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DaiKou type conjugate gradient methods with a line search only using gradient
Journal of Inequalities and Applications volumeÂ 2017, ArticleÂ number:Â 66 (2017)
Abstract
In this paper, the DaiKou type conjugate gradient methods are developed to solve the optimality condition of an unconstrained optimization, they only utilize gradient information and have broader application scope. Under suitable conditions, the developed methods are globally convergent. Numerical tests and comparisons with the PRP+ conjugate gradient method only using gradient show that the methods are efficient.
1 Introduction
Consider the following problem of finding \(x\in R^{n}\) such that
where \(g:R^{n}\rightarrow R^{n}\) is continuous. Throughout this paper, problem (1) corresponds to the firstorder optimality condition of the unconstrained optimization
where \(f:R^{n}\rightarrow R\) is the function whose gradient is g.
Conjugate gradient methods are very efficient in solving large scale problem (2), if f is known, due to their simple iteration and their low memory requirements. For any given starting point \(x_{0}\in R^{n}\), an iterative sequence \(\{ x_{k}\}\) is generated by the following form:
where \(\alpha_{k}\) is a steplength obtained by some line search, and \(d_{k}\) is a search direction generated by
where \(g_{k}=g(x_{k})\). Different choices of the parameter \(\beta_{k}\) in (4) lead to different nonlinear conjugate gradient methods. The FletcherReeves [1], HestenesStiefel [2], PolakRibiÃ©rePolyak [3, 4], DaiYuan [5] and LiuStorey [6] formulas, and so on, are wellknown formulas for \(\beta_{k}\). Particularly, conjugate gradient methods with the following (sufficient) descent condition
are very important and are always more efficient.
Recently, Dai and Kou [7] designed a family of conjugate gradient methods for the unconstrained nonlinear problems, the corresponding search direction is close to the direction of the scaled memoryless BFGS method. More importantly, they satisfied the sufficient descent condition (5). Numerical experiments illustrated that the DaiKou type conjugate gradient methods are more efficient than the HagerZhang type methods [8] presented by Hager and Zhang [8, 9]. For other descent conjugate gradient methods proposed by researchers, please see [7, 9â€“11] and the references therein.
For conjugate gradient methods, line search plays an important role for the global convergence. In general, the weak Wolfe line search,
where \(0<\delta<\sigma<1\), was used to obtain the steplength \(\alpha _{k}\). Hager and Zhang [9] showed that the first condition (6) may never be satisfied due to the existence of the numerical errors (see also [7]). Thus, in order to avoid the numerical drawback of the weak Wolfe line search, they proposed approximate Wolfe conditions [8, 9], which was a combination of the weak Wolfe line search and
where \(0<\delta<1/2\) and \(\delta<\sigma<1\). Numerical tests showed that the combined line search performed well, but there is no theory to guarantee the global convergence. Then Dai and Kou proposed an improved Wolfe line search, that is, the steplength \(\alpha_{k}\) satisfied (7) and
where \(0<\delta<\sigma<1\), \(\epsilon>0\) is a constant parameter and \(\{\eta_{k}\}\) is a positive sequence satisfying \(\sum_{k\geq0}\eta_{k}<+\infty\). With the improved Wolfe line search, the global convergence of DaiKou type conjugate gradient methods was guaranteed.
Although the HagerZhang type and DaiKou type conjugate gradient methods are efficient in solving problem (2), during the implementation of the methods, function evaluations are required. The goal of this paper is to solve problem (1) which is more general and includes some nonlinear equations, such as boundary value problems [12]. So, we hope to improve the DaiKou type conjugate gradient methods to directly solve problem (1) and retain their high numerical efficiency. More recently, Dong [13] embedded an Armijotype line search only using gradient into the PRP+ conjugate gradient method [14] to solve problem (1), the steplength \(\alpha_{k}\) satisfied
where \(\mu_{k}\) is a determined real number and \(0<\sigma<1\). The line search allowed small choices of \(\alpha_{k}\). In order to avoid this drawback, Dong [15] considered the following line search:
where \(0<\delta<\sigma<1\). Motivated by the work of [15], we embed the line search (11) into the DaiKou type conjugate gradient methods, then the improved methods of this paper have several advantages. They have the positive features of the DaiKou type methods for problem (2), they can be used to solve the nonlinear optimization (2) only requiring gradient information, and they can be used to solve some systems of nonlinear equations, such as those arising in boundary value problems and others.
The rest of this paper is organized as follows. In the next section, we simply review the DaiKou type conjugate gradient methods for unconstrained minimization and develop them to solve problem (1). In SectionÂ 3, we prove the global convergence of the improved methods under some suitable conditions. In SectionÂ 4, we select two classes of test problems to test the improved methods. One class is composed of test problems from the CUTEst test environment, and the other class is composed of some boundary value problems. The numerical performance is used to confirm their broader application and to compare with that of the PRP+ conjugate gradient method in [13]. Finally, some conclusions are given in SectionÂ 5.
2 Algorithm
In this section, we describe the details of the proposed methods. First, we briefly review the DaiKou type conjugate gradient methods in the setting of unconstrained minimization (2). We have mentioned above that nonlinear conjugate gradient methods are identified by the definitions of the parameter \(\beta_{k}\) in (4). For the family of DaiKou type conjugate gradient methods, the parameter \(\beta_{k}\) is defined as
Here,
where \(y_{k1}=g_{k}g_{k1}\), \(s_{k1}=\alpha _{k1}d_{k1}=x_{k}x_{k1}\), \(\tau_{k1}\) is a parameter corresponding to the scaling parameter in the scaled memoryless BFGS method, and \(\eta\in[0,1)\). The parameters \(\beta_{k}\) in the DaiLiao type methods [16] and the HagerZhang type methods [9] are special cases of formula (13). If \(\tau _{k1}\) is specially defined as
with \(\lambda\in[0,1]\) and
then the DaiKou type conjugate gradient methods satisfy the sufficient descent conditionÂ (5).
The DaiKou type methods are very efficient in solving the unconstrained minimization, so we hope they can be used to solve problem (1) only requiring gradient information. Now we describe the improved methods in detail.
Algorithm 2.1
 Step 0.:

Choose \(x_{0}\in R^{n}\), constants \(\sigma\in(0,1)\), \(\delta\in (0,\sigma)\), \(\lambda\in[0,1]\), \(\eta\in[0,1)\), \(\varepsilon>0\). Set \(g_{0}:=g(x_{0})\) and \(k:=0\).
 Step 1.:

If \(\g_{k}\_{\infty}\leq\varepsilon\), then stop.
 Step 2.:

Generate the search direction \(d_{k}\) by (4) with \(\beta_{k}\) from (12), where \(\tau_{k1}\) is defined byÂ (14).
 Step 3.:

Find \(\alpha_{k}\) such that condition (11) holds, then compute the new iterate \(x_{k+1}=x_{k}+\alpha_{k} d_{k}\). Set \(k:=k+1\) and go to Step 1.
In Step 3, the steplength \(\alpha_{k}\) is determined following the inexact line search strategies of Algorithm 2.6 in [17]. Detailed steps are described in the following line search algorithm.
Algorithm 2.2
 Step 0.:

Set \(u=0\) and \(v=+\infty\). Choose \(\alpha>0\). Set \(j:=0\).
 Step 1.:

If Î± does not satisfy
$$g(x_{k}+\alpha d_{k})^{T} d_{k}\leq \delta g_{k}^{T} d_{k}, $$then set j:=j+1, and go to Step 2. If Î± does not satisfy
$$\sigma g_{k}^{T} d_{k}\leq g(x_{k}+ \alpha d_{k})^{T} d_{k}, $$then set \(j:=j+1\), and go to Step 3. Otherwise, set \(\alpha_{k}:=\alpha\), and return.
 Step 2.:

Set \(v=\alpha\), \(\alpha=(u+v)/2\). Then go to Step 1.
 Step 3.:

Set \(u=\alpha\), \(\alpha=2u\). Then go to Step 1.
The choice of the initial steplength is important for a line search. For conjugate gradient methods, it is important to make an initial guess of the steplength by utilizing the current iterative information about the problem. So, in Algorithm 2.2, we choose the initial steplength \(\alpha=1/\g_{0}\\) if \(k=0\), and \(\alpha=\alpha _{k1}g_{k1}^{T} d_{k1}/y_{k1}^{T} d_{k1}\) if \(k\geq1\).
3 Convergence analysis
Assumption 1
Assume that \(f: R^{n}\rightarrow R\) is bounded below, that is, \(f(x)>\infty\) for all \(x\in R^{n}\), and f is continuously differentiable. Its gradient \(g: R^{n}\rightarrow R^{n}\) is LLipschitz continuous, that is, there exists a constant \(L>0\) such that
AssumptionÂ 1 implies that there exists a positive constant Î³Ì‚ such that
Lemma 3.1
Assume that \(g: R^{n}\rightarrow R^{n}\) satisfies AssumptionÂ 1. If \(d_{0}=g_{0}\) and \(d_{k1}^{T} y_{k1} \neq0\) for all \(k \geq1\), then
Proof
Since \(d_{0}=g_{0}\), we have \(g_{0}^{T} d_{0}=\g_{0}\^{2}\), which satisfies (19). If
from LemmaÂ 2.3 in [5], we have the result that
And if
it is easy to know that
The proof is complete.â€ƒâ–¡
Lemma 3.2
Suppose that \(f:R^{n}\rightarrow R\) is bounded below along the ray \(\{ x_{k}+\alpha d_{k}  \alpha>0\}\), its gradient g is continuous, \(d_{k}\) is a search direction at \(x_{k}\), and \(g_{k}^{T} d_{k}<0\). Then if \(0<\delta <\sigma<1\), there exists \(\alpha_{k}>0\) satisfying the line search (11).
Proof
Define \(\phi(\alpha)=f(x_{k}+\alpha d_{k})\) and \(\psi(\alpha )=f(x_{k})+\alpha\delta g_{k}^{T} d_{k}\). Since \(\phi(\alpha)\) is bounded below for all \(\alpha>0\), \(0<\delta <1\) and \(g_{k}^{T} d_{k}<0\), the functions \(\phi(\alpha)\) and \(\psi(\alpha )\) must intersect at at least one point. Let \(\alpha_{k}^{*}>0\) be the smallest intersecting value of Î±, i.e.,
Since f is continuously differentiable, by the mean value theorem, there exists \(\alpha_{k}\in(0,\alpha_{k}^{*})\) such that
By combining (20) and (21), we obtain
Furthermore,
since \(0<\delta<\sigma<1\) and \(g_{k}^{T} d_{k}<0\).â€ƒâ–¡
Lemma 3.3
Assume that \(g:R^{n}\rightarrow R^{n}\) is monotone on the interval \(\{ x_{k}+\alpha d_{k} : 0 \leq\alpha\leq\alpha_{k}\}\), where \(\alpha_{k}\) satisfies the line search (11), then the following inequality holds:
where \(f:R^{n}\rightarrow R\) is the function whose gradient is g.
Proof
Since g is monotone on the interval \(\{x_{k}+\alpha d_{k} : 0 \leq\alpha \leq\alpha_{k}\}\), then
Since \(\alpha\leq\alpha_{k}\), it is not difficult to get that
Applying this inequality to the following relation
yields inequality (24).â€ƒâ–¡
Now, we state the Zoutendijk condition [18] for the line search (11).
Lemma 3.4
Assume that \(g:R^{n}\rightarrow R^{n}\) satisfies AssumptionÂ 1. Consider any iterative method in the form (3), where \(d_{k}\) is a descent direction and \(\alpha_{k}\) satisfies the line search (11), then
Proof
It follows from the CauchySchwarz inequality, the Lipschitz condition (17) and the line search (11) that
Then we have
The formula with (24) implies that
Summing (28) over k and noting that f is bounded below, we have that the desired result holds.â€ƒâ–¡
Now we discus the convergence properties of Algorithm 2.1. In the following, we will prove that if the gradient \(g:R^{n}\rightarrow R^{n}\) is Î¼strongly monotone, that is, there exists a constant \(\mu >0\) such that
Algorithm 2.1 is globally convergent with \(\lim_{k\rightarrow \infty}\g_{k}\=0\), and for more general gradient \(g:R^{n}\rightarrow R^{n}\), Algorithm 2.1 is convergent in the sense that \(\liminf_{k\rightarrow\infty}\g_{k}\=0\).
Theorem 3.1
Assume that \(g:R^{n}\rightarrow R^{n}\) satisfies AssumptionÂ 1 and is Î¼strongly monotone. The sequence \(\{x_{k}\}\) is generated by Algorithm 2.1, then
Proof
It follows from (17) and (29) that
By (31) and (32), it is easy to see that
Then we have that
Consequently, we have that
Furthermore,
Then
where \(\zeta=1+\max\{\frac{(2\lambda)L^{2}}{\mu^{2}}+\frac {(2+\lambda)L}{\mu},\eta\}\).
By Lemmas 3.1 and 3.4, we have that
It follows from this and (35) that
which implies the desired result.â€ƒâ–¡
Theorem 3.2
Assume that \(g:R^{n}\rightarrow R^{n}\) satisfies AssumptionÂ 1. Then Algorithm 2.1 is convergent in the sense that
Proof
We prove the theorem by contradiction. Assume that both \(g_{k}\neq0\) for all k and \(\liminf_{k\rightarrow\infty} \g_{k}\>0\), then there must exist some \(\gamma>0\) such that
then \(d_{k}\neq0\), otherwise LemmaÂ 3.1 would imply \(g_{k}=0\).
It follows from (37), LemmaÂ 3.1 and LemmaÂ 3.4 that
and
where \(\bar{c}=\min\{\frac{3}{4},1\eta\}\), then we have that
This means that there exists a positive integer N, for all \(k\geq N\),
It follows from LemmaÂ 3.1, (11) and (37) that
It follows from (18), (33), (34), (40), (41) and the LLipschitz continuity of g that, for all \(k\geq N\),
Define \(u_{k}=d_{k}/\d_{k}\\), then similarly to the proof of LemmaÂ 4.3 in [7], we can get the result that
Then it follows from (38) and (43) that
From AssumptionÂ 1 and LemmaÂ 3.3, we know that the generated sequence \(\{x_{k}\}\) is bounded, then there exists some positive constant Î³Ì„ such that
By using inequalities (42), (44) and (45), we can get the desired result similarly to the proof of items II and III of TheoremÂ 3.2 in [9].â€ƒâ–¡
4 Numerical experiments
In this section, we did some numerical experiments to test the performance of the proposed method and compared it with the PRP+ conjugate gradient method in [13]. All codes were written in Matlab and run on a notebook computer with an Intel(R) Core(TM) i55200U 2.20Â GHz CPU, 8.00Â GB of RAM and Linux operation system Ubuntu 12.04. All test problems were drawn from the CUTEst test library [19, 20] and the literature [12]. For the test problems from the CUTEst test library, we particularly chose the unconstrained optimization problems whose dimensions were at least 50. Different from the work in the literature such as [5, 7], we solved them only using gradient information. In order to confirm the broader application scope of the proposed method, some boundary value problems were selected from [12]. See ChapterÂ 1 in [21] for the background of the boundary value problems.
In practical implementations, the stopping criterion used was \(\g_{k}\_{\infty}\leq10^{3}\). For the proposed method in this paper, the values of Ïƒ and Î´ in the line search (11) were taken to be 0.9 and 0.0001, respectively, \(\lambda=0.5\), and \(\eta=0.5\). For the PRP+ conjugate gradient, all the initial values came from the reference [13].
The numerical results are reported in TablesÂ 1 and 2, where Name, Dim, Iter, Ng and CPU represent the name of the test problem, the dimension, the number of iterations, the number of gradient evaluations and the CPU time elapsed in seconds, respectively. â€˜â€™Â means the method failed to achieve the prescribed accuracy when the number of iterations exceeded 50,000 or the gradient function generated â€˜NaNâ€™. The performances of the two methods were evaluated using the profiles of Dolan and MorÃ¨ [22]. That is, we plotted the fraction P of the test problems for which each of the two methods was within a factor Ï„. In the performance profiles, the top curve represents the most robust one within the same factor Ï„, and the left curve represents the fastest one to solve the same percentage of test problems. FiguresÂ 13 show the performance profiles for test problems from the CUTEst library relating to the number of iterations, the number of gradient evaluations and the CPU time, respectively. FiguresÂ 46 show the performance profiles for some boundary value problems. These figures reveal that, for the test problems, the proposed method is more efficient and robust than the PRP+ conjugate gradient method. Consequently, the improved method not only can solve problems only referring to gradient information but also inherits the good numerical performance of the DaiKou type conjugate gradient methods.
5 Conclusions
In this paper, we discussed the improved DaiKou type conjugate gradient methods only using gradient information. They inherited the advantages of the DaiKou type conjugate gradient methods for solving the unconstrained minimization problems, but had broader application scope. Moreover, the problem considered in this paper can be viewed as the nonlinear equation
with \(F=g\). While the convergence analysis of this paper needed some assumptions of the function f whose gradient is g, our further investigation is to avoid the function f and to solve general nonlinear equation (46) using different strategies from those of this paper and literature [23â€“25].
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Acknowledgements
The authors are very grateful to the associate editor and reviewers for their valuable suggestions which have greatly improved the paper. This work was partially supported by the National Natural Science Foundation of China (No. 11471102) and the Key Basic Research Foundation of the Higher Education Institutions of Henan Province (No. 16A110012).
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Huang, Y., Liu, C. DaiKou type conjugate gradient methods with a line search only using gradient. J Inequal Appl 2017, 66 (2017). https://doi.org/10.1186/s136600171341z
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DOI: https://doi.org/10.1186/s136600171341z