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A modified feasible semi-smooth asymptotically Newton method for nonlinear complementarity problems
- Changfeng Ma^{1}Email author,
- Baoguo Chen^{1} and
- Shaojun Pan^{1}
https://doi.org/10.1186/s13660-016-1174-1
© Ma et al. 2016
Received: 30 August 2016
Accepted: 12 September 2016
Published: 22 September 2016
Abstract
In this paper, a modified feasible semi-smooth asymptotically Newton method for nonlinear complementarity problems is proposed. The global convergence of the method and superlinear convergence are proved under some suitable assumptions. Numerical experiments are included to highlight the efficacy of the modified algorithm.
Keywords
MSC
1 Introduction
- (H1)
the function Φ is semi-smooth,
- (H2)
the function Ψ is continuously differentiable on \(\Re^{n}\).
The remainder of the paper is organized as follows: In the next section we state the modified algorithm and some useful results which will be used in subsequent analysis. In Section 3, we analysis the convergence of the algorithm described in the Section 2. Some numerical experiments are report in Section 4.
2 The algorithm and preliminaries
In order to analyze the convergence of algorithm, we describe some lemmas that are used in our subsequent analysis. A function F is said to be BD-regular at x if the generalized Jacobian V \(\in \partial_{B} F(x)\) is nonsingular. The concept of semi-smoothness was first introduction by Mifflin [6], and it then was extended by Qi and Sun [7]. We can obtain the properties of semi-smooth function from [8].
Lemma 1
Lemma 2
Lemma 3
Let \(\Phi: \Re^{n}\rightarrow\Re^{n}\) be a locally Lipschitz function. Suppose that Φ is BD-regular at x \(\in \Re^{n}\), then there exist a neighborhood \(\mathcal{N}(x)\) of x and a positive constant M such that for any \(y \in\mathcal {N}(x)\) and \(V\in \partial_{B}\Phi(y) \), V is nonsingular and \(\Vert V^{-1}\Vert \leq M\).
The next lemma summarizes the properties of the projection operator, which are very important in our subsequence analysis.
Lemma 4
- (a)
\(\Vert \Pi_{X}(y)-\Pi_{X}(z)\Vert \leq \Vert y-z\Vert \) for any \(y, z\in\Re^{n}\).
- (b)
For each y \(\in\Re^{n}\), \((\Pi_{X}(y)-y)^{T}(\Pi_{X}(y)-x)\leq0\) for any \(x \in X\).
- (c)
For each y, z \(\in\Re^{n}\), \(\Vert \Pi_{X}(y)-\Pi_{X}(z)\Vert ^{2}\leq(y-z)^{T}(\Pi_{X}(y)-\Pi_{X}(z))\).
Here X is a nonempty closed convex subset of \(\Re^{n}_{+}\).
It is shown in Lemma 4 that the projection operator \(\Pi_{X}(\cdot)\) is nonexpansive, that is, we have the property (a), thus the projection operator \(\Pi_{X}(\cdot)\) is globally Lipschitz continuous on X. The proof for detail and the more properties about the projection operator can be found in [9]. Here a direction d is said to be a descent direction of the function \(f(x)\) at x if and only if \(\nabla f(x)^{T}d<0\). The following two lemmas are very important in the proof of convergence and superlinear convergence, the proof of these two lemmas can be found in [4].
Lemma 5
Lemma 6
Lemma 5 shows that the projected Newton direction \(\bar{d}_{N}\) is a descent direction of \(\Psi(x)\) at x. In the next proposition we show that the descent of \(\Psi(x)\) at x along the projected Newton direction \(\bar{d}_{N}\) is bounded below.
Proposition 1
Proof
In this section, we recall some useful results which will be used later on first, now we give our modified feasible semi-smooth asymptotically Newton method for solving NCPs.
Algorithm 1
(A feasible asymptotically Newton method)
Step 1. Choose parameters ρ, \(\eta\in(0,1)\), \(h>2\), \(0<\sigma<\frac{1}{h}\), \(p_{2}>2\), \(p_{1}>0\), \(\epsilon>0\). Let \(x_{0}\in\) \(\Re_{+}^{n}\) be an arbitrary initial point. Set \(k : =0\).
Step 2. Choose \(V_{k}\in\partial_{B}\Phi(x_{k})\), and compute \(\nabla\Psi(x_{k})=V_{k}^{T}\Phi(x_{k})\).
Step 6. Set \(k:=k+1\), and go to Step 3.
The analysis for the global convergence and the convergence rate of the algorithm is reported in the next section.
Remark 1
3 The convergence analysis of the algorithm
In this section, we consider the convergence of the algorithm which describe in Section 2. First of all we consider the global convergence of Algorithm 1, then analysis the convergent rate and we see that Algorithm 1 is superlinear. The following theorem shows that Algorithm 1 is well defined.
Theorem 1
Suppose that \(\{x_{k}\}\) is a sequence generated by Algorithm 1. Then any accumulation point of \(\{x_{k}\}\) is a solution of the problem (2).
Proof
It is easy to see from the proof of Theorem 1 that the Algorithm 1 is always well defined. We begin with a non-solution point, Algorithm 1 always going to the stage that (6) is satisfied. In other words, \(m_{k}\) always will be a finite number, that is, Algorithm 1 is well defined. In the next theorem we analyze the superlinear convergence of Algorithm 1.
Theorem 2
Suppose that \({x_{k}}\) is a sequence generator by Algorithm 1, and \(x^{*}\) is an accumulation point of \({x_{k}}\), a solution of \(\Phi(x^{*})=0\). If \(\Phi(x)\) is BD-regular at \(x^{*}\), then the sequence \(\{x^{k}\}\) generalized by Algorithm 1 converges to \(x^{*}\) superlinearly.
Proof
4 Numerical experiments
In this section we present some numerical experiments for the algorithm proposed in Section 2. The algorithm was implemented in Matlab and run in a Matlab 7.0.1 workstation.
Example 1
Numerical results for Example 1
SP | Iter | FV | CPU |
---|---|---|---|
(0,…,0) | 6 | 9.5727e-013 | 0.0620 |
(0.5,…,0.5) | 5 | 2.9598e-013 | 0.0150 |
(−0.5,…,−0.5) | 6 | 9.5727e-013 | 0.0310 |
(1,…,1) | 6 | 7.0301e-024 | 0.0160 |
(−1,…,−1) | 6 | 9.5727e-013 | 0.0320 |
(−100,…,−100) | 6 | 9.5727e-013 | 0.0320 |
(10,…,10) | 9 | 2.0099e-015 | 0.0160 |
(50,…,50) | 10 | 8.2935e-019 | 0.0160 |
(100,…,100) | 10 | 2.4238e-015 | 0.0160 |
(1000,…,1000) | 11 | 2.9654e-022 | 0.0150 |
Example 2
Numerical results for Example 2 ( \(\pmb{n=100}\) )
SP | Iter | FV | CPU |
---|---|---|---|
(0,…,0) | 10 | 9.7021e-013 | 0.2500 |
(0.5,…,0.5) | 16 | 2.2473e-016 | 0.2810 |
(−0.5,…,−0.5) | 7 | 1.6175e-016 | 0.1570 |
(1,…,1) | 17 | 2.2481e-016 | 0.2970 |
(−1,…,−1) | 7 | 1.6173e-016 | 0.1410 |
(10,…,10) | 20 | 2.2869e-014 | 0.3430 |
(−10,…,−10) | 7 | 1.6170e-016 | 0.1400 |
(100,…,100) | 23 | 9.1849e-013 | 0.4060 |
(−100,…,−100) | 7 | 1.6170e-016 | 0.1410 |
Example 3
Numerical results for Example 3
SP | n = 32 | n = 100 | ||||
---|---|---|---|---|---|---|
Iter | FV | CPU | Iter | FV | CPU | |
(0,…,0) | 7 | 1.4147e-016 | 0.0470 | 8 | 4.2376e-014 | 0.2190 |
(1,…,1) | 9 | 2.7573e-021 | 0.0470 | 10 | 3.3580e-014 | 0.1870 |
(−1,…,−1) | 6 | 3.1293e-020 | 0.0310 | 6 | 4.3639e-020 | 0.0940 |
(−10,…,−10) | 6 | 3.8994e-021 | 0.0310 | 6 | 4.1741e-021 | 0.0940 |
(10,…,10) | 12 | 5.2178e-017 | 0.0630 | 14 | 5.2324e-022 | 0.2500 |
(100,…,100) | 15 | 1.0121e-013 | 0.0620 | 17 | 8.1853e-018 | 0.3130 |
(1000,…,1000) | 19 | 1.2999e-021 | 0.0630 | 20 | 1.7651e-014 | 0.3590 |
Example 4
Numerical results for Example 4
SP | Iter | FV | CPU |
---|---|---|---|
(0,…,0) | 21 | 1.0292e-015 | 0.0310 |
(0.5,…,0.5) | 18 | 1.0960e-016 | 0.0630 |
(1,…,1) | 18 | 9.5526e-020 | 0.0320 |
(3,…,3) | 40 | 5.4668e-018 | 0.0470 |
(5,…,5) | 100 | 3.3390e-009 | 0.0620 |
(2,1,0,1,2) | 21 | 2.4842e-018 | 0.0310 |
(0,1,0,1,0) | 21 | 4.7668e-018 | 0.0320 |
(0.5,1,0.5,2,0) | 20 | 4.3317e-014 | 0.0310 |
(1,2,3,4,5) | 29 | 8.5428e-016 | 0.0470 |
Example 5
Numerical results for Example 5
SP | Iter | FV | CPU | \(\boldsymbol {\bar{x}}\) |
---|---|---|---|---|
(0,…,0) | 5 | 9.5495e-018 | 0.0160 | (1.2247,0,0,0.5000) |
(1,…,1) | 9 | 8.9335e-021 | 0.0320 | (1.2247,0,0,0.5000) |
(−1,…,−1) | 6 | 1.0947e-018 | 0.0160 | (1.2247,0,0,0.5000) |
(10,…,10) | 14 | 1.2045e-025 | 0.0310 | (1.2247,0,0,0.5000) |
(−10,…,−10) | 11 | 7.3075e-016 | 0.0160 | (1.2247,0,0,0.5000) |
(100,…,100) | 17 | 2.4086e-023 | 0.0320 | (1.2247,0,0,0.5000) |
(−100,…,−100) | 6 | 2.4643e-023 | 0.0310 | (1.2247,0,0,0.5000) |
(10^{3},…,10^{3}) | 26 | 2.1254e-018 | 0.0310 | (1.2247,0,0,0.5000) |
(−10^{3},…,−10^{3}) | 6 | 1.9403e-021 | 0.0160 | (1.2247,0,0,0.5000) |
Example 6
Numerical results for Example 6
SP | Iter | FV | CPU | \(\boldsymbol {\bar{x}}\) |
---|---|---|---|---|
(0,…,0) | 1 | 0 | 0.0160 | (0.0000,0.0000,0.0000,0.0000) |
(1,…,1) | 4 | 6.2183e-016 | 0.0470 | (0.9508,0.0000,0.0000,0.0000) |
(5,…,5) | 6 | 0 | 0.0160 | (2.9845,0.0000,0.0000,0.0000) |
(10,…,10) | 4 | 7.2927e-032 | 0.0150 | (2.9964,0.0000,0.0000,0.0000) |
(30,…,30) | 7 | 0 | 0.0310 | (2.8521,0.0000,0.0000,0.0000) |
(60,…,60) | 5 | 0 | 0.0160 | (3.0000,0.0000,0.0000,0.0000) |
5 Conclusions
In this paper, based on the semi-smoothing asymptotically Newton method, we present a modified feasible semi-smooth asymptotically Newton method for nonlinear complementarity problems. We can achieve the global convergence and the local superlinear convergence with several mild assumptions. The numerical experiments reported in Section 4 show that the modified algorithm is effective.
Declarations
Acknowledgements
This work is supported by Fujian Natural Science Foundation (Grant No. 2016J01005).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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