1 Introduction

In this paper, we deal with the following problem:

$$\begin{aligned} \text {Find a vector}~ w\in B ~ \text {such that}~ \vartheta (w) =0, \end{aligned}$$
(1)

where the mapping \(\vartheta \) is from the Euclidean space \({\mathbb {R}}^n\) onto itself. Also, \(\vartheta \) is monotone and continuous and the set B is a nonempty closed and convex subset of \({\mathbb {R}}^n\). The problem (1) is called a convex constraint nonlinear equation and we denote the solution set of it by \(Sol (B,\vartheta )\).

The convex constraint nonlinear problem is motivated by several applications in various fields such as financial forecasting problems [9], learning constrained neural networks [8], economic equilibrium problems [14], nonlinear compressed sensing [6], chemical equilibrium systems [27], phase retrieval [7], power flow equations [33], and non-negative matrix factorisation [4, 24]. The fast local superlinear convergence property of the Newton method, quasi-Newton method, Levenberge–Marquardt method, as well as a variety of their variants [10,11,12, 30] have made them appealing for solving (1). However, to apply these methods, a linear system of equations must be computed using a Jacobian matrix or its approximation. To overcome this drawback, many authors have suggested derivative-free methods that does not require or approximate the Jacobian matrix [18]. Our focus in this present research will be on derivative-free methods based on one of the first-order optimization methods. The method of interest is the conjugate gradient (CG) method which is well known for solving large-scale unconstrained optimization problems due to its simplicity and low storage.

In recent years, motivated by the projection scheme proposed by Solodov and Svaiter [31], several derivative-free methods have been proposed for solving (1). For example, Liu and Feng [25] introduced a derivative-free projection method which converges to the solution of the convex constraint problem (1). The proposed scheme involves only one projection per iteration with a monotone and Lipschitz continuity assumption imposed on the underlying mapping. Their proposed method can be viewed as a modification of the well-known Dai–Yuan CG method for unconstrained optimization. Also, Ibrahim et al. [19] proposed a derivative-free projection method for solving the nonlinear equation (1). The method combines the projection technique and the LS–FR CG method proposed by Djordjević [15]. At each iteration, the proposed method does not store any matrix. With the aid of the projection scheme, several other derivative-free methods have been developed. Interested readers can refer to [1,2,3, 20,21,22, 28] as well as references therein.

In this present work, based on the Fletcher and Reeves (FR) CG method for unconstrained optimization, we introduced a derivative-free type iterative method for solving the constraint problem (1). The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

Our paper is organized as follows: in Sect. 2, we review some definitions which are used in the sequel. Also, the proposed search direction is presented and its global convergence result is established. Section 3 is devoted for the numerical experiment where numerical results are reported for several examples. In the last section, conclusions and discussions are given.

Notation. Throughout this article except stated otherwise, \(\Vert \cdot \Vert \) stands for Euclidean norm on \({\mathbb {R}}^n\). In addition, for a nonempty closed and convex set \(B\in {\mathbb {R}}^n\), \(P_B[\cdot ]\) is the projection mapping from \({\mathbb {R}}^n\) onto B given by \(P_{B}[w]=\arg \min \{\Vert w-y\Vert \ : y\in B\}.\)

2 Algorithm and convergence analysis

Let \(\vartheta : {\mathbb {R}}^n \rightarrow {\mathbb {R}}^n\) be a monotone and continuous nonlinear function, and let B be a nonempty closed and convex subset of \({\mathbb {R}}^n\). Recall that \(\vartheta \) is said to be:

  1. 1.

    monotone if:

    $$\begin{aligned} \forall y_1,y_2\in {\mathbb {R}}^n, ~~(\vartheta (y_1)-\vartheta (y_2))^T(y_1-y_2) \ge 0. \end{aligned}$$
    (2)
  2. 2.

    L-Lipschitz continuous, with \(L>0\) if:

    $$\begin{aligned} \forall y_1,y_2\in {\mathbb {R}}^n, ~~\Vert \vartheta (y_1)-\vartheta (y_2)\Vert \le L\Vert y_1-y_2\Vert . \end{aligned}$$
    (3)

Next, we propose an algorithm based on a modified FR CG method. The modification is done on the FR CG parameter and by extension, the direction. Generally, a CG method for solving (1) generates a sequence of iterates from an initial guess \(w_0\) via the formula:

$$\begin{aligned} w_{k+1}:=w_k +\alpha _k d_k, \end{aligned}$$

where \(\alpha _k\) is the stepsize computed using a suitable line search procedure and \(d_k\) is the search direction defined as:

$$\begin{aligned} d_0:=-\vartheta (w_0),~d_k:=-\vartheta (w_k)+\beta _kd_{k-1},~k\ge 1. \end{aligned}$$
(4)

\(\beta _k\) is called the CG parameter and \(\vartheta (w_k)\) is the function evaluation of \(\vartheta \) at \(w_k\). One of the properties required to establish the convergence of an algorithm for finding approximate solution to (1) is the sufficient descent property of the direction. We say that a direction is sufficiently descent if:

$$\begin{aligned} \vartheta (w_k)^T d_k=-\Vert \vartheta (w_k)\Vert ^2. \end{aligned}$$
(5)

In this paper, to solve the problem (1), we consider the following FR-type direction given by :

$$\begin{aligned} d_0:=-\vartheta (w_0),~d_k:=-\vartheta (w_k)+\beta _k^{FR}d_{k-1},~k\ge 1, \end{aligned}$$
(6)

where:

$$\begin{aligned} \beta _k^{FR}:=\frac{\Vert \vartheta (w_k)\Vert ^2}{\Vert \vartheta (w_{k-1})\Vert ^2}. \end{aligned}$$
(7)

It can be observed that as \(\Vert \vartheta (w_{k-1})\Vert \rightarrow 0\), the FR parameter may fail to be defined. To maintain well-defineness of the parameter, we replace \(\Vert \vartheta (w_{k-1})\Vert \) with \(\mu \Vert \vartheta (w_{k})\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2\) and define a modified FR parameter \( \beta _k^{MFR}\) as:

$$\begin{aligned} \beta _k^{MFR}:=\frac{\Vert \vartheta (w_k)\Vert ^2}{\mu \Vert \vartheta (w_{k})\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}. \end{aligned}$$
(8)

To make the direction (6) with the parameter defined in (8) descent, we introduce a new term to the direction as follows:

$$\begin{aligned} d_0:=-\vartheta (w_0),~d_k:=-\vartheta (w_k)+\beta _k^{MFR}d_{k-1}-\varrho _k \vartheta (w_k),~k\ge 1, \end{aligned}$$
(9)

where:

$$\begin{aligned} \varrho _k:=\frac{\vartheta (w_k)^T d_{k-1}}{\mu \Vert \vartheta (w_{k})\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}{.} \end{aligned}$$
(10)

Remark 2.1

Note that the term \(-\varrho _k \vartheta (w_k)\) was specifically introduced, so that the direction defined by (9) satisfies the sufficient descent condition (see Lemma 2.5).

In what follows, we give a systematic description of the proposed algorithm for finding approximate solutions to problem (1).

figure a

In what follows, we show that the sequence generated by the algorithm converges. In this case, we will require the following conditions and results from the Lemmas.

Condition 2.2

The constraint set \(B \subseteq {\mathbb {R}}^n\) is nonempty, closed, and convex set.

Condition 2.3

The mapping \(\vartheta \) is monotone and L-Lipschitz continuous on \({\mathbb {R}}^n\).

Condition 2.4

The solution set of the convex constraint problem is nonempty, that is \(Sol(B, \vartheta ) \ne \emptyset \).

Lemma 2.5

The search direction defined by (8)–(10) satisfies the sufficient descent condition.

Proof

For \(k=0\), we have:

$$\begin{aligned} \vartheta (w_0)^T d_0=-\Vert \vartheta (w_0)\Vert ^2. \end{aligned}$$

For \(k\ge 1\), then by (9) and (10):

$$\begin{aligned} \vartheta (w_k)^T d_k&=-\Vert \vartheta (w_k)\Vert ^2+ \frac{\Vert \vartheta (w_k)\Vert ^2}{\mu \Vert \vartheta (w_{k})\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}\vartheta (w_k)^T d_{k-1}\\&\quad -\frac{\vartheta (w_k)^T d_{k-1}}{\mu \Vert \vartheta (w_{k})\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}\Vert \vartheta (w_k)\Vert ^2\\&=-\Vert \vartheta (w_k)\Vert ^2. \end{aligned}$$

\(\square \)

Lemma 2.6

Let Conditions 2.2,  2.3, and  2.4 be fulfilled. If \(\{\varphi _k\}\) and \(\{w_k\}\) are given by (11) and (13) in Algorithm 1, then:

$$\begin{aligned} \alpha _k\ge \min \left\{ \varpi ,\frac{\rho \Vert \vartheta (w_k)\Vert ^2}{(L+t\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert )\Vert d_k\Vert ^2}\right\} .\end{aligned}$$
(14)

Proof

If \(\alpha _k\ne \varpi \), then \(\alpha _k^{'}=\frac{\alpha _k}{\rho }\) does not satisfy (12), that is:

$$\begin{aligned} { - \vartheta (w_k+\alpha _k^{'}d_k)^T d_k <t\alpha _k^{'}\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert \Vert d_k\Vert ^2.} \end{aligned}$$
(15)

Now, from Condition 2.3, (5), and (15):

$$\begin{aligned} \begin{aligned} \Vert \vartheta (w_k)\Vert ^2&\le -\vartheta (w_k)^T d_k\\&=(\vartheta (w_k+\alpha _k^{'}d_k)-\vartheta _k)^T d_k-\vartheta (w_k+\alpha _k^{'}d_k)^T d_k\\&\le L\alpha _k^{'}\Vert d_k\Vert ^2+t\alpha _k^{'}\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert \Vert d_k\Vert ^2\\&=(L+t\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert )\alpha _k\rho ^{-1}\Vert d_k\Vert ^2. \end{aligned} \end{aligned}$$

Therefore:

$$\begin{aligned} \alpha _k\ge \frac{\rho \Vert \vartheta (w_k)\Vert ^2}{(L+t\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert )\Vert d_k\Vert ^2}. \end{aligned}$$

\(\square \)

Lemma 2.7

Let Condition 2.2,  2.3, and  2.4 be fulfilled, then the sequences \(\{\varphi _k\}\) and \(\{w_k\}\) defined by (11) and (13) are bounded. Furthermore:

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert w_k-\varphi _k\Vert =\lim _{k\rightarrow \infty }\alpha _k\Vert d_k\Vert =0 \end{aligned}$$
(16)

and

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert w_{k+1}-w_k\Vert =0. \end{aligned}$$
(17)

Proof

Let \({\tilde{w}}\in Sol(B,\vartheta )\), and then, by Condition 2.3:

$$\begin{aligned} \vartheta (\varphi _k)^T(w_k - {\tilde{w}}) \ge \vartheta (\varphi _k)^T(w_k - \varphi _k). \end{aligned}$$
(18)

Likewise from (12) and \(t>0\):

$$\begin{aligned} \vartheta (\varphi _k)^T(w_k - \varphi _k) \ge t\alpha _{k}^2\Vert \vartheta (\varphi _k)\Vert \Vert d_k\Vert ^2\ge 0. \end{aligned}$$
(19)

Now:

$$\begin{aligned} \Vert w_{k+1}- {\tilde{w}}\Vert ^2&= \Vert P_{B}[w_k-\eta \gamma _k \vartheta (\varphi _k)]-P_{B}({\tilde{w}})\Vert ^2\nonumber \\&\le \Vert w_k-\eta \gamma _k \vartheta (\varphi _k)-{\tilde{w}}\Vert ^2\nonumber \\&= \Vert w_k - {\tilde{w}}\Vert ^2-2\eta \gamma _k \vartheta (w_k)^T (w_k - {\tilde{w}}) +\Vert \eta \gamma _k \vartheta (\varphi _k)\Vert ^2\nonumber \\&=\Vert w_k - {\tilde{w}}\Vert ^2-2\eta \frac{ \vartheta (\varphi _k)^T (w_k -\vartheta (w_k))}{\Vert \vartheta (\varphi _k)\Vert ^2} \vartheta (\varphi _k)^T (w_k - {\tilde{w}})+\eta ^2 \left( \frac{ \vartheta (\varphi _k)^T (w_k -\varphi _k)}{\Vert \vartheta (\varphi _k)\Vert } \right) ^2\nonumber \\&\le \Vert w_k - {\tilde{w}}\Vert ^2-2\eta \frac{ (\vartheta (\varphi _k)^T (w_k -\varphi _k))^2}{\Vert \vartheta (\varphi _k)\Vert ^2} +\eta ^2\left( \frac{ \vartheta (\varphi _k)^T (w_k -\varphi _k)}{\Vert \vartheta (\varphi _k)\Vert }\right) ^2\nonumber \\&\le \Vert w_k - {\tilde{w}}\Vert ^2-\eta (2-\eta )\left( \frac{ \vartheta (\varphi _k)^T (w_k -\varphi _k)}{\Vert \vartheta (\varphi _k)\Vert }\right) ^2\nonumber \\&\le \Vert w_k - {\tilde{w}}\Vert ^2-\eta (2-\eta )t^2\Vert w_k-\varphi _k\Vert ^4. \end{aligned}$$
(20)

We can conclude from (20) that:

  • the sequence \(\{\Vert w_k - {\tilde{w}}\Vert \}\) is convergent.

  • for all k:

    $$\begin{aligned} \Vert w_{k}- {\tilde{w}}\Vert ^2 \le \Vert w_0 - {\tilde{w}}\Vert ^2. \end{aligned}$$
    (21)

  • for all k, \(\{w_k\}\) is bounded. That is, there is a \(c_1>0\), such that:

    $$\begin{aligned} \Vert w_k\Vert \le c_1. \end{aligned}$$
    (22)

  • From Condition 2.3, (21) and letting \(\ell _1=L\Vert w_{0}- {\tilde{w}}\Vert :\)

    $$\begin{aligned} \Vert \vartheta (w_k)\Vert =\Vert \vartheta (w_k)- \vartheta ({\tilde{w}})\Vert \le L\Vert w_{k}- {{\tilde{w}}}\Vert \le L\Vert w_{0}- {\tilde{w}}\Vert =\ell _1. \end{aligned}$$
    (23)

Now, by monotonicity of \(\vartheta \), we have that:

$$\begin{aligned} \vartheta (\varphi _k)^T (w_k - \varphi _k)\le \vartheta (w_k)^T (w_k - \varphi _k). \end{aligned}$$
(24)

From (11), (19), (24), and the Cauchy-Schwartz inequality:

$$\begin{aligned} \Vert \vartheta (\varphi _k)\Vert \Vert w_k-\varphi _k\Vert \ge \vartheta (\varphi _k)^T(w_k-\varphi _k) \ge t\alpha _{k}^2\Vert \vartheta (\varphi _k)\Vert \Vert d_k\Vert ^2=t\Vert \vartheta (\varphi _k) \Vert \Vert w_k-\varphi _k\Vert ^2. \end{aligned}$$

Thus, we have:

$$\begin{aligned} t\Vert {w_k}-\varphi _k\Vert \le 1. \end{aligned}$$
(25)

From (25) and the reverse triangle inequality:

$$\begin{aligned} t(\Vert \varphi _k\Vert -\Vert w_k\Vert )\le t\Vert \varphi _k-w_k\Vert \le {1}. \end{aligned}$$

Combining the above inequality with (22) and (23):

$$\begin{aligned} \Vert \varphi _k\Vert&\le {\frac{1}{t}+\Vert w_k\Vert }\\&\le {\frac{1}{t}+c_1}. \end{aligned}$$

Hence, \(\{\varphi _k\}\) is bounded and so is \(\{\varphi _k-{\tilde{w}}\}\). That is, there is an \(\epsilon _1>0\), such that:

$$\begin{aligned} \Vert \varphi _k-{\tilde{w}}\Vert \le \epsilon _1. \end{aligned}$$

By Condition 2.3:

$$\begin{aligned} \Vert \vartheta (\varphi _k)\Vert =\Vert \vartheta (\varphi _{k})- \vartheta ({\tilde{w}})\Vert \le L\Vert \varphi _{k}- {\tilde{w}}\Vert \le L\epsilon _1. \end{aligned}$$

Also, by (20):

$$\begin{aligned} \eta (2-\eta )t^2\Vert w_k-\varphi _k\Vert ^4\le \Vert w_k-{\tilde{w}}\Vert ^2-\Vert w_{k+1}-{\tilde{w}}\Vert ^2, \end{aligned}$$

and

$$\begin{aligned} \eta (2-\eta )t^2\sum _{k=0}^{\infty }\Vert w_k-\varphi _k\Vert ^4\le \sum _{k=0}^{\infty }(\Vert w_k-{\tilde{w}}\Vert ^2-\Vert w_{k+1}-{\tilde{w}}\Vert ^2)= \Vert w_{0}-{\tilde{w}}\Vert ^2<\infty . \end{aligned}$$
(26)

Inequality (26) implies that:

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert w_k-\varphi _k\Vert =0. \end{aligned}$$

However, using (13) and the Cauchy–Schwartz inequality:

$$\begin{aligned} \Vert w_{k+1}-w_k\Vert =\Vert P_{B}[w_k-\eta \gamma _k \vartheta (\varphi _{k})]-P_{B}(w_k)\Vert \le \Vert w_k-\eta \gamma _k \vartheta (\varphi _{k})-w_k\Vert =\Vert \eta \gamma _k \vartheta (\varphi _{k})\Vert \le \eta \Vert w_k-\varphi _k\Vert . \end{aligned}$$

Therefore:

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert w_{k+1}-w_k\Vert =\lim _{k\rightarrow \infty }\alpha _k\Vert d_k\Vert =0. \end{aligned}$$

\(\square \)

Lemma 2.8

Let the search direction \(d_k\) be defined by (9)–(10), and then, for all \(k\ge 0:\)

$$\begin{aligned} \Vert d_k\Vert \le N_1,~N_1>0. \end{aligned}$$
(27)

Proof

It is obvious that for \(k=0\),

$$\begin{aligned} \Vert d_0\Vert =\Vert \vartheta (w_0)\Vert \le \ell _1. \end{aligned}$$

Now, for \(k\ge 1,\) from (9), (10), (22), and (23):

$$\begin{aligned} \Vert d_k\Vert&\le \left\| -\vartheta (w_k) +\frac{\Vert \vartheta (w_k)\Vert ^2}{\mu \Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}d_{k-1} -\frac{\vartheta (w_k)^T d_{k-1}}{\mu \Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}\vartheta (w_k)\right\| \nonumber \\&\le \Vert \vartheta (w_k)\Vert +\frac{\Vert \vartheta (w_k)\Vert ^2}{\mu \Vert \vartheta (w_k) \Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2}\Vert d_{k-1}\Vert +\frac{\Vert \vartheta (w_k) \Vert \Vert d_{k-1}\Vert }{\mu \Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert +\Vert \vartheta (w_{k-1})\Vert ^2} \Vert \vartheta (w_k)\Vert \nonumber \\&\le \Vert \vartheta (w_k)\Vert +\frac{\Vert \vartheta (w_k)\Vert ^2}{\mu \Vert \vartheta (w_k) \Vert \Vert d_{k-1}\Vert }\Vert d_{k-1}\Vert +\frac{\Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert }{\mu \Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert }\Vert \vartheta (w_k)\Vert \nonumber \\&=\Vert \vartheta (w_k)\Vert +2\frac{\Vert \vartheta (w_k)\Vert ^2\Vert d_{k-1}\Vert }{\mu \Vert \vartheta (w_k)\Vert \Vert d_{k-1}\Vert }\nonumber \\&=\Vert \vartheta (w_k)\Vert +2\frac{\Vert \vartheta (w_k)\Vert }{\mu }\nonumber \\&\le \ell _1 + 2\frac{\ell _1}{\mu }. \end{aligned}$$
(28)

Letting \(N_1:= \ell _1 + 2\frac{\ell _1}{\mu }\), (27) is obtained. \(\square \)

Theorem 2.9

Let Condition 2.2, 2.3, and 2.4 be fulfilled. If \(\{w_k\}\) is a sequence defined by (13), then:

$$\begin{aligned} \displaystyle \liminf _{k\rightarrow \infty }\Vert \vartheta (w_k)\Vert =0. \end{aligned}$$
(29)

Proof

Suppose that \(\displaystyle \liminf _{k\rightarrow \infty }\Vert \vartheta (w_k)\Vert \ne 0,\) and then, we can find a positive constant \(\nu _1\) for which \(\Vert \vartheta (w_k)\Vert \ge \nu _1.\) By the sufficient descent condition (5) and the Cauchy–Schwartz inequality:

$$\begin{aligned} \Vert d_k\Vert \ge \Vert \vartheta (w_k)\Vert \ge \nu _1. \end{aligned}$$
(30)

Combining (16) and (30), we have:

$$\begin{aligned} \lim _{k\rightarrow \infty }\alpha _k=0. \end{aligned}$$
(31)

However, by (14):

$$\begin{aligned} \alpha _k\ge \min \left\{ \varpi , \frac{\rho \Vert \vartheta (w_k)\Vert ^2}{{(L+t\Vert \vartheta (w_k +\alpha _k^{'}d_k)\Vert )\Vert d_k\Vert ^2}}\right\} \ge \min \left\{ \varpi ,\frac{\rho \nu _1^2}{(L+L\epsilon _1 t)N_1^2}\right\} >0,\end{aligned}$$

which contradicts (31). Therefore: \(\displaystyle \liminf _{k\rightarrow \infty }\Vert \vartheta (w_k)\Vert =0\). \(\square \)

3 Numerical experiments on monotone operator equations

In this section, we give some numerical illustrations of Algorithm 1 called modified FR derivative-free (MFRDF) method by solving nonlinear monotone operator equations. We consider numerical comparison of MFRDF method with MPCGM method proposed in [32] and Algorithm 2.1 proposed in [17]. For the numerical illustrations, we use the proposed method and the compared methods to solve the test problems given in Table 1. For the control parameters, we choose \(\varpi = 1, \eta = 1.8\), \(\rho = 0.8\) \(\mu =1.3\), and \(t=10^{-4}\) for the MFRDF Algorithm, and for the compared methods, we set the same values of parameters as it appears in their respective papers. Additionally, for each algorithm, we take the stopping criteria to be \(\Vert \vartheta (w_k) \Vert \le 10^{-5}.\)

Table 1 List of test problems with references
Full size table
Fig. 1
figure 1

Performance profiles for the number of iterations

Full size image
Fig. 2
figure 2

Performance profiles for the CPU time (in seconds)

Full size image

In the numerical experiments, we consider six different initial points \(x_1=(0.1, 0.1, \dots , 0.1)^T, x_2=(0.2, 0.2, \dots , 0.2)^T, x_3=(0.5, 0.5, \dots , 0.5)^T, x_4=(1.2, 1.2, \dots , 1.2)^T, x_5=(1.5, 1.5, \dots 1.5)^T, x_6=(2, 2, \dots , 2)^T\) and five dimensions ranging from 1000 to 100, 000. Therefore, we performed 300 cases for the experiments. The results of the experiments in terms of number of functions evaluations, CPU elapsed time, and number of iterations can be seen at https://documentcloud.adobe.com/link/review?uri=urn:aaid:scds:US:c6829e5f-3f54-46d2-ab38-e10a2a1dd2f7.

It can be seen from the results in the table of the compared algorithms that the proposed MFRDF method performs better than the compared methods in terms of number of iterations, elapsed CPU time, and number of function evaluations. Moreover, to visualize the extent of performance of the proposed method in comparison with the MPCGM and Algorithm 2.1, we adopt the performance profiles from [16]. It can be seen, respectively, from Figs. 1, 2, and 3 that MFRDF has more than \(75 \%\) performance in terms of number of iterations, about \(72 \%\) in terms of CPU time, and \(55 \%\) success in terms of number of function evaluations.

Fig. 3
figure 3

Performance profiles for the number of function evaluations

Full size image

4 Conclusion

This paper proposed an FR-like derivative-free algorithm combined with the projection technique for solving nonlinear monotone operator equations. The proposed search direction is bounded and satisfies the sufficient descent condition. Useful assumptions were considered to establish the global convergence. The results obtained from the numerical experiments illustrate the strength and efficiency of the proposed algorithm in contrast with the existing algorithms.