## Abstract

In this paper, we develop a new hybrid conjugate gradient method that inherits the features of the Liu and Storey (LS), Hestenes and Stiefel (HS), Dai and Yuan (DY) and Conjugate Descent (CD) conjugate gradient methods. The new method generates a descent direction independently of any line search and possesses good convergence properties under the strong Wolfe line search conditions. Numerical results show that the proposed method is robust and efficient.

## Introduction

In this paper, we consider solving the unconstrained optimization problem

where \(x \in {\mathbb {R}}^n\) is an *n*-dimensional real vector and \(f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is a smooth function, using a nonlinear conjugate gradient method. Optimization problems arise naturally in problems from many scientific and operational applications (see e.g. [12, 19,20,21,22, 35, 36], among others).

To solve problem (1), a nonlinear conjugate gradient method starts with an initial guess, \(x_{0}\in {\mathbb {R}}^n\), and generates a sequence \(\{x_{k}\}_{k=0}^{\infty }\) using the recurrence

where the step size \(\alpha _{k}\) is a positive parameter and \(d_{k}\) is the search direction defined by

The scalar \(\beta _{k}\) is the conjugate gradient update coefficient and \(g_{k}=\nabla f(x_{k})\) is the gradient of *f* at \(x_{k}.\) In finding the step size \(\alpha _{k},\) the inexpensive line searches such as the weak Wolfe line search

the strong Wolfe line search

or the generalized Wolfe conditions

where \(0< \delta< \sigma <1\) and \(\sigma _{1}\ge 0\) are constants, are often used. Generally, conjugate gradient methods differ by the choice of the coefficient \(\beta _{k}.\) Well-known formulas for \(\beta _{k}\) can be divided into two categories. The first category includes Fletcher and Reeves (FR) [11], Dai and Yuan (DY) [6] and Conjugate Descent (CD) [10]:

where \(\Vert \cdot \Vert\) denotes the Euclidean norm and \(y_{k-1}=g_{k}-g_{k-1}.\) These methods have strong convergence properties. However, since they are very often susceptible to jamming, they tend to have poor numerical performance. The other category includes Hestenes and Stiefel (HS) [16], Polak-Ribi\(\grave{\text {e}}\)re-Polyak (PRP) [28, 29] and Liu and Storey (LS) [26]:

Although these methods may fail to converge, they have an in-built automatic restart feature which helps them avoid jamming and hence makes them numerically efficient [5].

In view of the above stated drawbacks and advantages, many researchers have proposed hybrid conjugate gradient methods that combine different \(\beta _{k}\) coefficients so as to limit the drawbacks and maximize in the advantages of the original respective conjugate gradient methods. For instance, Touati-Ahmed and Storey [31] suggested one of the first hybrid method where the coefficient \(\beta _{k}\) is given by

The authors proved that \(\beta _{k}^{TS}\) has good convergence properties and numerically outperforms both the \(\beta _{k}^{FR}\) and \(\beta _{k}^{PRP}\) methods. Alhawarat et al. [3] introduced a hybrid conjugate gradient method in which the conjugate gradient update coefficient is computed as

where \(\mu _{k}\) is defined as

The authors proved that the method possesses global convergence property when weak Wolfe line search is employed. Moreover, numerical results demonstrate that the proposed method outperforms both the CG-Descent 6.8 [14] and CG-Descent 5.3 [13] methods on a number of benchmark test problems.

In [5], Babaie-Kafaki gave a quadratic hybridization of \(\beta _{k}^{FR}\) and \(\beta _{k}^{PRP}\), where

and the hybridization parameter \(\theta ^{\pm }_{k}\) is taken from the roots of the quadratic equation

that is

and

Thus, the author suggested two methods \(\beta _{k}^{HQ+}\) and \(\beta _{k}^{HQ-}\), corresponding to \(\theta ^{+}\) and \(\theta ^{-},\) respectively, with numerical results showing \(\beta _{k}^{HQ-}\) to be more efficient than \(\beta _{k}^{HQ+}\).

More recently, Salih et al. [30] presented another hybrid conjugate gradient method defined by

The authors showed that the \(\beta _{k}^{YHM}\) method satisfies the sufficient descent condition and possesses global convergence property under the strong Wolfe line search. In 2019, Faramarzi and Amini [9] introduced a spectral conjugate gradient method defined as

with the spectral search direction given as

where

with *r* and *D* being positive constants. The authors suggested computing the spectral parameter \(\theta _{k}\) as

or

where \(\eta\) and \(\tau\) are constants such that \(\frac{1}{4}+\eta \le \theta _{k}\le \tau ,\)

and

Convergence of this method is established under the strong Wolfe conditions. For more conjugate gradient methods, the reader is referred to the works of [1, 2, 8, 15, 17, 18, 23,24,25, 27, 32, 33].

In this paper, we suggest another new hybrid conjugate gradient method that inherits good computational efforts of \(\beta _{k}^{LS}\) and \(\beta _{k}^{HS}\) methods and also nice convergence properties of \(\beta _{k}^{DY}\) and \(\beta _{k}^{CD}\) methods. This proposed method is presented in the next section, and the rest of the paper is structured as follows. In Sect. 3, we show that the proposed method satisfies the descent condition for any line search and also present its global convergence analysis under the strong Wolfe line search. Numerical comparison with respect to performance profiles of Dolan-Morè [7] and conclusion is presented in Sects. 4 and 5, respectively.

## A new hybrid conjugate gradient method

In [32], a variant of the \(\beta _k^{PRP}\) method is proposed, where the coefficient \(\beta _{k}\) is computed as

This method inherits the good numerical performance of the PRP method. Moreover, Huang et al. [17] proved that the \(\beta _k^{WYL}\) method satisfies the sufficient descent property and established that the method is globally convergent under the strong Wolfe line search if the parameter \(\sigma\) in (5) satisfies \(\sigma < \frac{1}{4}.\) Yao et al. [34] extended this idea to the \(\beta _k^{HS}\) method and proposed the update

The authors proved that the method is globally convergent under the strong Wolfe line search with the parameter \(\sigma <\frac{1}{3}\). In Jian et al. [18], a hybrid of \(\beta _{k}^{DY}, \beta _{k}^{FR}, \beta _{k}^{WYL}\ {\text {and}}\ \beta _{k}^{YWH}\) is proposed by introducing the update

with

Independent of the line search, the method generates a descent direction at every iteration. Furthermore, its global convergence is established under the weak Wolfe line search.

Now, motivated by the ideas of Jian et al. [18], in this paper we suggest a hybrid conjugate gradient method that inherits the strengths of the \(\beta _{k}^{LS},\ \beta _{k}^{HS},\ \beta _{k}^{DY}\) and \(\beta _{k}^{CD}\) methods by introducing \(\beta _{k}^{PKT}\) as

with direction \(d_{k}\) defined as

Now, with \(\beta _{k}\) and \(d_{k}\) defined as in (7) and (8), respectively, we present our hybrid conjugate gradient algorithm below.

## Global convergence of the proposed method

The following standard assumptions which have been used extensively in the literature are necessary to analyse the global convergence properties of our hybrid method.

###
**Assumption 3.1**

The level set

is bounded, where \(x_{0}\in {\mathbb {R}}^n\) is the initial guess of the iterative method (2).

### Assumption 3.2

In some neighbourhood *N* of *X*, the objective function *f* is continuous and differentiable, and its gradient is Lipschitz continuous, that is, there exists a constant \(L>0\) such that

If Assumptions 3.1 and 3.2 hold, then there exists a positive constant \(\varsigma\) such that

###
**Lemma 3.1**

*Consider the sequence*
\(\{g_{k}\}\)
* and*
\(\{d_{k}\}\)
* generated by Algorithm 1. Then, the sufficient descent condition*

*holds.*

###
*Proof*

If \(k=0\) or \(|g_{k}^{T}g_{k-1}|\ge 0.2 \Vert g_{k}\Vert ^{2},\) then the search direction \(d_{k}\) is given by

This gives

Otherwise, the search direction \(d_{k}\) is given by

Now, if we pre-multiply Eq. (11) by \(g_{k}^T,\) we get

Therefore, the new method satisfies the sufficient descent property (10) for all *k*. \(\square\)

###
**Lemma 3.2**

*Suppose that Assumptions *3.1* and*3.2* hold. Let the sequence *\(\{x_{k}\}\)* be generated by *(2)* and the search direction*\(d_{k}\)* be a descent direction. If*\(\alpha _{k}\)* is obtained by the strong Wolfe line search, then the Zoutendijk condition*

*holds*.

###
**Lemma 3.3**

*For any*\(k\ge 1,\)* the relation*\(0< \beta _{k}^{PKT}\le \beta _{k}^{CD}\)* always holds*.

###
*Proof*

From (5) and (10), it follows that

and since \(0<\sigma <1,\) we have

Also, by descent condition (10), we get

implying

Therefore, from (7), (13) and (14), it is clear that \(\beta _{k}^{PKT}>0.\) Moreover, if \(0<g_{k}^{T}g_{k-1}<\Vert g_{k}\Vert ^{2},\) then

and since \(\max \{d_{k-1}^{T}y_{k-1},-g_{k-1}^{T}d_{k-1}\}\ge -g_{k-1}^{T}d_{k-1},\) we have

Hence, lemma is proved. \(\square\)

###
**Theorem 3.1**

*Suppose that Assumptions *
3.1
* and*
3.2
* hold. Consider the sequences*
\(\{g_{k}\}\)
* and*
\(\{d_{k}\}\)
* generated by Algorithm 1. Then*

###
*Proof*

Assume that (15) does not hold. Then there exists a constant \(r>0\) such that

Letting \(\xi _{k}= 1+\beta _{k}^{PKT}\frac{d_{k-1}^{T}g_{k}}{\Vert g_{k}\Vert ^{2}},\) it follows from (8) that

Squaring both sides gives

Now, dividing by \((g_{k}^{T}d_{k})^{2}\) and applying the descent condition \(g_{k}^{T}d_{k}= -\Vert g_{k}\Vert ^{2}\) yields

Since \(\beta _{k}^{PKT}\le \beta _{k}^{CD}=\frac{\Vert g_{k}\Vert ^{2}}{-g_{k-1}^{T}d_{k-1}},\) we obtain

Noting that

and using (17) recursively yields

From (16), we have

Thus,

which implies that

This contradicts the Zoutendjik condition (12), concluding the proof. \(\square\)

## Numerical results

In this section, we analyse the numerical efficiency of our proposed \(\beta _k^{PKT}\) method, herein denoted PKT, by comparing its performance to that of Jian et al. [18], herein denoted N, and that of Alhawarat et al. [3], herein denoted AZPRP, on a set of 55 unconstrained test problems selected from [4]. We stop the iterations if either \(\Vert g_{k}\Vert \le 10^{-5}\) or a maximum of 10,000 iterations is reached. All the algorithms are coded in MATLAB R2019a. The authors for both N and AZPRP methods suggested that their algorithms will have optimal performance if they are implemented with the generalized Wolfe line search conditions (6) with the following choice of parameters: \(\sigma =0.1, \ \delta = 0.0001 \ {\text {and}}\ \sigma _{1}=1-2\delta\) for N method and \(\sigma =0.4, \ \delta = 0.0001 \ {\text {and}}\ \sigma _{1}=0.1\) for AZPRP method. Hence for our numerical experiments, we set the parameters of N and AZPRP as determined in the respective papers. For the PKT method, we implemented the strong Wolfe line search conditions with \(\delta = 0.0001 \ {\text {and}}\ \sigma =0.05.\)

Numerical results are presented in Table 1, where “Function” denotes name of test problem, “Dim” denotes dimension of test problem,“NI” denotes number of iterations, “FE” denotes number of function evaluations, “GE” denotes number of gradient evaluations, “CPU” denotes CPU time in seconds and “-” means that the method failed to solve the problem within 10,000 iterations. The bolded figures show the best performer for each problem. From Table 1, we observe that the PKT and AZPRP methods successfully solved all the problems, whereas the N method failed to solve one problem within 10,000 iterations. Moreover, the numerical results in the table indicate that the new PKT method is competitive as it is the best performer for a significant number of problems.

To further illustrate the performance of the three methods, we adopted the performance profile tool proposed by Dolan and Morè [7]. This tool evaluates and compares the performance of \(n_{s}\) solvers running on a set of \(n_{p}\) problems. The comparison between the solvers is based on the performance ratio

where \(f_{p,s}\) denotes either number of functions (gradient) evaluations, number of iterations or CPU time required by solver *s* to solve problem *p*. The overall evaluation of the performance of the solvers is then given by the performance profile function

where \(\tau \ge 0.\) If solver *s* fails to solve a problem *p*, we set the ratio \(r_{p,s}\) to some sufficiently large number.

The corresponding profiles are plotted in Figs. 1, 2, 3 and 4, where Fig. 1 shows the performance profile of number of iterations, Fig. 2 shows the performance profile of number of gradient evaluations, Fig. 3 shows the performance profile of function evaluations and Fig. 4 shows the performance profile of CPU time. The figures illustrate that the new method outperforms the AZPRP and N conjugate gradient methods.

## Conclusion

In this paper, we developed a new hybrid conjugate gradient method that inherits the features of the famous Liu and Storey (LS), Hestenes and Stiefel (HS), Dai and Yuan (DY) and Conjugate Descent (CD) conjugate gradient methods. The global convergence of the proposed method was established under the strong Wolfe line search conditions. We compared the performance of our method with those of Jian et al. [18] and Alhawarat et al. [3] on a number of benchmark unconstrained optimization problems. Evaluation of performance based on the tool of Dolan-Mor\(\acute{e}\) [7] showed that the proposed method is both efficient and effective.

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Kaelo, P., Mtagulwa, P. & Thuto, M.V. A globally convergent hybrid conjugate gradient method with strong Wolfe conditions for unconstrained optimization.
*Math Sci* **14, **1–9 (2020). https://doi.org/10.1007/s40096-019-00310-y

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### Keywords

- Unconstrained optimization
- Global convergence
- Sufficient descent
- Strong Wolfe conditions

### Mathematics Subject Classification

- 90C30
- 65K05
- 90C06