Three-term derivative-free projection method for solving nonlinear monotone equations

Abstract

In this paper, a three-term derivative-free projection method is proposed for solving nonlinear monotone equations. Under some appropriate conditions, the global convergence and R-linear convergence rate of the proposed method are analyzed and proved. With no need of any derivative information, the proposed method is able to solve large-scale nonlinear monotone equations. Numerical comparisons show that the proposed method is effective.

Keywords

Nonlinear monotone equations / conjugate gradient method / derivative-free projection method / global convergence / R-linear convergence rate

1 Introduction

In this paper, we consider solving the nonlinear monotonic system of equations:

(1.1)F\left(x\right)=0,

where F:{\mathbb{R}}^n\to {\mathbb{R}}^n is continuous nonlinear mapping. The monotonicity of Fmeans:

(1.2)⟨F\left(x\right)-F\left(y\right),x-y⟩⩾0,\mathrm{\forall }x,y\in {\mathbb{R}}^n.

Nonlinear monotone equations are originated from many scientific and practical problems, such as first-order necessity conditions for unconstrained convex optimization problems and subproblems of generalized proximity point algorithms [9], some monotone variational inequality problems [20], chemical equilibrium problems [14], economic equilibrium problems [6], and power equations [4]. Therefore, many algorithms have been proposed to solve nonlinear monotone systems of equations, among which Newton’s Method and Quasi-Newton Methods [4, 5, 11, 17, 21-23] are well known by researchers due to their excellent convergence speed. However, these algorithms are not suitable for solving large-scale non-smooth systems of equations, because they need to solve a linear system of equations using the Jacobian matrix of the objective function or an approximate Jacobian matrix at each iteration to determine the next search direction.

In recent years, in order to efficiently solve large-scale systems of equations, many researchers have promoted the nonlinear conjugate gradient method under the hyperplane projection technique [17]. For example, Cheng [3] combined the well-known PRP method [15, 16] and hyperplane projection technique to study the derivative-free PRP-type projection method, and proved the global convergence for nonlinear monotone systems of equations without requiring the objective function to be differentiable. Li et al. [12] and Ahookhosh et al. [1] further studied three derivative-free PRP-type projection algorithms. Wu et al. [18] established a derivative-free three-term HS projection algorithm based on the well-known HS conjugate gradient method [8] and hyperplane projection technique. The algorithm inherits the advantages of the HS conjugate gradient method with small storage and does not need to calculate any derivatives, thus it can solve large-scale non-smooth nonlinear monotonic systems of equations. Xiao and Zhu [19], Liu and Li [13] proposed a derivative-free CG_DESCENT type projection method based on the CG_DESCENT method [7] and hyperplane projection technique, and achieved good numerical results and proved the global convergence of the method under appropriate conditions.

Recently, Andrei [2] proposed a three-term nonlinear conjugate gradient method (denoted as TTCG method) based on the well-known CG_DESCENT method. This method is a modified memoryless BFGS algorithm whose search direction satisfies the descent and D-L conjugacy conditions. Numerous numerical experiments have shown that the TTCG method is more efficient than the well-known CG_DESCENT method. To solve the large-scale system of equations problem more effectively, this paper extends the TTCG method based on the hyperplane projection technique and establishes a three-term derivative-free projection algorithm. Under appropriate assumptions, we prove the global convergence and R-linear convergence rate of the algorithm.

2 Derivative-free projection method

The iterative method is one of the most used methods for solving nonlinear systems of equations problems. It generates an iterative column of points \left\{x_k\right\}, which is usually based on the following unconstrained optimization problem, namely

(2.1)\underset{x\in {\mathbb{R}}^n}{min}f\left(x\right),

where f\left(x\right)=\frac{1}{2}{\Vert F\left(x\right)\Vert }^2. Various iterative methods based on solving Problem (2.1) have been widely studied and applied in practice. However, the stable point of Problem (2.1) is not necessarily the solution of Problem (1.1). Moreover, in many practical problems, the mapping F is usually non-smooth.

To solve Problem (1.1) directly, Solodov and Svaiter [17] discussed the hyperplane projection method. Let x_k and d_k be the current iteration point and the search direction, respectively. A certain line search was used to obtain the step {\alpha }_k so that the point z_k=x_k+{\alpha }_k{d}_k is obtained such that:

F^{\mathrm{T}}\left(z_k\right)\left(x_k-z_k\right)>0.

If F\left(\tilde{x}\right)=0 for some point \tilde{x}, then it follows from the monotonicity of F:

F^{\mathrm{T}}\left(z_k\right)\left(\tilde{x}-z_k\right)⩽0.

Thus, the hyperplane H_k=\left\{F^{\mathrm{T}}\left(z_k\right)\left(x-z_k\right)=0\right\} strictly separates the current iteration point x_k from the solution of the nonlinear monotonic system of equations. Using the hyperplane, the next iteration point x_{k+1} is obtained:

(2.2)x_{k+1}=x_k-\frac{F^{\mathrm{T}}\left(z_k\right)\left(x_k-z_k\right)}{{\Vert F\left(z_k\right)\Vert }^2}F\left(z_k\right).

The following is a three-term derivative-free projection algorithm for solving a nonlinear monotonic system of equations based on the TTCG method and the hyperplane projection method.

Algorithm 2.1

Step 0: Given the initial point x_0\in {\mathbb{R}}^n and the constant \sigma \in \left(0,1\right),t⩾0,\beta >0. If \Vert F_0\Vert =0, stop.

Step 1: Set d_0\in -F_0, put k:=0.

Step 2: Calculate the step {\alpha }_k={\beta }^{m_k} such that m_k satisfies the following equation for the smallest non-negative integer m:

(2.3)-F^{\mathrm{T}}\left(x_k+{\alpha }_k{d}_k\right)d_k⩾\sigma {\alpha }_k\Vert F\left(x_k+{\alpha }_k{d}_k\right)\Vert \cdot {\Vert d_k\Vert }^2.

Step 3: Let z_k=x_k+{\alpha }_k{d}_k. If \Vert F\left(z_k\right)\Vert =0, stop. Otherwise, use (2.2) to solve for x_{k+1}.

Step 4: If \Vert F_{k+1}\Vert =0, stop. Otherwise, calculate the search direction:

(2.4)d_{k+1}=-F_{k+1}+{\beta }_{k+1}{d}_k+{\theta }_{k+1}\left(d_k+y_k\right),

where {\beta }_{k+1}=\frac{1}{d_k^{\mathrm{T}}{\omega }_k}{\left(y_k-t{d}_k\frac{{\Vert y_{\text{k}}\Vert }^2}{d_k^{\mathrm{T}}{\omega }_k}\right)}^{\mathrm{T}}{F}_{k+1}, {\theta }_{k+1}=-\frac{F_{k+1}^{\mathrm{T}}{d}_k}{d_k^{\mathrm{T}}{\omega }_k}, y_k=-F_{k+1}-F_k, {\omega }_k=y_k+t_k{d}_k, t_k = 1+max\left\{0,\frac{d_k^{\mathrm{T}}{y}_k}{d_k^{\mathrm{T}}{d}_k}\right\}.

Step 5: Set k:=k+1, turn to Step 2.

The following theorem shows that Algorithm 2.1 is a descending iterative algorithm when F is the gradient of the real-valued function f:{\mathbb{R}}^n\to \mathbb{R}.

Theorem 2.1　 Let the sequence \left\{F_k\right\} and \left\{d_k\right\} be generated by Algorithm 2.1, then

(2.5)F_k^{\mathrm{T}}{d}_k⩽-{\Vert F_k\Vert }^2,\mathrm{\forall }k⩾0.

Proof　According to the definition of parameter{\omega }_k, it can be seen that

(2.6)d_k^{\mathrm{T}}{\omega }_k=d_k^{\mathrm{T}}{y}_k+t_k{\Vert d_k\Vert }^2=d_k^{\mathrm{T}}{y}_k+{\Vert d_k\Vert }^2+max\left\{0,-d_k^{\mathrm{T}}{y}_k\right\}⩾{\Vert d_k\Vert }^2.

Multiply both ends of (2.4) by F_{k+1}^{\mathrm{T}} and use (2.6) to obtain

\begin{array}{rl}{F}_{k+1}^{\mathrm{T}}{d}_{k-1}& =-{\Vert F_{k+1}\Vert }^2+{\beta }_{k+1}{F}_{k+1}^{\mathrm{T}}{d}_k+{\theta }_{k+1}\left(F_{k+1}^{\mathrm{T}}{d}_k+F_{k+1}^{\mathrm{T}}{y}_k\right)\\ & =-{\Vert F_{k+1}\Vert }^2-\left(1+\frac{t{\Vert y_{\text{k}}\Vert }^2}{d_k^{\mathrm{T}}{\omega }_k}\right)\frac{{\left(F_{k+1}^{\mathrm{T}}{d}_k\right)}^2}{d_k^{\mathrm{T}}{\omega }_k}\\ & ⩽-{\Vert F_{k+1}\Vert }^2.\end{array}

Known that d_0=-F_0, then F_k^{\mathrm{T}}{d}_0=-{\Vert F_0\Vert }^2. The conclusion is proved.

Note 2.1　According (2.5) and (2.6), we know that

d_k^{\mathrm{T}}{\omega }_k⩾{\Vert F_k\Vert }^2.

Since that, before Algorithm 2.1 stops, the parameter {\omega }_k and {\theta }_k is always meaningful.

3 Global convergences

To prove the global convergence of Algorithm 2.1, assume that the mapping F satisfies:

(H) Let the mapping F be Lipschitz continuous, i.e., there exists a constant L>0 such that

(3.1)\Vert F\left(x\right)-F\left(y\right)\Vert ⩽L\Vert x-y\Vert ,\mathrm{\forall }x,y\in {\mathbb{R}}^n.

Lemma 3.1　 Let the mapping F satisfy (H). The sequence \left\{F_k\right\} and \left\{d_k\right\} are generated by Algorithm 2.1, then the step factor {\alpha }_k satisfies

(3.2){\alpha }_k⩾\frac{\beta }{L+\sigma \Vert F\left(x_k+{\beta }^{-1}{\alpha }_k{d}_k\right)\Vert }\cdot \frac{{\Vert F_k\Vert }^2}{{\Vert d_k\Vert }^2}.

Proof　If {\alpha }_k\ne 1, it follows from the definition of {\alpha }_k that {\beta }^{-1}{\alpha }_k does not satisfy the line search condition (2.3). Thus,

According to (2.5) and (3.1), it follows that

\begin{array}{rl}{\Vert F_k\Vert }^2& ⩽-F_k^{\mathrm{T}}{d}_k={\left[F\left(x_k+{\beta }^{-1}{\alpha }_k{d}_k\right)-F\left(x_k\right)\right]}^{\mathrm{T}}{d}_k-F{\left(x_k+{\beta }^{-1}{\alpha }_k{d}_k\right)}^{\mathrm{T}}{d}_k\\ & ⩽L{\beta }^{-1}{\alpha }_k{\Vert d_k\Vert }^2+\sigma {\beta }^{-1}{\alpha }_k\Vert F\left(x_k+{\beta }^{-1}{\alpha }_k{d}_k\right)\Vert \cdot {\Vert d_k\Vert }^2.\end{array}

So, it can be concluded that

{\alpha }_k⩾\frac{\beta }{L+\sigma \Vert F\left(x_k+{\beta }^{-1}{\alpha }_k{d}_k\right)\Vert }\cdot \frac{{\Vert F_k\Vert }^2}{{\Vert d_k\Vert }^2}.

The lemma is proven.

The following lemma shows that sequence \left\{\Vert x_k-\tilde{x}\Vert \right\} is descending and convergent, where sequence \left\{x_k\right\} is generated by Algorithm 2.1 and point \tilde{x} is an arbitrary solution to Problem (1.1). The proof process of this lemma can be found in Reference [17].

Lemma 3.2　 Assuming that mapping F satisfies (H) and sequence \left\{x_k\right\} and \left\{z_k\right\} are generated by Algorithm 2.1, then for any solution \tilde{x} to Problem (1.1), there is

(3.3){\Vert x_{k+1}-\tilde{x}\Vert }^2⩽{\Vert x_k-\tilde{x}\Vert }^2-{\Vert x_{k+1}-x_k\Vert }^2.

Especially, the sequence \left\{x_k\right\} is bounded and

(3.4)\underset{k\to \mathrm{\infty }}{lim}\Vert x_{k+1}-x_k\Vert =0.

Note 3.1　According to (2.2) and (2.3), we know that

\Vert x_{k+1}-x_k\Vert =\frac{{\alpha }_k\left|F^{\mathrm{T}}\left(z_k\right)d_k\right|}{{\Vert F\left(z_k\right)\Vert }^2}\cdot \Vert F\left(z_k\right)\Vert ⩾\sigma {\Vert {\alpha }_k{d}_k\Vert }^2.

According to (3.4), we know that

(3.5)\underset{k\to \mathrm{\infty }}{lim}\Vert {\alpha }_k{d}_k\Vert =0.

Lemma 3.3　 Assuming that the mapping F satisfies (H), the sequences\left\{d_k\right\} and \left\{F_k\right\} are generated by Algorithm 2.1, then there exists a positive real number M such that

(3.6)\Vert d_k\Vert ⩽M,\mathrm{\forall }k⩾0.

Proof　For a certain solution \tilde{x} of Problem (1.1) satisfies

\Vert F_k\Vert =\Vert F\left(x_k\right)-F\left(\tilde{x}\right)\Vert ⩽L\Vert x_k-\tilde{x}\Vert .

Then, applying the boundedness of sequence \left\{x_k\right\}, we can see that there is a normal number \mu, which makes

(3.7)\Vert F_k\Vert ⩽\mu ,\mathrm{\forall }k⩾0.

By using (2.2) and (3.1), as well as the Cauchy-Schwartz inequality and the definition of step {\alpha }_k, it can be obtained that

(3.8)\Vert y_k\Vert ⩽L\Vert x_{k+1}-x_k\Vert ⩽\frac{L{\Vert F\left(z_k\right)\Vert }^2\cdot \Vert x_k-z_k\Vert }{{\Vert F\left(z_k\right)\Vert }^2}=L\Vert {\alpha }_k{d}_k\Vert ⩽L\beta \Vert d_k\Vert .

According to (2.6), (3.7), (3.8) and Cauchy-Schwartz inequality, it can be obtained that

\begin{array}{rl}\left|{\beta }_{k+1}\right|& ⩽\frac{\Vert F_{k+1}\Vert \cdot \Vert y_k\Vert }{d_k^{\mathrm{T}}{\omega }_k}+\frac{t{\Vert y_k\Vert }^2\cdot \Vert F_{k+1}\Vert \cdot \Vert d_k\Vert }{{\left(d_k^{\mathrm{T}}{\omega }_k\right)}^2}\\ & ⩽\frac{L\beta \Vert F_{k+1}\Vert \cdot \Vert d_k\Vert }{{\Vert d_k\Vert }^2}+\frac{t{L}^2{\beta }^2{\Vert d_k\Vert }^2\cdot \Vert F_{k+1}\Vert \cdot \Vert d_k\Vert }{{\Vert d_k\Vert }^4}\\ & ⩽\frac{L\beta \mu }{\Vert d_k\Vert }+\frac{t{\beta }^2{L}^2\mu }{\Vert d_k\Vert }.\end{array}

\left|{\theta }_{k+1}\right|⩽\frac{\Vert F_{k+1}\Vert \cdot \Vert d_k\Vert }{{\Vert d_k\Vert }^2}⩽\frac{\mu }{\Vert d_k\Vert }.

Based on the above two inequalities and (2.4), (3.8), it can be concluded that

\begin{array}{rl}\Vert d_{k+1}\Vert & ⩽\Vert F_{k+1}\Vert +\left|{\beta }_{k+1}\right|\cdot \Vert d_k\Vert +\left|{\theta }_{k+1}\right|\left(\Vert d_k\Vert +\Vert y_k\Vert \right)\\ & ⩽\mu +\left(\beta L\mu +t{\beta }^2{L}^2\mu \right)+\mu \left(1+L\beta \right).\end{array}

Since d_0=-F_0, take M=\mu +\left(\beta L\mu +t{\beta }^2{L}^2\mu \right)+\mu \left(1+L\beta \right), the conclusion is proved.□

Theorem 3.1　 If the mapping F satisfies (H), the sequences \left\{d_k\right\} and \left\{F_k\right\} are generated by Algorithm 2.1, then

(3.9)\underset{k\to \mathrm{\infty }}{liminf}\Vert F_k\Vert =0.

Proof　Assuming (3.9) doesn’t hold, there exists a normal number \epsilon such that

(3.10)\Vert F_k\Vert ⩾\epsilon ,\mathrm{\forall }k⩾0.

From (3.2), (3.6) and (3.7), it can be found that

{\alpha }_k\Vert d_k\Vert ⩾\frac{\beta {\Vert F_k\Vert }^2}{\left(L+\sigma \Vert F\left(x_k+{\beta }^{-1}{\alpha }_k\right)\Vert \cdot \Vert d_k\Vert \right)}⩾\frac{\beta {\epsilon }^2}{\left(L+\sigma \mu \right)M}>0.

Obviously, it is contradictory with (3.5). Therefore, the hypothesis does not hold and the original proposition holds.

4 \mathit{R}-Linear convergence rate

To further prove the R-linear convergence rate of Algorithm 2.1, it is necessary to assume that the mapping F satisfies:

(G) Let \mathrm{\forall }\tilde{x}\in \mathrm{\Omega }, there exist positive constants \rho \in \left(0,1\right) and \nu >0 such that

(4.1)\rho \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(x,\mathrm{\Omega }\right)⩽{\Vert F_k\Vert }^2,\mathrm{\forall }x\in N\left(\tilde{x},\nu \right),

where \mathrm{\Omega } is the set of solutions of Problem (1.1)，N\left(\tilde{x},\nu \right)=\{x\in {\mathbb{R}}^n|\Vert x-\tilde{x}\Vert ⩽\nu \}，\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(x,\mathrm{\Omega }\right) denotes the distance of point x from the solution set \mathrm{\Omega }.

Theorem 4.1　 Suppose that the mapping F satisfies (H) and (G), the sequence \left\{x_k\right\} is generated by Algorithm 2.1, then the sequence \left\{dist\left(x_k,\mathrm{\Omega }\right)\right\} is Q-linearly convergent to 0. Thus, the sequence \left\{x_k\right\} is R-linearly convergent.

Proof　Let {\omega }_k=\arg min\left\{\Vert x_k-\omega \Vert |\omega \in \mathrm{\Omega }\right\}. we can know {\omega }_k is the solution closest to point x_k in solution set \mathrm{\Omega }, i.e.,

(4.2)\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x_{k+1},\mathrm{\Omega })=\Vert x_k-{\omega }_k\Vert .

By using (2.5) and Cauchy-Schwartz inequality, it can be obtained that

(4.3)\Vert d_k\Vert ⩾\Vert F_k\Vert .

Known that {\omega }_k\in \mathrm{\Omega }, using (3.3), (4.1)‒(4.3) and Note 3.1, we can get

\begin{array}{rl}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x_{k+1},\mathrm{\Omega }{)}^2& ⩽{\Vert x_{k+1}-{\omega }_k\Vert }^2⩽{\Vert x_k-{\omega }_k\Vert }^2-{\Vert x_{k+1}-x_k\Vert }^2\\ & =dist(x_k,\mathrm{\Omega }{)}^2-{\Vert x_{k+1}-x_k\Vert }^2\\ & ⩽\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x_k,\mathrm{\Omega }{)}^2-{\sigma }^2{\Vert {\alpha }_k{d}_k\Vert }^2\\ & ⩽\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x_k,\mathrm{\Omega }{)}^2-{\sigma }^2{\alpha }_k^4{\Vert F_k\Vert }^4\\ & ⩽\left(1-{\sigma }^2{\alpha }_k^4{\rho }^2\right)\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x_k,\mathrm{\Omega }{)}^2.\end{array}

This shows that the sequence \left\{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\left(x_k,\mathrm{\Omega }\right)\right\} is Q-linearly convergent to 0. Since 1-{\sigma }^2{\alpha }_k^4{\rho }^2\in \left(0,1\right), the sequence \left\{x_k\right\} is R-linearly convergent.

5 Numerical tests

Algorithm 2.1 is tested numerically below and compared numerically with the MPRP-based method [12] and the MHS-based method [18]. The program code is written using MATLAB 7.0, running on an Intel Core i5-4590 processor (quad-core, 3.30 GHz), 8.0 GB of internal memory, and the Windows 7 operating system. The selected test problems were taken as follows, from the literature [22, 3, 10], respectively.

Problem 5.1 [22]　 F is defined by the following equation:

F\left(x\right)=A\left(x\right)+g\left(x\right)-1,

where g\left(x\right)={\left({\text{e}}^{x_1},{\text{e}}^{x_2},\dots ,{\text{e}}^{x_n}\right)}^{\mathrm{T}} and

A=\left(\begin{array}{ccccc}2& -1& & & \\ -1& 2& -1& & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & -1\\ & & & -1& 2\end{array}\right).

Problem 5.2 [3]　 F is defined by the following equation:

F\left(x\right)=A\left(x\right)-1,

where

A=\left(\begin{array}{ccccc}2.5& 1& & & \\ 1& 2.5& 1& & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \ddots & 1\\ & & & 1& 2.5\end{array}\right).

Problem 5.3 [10]　 F is defined by the following equation:

F\left(x\right)={\left(F_1\left(x\right),F_2\left(x\right),\dots ,F_n\left(x\right)\right)}^{\mathrm{T}},

where

\begin{array}{rl}& F_1\left(x\right)=x_1+{\text{e}}^{\cos \left(\frac{x_1+x_2}{n+1}\right)},\\ & F_i\left(x\right)=x_i+{\text{e}}^{\cos \left(\frac{x_{i-1}+x_i+x_{i+1}}{n+1}\right)},i=2,3,\dots ,n-1.\\ & F_n\left(x\right)=x_n+{\text{e}}^{\cos \left(\frac{x_{n-1}+x_n}{n+1}\right)}.\end{array}

The parameter value in Algorithm 2.1 is \sigma =0.01,t=2,\beta =0.5. The termination conditions for all programs are \Vert F\left(x_k\right)\Vert ⩽1.0\times {10}^{-5} or \Vert F\left(z_k\right)\Vert ⩽1.0\times {10}^{-5}.

In numerical tests, the rand function in MATLAB software is used to randomly generate a fixed initial point within \left(-1,0\right),\left(0,1\right),\left(-1,1\right),\left(-2,0\right),\left(0,2\right),\left(-5,5\right),\left(-10,10\right), and calculate three test problems. The numerical results are shown in Tab.1‒Tab.7. In Tab.1‒Tab.7, “Dim” represents the dimension of the test problem, “NI” represents the number of iterations, and “NF” represents the number of calculations for F, “CPU ” represents CPU time (in seconds).

Tab.1‒Tab.7 indicate that Algorithm 2.1, PRP-based method, and HS-based method can successfully solve the given test problem starting from randomly generated initial points. For Problems 5.1 and 5.2, the numerical effects of the three test methods are comparable; However, for Problem 5.3, the performance of Algorithm 2.1 is significantly better than that of PRP-based and HS-based methods.

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