A Scaled Conjugate Gradient Method for Solving Monotone Nonlinear Equations with Convex Constraints

Abstract

Based on the Scaled conjugate gradient (SCALCG) method presented by Andrei (2007) and the projection method presented by Solodov and Svaiter, we propose a SCALCG method for solving monotone nonlinear equations with convex constraints. SCALCG method can be regarded as a combination of conjugate gradient method and Newton-type method for solving unconstrained optimization problems. So, it has the advantages of the both methods. It is suitable for solving large-scale problems. So, it can be applied to solving large-scale monotone nonlinear equations with convex constraints. Under reasonable conditions, we prove its global convergence. We also do some numerical experiments show that the proposed method is efficient and promising.

1. Introduction

In this paper, we consider the following convex constrained monotone equations: 𝐹(𝑥)=0,𝑥∈Ω, (1) where 𝐹∶𝑅^𝑛→𝑅^𝑛 is a continuous and monotone function. The feasible region Ω is a nonempty closed convex set. Monotone means that 𝐹(𝑥)−𝐹𝑦,𝑥−𝑦≥0,∀𝑥,𝑦∈𝑅^𝑛. (2)

The algorithms of solving monotone nonlinear equations 𝐹(𝑥)=0 have strong relationship to algorithms of solving optimization problems. It’s known that the function 𝑓(𝑥) is strictly function is equivalent to that the vector function ∇𝑓 is strictly monotone which means (∇𝑓(𝑥)−∇𝑓(𝑦))^𝑇(𝑥−𝑦)>0, and the definition of monotone nonlinear equations is same to this. The strictly convex function must exists unique minimum point, so the minimum point is a stable point of the convex functions, namely, the point which the gradient vector ∇𝑓(𝑥)=0. The monotone vector function 𝐹(𝑥)=0 can be seen as a gradient vector of some strictly convex function. There exists strictly convex function 𝑓(𝑥), satisfying ∇𝑓(𝑥)=𝐹(𝑥), Therefore, solving min𝑓(𝑥) is equivalent to solving 𝐹(𝑥)=0.

Nonlinear monotone equations arise in wide variety of applications, such as subproblems in the generalized proximal algorithms with Bergman distances [1]. In power engineering, the operations of a power system are described by a system of nonlinear equations, called the power flow equations, which are constrained by some operating constraints.

It has received much attention for the unconstrained nonlinear monotone equations [2–5]. Solodov and Svaiter [2] proposed a Newton-type method and a good property of the method is that the whole sequence of iterates converges to a solution of the system without any regularity assumptions. Under some weaker conditions, Zhou and Toh [4] showed that the Solodov and Svaiter’s method is super linear convergence. Zhou and Li [5, 6] extended Solodov and Svaiter’s projection method to the BFGS method and limited memory BFGS method. Zhang and Zhou [3] combined the spectral gradient method and the projection method of Solodov and Svaiter, proposed a spectral gradient projection method. Wang et al. [7] extended Solodov and Svaiter’s projection method to solve monotone equations with convex constraints. Yu et al. [8] proposed a spectral gradient projection algorithm for monotone nonlinear equations with convex constraints by combining a modified spectral gradient method and the projection method. A good property of the method is that the linear system is not necessary at each iteration. Xiao and Zhu [9] extended CG_DESCENT to solve large-scale nonlinear convex constrained monotone equations in compressive sensing by combining with the projection method of Solodov and Svaiter. At each iteration, the proposed method is not necessary to compute the Jacobian information or store any matrix.

This paper is organized as follows. In Section 2, we propose a SCALCG method for solving monotone nonlinear equation with convex constraints. Under reasonable conditions, we prove its global convergence in Section 3. In Section 4, we do some numerical experiments show that our method are efficient and promising.

2. The Method

In this section, we propose our method. At first, we simply review the SCALCG method presented by Andrei [10] for the following unconstrained optimization problems. min𝑓(𝑥),𝑥∈𝑅^𝑛, (3) where 𝐹∶𝑅^𝑛→𝑅 is a continuously differentiable function, 𝑔_𝑘 is its gradient at point 𝑥_𝑘.

The method of Andrei generate a sequence {𝑥_𝑘} of approximations to the minimum 𝑥^∗ of 𝑓, in which 𝑥_𝑘+1=𝑥_𝑘+𝛼_𝑘𝑑_𝑘,𝑑_𝑘+1=−𝜃_𝑘+1𝑔_𝑘+1+𝜃_𝑘+1𝑔^𝑇_𝑘+1𝑠_𝑘𝑦^𝑇_𝑘𝑠_𝑘𝑦_𝑘−1+𝜃_𝑘+1𝑦^𝑇_𝑘𝑦_𝑘𝑦^𝑇_𝑘𝑠_𝑘𝑔^𝑇_𝑘+1𝑠_𝑘𝑦^𝑇_𝑘𝑠_𝑘−𝜃_𝑘+1𝑔^𝑇_𝑘+1𝑦_𝑘𝑦^𝑇_𝑘𝑠_𝑘𝑠_𝑘, (4)(5) where 𝜃_𝑘+1=𝑠^𝑇_𝑘𝑠_𝑘/𝑦^𝑇_𝑘𝑠_𝑘.

Based on the SCALCG method, we now introduce our method for solving (1). Inspired by (5), we define 𝑑_𝑘 as 𝑑_𝑘=−𝐹₀,𝑘=0,−𝜃_𝑘𝐹_𝑘+𝜃_𝑘𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1𝑦_𝑘−1−1+𝜃_𝑘𝑦^𝑇_𝑘−1𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1−𝜃_𝑘𝐹^𝑇_𝑘𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1𝑠_𝑘−1,𝑘≥1, (6) where 𝑦_𝑘=𝛾_𝑘+𝜆_𝑘𝑡_𝑘‖𝐹_𝑘‖𝑑_𝑘, 𝛾_𝑘=𝐹_𝑘+1−𝐹_𝑘, 𝜆_𝑘=1+‖𝐹_𝑘‖⁻¹max{0,−⟨𝛾_𝑘,𝑡_𝑘𝑑_𝑘⟩/‖𝑡_𝑘𝑑_𝑘‖²}, 𝑠_𝑘=𝑧_𝑘−𝑥_𝑘=𝑡_𝑘𝑑_𝑘, 𝑡_𝑘 is a step length which will be defined later. The definition of 𝑦_𝑘 is similar to the one in [9].

Lemma 1. Let {𝑑_𝑘} be generated by (6), then for any 𝑘, we have 𝐹^𝑇_𝑘𝑑_𝑘<0. (7)

Proof. If 𝑘=0, we have 𝐹^𝑇₀𝑑₀=−‖𝐹₀‖²<0.
If 𝑘≥1, we obtain 𝐹^𝑇_𝑘𝑑_𝑘=−𝜃_𝑘𝐹_𝑘²+𝜃_𝑘𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹^𝑇_𝑘𝑦_𝑘−1−1+𝜃_𝑘𝑦_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1−𝜃_𝑘𝐹^𝑇_𝑘𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1=1𝑦^𝑇_𝑘−1𝑠_𝑘−1²−𝜃_𝑘𝐹_𝑘²𝑦^𝑇_𝑘−1𝑠_𝑘−1²+𝜃_𝑘𝐹^𝑇_𝑘𝑠_𝑘−1𝐹^𝑇_𝑘𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1−𝐹^𝑇_𝑘𝑠_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1−𝜃_𝑘𝑦_𝑘−1²𝐹^𝑇_𝑘𝑠_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1+𝜃_𝑘𝐹^𝑇_𝑘𝑦_𝑘−1𝐹^𝑇_𝑘𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1=1𝑦^𝑇_𝑘−1𝑠_𝑘−1²−𝜃_𝑘𝐹_𝑘²𝑦^𝑇_𝑘−1𝑠_𝑘−1²−𝜃_𝑘𝑦_𝑘−1²𝐹^𝑇_𝑘𝑠_𝑘−1²+2𝜃_𝑘𝐹^𝑇_𝑘𝑠_𝑘−1𝐹^𝑇_𝑘𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1−𝐹^𝑇_𝑘𝑠_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1, (8) where 2𝐹^𝑇_𝑘𝑠_𝑘−1𝐹^𝑇_𝑘𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1=2𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹^𝑇_𝑘⋅𝐹^𝑇_𝑘𝑠_𝑘−1𝑦_𝑘−1≤𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹_𝑘²+𝐹^𝑇_𝑘𝑠_𝑘−1𝑦_𝑘−1²=𝑦^𝑇_𝑘−1𝑠_𝑘−1²𝐹_𝑘²+𝐹^𝑇_𝑘𝑠_𝑘−1²𝑦_𝑘−1². (9) So, we have 𝐹^𝑇_𝑘𝑑_𝑘𝐹≤−^𝑇𝑘𝑠_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1. (10) By the definition of 𝜆_𝑘, the following inequality holds 𝜆_𝑘≥1−𝐹_𝑘⁻¹𝛾_𝑘,𝑡_𝑘𝑑_𝑘𝑡_𝑘𝑑_𝑘². (11) So, we obtain 𝑦^𝑇_𝑘𝑠_𝑘=𝛾_𝑘+𝜆_𝑘𝐹_𝑘𝑠_𝑘,𝑠_𝑘=𝛾_𝑘,𝑠_𝑘+𝜆_𝑘𝐹_𝑘𝑠_𝑘²≥𝛾_𝑘,𝑠_𝑘+𝐹_𝑘𝑠_𝑘²−𝛾_𝑘,𝑠_𝑘=𝐹_𝑘𝑠_𝑘²>0. (12) It can be seen that 𝐹^𝑇_𝑘𝑑_𝑘𝐹≤−^𝑇𝑘𝑠_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1<0. (13)

The steps of our method are stated as follows.

Algorithm 2. Consider the following steps.
Step  0. Choose an initial point 𝑥₀∈Ω, and constants 𝜀>0, 𝜎∈(0,1), 𝜌∈(0,1), 𝜉>0 Set 𝑘∶=0.
Step 1. Stop if ‖𝐹_𝑘‖≤𝜀. Otherwise, compute 𝑑_𝑘 by (6).
Step 2. Let 𝑡_𝑘=max{𝜉𝜌^𝑖∶𝑖=0,1,2,…} which satisfies −𝐹𝑥_𝑘+𝑡_𝑘𝑑_𝑘,𝑑_𝑘≥𝜎𝑡_𝑘𝑑_𝑘². (14) Let 𝑧_𝑘=𝑥_𝑘+𝑡_𝑘𝑑_𝑘.
Step 3. Compute 𝑥_𝑘+1=𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘, (15) where 𝛼_𝑘=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝐹𝑧_𝑘². (16)
Step 4. Let 𝑘∶=𝑘+1. Go to Step 1.

3. Convergence Analysis

In this section, we establish the global convergence of Algorithm 2. For our purpose, we assume that 𝐹 satisfies the following assumptions.

Condition A. Consider the following.(1)The mapping 𝐹 is Lipchitz continuous, it means that it satisfies 𝐹(𝑥)−𝐹𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈Ω. (17)(2)The solution set of (1), denoted by 𝑆, is nonempty.

Lemma 3. Algorithm 2 is well defined.

Proof. We just need prove that Step 2 is well defined in Algorithm 2. We take the limit of the both sides of (14), we have lim_𝑡𝑘→0−𝐹𝑥_𝑘+𝑡_𝑘𝑑_𝑘,𝑑_𝑘=lim_𝑡𝑘→0−𝐹^𝑇_𝑘𝑑_𝑘>0,lim_𝑡𝑘→0𝜎𝑡_𝑘𝑑_𝑘²=0. (18) So Algorithm 2 is well defined.

Lemma 4. Suppose Condition A hold, the step length 𝑡_𝑘 satisfies 𝑡_𝑘≥min𝜉,𝜌𝛿()𝐹𝐿+𝜎𝑑^𝑘2_𝑘². (19)

Proof. If the algorithm stops at some iteration 𝑘 then ‖𝐹_𝑘‖=0, so that 𝑥_𝑘 is a solution of (1). From now on, we assume that 𝐹_𝑘≠0 for any 𝑘. It is easy to see that 𝑑_𝑘≠0 from (7).
If 𝑡_𝑘≠𝜉, by the line search process, we know that 𝑡^′_𝑘=𝜌⁻¹𝑡_𝑘 does not satisfies (14), that is −𝐹𝑧^′_𝑘,𝑑_𝑘≤𝜎𝑡^′_𝑘𝑑_𝑘², (20) where 𝑧^′_𝑘=𝑥_𝑘+𝑡^′_𝑘𝑑_𝑘.
From (7), we know 𝐹^𝑇_𝑘𝑑_𝑘<0, (21) So, for any 𝑘≥0, there exists a positive number 𝛿>0, such that 𝐹^𝑇_𝑘𝑑_𝑘≤−𝛿‖𝐹_𝑘‖².
From (7) and condition (1), we have 𝛿𝐹_𝑘²≤−𝐹^𝑇_𝑘𝑑_𝑘=−𝐹𝑥_𝑘,𝑑_𝑘=𝐹𝑧^′_𝑘−𝐹𝑥_𝑘,𝑑_𝑘−𝐹𝑧^′_𝑘,𝑑_𝑘≤𝐿𝑧^′_𝑘−𝑥_𝑘𝑑_𝑘+𝜎𝑡^′_𝑘𝑑_𝑘²=𝐿𝑡^′_𝑘𝑑_𝑘²+𝜎𝑡^′_𝑘𝑑_𝑘²=𝜌⁻¹(𝐿+𝜎)𝑡_𝑘𝑑_𝑘². (22) So we get 𝑡_𝑘≥𝜌𝛿()𝐹𝐿+𝜎𝑑^𝑘2_𝑘²,𝑡_𝑘≥min𝜉,𝜌𝛿()𝐹𝐿+𝜎𝑑^𝑘2_𝑘². (23)

Lemma 5. Suppose Condition A hold and 𝑥∈𝑆, the sequence {𝑥_𝑘} is generated by Algorithm 2. Then the sequence {‖𝐹_𝑘‖} is bounded. That means for all 𝑘≥0, there exists a positive 𝑀>0, such that 𝐹𝑥_𝑘≤𝑀. (24)

Proof. From (2), we have 𝐹𝑧_𝑘,𝑥_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝑧_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑧_𝑘,𝑧_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑧_𝑘−𝐹(𝑥),𝑧_𝑘−𝑥>𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘. (25) From the non-expansiveness of the projection operator, it holds 𝑥_𝑘+1−𝑥²=𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘−𝑥²=𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘−𝑃_Ω(𝑥)²≤𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘−𝑥²=𝑥_𝑘−𝑥²−2𝛼_𝑘𝐹𝑧_𝑘,𝑥_𝑘−𝑥+𝛼²_𝑘𝐹𝑧_𝑘²≤𝑥_𝑘−𝑥²−2𝛼_𝑘𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝛼²_𝑘𝐹𝑧_𝑘²=𝑥_𝑘−𝑥²𝐹𝑧−2^𝑘,𝑥_𝑘−𝑧_𝑘𝐹𝑧_𝑘²×𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘²𝐹𝑧_𝑘⁴𝐹𝑧_𝑘²=𝑥_𝑘−𝑥²−𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘²𝐹𝑧_𝑘²≤𝑥_𝑘−𝑥². (26) It is easy to see 𝑥_𝑘+1−𝑥²≤𝑥_𝑘−𝑥²≤𝑥_𝑘−1−𝑥²≤𝑥_𝑘−2−𝑥²≤⋯≤𝑥₀−𝑥². (27) Since 𝐹(𝑥) is Lipchitz continuous, we get 𝐹𝑥_𝑘=𝐹𝑥_𝑘−𝐹(𝑥)≤𝐿𝑥_𝑘−𝑥≤𝐿𝑥₀−𝑥. (28) Let 𝑀=𝐿‖𝑥₀−𝑥‖, then (46) is established.

Lemma 6. Suppose Condition A hold, and the sequence {𝑥_𝑘} and {𝑧_𝑘} are generated by Algorithm 2. Then, −𝐹(𝑧_𝑘) is a decent direction of the function (1/2)‖𝑥−𝑥‖² at the point 𝑥_𝑘, where 𝑥∈𝑆.

Proof. The gradient of the function (1/2)‖𝑥−𝑥‖² is 𝑔_𝑘=𝑥_𝑘−𝑥.
From (2), it can be seen that 𝐹𝑧_𝑘,𝑥_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝑧_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑧_𝑘,𝑧_𝑘−𝑥=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑧_𝑘−𝐹(𝑥),𝑧_𝑘−𝑥>0. (29) So, we obtain −𝐹𝑧_𝑘,𝑥_𝑘−𝑥<0. (30)

Lemma 7. Suppose Condition A hold, and the sequence {𝑥_𝑘} and {𝑧_𝑘} are generated by Algorithm 2. Then we have the following:(1){𝑥_𝑘} and {𝑧_𝑘} are bounded.(2)lim_𝑘→∞(𝑥_𝑘−𝑧_𝑘)=0.
Particularly, we have lim_𝑘→∞𝑡_𝑘𝑑_𝑘=0. (31)(3)lim_𝑘→∞(𝑥_𝑘−𝑥_𝑘+1)=0.

Proof. (1) From (26), we have 𝑥_𝑘+1−𝑥²≤𝑥_𝑘−𝑥²≤𝑥_𝑘−1−𝑥²≤𝑥_𝑘−2−𝑥²≤⋯≤𝑥₀−𝑥². (32) So the sequence {𝑥_𝑘} is bounded.
From (2), (14), and (24), we get 𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘=𝐹𝑧_𝑘,−𝑡_𝑘𝑑_𝑘=−𝑡_𝑘𝐹𝑧_𝑘,𝑑_𝑘≥𝜎𝑡²_𝑘𝑑_𝑘²=𝜎𝑥_𝑘−𝑧_𝑘²,𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘=𝐹𝑧_𝑘−𝐹𝑥_𝑘,𝑥_𝑘−𝑧_𝑘+𝐹𝑥_𝑘,𝑥_𝑘−𝑧_𝑘≤𝐹𝑥_𝑘𝑥_𝑘−𝑧_𝑘≤𝑀𝑥_𝑘−𝑧_𝑘. (33) So, the following inequality holds 𝜎𝑥_𝑘−𝑧_𝑘²≤𝐹𝑥_𝑘𝑥_𝑘−𝑧_𝑘. (34) That is, 𝑥_𝑘−𝑧_𝑘≤𝑀𝜎. (35) So, the sequence {𝑧_𝑘} is bounded.

(2) From (26), we obtain 𝑥_𝑘+1−𝑥²≤𝑥_𝑘−𝑥²−𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘²𝐹𝑧_𝑘²≤𝑥_𝑘−𝑥²−𝜎²𝑥_𝑘−𝑧_𝑘⁴𝐹𝑧_𝑘². (36)

Since the function 𝐹(𝑥) is continuous, and the sequence {𝑧_𝑘} is bounded, so the sequence ‖𝐹(𝑧_𝑘)‖ is bounded, that is for all 𝑘≥0, that exists a positive 𝑀₁>0, such that ‖𝐹(𝑧_𝑘)‖≤𝑀₁. Then, we get 𝑥_𝑘−𝑧_𝑘⁴≤𝑀²₁𝜎²𝑥_𝑘−𝑥²−𝑥_𝑘+1−𝑥²,_∞𝑘=0𝑥_𝑘−𝑧_𝑘⁴≤^∞_𝑘=0𝑀²₁𝜎²𝑥_𝑘−𝑥²−𝑥_𝑘+1−𝑥²<+∞. (37) So, we have lim_𝑘→∞𝑥_𝑘−𝑧_𝑘=0. (38)

Particularly, we obtain lim_𝑘→∞𝑡_𝑘𝑑_𝑘=lim_𝑘→∞𝑥_𝑘−𝑧_𝑘=0. (39)

(3) From the non-expansiveness of the projection operator, it holds 𝑥_𝑘−𝑥_𝑘+1=𝑥_𝑘−𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘=𝑃_Ω𝑥_𝑘−𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘≤𝑥_𝑘−𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘=𝛼_𝑘𝐹𝑧_𝑘=𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝐹𝑧_𝑘²𝐹𝑧_𝑘≤𝑥_𝑘−𝑧_𝑘. (40) So, we obtain lim_𝑘→∞𝑥_𝑘−𝑥_𝑘+1=0. (41)

Theorem 8. Suppose Condition A hold, and the sequence {𝑥_𝑘} is generated by Algorithm 2. Then, we have lim_𝑘→∞inf𝐹_𝑘=0. (42)

Proof. If (42) does not hold, for any 𝑘≥0, there exist 𝜀>0, such that 𝐹_𝑘≥𝜀. (43) From the nonexpansiveness of the projection operator, it holds 𝑥_𝑘+1−𝑥_𝑘=𝑃_Ω𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘−𝑥_𝑘≤𝑥_𝑘−𝛼_𝑘𝐹𝑧_𝑘−𝑥_𝑘=𝛼_𝑘𝐹𝑧_𝑘. (44) By the definition of 𝛼_𝑘 and Cauchy-Schwartz inequality, we have 𝑥_𝑘+1−𝑥_𝑘≤𝐹𝑧_𝑘,𝑥_𝑘−𝑧_𝑘𝐹𝑧_𝑘²𝐹𝑧_𝑘≤𝐹𝑧_𝑘𝑥_𝑘−𝑧_𝑘𝐹𝑧_𝑘²𝐹𝑧_𝑘=𝑥_𝑘−𝑧_𝑘=𝑡_𝑘𝑑_𝑘. (45) By the definition of 𝑦_𝑘, assumption (1) and (45), we obtain 𝑦_𝑘=𝛾_𝑘+𝜆_𝑘𝑡_𝑘𝐹_𝑘𝑑_𝑘≤𝛾_𝑘+𝜆_𝑘𝐹_𝑘𝑡_𝑘𝑑_𝑘=𝛾_𝑘+1+𝐹_𝑘⁻¹𝛾max0,−^𝑘,𝑡_𝑘𝑑_𝑘𝑡_𝑘𝑑_𝑘²𝐹_𝑘𝑡_𝑘𝑑_𝑘≤𝛾_𝑘+1+𝐹_𝑘⁻¹𝛾_𝑘,𝑡_𝑘𝑑_𝑘𝑡_𝑘𝑑_𝑘²𝐹_𝑘𝑡_𝑘𝑑_𝑘≤𝛾_𝑘+1+𝐹_𝑘⁻¹𝛾_𝑘𝑡_𝑘𝑑_𝑘𝑡_𝑘𝑑_𝑘²𝐹_𝑘𝑡_𝑘𝑑_𝑘=2𝛾_𝑘+𝐹_𝑘𝑡_𝑘𝑑_𝑘=2𝐹𝑥_𝑘+1−𝐹𝑥_𝑘+𝐹_𝑘𝑡_𝑘𝑑_𝑘≤2𝐿𝑥_𝑘+1−𝑥_𝑘+𝐹_𝑘𝑡_𝑘𝑑_𝑘≤2𝐿𝑡_𝑘𝑑_𝑘+𝑀𝑡_𝑘𝑑_𝑘=(2𝐿+𝑀)𝑡_𝑘𝑑_𝑘. (46)
From (12), we get 𝑦^𝑇_𝑘−1𝑠_𝑘−1≥𝐹_𝑘−1𝑠_𝑘−1²≥𝜀𝑠_𝑘−1². (47)
From (46), we have 𝑦_𝑘−1≤(2𝐿+𝑀)𝑠_𝑘−1,𝜃_𝑘=𝑠^𝑇_𝑘−1𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1≤𝑠_𝑘−1²𝜀𝑠_𝑘−1²=1𝜀. (48)
From (7), we get 𝐹_𝑘𝑑_𝑘≤−𝛿𝐹_𝑘². (49) So, we obtain 𝛿𝐹_𝑘²≤−𝐹_𝑘𝑑_𝑘≤𝐹_𝑘𝑑_𝑘. (50) That is, 𝑑_𝑘≥𝛿𝐹_𝑘≥𝛿𝜀. (51) From (6), (24), (43), and (47), we have 𝑑_𝑘≤𝜃_𝑘𝐹_𝑘+𝜃_𝑘𝐹_𝑘𝑠_𝑘−1𝑦_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1+1+𝜃_𝑘𝑦_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1𝐹_𝑘𝑠_𝑘−1²𝑦^𝑇_𝑘−1𝑠_𝑘−1+𝜃_𝑘𝐹_𝑘𝑦_𝑘−1𝑠_𝑘−1𝑦^𝑇_𝑘−1𝑠_𝑘−1≤1𝜀𝐹_𝑘+1𝜀𝐹_𝑘𝑠_𝑘−1(2𝐿+𝑀)𝑠_𝑘−1𝜀𝑠_𝑘−1²+11+𝜀(2𝐿+𝑀)²𝑠_𝑘−1²𝜀𝑠_𝑘−1²𝐹_𝑘𝑠_𝑘−1²𝜀𝑠_𝑘−1²+1𝜀𝐹_𝑘(2𝐿+𝑀)𝑠_𝑘−1²𝜀𝑠_𝑘−1²=1𝜀𝐹_𝑘+1𝜀²(2𝐿+𝑀)𝐹_𝑘+()1+2𝐿+𝑀²𝜀²1𝜀𝐹_𝑘+(2𝐿+𝑀)𝜀²𝐹_𝑘≤2𝜀+2(2𝐿+𝑀)𝜀²+(2𝐿+𝑀)²𝜀³𝑀. (52)
Let (2/𝜀+2(2𝐿+𝑀)/𝜀²+(2𝐿+𝑀)²/𝜀³)𝑀=𝐶, then for all 𝑘≥0, we have 𝑑_𝑘≤𝐶. (53) From (19), (43), and (53), it can be seen that 𝑡_𝑘𝑑_𝑘≥min𝜉,𝜌𝛿𝐹𝐿+𝜎𝑑^𝑘2_𝑘²𝑑_𝑘=min𝜉𝑑_𝑘,𝜌𝛿𝐹𝐿+𝜎𝑑^𝑘2_𝑘≥min𝜉𝛿𝜀,𝜌𝛿𝜀²𝐶().𝐿+𝜎 (54) The last inequality yields a contradiction with (31), so (42) holds.

4. Numerical Experiments

In this section, we do some numerical experiments to test the performance of Algorithm 2 on the following two problems. The algorithm was coded in Matlab and run on a personal computer with a 2.3 GHZ CPU and 2 GB memory and Windows XP operating system.

For each test problem, the termination condition is 𝐹𝑥_𝑘≤10⁻⁵. (55) We set 𝜉=1, 𝜌=0.1, 𝜎=0.0001. We test both problems with the number of variables 𝑛=100, 500, 1000, 2000, and 5000 and start form different initial points. The meaning of the columns in Tables 1 and 2 is stated as follows. “Dim” means the dimension of the problem, “Init” means the initials points, “Iter” means the number of iterations, “Time” stands for CPU time in seconds, and “Fn” stands for the final norm of equations.

Table 1 

Test results for Problem 9 with given initial points.

Table 2 

Test results for Problem 10 with given initial points.

Problem 9. The 𝐹 is taken as 𝐹(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥))^Τ, where 𝑓_𝑖(𝑥)=𝑒^𝑥𝑖−2,𝑖=1,2,…,𝑛,Ω=𝑅^𝑛₊. (56)

Problem 10. The 𝐹 is taken as 𝐹(𝑥)=(𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑛(𝑥))^Τ, where 𝑓_𝑖(𝑥)=2𝑥_𝑖−sin𝑥_𝑖−1,𝑖=1,2,…,𝑛,Ω=𝑅^𝑛₊. (57)

Tables 1 and 2 show that our method is efficient. It is suitable for solving large-scale monotone equations with convex constraints.

5. Conclusions

In this paper, we have proposed a SCALCG method for solving nonlinear monotone equations with convex constraints. Under some wild conditions, we proved its global convergence.

Preliminary numerical experiments have illustrated that the proposed method works well for Problems 9 and 10.

Acknowledgment

This work has been supported by Scientific Research Fund of Hunan Provincial Education Department [12C0664].