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A family of conjugate gradient methods for largescale nonlinear equations
Journal of Inequalities and Applications volumeÂ 2017, ArticleÂ number:Â 236 (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.
1 Introduction
Consider the following nonlinear equations problem of finding \(x\in C\) such that
where \(F: R^{n}\rightarrow R^{n}\) is a continuous nonlinear mapping and C is a nonempty closed convex set of \(R^{n}\). The problem finds wide applications in areas such as ballistic trajectory computation and vibration systems [1, 2], the power flow equations [3â€“5], economic equilibrium problem [6â€“8], etc.
Generally, there are two categories of solution methods for solving the problem. The first one is firstorder methods including the trust region method, the LevenbergMarquardt method and the projection method. The second one is secondorder methods including the Newton method and quasiNewton method. For the first method, Zhang et al. [9] proposed a spectral gradient method for problem (1.1) with \(C=R^{n}\), and Wang et al. [3] proposed a projection method for problem (1.1). Later, Yu et al. [10] proposed a spectral gradient projection method for constrained nonlinear equations. Compared with the projection method in [3], the methods [9, 10] need the Lipschitz continuity of the underlying mapping \(F(\cdot)\), but the former needs to solve a linear equation at each iteration, and its variants [11, 12] also inherit the shortcoming. Different from the above, in this paper, we consider the conjugate gradient method for solving the nonmonotone problem (1.1). To this end, we briefly review the wellknown conjugate gradient method for the unconstrained optimization problem
The conjugate gradient method generates the sequence of iterates recurrently by
where \(x_{k}\) is the current iterate, \(\alpha_{k}>0\) is the stepsize determined by some line search, and \(d_{k}\) is the search direction defined by
in which \(g_{k}=\nabla f(x_{k})\) and \(\beta_{k}>0\) is a parameter. The famous conjugate gradient methods include the FletcherReeves (FR) method, the PolakRibiÃ¨rePolyak (PRP) method, the LiuStorey (LS) method, the DaiYuan (DY) method. Recently, Sun et al. [13] and Li et al. [14] proposed two variants of PRP method which possesses the following property:
where \(t>0\) is a constant.
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. SectionÂ 2 describes the new method and presents its convergence. The numerical results are reported in SectionÂ 3. Some concluding remarks are drawn in the last section.
2 Algorithm and convergence analysis
Throughout this paper, we assume that the mapping \(F(\cdot)\) is monotone, or more generally pseudomonotone, on \(R^{n}\) in the sense of Karamardian [15]. That is, it satisfies that
where \(\langle\cdot,\cdot\rangle\) denotes the usual inner product in \({R}^{n}\). Further, we use \(P_{C}(x)\) to denote the projection of point \(x\in R^{n}\) onto the convex set C, which satisfies the following property:
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.:

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

Compute
$$ x_{k+1}=P_{H_{k}}\bigl[x_{k} \xi_{k}F(y_{k})\bigr], $$(2.6)where
$$H_{k}=\bigl\{ x\in R^{n}h_{k}(x)\leq0\bigr\} , $$with
$$ h_{k}(x)=\bigl\langle F(y_{k}),xy_{k} \bigr\rangle , $$(2.7)and
$$\xi_{k}=\frac{\langle F(y_{k}), x_{k}y_{k}\rangle}{ \Vert F(y_{k}) \Vert ^{2}}. $$Set \(k:=k+1\) and go to Step 1.
Obviously, Algorithm 2.1 is different from the methods in [13, 14].
Now, we give some comment on the searching direction \(d_{k}\) defined by (2.3). We claim that it is derived from Schmidt orthogonalization. In fact, in order to make \(d_{k}=F(x_{k})+\beta_{k}d_{k1}\) satisfy the property
we only need to ensure that \(\beta_{k}d_{k1}\) is vertical to \(F(x_{k})\). As a matter of fact, by Schmidt orthogonalization, we have
Equality (2.8) together with the CauchySchwarz inequality implies that \(\Vert d_{k} \Vert \geq \Vert F(x_{k}) \Vert \). In addition, by (2.3) and (2.4), we have
Therefore, for all \(k\geq0\), it holds that
Furthermore, it is easy to see that the line search (2.5) is well defined if \(F(x_{k})\neq0\).
For parameter \(\beta_{k}\) defined by (2.4), it has many choices such as \(\beta_{k}^{\mathrm{S1}}= \Vert F(x_{k}) \Vert / \Vert d_{k1} \Vert \), or [13, 14]
From the structure of \(H_{k}\), the orthogonal projection onto \(H_{k}\) has a closedform expression. That is,
Lemma 2.1
For function \(h_{k}(x)\) defined by (2.7), let \(x^{*}\in S^{*}\), it holds that
In particular, if \(x_{k}\neq y_{k}\), then \(h_{k}(x_{k})>0\).
Proof
From \(x_{k}y_{k}=\alpha_{k}d_{k}\) and the line search (2.5), we have
On the other hand, from condition (2.1), we can obtain
This completes the proof.â€ƒâ–¡
LemmaÂ 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 LemmaÂ 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
First, we show that the sequences \(\{x_{k}\}\) and \(\{y_{k}\}\) are both bounded. In fact, it follows from \(x^{*}\in H_{k}\), (2.1), (2.2) and (2.6) that
Thus the sequence \(\{ \Vert x_{k}x^{*} \Vert \}\) is decreasing and convergent, and hence the sequence \(\{x_{k}\}\) is bounded, and from (2.9), the sequence \(\{d_{k}\}\) is also bounded. Then, by \(y_{k}=x_{k}+\alpha_{k} d_{k}\), the sequence \(\{y_{k}\}\) is also bounded. Then, by the continuity of \(F(\cdot)\), there exists constant \(M>0\) such that \(\Vert F(y_{k}) \Vert \leq M\) for all k. So,
from which we can deduce that
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 xÌ„ 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 toÂ xÌ„.
If \(\liminf_{k\rightarrow\infty} \Vert d_{k} \Vert >0\), from (2.9) again, we have
By (2.12), it holds that
Therefore, from the line search (2.5), we have
Since \(\{x_{k}\}\) and \(\{d_{k}\}\) are both bounded, then letting \(k\rightarrow\infty\) in (2.15) yields that
where \(\bar{x},\bar{d}\) are limits of corresponding sequences. In addition, from (2.8) and (2.13), we get
Obviously, (2.16) contradicts (2.17). This completes the proof.â€ƒâ–¡
3 Numerical results
In this section, numerical results are provided to substantiate the efficacy of the proposed method. The codes are written in Mablab R2010a and run on a personal computer with 2.0Â GHz CPU processor. For comparison, we also give the numerical results of the spectral gradient method (denoted by SGM) in [9], the conjugate gradient method (denoted by CGM) in [16]. The parameters used in Algorithm 2.1 are set as \(t=1,\sigma=0.01,\rho=0.5,\beta =1\), and
For SGM, we set \(\beta=0.4,\sigma=0.01, r=0.001\). For CGM, we choose \(\rho= 0.1, \sigma=10^{4}\) and \(\xi=1\). Furthermore, the stopping criterion is set as \(\Vert F(x_{k}) \Vert \leq10^{6}\) for all the tested methods.
Problem 1
The mapping \(F(\cdot)\) is taken as \(F(x)=(f_{1}(x), f_{2}(x),\ldots,f_{n}(x))^{\top}\), where
Obviously, this problem has a unique solution \(x^{*}=(0,0,\ldots,0)^{\top}\).
Problem 2
The mapping \(F(\cdot)\) is taken as \(F(x)=(f_{1}(x), f_{2}(x),\ldots,f_{n}(x))^{\top}\), where
Problem 3
The mapping \(F(x): {R}^{4}\rightarrow{R}^{4}\) is given by
This problem has a degenerate solution \(x^{*}=(2,0,1,0)\).
For Problem 1, the initial point is set as \(x_{0}=(1,1,\ldots ,1)\), and TableÂ 1 gives the numerical results by Algorithm 2.1 with different dimensions, where Iter. denotes the iteration number and CPU denotes the CPU time in seconds when the algorithm terminates. TableÂ 2 lists the numerical results of Problem 2 with different initial points. The numerical results given in TableÂ 1 and TableÂ 2 show that the proposed method is efficient for solving the given two test problems.
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.
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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).
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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.
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Feng, D., Sun, M. & Wang, X. A family of conjugate gradient methods for largescale nonlinear equations. J Inequal Appl 2017, 236 (2017). https://doi.org/10.1186/s1366001715100
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DOI: https://doi.org/10.1186/s1366001715100