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Global Convergence of a Modified Spectral Conjugate Gradient Method
A modified spectral PRP conjugate gradient method is presented for solving unconstrained optimization problems. The constructed search direction is proved to be a sufficiently descent direction of the objective function. With an Armijo-type line search to determinate the step length, a new spectral PRP conjugate algorithm is developed. Under some mild conditions, the theory of global convergence is established. Numerical results demonstrate that this algorithm is promising, particularly, compared with the existing similar ones.
Recently, it is shown that conjugate gradient method is efficient and powerful in solving large-scale unconstrained minimization problems owing to its low memory requirement and simple computation. For example, in [1–17], many variants of conjugate gradient algorithms are developed. However, just as pointed out in , there exist many theoretical and computational challenges to apply these methods into solving the unconstrained optimization problems. Actually, 14 open problems on conjugate gradient methods are presented in . These problems concern the selection of initial direction, the computation of step length, and conjugate parameter based on the values of the objective function, the influence of accuracy of line search procedure on the efficiency of conjugate gradient algorithm, and so forth.
The general model of unconstrained optimization problem is as follows: where is continuously differentiable such that its gradient is available. Let denote the gradient of at , and let be an arbitrary initial approximate solution of (1.1). Then, when a standard conjugate gradient method is used to solve (1.1), a sequence of solutions will be generated by where is the steplength chosen by some line search method and is the search direction defined by where is called conjugacy parameter and denotes the value of . For a strictly convex quadratical programming, can be appropriately chosen such that and are conjugate with respect to the Hessian matrix of the objective function. If is taken by where stands for the Euclidean norm of vector, then (1.2)–(1.4) are called Polak-Ribiére-Polyak (PRP) conjugate gradient method (see [8, 18]).
It is well known that PRP method has the property of finite termination when the objective function is a strong convex quadratic function combined with the exact line search. Furthermore, in , for a twice continuously differentiable strong convex objective function, the global convergence has also been proved. However, it seems to be nontrivial to establish the global convergence theory under the condition of inexact line search, especially for a general nonconvex minimization problem. Quite recently, it is noticed that there are many modified PRP conjugate gradient methods studied (see, e.g., [10–13, 17]). In these methods, the search direction is constructed to possess the sufficient descent property, and the theory of global convergence is established with different line search strategy. In , the search direction is given by where Similar to the idea in , a new spectral PRP conjugate gradient algorithm will be developed in this paper. On one hand, we will present a new spectral conjugate gradient direction, which also possess the sufficiently descent feature. On the other hand, a modified Armijo-type line search strategy is incorporated into the developed algorithm. Numerical experiments will be used to make a comparison among some similar algorithms.
The rest of this paper is organized as follows. In the next section, a new spectral PRP conjugate gradient method is proposed. Section 3 will be devoted to prove the global convergence. In Section 4, some numerical experiments will be done to test the efficiency, especially in comparison with the existing other methods. Some concluding remarks will be given in the last section.
2. New Spectral PRP Conjugate Gradient Algorithm
In this section, we will firstly study how to determine a descent direction of objective function.
Let be the current iterate. Let be defined by where is specified by (1.4) and
We first prove that is a sufficiently descent direction.
Proof. Firstly, for , it is easy to see that (2.3) is true since .
Secondly, assume that holds for when . Then, from (1.4), (2.1), and (2.2), it follows that Thus, (2.3) is also true with replaced by . By mathematical induction method, we obtain the desired result.
From Lemma 2.1, it is known that is a descent direction of at . Furthermore, if the exact line search is used, then ; hence In this case, the proposed spectral PRP conjugate gradient method reduces to the standard PRP method. However, it is often that the exact line search is time-consuming and sometimes is unnecessary. In the following, we are going to develop a new algorithm, where the search direction is chosen by (2.1)-(2.2) and the stepsize is determined by Armijio-type inexact line search.
Algorithm 2.2 (Modified Spectral PRP Conjugate Gradient Algorithm). We have the following steps.
Step 1. Given constants , , , . Choose an initial point . Let .Step 2. If , then the algorithm stops. Otherwise, compute by (2.1)-(2.2), and go to Step 3.Step 3. Determine a steplength such that Step 4. Set , and . Return to Step 2.
Since is a descent direction of at , we will prove that there must exist such that satisfies the inequality (2.7).
Proposition 2.3. Let be a continuously differentiable function. Suppose that is a descent direction of at . Then, there exists such that where , is the gradient vector of at , , and are given constant scalars.
Proof. Actually, we only need to prove that a step length is obtained in finitely many steps. If it is not true, then for all sufficiently large positive integer , we have Thus, by the mean value theorem, there is a such that It reads When , it is obtained that From , it follows that . This contradicts the condition that is a descent direction.
Remark 2.4. From Proposition 2.3, it is known that Algorithm 2.2 is well defined. In addition, it is easy to see that more descent magnitude can be obtained at each step by the modified Armijo-type line search (2.7) than the standard Armijo rule.
3. Global Convergence
In this section, we are in a position to study the global convergence of Algorithm 2.2. We first state the following mild assumptions, which will be used in the proof of global convergence.
Assumption 3.1. The level set is bounded.
Assumption 3.2. In some neighborhood of , is continuously differentiable and its gradient is Lipschitz continuous, namely, there exists a constant such that
Since is decreasing, it is clear that the sequence generated by Algorithm 2.2 is contained in a bounded region from Assumption 3.1. So, there exists a convergent subsequence of . Without loss of generality, it can be supposed that is convergent. On the other hand, from Assumption 3.2, it follows that there is a constant such that Hence, the sequence is bounded.
In the following, we firstly prove that the stepsize at each iteration is large enough.
Lemma 3.3. With Assumption 3.2, there exists a constant such that the following inequality holds for all sufficiently large.
Proof. Firstly, from the line search rule (2.7), we know that .
If , then we have . The reason is that implies that which contradicts (2.3). Therefore, taking , the inequality (3.3) holds.
If , then the line search rule (2.7) implies that does not satisfy the inequality (2.7). So, we have
Since where satisfies and the last inequality is from (3.2), it is obtained that due to (3.5) and (3.1). It reads that is, Therefore, From Lemma 2.1, it follows that
Taking then the desired inequality (3.3) holds.
Proof. From the line search rule (2.7) and Assumption 3.1, there exists a constant such that
Then, from Lemma 2.1, we have
Therefore, the first conclusion is proved.
Since the series is convergent. Thus,
The second conclusion (3.14) is obtained.
In the end of this section, we come to establish the global convergence theorem for Algorithm 2.2.
Proof. Suppose that there exists a positive constant such that
for all . Then, from (2.1), it follows that
Dividing by in the both sides of this equality, then from (1.4), (2.3), (3.1), and (3.21), we obtain
From (3.14) in Lemma 3.4, it follows that
Thus, there exists a sufficient large number such that for , the following inequalities
Therefore, for , where is a nonnegative constant.
The last inequality implies which contradicts the result of Lemma 3.4.
The global convergence theorem is established.
4. Numerical Experiments
In this section, we will report the numerical performance of Algorithm 2.2. We test Algorithm 2.2 by solving the 15 benchmark problems from  and compare its numerical performance with that of the other similar methods, which include the standard PRP conjugate gradient method in , the modified FR conjugate gradient method in , and the modified PRP conjugate gradient method in . Among these algorithms, either the updating formula or the line search rule is different from each other.
All codes of the computer procedures are written in MATLAB 7.0.1 and are implemented on PC with 2.0 GHz CPU processor, 1 GB RAM memory, and XP operation system.
The parameters are chosen as follows:
In Tables 1 and 2, we use the following denotations: Dim: the dimension of the objective function;GV: the gradient value of the objective function when the algorithm stops;NI: the number of iterations;NF: the number of function evaluations;CT: the run time of CPU;mfr: the modified FR conjugate gradient method in ; prp: the standard PRP conjugate gradient method in ;msprp: the modified PRP conjugate gradient method in ;mprp: the new algorithm developed in this paper.
From the above numerical experiments, it is shown that the proposed algorithm in this paper is promising.
In this paper, a new spectral PRP conjugate gradient algorithm has been developed for solving unconstrained minimization problems. Under some mild conditions, the global convergence has been proved with an Armijo-type line search rule. Compared with the other similar algorithms, the numerical performance of the developed algorithm is promising.
The authors would like to express their great thanks to the anonymous referees for their constructive comments on this paper, which have improved its presentation. This work is supported by National Natural Science Foundation of China (Grant nos. 71071162, 70921001).
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