publisher Site | Google Scholar
23. Y. B. Zhao and D. Li, “A globally and locally superlinearly convergent non-interior-point algorithm for P0 LCPs,” SIAM Journal on Optimization, vol. 13, no. 4, pp. 1196–1221, 2003. View at: Publisher Site | Google Scholar
24. Z.-H. Huang, L. Qi, and D. Sun, “Sub-quadratic convergence of a smoothing Newton algorithm for the P0- and monotone LCP,” Mathematical Programming, vol. 99, no. 3, pp. 423–441, 2004. View at: Publisher Site | Google Scholar
25. S. Engelke and C. Kanzow, “Improved non-interior continuation methods for the solution of linear programming,” Numerische Mathematik, vol. 90, no. 3, pp. 487–507, 2002. View at: Publisher Site | Google Scholar
26. B. Chen and N. Xiu, “A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions,” SIAM Journal on Optimization, vol. 9, no. 3, pp. 605–623, 1999. View at: Publisher Site | Google Scholar
27. L. Qi, D. Sun, and G. Zhou, “A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems,” Mathematical Programming, vol. 87, no. 1, pp. 1–35, 2000. View at: Publisher Site | Google Scholar
28. Z. H. Huang, J. Han, and Z. Chen, “A predictor-corrector smoothing Newton method, based on a new smoothing function, for solving the NCP with a P0 function,” Journal of Optimization Theory and Applications, vol. 117, no. 1, pp. 39–68, 2003. View at: Publisher Site | Google Scholar
29. Z.-H. Huang, “Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms,” Mathematical Methods of Operations Research, vol. 61, no. 1, pp. 41–55, 2005. View at: Publisher Site | Google Scholar
30. Z.-H. Huang and L. Qi, “Formulating an n-person noncooperative game as a tensor complementarity problem,” Computational Optimization and Applications, vol. 66, no. 3, pp. 557–576, 2017. View at: Publisher Site | Google Scholar
31. H.-D. Qi, “A regularized smoothing Newton method for box constrained variational inequality problems with P0-functions,” SIAM Journal on Optimization, vol. 10, no. 2, pp. 315–330, 2000. View at: Publisher Site | Google Scholar
32. X. Chen and P. Tseng, “Non-interior continuation methods for solving semidefinite complementarity problems,” Mathematical Programming, vol. 95, no. 3, pp. 431–474, 2003. View at: Publisher Site | Google Scholar
33. Z. Huang and J. Han, “Non-interior continuation method for solving the monotone semidefinite complementarity problem,” Applied Mathematics and Optimization, vol. 47, no. 3, pp. 195–211, 2003. View at: Publisher Site | Google Scholar
34. Z. H. Huang, Y. Zhang, and W. Wu, “A smoothing-type algorithm for solving system of inequalities,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 355–363, 2008. View at: Publisher Site | Google Scholar
35. L. Kong, J. Sun, and N. Xiu, “A regularized smoothing Newton method for symmetric cone complementarity problems,” SIAM Journal on Optimization, vol. 19, no. 3, pp. 1028–1047, 2008. View at: Publisher Site | Google Scholar
36. Z.-H. Huang and T. Ni, “Smoothing algorithms for complementarity problems over symmetric cones,” Computational Optimization and Applications, vol. 45, no. 3, pp. 557–579, 2010. View at: Publisher Site | Google Scholar
37. N. Lu and Z.-H. Huang, “A smoothing Newton algorithm for a class of non-monotonic symmetric cone linear complementarity problems,” Journal of Optimization Theory and Applications, vol. 161, no. 2, pp. 446–464, 2014. View at: Publisher Site | Google Scholar
38. Z.-H. Huang and J. Sun, “A smoothing Newton algorithm for mathematical programs with complementarity constraints,” Journal of Industrial & Management Optimization, vol. 1, no. 2, pp. 153–170, 2005. View at: Publisher Site | Google Scholar
39. Y. Chen and Z. Wan, “A new smoothing method for mathematical programs with complementarity constraints based on logarithm-exponential function,” Mathematical Problems in Engineering, vol. 2018, Article ID 5056148, 11 pages, 2018. View at: Publisher Site | Google Scholar
40. X. Jiang and Y. Zhang, “A smoothing-type algorithm for absolute value equations,” Journal of Industrial & Management Optimization, vol. 9, no. 4, pp. 789–798, 2013. View at: Publisher Site | Google Scholar
41. Z. Huang, “Sufficient conditions on nonemptiness and boundedness of the solution set of the P0 function nonlinear complementarity problem,” Operations Research Letters, vol. 30, no. 3, pp. 202–210, 2002. View at: Publisher Site | Google Scholar
42. Z. Huang, S. Hu, and J. Han, “Convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search,” Science in China Series A: Mathematics, vol. 52, no. 4, pp. 833–848, 2009. View at: Publisher Site | Google Scholar
43. S.-L. Hu, Z.-H. Huang, and P. Wang, “A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems,” Optimization Methods and Software, vol. 24, no. 3, pp. 447–460, 2009. View at: Publisher Site | Google Scholar
44. T. Ni and P. Wang, “A smoothing-type algorithm for solving nonlinear complementarity problems with a non-monotone line search,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 2207–2214, 2010. View at: Publisher Site | Google Scholar
45. J. Zhu, H. Liu, and C. Liu, “A family of new smoothing functions and a nonmonotone smoothing Newton method for the nonlinear complementarity problems,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 647–662, 2011. View at: Publisher Site | Google Scholar
46. T. Dehghan Niri, M. Heydari, and M. M. Hosseini, “An improvement of adaptive cubic regularization method for unconstrained optimization problems,” International Journal of Computer Mathematics, pp. 1–17, 2020. View at: Publisher Site | Google Scholar
47. T. Dehghan Niri, M. Heydari, and M. M. Hosseini, “Correction of trust region method with a new modified Newton method,” International Journal of Computer Mathematics, vol. 97, no. 5, pp. 1118–1132, 2020. View at: Publisher Site | Google Scholar
48. S. Huang, Z. Wan, and J. Zhang, “An extended nonmonotone line search technique for large-scale unconstrained optimization,” Journal of Computational and Applied Mathematics, vol. 330, pp. 586–604, 2018. View at: Publisher Site | Google Scholar
49. S. Huang and Z. Wan, “A new nonmonotone spectral residual method for nonsmooth nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 313, pp. 82–101, 2017. View at: Publisher Site | Google Scholar
50. S. Huang, Z. Wan, and X. Chen, “A new nonmonotone line search technique for unconstrained optimization,” Numerical Algorithms, vol. 68, no. 4, pp. 671–689, 2015. View at: Publisher Site | Google Scholar
51. G. Q. Wang, Y. J. Yue, and X. Z. Cai, “A weighted-path-following method for monotone horizontal linear complementarity problem,” Advances in Soft Computing, Springer, Berlin, Germany, 2009. View at: Publisher Site | Google Scholar
52. F. Gurtuna, C. Petra, F. A. Potra, O. Shevchenko, and A. Vancea, “Corrector-predictor methods for sufficient linear complementarity problems,” Computational Optimization and Applications, vol. 48, no. 3, pp. 453–485, 2011. View at: Publisher Site | Google Scholar
53. T. Dehghan Niri, S. A. Shahzadeh Fazeli, and M. Heydari, “A two-step improved Newton method to solve convex unconstrained optimization problems,” Journal of Applied Mathematics and Computing, vol. 62, no. 1-2, pp. 37–53, 2020. View at: Publisher Site | Google Scholar
54. T. Dehghan Niri, M. M. Hosseini, and M. Heydari, “An efficient improvement of the Newton method for solving nonconvex optimization problems,” Computational Methods for Differential Equations, vol. 7, no. 1, pp. 69–85, 2019. View at: Google Scholar
55. T. Dehghan Niri, M. M. Hosseini, and M. Heydari, “A new modified trust region algorithm for solving unconstrained optimization problems,” Journal of Mathematical Extension, vol. 12, no. 4, pp. 115–135, 2018. View at: Google Scholar

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