Research Article  Open Access
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 Newtontype method for solving unconstrained optimization problems. So, it has the advantages of the both methods. It is suitable for solving largescale problems. So, it can be applied to solving largescale 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: where is a continuous and monotone function. The feasible region is a nonempty closed convex set. Monotone means that
The algorithms of solving monotone nonlinear equations have strong relationship to algorithms of solving optimization problems. It’s known that the function is strictly function is equivalent to that the vector function is strictly monotone which means , 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 . The monotone vector function can be seen as a gradient vector of some strictly convex function. There exists strictly convex function , satisfying , Therefore, solving is equivalent to solving .
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 Newtontype 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 largescale 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. where is a continuously differentiable function, is its gradient at point .
The method of Andrei generate a sequence of approximations to the minimum of , in which where .
Based on the SCALCG method, we now introduce our method for solving (1). Inspired by (5), we define as where , , , , is a step length which will be defined later. The definition of is similar to the one in [9].
Lemma 1. Let be generated by (6), then for any , we have
Proof. If , we have .
If , we obtain
where
So, we have
By the definition of , the following inequality holds
So, we obtain
It can be seen that
The steps of our method are stated as follows.
Algorithm 2. Consider the following steps.
Step 0. Choose an initial point , and constants , , , Set .
Step 1. Stop if . Otherwise, compute by (6).
Step 2. Let which satisfies
Let .
Step 3. Compute
where
Step 4. Let . Go to Step 1.
3. Convergence Analysis
In this section, we establish the global convergence of Algorithm 2. For our purpose, we assume that satisfies the following assumptions.
Condition A. Consider the following.(1)The mapping is Lipchitz continuous, it means that it satisfies (2)The solution set of (1), denoted by , 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 So Algorithm 2 is well defined.
Lemma 4. Suppose Condition A hold, the step length satisfies
Proof. If the algorithm stops at some iteration then , so that is a solution of (1). From now on, we assume that for any . It is easy to see that from (7).
If , by the line search process, we know that does not satisfies (14), that is
where .
From (7), we know
So, for any , there exists a positive number , such that .
From (7) and condition (1), we have
So we get
Lemma 5. Suppose Condition A hold and , the sequence is generated by Algorithm 2. Then the sequence is bounded. That means for all , there exists a positive , such that
Proof. From (2), we have From the nonexpansiveness of the projection operator, it holds It is easy to see Since is Lipchitz continuous, we get Let , then (46) is established.
Lemma 6. Suppose Condition A hold, and the sequence and are generated by Algorithm 2. Then, is a decent direction of the function at the point , where .
Proof. The gradient of the function is .
From (2), it can be seen that
So, we obtain
Lemma 7. Suppose Condition A hold, and the sequence and are generated by Algorithm 2. Then we have the following:(1) and are bounded.(2).
Particularly, we have
(3).
Proof. (1) From (26), we have
So the sequence is bounded.
From (2), (14), and (24), we get
So, the following inequality holds
That is,
So, the sequence is bounded.
(2) From (26), we obtain
Since the function is continuous, and the sequence is bounded, so the sequence is bounded, that is for all , that exists a positive , such that . Then, we get So, we have
Particularly, we obtain
(3) From the nonexpansiveness of the projection operator, it holds So, we obtain
Theorem 8. Suppose Condition A hold, and the sequence is generated by Algorithm 2. Then, we have
Proof. If (42) does not hold, for any , there exist , such that
From the nonexpansiveness of the projection operator, it holds
By the definition of and CauchySchwartz inequality, we have
By the definition of , assumption (1) and (45), we obtain
From (12), we get
From (46), we have
From (7), we get
So, we obtain
That is,
From (6), (24), (43), and (47), we have
Let , then for all , we have
From (19), (43), and (53), it can be seen that
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 We set , , . We test both problems with the number of variables , 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.


Problem 9. The is taken as , where
Problem 10. The is taken as , where
Tables 1 and 2 show that our method is efficient. It is suitable for solving largescale 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].
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Copyright
Copyright © 2013 Sheng Wang and Hongbo Guan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.