Mathematical Problems in Engineering

Volume 2018, Article ID 4729318, 11 pages

https://doi.org/10.1155/2018/4729318

## A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search Technique for Large-Scale Minimization Optimization Models

^{1}College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China^{2}School of Mathematics and Statistics, Baise University, Baise, Guangxi 533000, China^{3}Business School, Guangxi University, Nanning, Guangxi 530004, China^{4}Thai Nguyen University of Economics and Business Administration, Thai Nguyen, Vietnam

Correspondence should be addressed to Xiangrong Li; moc.361@86ilrx

Received 17 October 2017; Accepted 27 November 2017; Published 15 January 2018

Academic Editor: Guillermo Cabrera-Guerrero

Copyright © 2018 Xiangrong Li et al. 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.

#### Abstract

This paper gives a modified Hestenes and Stiefel (HS) conjugate gradient algorithm under the Yuan-Wei-Lu inexact line search technique for large-scale unconstrained optimization problems, where the proposed algorithm has the following properties: the new search direction possesses not only a sufficient descent property but also a trust region feature; the presented algorithm has global convergence for nonconvex functions; the numerical experiment showed that the new algorithm is more effective than similar algorithms.

#### 1. Introduction

Consider the minimization optimization models defined by where the function and There exist many good algorithms for (1), such as the quasi-Newton methods [1] and the conjugate gradient methods [2–5], where the iterative formula of the conjugate gradient algorithm for (1) is designed bywhere is the th iterative point, is the steplength, and is the so-called conjugate gradient search direction withwhere is a scalar determined from different conjugate gradient formulas and the HS method [3] is one of the most well-known conjugate gradient methods, which is where and The HS method has good numerical results for (1); however, the convergent theory is not interesting especially for the nonconvex function. At present, there exist many good conjugate gradients (see [6–8], etc.). Yuan, Wei, and Lu [9] gave a modified weak Wolfe-Powell (we called it YWL) line search for steplength designed bywhere , , and denotes the Euclidean norm. It is well known that there exist two open problems which are the global convergence of the normal BFGS method and the global convergence of the PRP method for nonconvex functions under the inexact line search technique, where the first problem is regarded as one of the most difficult one thousand mathematical problems of the 20th century [10]. Yuan et al. [9] partly solved these two open problems under the YWL technique, and the numerical performance shows that the YWL technique is more competitive than the normal weak Wolfe-Powell technique. Further study work can be found in their paper [11]. By (5), it is not difficult to see that the YWL conditions are equivalent to the weak Wolfe-Powell (WWP) conditions if holds, which implies that the YWL technique includes the WWP technique in some sense. Motivated by the above observations, we will make a further study and propose a new algorithm for (1). The main features of this paper are as follows:(i)A modified HS conjugate gradient formula is given, which has not only a sufficient descent property but also a trust region feature.(ii)The global convergence of the given HS conjugate gradient algorithm for nonconvex functions is established.(iii)Numerical results show that the new HS conjugate gradient algorithm under the YWL line search technique is better than the normal weak Wolfe-Powell technique.

This paper is organized as follows. In Section 2, a modified HS conjugate gradient algorithm is introduced. The global convergence of the given algorithm for nonconvex functions is established in Section 3 and numerical results are reported in Section 4.

#### 2. Motivation and Algorithm

The nonlinear conjugate gradient algorithm is simple and has low memory requirement properties and is very effective for large-scale optimization problems, where the HS method is one of the most effective methods. However, the normal HS method has good numerical performance but fails in the convergence of nonconvex functions under the inexact line search technique. In order to overcome this shortcoming, a modified HS formula is defined by where and , , and are positive constants. This formula is inspired by the idea of these two papers [6, 8]. In recent years, lots of scholars like to study the three-term conjugate gradient formula because of its good properties [7]. In the next section, we will prove that the new formula possesses not only a sufficient descent property but also a trust region feature. The sufficient descent property is good for the convergence and the trust region makes the convergence easy to prove. Now, we give the steps of the proposed algorithm as follows.

*Algorithm 1 (the modified three-term HS conjugate gradient algorithm (M-TT-HS-A)). ****Step 1*. Let and .*Step 2*. If holds, stop.*Step 3*. Find by the YWL line search satisfying (5) and (6).*Step 4*. Let .*Step 5*. Compute the direction by (7).*Step 6*. The algorithm stops if *Step 7*. Let and go to Step .

#### 3. Sufficient Descent Property, Trust Region Feature, and Global Convergence

This section will prove some properties of Algorithm 1.

Lemma 2. *The search direction is designed by (7); the following two relations hold:where is a constant.*

*Proof. *If , it is easy to have (8) and (9). If , by formula (7), we have Then, (8) holds as well as (9) by letting This completes the proof.

Inequality (8) shows that the new formula has a sufficient descent property and inequality (9) proves that the new formula possesses a trust region feature. Both of these properties (8) and (9) are good theory characters and they play an important role in the global convergence of a conjugate gradient algorithm. The following global convergence theory will explain all this.

The following general assumptions are needed.

*Assumption A. *(i) The defined level set is bounded.

(ii) The objective function is bounded below, twice continuously differentiable, and is Lipschitz continuous; namely, the following inequality is true:where is the Lipschitz constant.

By Lemma 2 and Assumption A, similar to [9], it is not difficult to show that the YWL line search technique is reasonable and Algorithm 1 is well defined. Here, we do not state it anymore. Now, we prove the global convergence of Algorithm 1 for nonconvex functions.

Theorem 3. *Let Assumption A hold, and the iterate sequence is generated by M-TT-HS-A. Then, the relation is true.*

*Proof. *By (5), (8), and (9), we obtain Summing these inequalities for to and using Assumption A (ii) generate Inequality (14) implies that is true. By (6) and (8) again, we get Thus, the inequality holds, where the first inequality follows (8) and the last inequality follows (11). Then, we have By (15) and (18), we have Therefore, we get (12) and the proof is complete.

#### 4. Numerical Results Performance

This section will give numerical results of Algorithm 1 and the similar algorithms for comparing them. We will give another two algorithms for comparison; they are listed as follows.

*Algorithm 2 (the normal three-term formula [8] under the YWL technique). ****Step 1*. Let and *Step 2*. If holds, stop.*Step 3*. Find by the YWL line search satisfying (5) and (6).*Step 4*. Let .*Step 5*. Compute the direction by *Step 6*. The algorithm stops if *Step 7*. Let and go to Step .

*Algorithm 3 (the normal three-term formula [8] under the WWP technique). ****Step 1*. . Let and *Step 2*. If holds, stop.*Step 3*. Find by the WWP line search satisfying *Step 4*. Let .*Step 5*. Compute the direction by (20).*Step 6*. The algorithm stops if *Step 7*. Let and go to Step .

##### 4.1. Problems and Experiment

The following are some notes.

*Test Problems*. These problems and the related initial points are listed in Table 1; the detailed problems can be found in Andrei [12], and some papers also use these problems [13].