 Research
 Open Access
A family of conjugate gradient methods for largescale nonlinear equations
 Dexiang Feng^{1, 2},
 Min Sun^{3} and
 Xueyong Wang^{3}Email author
https://doi.org/10.1186/s1366001715100
© The Author(s) 2017
 Received: 10 July 2017
 Accepted: 11 September 2017
 Published: 22 September 2017
Abstract
In this paper, we present a family of conjugate gradient projection methods for solving largescale nonlinear equations. At each iteration, it needs low storage and the subproblem can be easily solved. Compared with the existing solution methods for solving the problem, its global convergence is established without the restriction of the Lipschitz continuity on the underlying mapping. Preliminary numerical results are reported to show the efficiency of the proposed method.
Keywords
 nonlinear equations
 conjugate gradient method
 projection method
 global convergence
MSC
 90C25
 94A08
1 Introduction
In this paper, motivated by the projection methods in [3, 11] and the conjugate gradient methods in [13, 14], we propose a new family of conjugate gradient projection methods for solving nonlinear problem (1.1). The new designed method is derivativefree as it does not need to compute the Jacobian matrix or its approximation of the underlying function (1.1). Further, the new method does not need to solve any linear equations at each iteration, thus it is suitable to solve large scale problem (1.1).
The remainder of this paper is organized as follows. Section 2 describes the new method and presents its convergence. The numerical results are reported in Section 3. Some concluding remarks are drawn in the last section.
2 Algorithm and convergence analysis
Now, we describe the new conjugate gradient projection method for nonlinear constrained equations.
Algorithm 2.1
 Step 0.:

Given an arbitrary initial point \(x_{0}\in R^{n}\), parameters \(0<\rho<1,\sigma>0,t>0,\beta>0,\epsilon>0\), and set \(k:=0\).
 Step 1.:

If \(\Vert F(x_{k}) \Vert <\epsilon\), stop; otherwise go to Step 2.
 Step 2.:

Computewhere \(\beta_{k}\) is such that$$ d_{k}= \textstyle\begin{cases}F(x_{k})& \text{if } k=0,\\  ( 1+\beta_{k}\frac{F(x_{k})^{\top}d_{k1}}{ \Vert F(x_{k}) \Vert ^{2}} )F(x_{k})+\beta_{k}d_{k1} &\text{if } k\geq1, \end{cases} $$(2.3)$$ \vert \beta_{k} \vert \leq t\frac{ \Vert F(x_{k}) \Vert }{ \Vert d_{k1} \Vert },\quad \forall k\geq1. $$(2.4)
 Step 3.:

Find the trial point \(y_{k}=x_{k}+\alpha_{k} d_{k}\), where \(\alpha_{k}=\beta\rho^{m_{k}}\) and \(m_{k}\) is the smallest nonnegative integer m such that$$ \bigl\langle F\bigl(x_{k}+\beta\rho^{m}d_{k} \bigr),d_{k}\bigr\rangle \geq\sigma\beta\rho^{m} \Vert d_{k} \Vert ^{2}. $$(2.5)
 Step 4.:

Computewhere$$ x_{k+1}=P_{H_{k}}\bigl[x_{k} \xi_{k}F(y_{k})\bigr], $$(2.6)with$$H_{k}=\bigl\{ x\in R^{n}h_{k}(x)\leq0\bigr\} , $$and$$ h_{k}(x)=\bigl\langle F(y_{k}),xy_{k} \bigr\rangle , $$(2.7)Set \(k:=k+1\) and go to Step 1.$$\xi_{k}=\frac{\langle F(y_{k}), x_{k}y_{k}\rangle}{ \Vert F(y_{k}) \Vert ^{2}}. $$
Obviously, Algorithm 2.1 is different from the methods in [13, 14].
Lemma 2.1
Proof
Lemma 2.1 indicates that the hyperplane \(\partial H_{k}=\{x\in R^{n}h_{k}(x)=0\}\) strictly separates the current iterate from the solutions of problem (1.1) if \(x_{k}\) is not a solution. In addition, from Lemma 2.1, we also can derive that the solution set \(S^{*}\) of problem (1.1) is included in \(H_{k}\) for all k.
Certainly, if Algorithm 2.1 terminates at step k, then \(x_{k}\) is a solution of problem (1.1). So, in the following analysis, we assume that Algorithm 2.1 always generates an infinite sequence \(\{x_{k}\}\). Based on the lemma, we can establish the convergence of the algorithm.
Theorem 2.1
If F is continuous and condition (2.1) holds, then the sequence \(\{x_{k}\}\) generated by Algorithm 2.1 globally converges to a solution of problem (1.1).
Proof
If \(\liminf_{k\rightarrow\infty} \Vert d_{k} \Vert =0\), then from (2.9) it holds that \(\liminf_{k\rightarrow\infty} \Vert F(x_{k}) \Vert =0\). From the boundedness of \(\{x_{k}\}\) and the continuity of \(F(\cdot)\), \(\{x_{k}\}\) has some accumulation point x̄ such that \(F(\bar{x})=0\). Then from (2.11), \(\{ \Vert x_{k}\bar{x} \Vert \}\) converges, and thus the sequence \(\{x_{k}\}\) globally converges to x̄.
3 Numerical results
Problem 1
Problem 2
Problem 3
Numerical results with different dimensions of Problem 1
Dimension  10  50  100  500  1,000 

Iter.  20  21  22  23  23 
CPU  0.3594  0.8750  1.7031  41.4844  257.1406 
Numerical results with different initial points of Problem 3
Initial point  Iter.  CPU  \(\boldsymbol {\Vert F(x_{k}) \Vert } \) 

(0,0,0,0)  20  0.9219  9.9320×10^{−7} 
(3,0,0,0)  22  0.8438  7.0942×10^{−7} 
(1,1,1,0)  41  1.6250  7.2649×10^{−7} 
(0,1,1,1)  28  1.1875  3.8145×10^{−7} 
(0,100,100,1)  34  1.4063  6.6320×10^{−7} 
4 Conclusion
In this paper, we extended the conjugate gradient method to nonlinear equations. The major advantage of the method is that it does not need to compute the Jacobian matrix or any linear equations at each iteration, thus it is suitable to solve largescale nonlinear constrained equations. Under mild conditions, the proposed method possesses global convergence.
In Step 4 of Algorithm 2.1, we have to compute a projection onto the intersection of the feasible set C and a halfspace at each iteration, which is equivalent to quadratic programming, quite timeconsuming work. Hence, how to remove this projection step is one of our future research topics.
Declarations
Acknowledgements
The authors thank anonymous referees for valuable comments and suggestions, which helped to improve the manuscript. This work is supported by the Natural Science Foundation of China (11671228).
Authors’ contributions
DXF and MS organized and wrote this paper. XYW examined all the steps of the proofs in this research and gave some advice. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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