A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4729318, 11 pages https://doi.org/10.1155/2018/4729318

Research Article A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search Technique for Large-Scale Minimization Optimization Models Xiangrong Li ,1 Songhua Wang,2 Zhongzhou Jin,3 and Hongtruong Pham4 1

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China 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 2

Correspondence should be addressed to Xiangrong Li; [email protected] 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. 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: (1) the new search direction possesses not only a sufficient descent property but also a trust region feature; (2) the presented algorithm has global convergence for nonconvex functions; (3) the numerical experiment showed that the new algorithm is more effective than similar algorithms.

1. Introduction Consider the minimization optimization models defined by min

{𝑓 (𝑥) | 𝑥 ∈ R𝑛 } ,

(2)

where 𝑥𝑘 is the 𝑘th iterative point, 𝛼𝑘 is the steplength, and 𝑑𝑘 is the so-called conjugate gradient search direction with 𝑑𝑘+1

{−𝑔 (𝑥𝑘+1 ) + 𝛽𝑘 𝑑𝑘 , if 𝑘 ≥ 1, ={ if 𝑘 = 0, −𝑔 (𝑥𝑘+1 ) , {

𝛽𝑘HS =

(1)

where the function 𝑓 : R𝑛 → R and 𝑓 ∈ 𝐶2 . 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 by 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘 𝑑𝑘 , 𝑘 = 0, 1, 2, . . . ,

where 𝛽𝑘 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 𝑇 𝑦𝑘 𝑔𝑘+1 , 𝑇 𝑑𝑘 𝑦𝑘

(4)

where 𝑦𝑘 = 𝑔𝑘+1 − 𝑔𝑘 , 𝑔𝑘+1 = ∇𝑓(𝑥𝑘+1 ) and 𝑔𝑘 = ∇𝑓(𝑥𝑘 ). 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 𝛼𝑘 designed by 𝑓 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) ≤ 𝑓𝑘 + 𝛿𝛼𝑘 𝑔𝑘𝑇 𝑑𝑘 + 𝛼𝑘 min [−𝛿1 𝑔𝑘𝑇 𝑑𝑘 , 𝛿

𝛼𝑘 󵄩󵄩 󵄩󵄩2 󵄩𝑑 󵄩 ] , 2 󵄩 𝑘󵄩

(5)

𝑇

𝑔 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) 𝑑𝑘 (3)

󵄩 󵄩2 ≥ 𝜎𝑔𝑘𝑇 𝑑𝑘 + min [−𝛿1 𝑔𝑘𝑇 𝑑𝑘 , 𝛿𝛼𝑘 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 ] ,

(6)

2

Mathematical Problems in Engineering

where 𝛿 ∈ (0, 1/2), 𝛿1 ∈ (0, 𝛿), 𝜎 ∈ (𝛿, 1), 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 −𝛿1 𝑔𝑘𝑇 𝑑𝑘 < 𝛿(𝛼𝑘 /2)‖𝑑𝑘 ‖2 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.

𝑑𝑘+1

(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

𝑇 𝑦𝑘 𝑑𝑘 − 𝑑𝑘𝑇 𝑔𝑘+1 𝑦𝑘 𝑔𝑘+1 { {−𝑔𝑘+1 + 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 , if 𝑘 ≥ 1, ={ 𝑑 𝑑 𝑦 + 𝑔 𝜓 + 2𝜓 + 𝜓 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩𝑦𝑘 󵄩󵄩 1 𝑘 2 𝑘 𝑘 𝑘 3 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 { if 𝑘 = 0, {−𝑔𝑘+1 ,

where 𝑦𝑘 = 𝑔𝑘+1 − 𝑔𝑘 and 𝜓1 , 𝜓2 , and 𝜓3 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. 𝑥1 ∈ R𝑛 , 𝜖 ∈ (0, 1), 𝛿 ∈ (0, 1/2), 𝛿1 ∈ (0, 𝛿), 𝜎 ∈ (𝛿, 1), 𝜓1 > 0, 𝜓2 > 0, 𝜓3 > 0. Let 𝑘 = 1 and 𝑑1 = −𝑔(𝑥1 ). Step 2. If ‖𝑔𝑘 ‖ ≤ 𝜖 holds, stop. Step 3. Find 𝛼𝑘 by the YWL line search satisfying (5) and (6). Step 4. Let 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘 𝑑𝑘 . Step 5. Compute the direction 𝑑𝑘+1 by (7). Step 6. The algorithm stops if ‖𝑔𝑘+1 ‖ ≤ 𝜖. Step 7. Let 𝑘 = 𝑘 + 1 and go to Step 3.

(7)

3. Sufficient Descent Property, Trust Region Feature, and Global Convergence This section will prove some properties of Algorithm 1. Lemma 2. The search direction 𝑑𝑘 is designed by (7); the following two relations hold: 󵄩 󵄩2 𝑔𝑘𝑇 𝑑𝑘 = − 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 ,

󵄩󵄩 󵄩󵄩 ∗󵄩 󵄩 󵄩󵄩𝑑𝑘 󵄩󵄩 ≤ 𝜓 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 ,

(8) (9)

∗

where 𝜓 > 0 is a constant. Proof. If 𝑘 = 1, it is easy to have (8) and (9). If 𝑘 ≥ 1, by formula (7), we have 𝑇 𝑇 𝑑𝑘+1 = 𝑔𝑘+1 [−𝑔𝑘+1 𝑔𝑘+1 𝑇 𝑦𝑘 𝑑𝑘 − 𝑑𝑘𝑇 𝑔𝑘+1 𝑦𝑘 𝑔𝑘+1 󵄩 󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 ] 𝜓1 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 + 2𝜓2 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 + 𝜓3 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 󵄩2 󵄩 = − 󵄩󵄩󵄩𝑔𝑘+1 󵄩󵄩󵄩

+

𝑇

𝑇 𝑇 𝑑𝑘 − 𝑑𝑘𝑇 𝑔 (𝑥𝑘+1 ) 𝑔𝑘+1 𝑦𝑘 𝑔 (𝑥𝑘+1 ) 𝑦𝑘 𝑔𝑘+1 + 2 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2 𝜓1 󵄩󵄩𝑑𝑘 󵄩󵄩 + 2𝜓2 󵄩󵄩𝑑𝑘 󵄩󵄩 󵄩󵄩𝑦𝑘 󵄩󵄩 + 󵄩󵄩𝑔𝑘 󵄩󵄩 + 𝜓3 󵄩󵄩𝑦𝑘 󵄩󵄩


3

󵄩2 󵄩 = − 󵄩󵄩󵄩𝑔𝑘+1 󵄩󵄩󵄩 ,

Proof. By (5), (8), and (9), we obtain

󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩󵄩 𝑑 = 󵄩󵄩 𝑘+1 󵄩󵄩 󵄩󵄩−𝑔 (𝑥𝑘+1 ) 󵄩󵄩 󵄩

𝑓 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) ≤ 𝑓 (𝑥𝑘 ) + 𝛿𝛼𝑘 𝑔𝑘𝑇 𝑑𝑘 + 𝛼𝑘 min [−𝛿1 𝑔𝑘𝑇 𝑑𝑘 , 𝛿

󵄩󵄩 𝑇 󵄩󵄩 𝑦𝑘 𝑑𝑘 − 𝑑𝑘𝑇 𝑔𝑘+1 𝑦𝑘 𝑔𝑘+1 󵄩 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 󵄩󵄩󵄩󵄩 𝜓1 󵄩󵄩𝑑𝑘 󵄩󵄩 + 2𝜓2 󵄩󵄩𝑑𝑘 󵄩󵄩 󵄩󵄩𝑦𝑘 󵄩󵄩 + 󵄩󵄩𝑔𝑘 󵄩󵄩 + 𝜓3 󵄩󵄩𝑦𝑘 󵄩󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑔𝑘+1 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩𝑔 (𝑥𝑘+1 )󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑔 (𝑥𝑘+1 )󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 + 󵄩 󵄩2 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 𝜓1 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 + 2𝜓2 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 + 𝜓3 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 1 󵄩 2 󵄩󵄩𝑔 (𝑥𝑘+1 )󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 󵄩 ≤ (1 + ) ≤ 󵄩󵄩󵄩𝑔𝑘+1 󵄩󵄩󵄩 + 󵄩 󵄩 󵄩󵄩 󵄩 𝜓2 2𝜓2 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 󵄩󵄩󵄩𝑦𝑘 󵄩󵄩󵄩 󵄩 󵄩 ⋅ 󵄩󵄩󵄩𝑔𝑘+1 󵄩󵄩󵄩 .

+

𝛼𝑘 󵄩󵄩 󵄩󵄩2 󵄩𝑑 󵄩 ] 2 󵄩 𝑘󵄩

≤ 𝑓 (𝑥𝑘 ) + 𝛿𝛼𝑘 𝑔𝑘𝑇 𝑑𝑘 − 𝛼𝑘 𝛿1 𝑔𝑘𝑇 𝑑𝑘

(13)

≤ 𝑓 (𝑥𝑘 ) + 𝛼𝑘 (𝛿 − 𝛿1 ) 𝑔𝑘𝑇 𝑑𝑘 󵄩 󵄩2 ≤ 𝑓 (𝑥𝑘 ) − 𝛼𝑘 (𝛿 − 𝛿1 ) 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 . Summing these inequalities for 𝑘 = 0 to ∞ and using Assumption A (ii) generate ∞

󵄩 󵄩2 ∑ (𝛾1 − 𝛾) 𝛼𝑘 (𝛿 − 𝛿1 ) 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 ≤ 𝑓 (𝑥1 ) − 𝑓∞ < +∞.

(14)

𝑘=0

(10)

Inequality (14) implies that

Then, (8) holds as well as (9) by letting 𝜓∗ ∈ [1 + 1/𝜓2 , +∞). This completes the proof.

𝑘→∞

󵄩 󵄩2 lim 𝛼𝑘 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 = 0

(15)

is true. By (6) and (8) again, we get 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 𝐿 0 = {𝑥 | 𝑓(𝑥) ≤ 𝑓(𝑥1 )} is bounded. (ii) The objective function 𝑓(𝑥) is bounded below, twice continuously differentiable, and is Lipschitz continuous; namely, the following inequality is true: 󵄩󵄩 󵄩 󵄩 󵄩 𝑛 󵄩󵄩𝑔 (𝑥) − 𝑔 (𝑦)󵄩󵄩󵄩 ≤ 𝐿 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 , 𝑥, 𝑦 ∈ R ,

(11)

where 𝐿 > 0 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 {𝑥𝑘 , 𝛼𝑘 , 𝑑𝑘 , 𝑔(𝑥𝑘 )} is generated by M-TT-HS-A. Then, the relation 󵄩 󵄩 lim 󵄩󵄩𝑔𝑘 󵄩󵄩 = 0 𝑘→∞ 󵄩 󵄩 is true.

(12)

𝑇

𝑔 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) 𝑑𝑘 ≥ 𝜎𝑔𝑘𝑇 𝑑𝑘 󵄩 󵄩2 + min [−𝛿1 𝑔𝑘𝑇 𝑑𝑘 , 𝛿𝛼𝑘 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 ]

(16)

≥ 𝜎𝑔𝑘𝑇 𝑑𝑘 . Thus, the inequality 󵄩 󵄩2 − (𝜎 − 1) 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 ≤ (𝜎 − 1) 𝑔𝑘𝑇 𝑑𝑘 𝑇

≤ [𝑔 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) − 𝑔 (𝑥𝑘 )] 𝑑𝑘 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩𝑔 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) − 𝑔 (𝑥𝑘 )󵄩󵄩󵄩 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩

(17)

󵄩 󵄩2 ≤ 𝛼𝑘 𝐿 󵄩󵄩󵄩𝑑𝑘 󵄩󵄩󵄩 holds, where the first inequality follows (8) and the last inequality follows (11). Then, we have 󵄩 󵄩2 󵄩 󵄩2 (1 − 𝜎) 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 (1 − 𝜎) 󵄩󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 (1 − 𝜎) ≥ . 𝛼𝑘 ≥ 2󵄩 2 󵄩󵄩 󵄩󵄩2 󵄩󵄩2 = ∗ 󵄩 𝐿 󵄩󵄩𝑑𝑘 󵄩󵄩 𝐿 (𝜓 ) 󵄩󵄩𝑔𝑘 󵄩󵄩 𝐿 (𝜓∗ )

(18)

By (15) and (18), we have 󵄩 󵄩2 lim 󵄩󵄩𝑔𝑘 󵄩󵄩󵄩 = 0.

𝑘→∞ 󵄩

(19)

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.

4

Mathematical Problems in Engineering 1

Algorithm 2 (the normal three-term formula [8] under the YWL technique).

0.9 0.8

Step 1. 𝑥1 ∈ R𝑛 , 𝜖 ∈ (0, 1), 𝛿 ∈ (0, 1/2), 𝛿1 ∈ (0, 𝛿), 𝜎 ∈ (𝛿, 1), 𝜓1 > 0, 𝜓2 > 0, 𝜓3 > 0. Let 𝑘 = 1 and 𝑑1 = −𝑔(𝑥1 ).

Step 3. Find 𝛼𝑘 by the YWL line search satisfying (5) and (6).

0.6 0.5 0.4 0.3

Step 4. Let 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘 𝑑𝑘 .

0.2

Step 5. Compute the direction 𝑑𝑘+1 by { {−𝑔 + 𝑑𝑘+1 = { 𝑘+1 { {−𝑔𝑘+1 ,

Pp : r(p, s) ≤ t

Step 2. If ‖𝑔𝑘 ‖ ≤ 𝜖 holds, stop.

0.7

𝑇 𝑦𝑘 𝑑𝑘 𝑔𝑘+1

− 𝑑𝑘𝑇 𝑔𝑘+1 𝑦𝑘 , 󵄩󵄩 󵄩󵄩2 󵄩󵄩𝑔𝑘 󵄩󵄩

0.1

if 𝑘 ≥ 1,

0

(20)

if 𝑘 = 0.

Algorithm 3 (the normal three-term formula [8] under the WWP technique). 𝑛

Step 1. 𝑥1 ∈ R , 𝜖 ∈ (0, 1), 𝛿 ∈ (0, 1/2), 𝜎 ∈ (𝛿, 1). Let 𝑘 = 1 and 𝑑1 = −𝑔(𝑥1 ). Step 2. If ‖𝑔𝑘 ‖ ≤ 𝜖 holds, stop.

2.5

3

3.5

4

4.5

5

t

Stop Rules. 𝐻𝑖𝑚𝑚𝑒𝑏𝑙𝑎𝑢 stop rule: if |𝑓(𝑥𝑘 )| > 1𝑒 − 5, let stop1 = |𝑓(𝑥𝑘 ) − 𝑓(𝑥𝑘+1 )|/|𝑓(𝑥𝑘 )|; otherwise, set stop1 = |𝑓(𝑥𝑘 ) − 𝑓(𝑥𝑘+1 )|. If stop1 < 1𝑒 − 5 or ‖𝑔(𝑥𝑘 )‖ < 1𝑒 − 6 holds, the program stops. Other Cases. The line search technique accepts 𝛼𝑘 if the searching number is more than 6 and the algorithm will stop if the total iteration number is larger than 800. The numerical results are listed in Table 2, where

Step 3. Find 𝛼𝑘 by the WWP line search satisfying

𝑇

2

Figure 1: NI performance of these methods.

Step 7. Let 𝑘 = 𝑘 + 1 and go to Step 3.

𝑔 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) 𝑑𝑘 ≥ 𝜎𝑔𝑘𝑇 𝑑𝑘 .

1.5

Algorithm 1 Algorithm 2 Algorithm 3

Step 6. The algorithm stops if ‖𝑔𝑘+1 ‖ ≤ 𝜖.

𝑓 (𝑥𝑘 + 𝛼𝑘 𝑑𝑘 ) ≤ 𝑓𝑘 + 𝛿𝛼𝑘 𝑔𝑘𝑇 𝑑𝑘 ,

1

(21)

“Number” is the tested problems number; “Dim.” is the problems dimension;

Step 4. Let 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘 𝑑𝑘 .

“NI” is the total iteration number;

Step 5. Compute the direction 𝑑𝑘+1 by (20).

“CPU” is the system CPU time in seconds;

Step 6. The algorithm stops if ‖𝑔𝑘+1 ‖ ≤ 𝜖.

“NFG” is the total number of functions and gradients.

Step 7. Let 𝑘 = 𝑘 + 1 and go to Step 3. 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]. Experiments. Codes are run on Intel(R) Xeon(R) CPU, E5507 @2.27 GHz, and 6.00 GB memory and Windows 7 operation system and written by MATLAB R2009a. Parameters. 𝛿 = 0.1, 𝛿1 = 0.05, 𝜎 = 0.9, and 𝜓1 = 𝜓2 = 𝜓3 = 0.001. Dimension. Large-scale dimensions 𝑛 = 3000, 6000, 12000, and 30000.

4.2. Results and Discussion. We use the tool of Dolan and Moré [14] to analyze the efficiency of the three given algorithms. Figures 1 and 2 show that the performance of Algorithm 1 is the best and that Algorithm 1 has the best robust property among those three methods and Algorithm 2 is better than Algorithm 3, which shows that the given formula (7) is competitive to the normal three-term conjugate gradient formula (20) and the YWL line search technique is more effective than the norm WWP technique, and all of these conclusions are coincident with the results of [9]. Algorithm 1 in Figure 3 is competitive to the other two algorithms and it has the best robust property. It is not difficult to see that Figure 3 shows that Algorithm 1 is not so good and we think the reason is formula (7) or the YWL technique since more information is needed and hence more CPU time is necessary.


5 Table 1: Test problems.

Number (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) (50) (51) (52)

Problem Extended Freudenstein and Roth Function Extended Trigonometric Function Extended Rosenbrock Function Extended Beale Function Raydan 1 Function Raydan 2 Function Diagonal 1 Function Diagonal 3 Function Hager Function Generalized Tridiagonal 1 Function Extended Tridiagonal 1 Function Extended Three Exponential Terms Function Diagonal 4 Function Diagonal 5 Function Extended Himmelblau Function Generalized PSC1 Function Extended PSC1 Function Extended Block Diagonal BD1 Function Extended Maratos Function Extended Cliff Function Extended Wood Function Extended Quadratic Penalty QP1 Function Extended Quadratic Penalty QP2 Function A Quadratic Function QF2 Function Extended EP1 Function Extended Tridiagonal-2 Function BDQRTIC Function (CUTE) ARWHEAD Function (CUTE) NONDIA (Shanno-78) Function (CUTE) DQDRTIC Function (CUTEr) EG2 Function (CUTE) DIXMAANA Function (CUTE) DIXMAANB Function (CUTE) DIXMAANC Function (CUTE) Partial Perturbed Quadratic Function Broyden Tridiagonal Function EDENSCH Function (CUTE) LIARWHD Function (CUTEr) DIAGONAL 6 Function DIXON3DQ Function (CUTE) DIXMAANF Function (CUTE) DIXMAANG Function (CUTE) DIXMAANH Function (CUTE) DIXMAANI Function (CUTE) DIXMAANJ Function (CUTE) DIXMAANK Function (CUTE) DIXMAANL Function (CUTE) DIXMAAND Function (CUTE) ENGVAL1 Function (CUTE) Extended DENSCHNB Function (CUTE) SINQUAD Function (CUTE) Partial Perturbed Quadratic PPQ2 Function

𝑥0 [0.5, −2, . . . , 0.5, −2] [0.2, 0.2, . . . , 0.2] [−1.2, 1, −1.2, 1, . . . , −1.2, 1] [1, 0.8, . . . , 1, 0.8] [1, 1, . . . , 1] [1, 1, . . . , 1] [1/𝑛, 1/𝑛, . . . , 1/𝑛] [1, 1, . . . , 1] [1, 1, . . . , 1] [2, 2, . . . , 2] [2, 2, . . . , 2] [0.1, 0.1, . . . , 0.1] [1, 1, . . . , 1, 1] [1.1, 1.1, . . . , 1.1] [1, 1, . . . , 1] [3, 0.1, . . . , 3, 0.1] [3, 0.1, . . . , 3, 0.1] [0.1, 0.1, . . . , 0.1] [1.1, 0.1, . . . , 1.1, 0.1] [0, −1, . . . , 0, −1] [−3, −1, −3, −1, . . . , −3, −1] [1, 1, . . . , 1] [1, 1, . . . , 1] [0.5, 0.5, . . . , 0.5] [1.5, 1.5, . . . , 1.5] [1, 1, . . . , 1] [1, 1, . . . , 1] [1, 1, . . . , 1] [−1, −1, . . . , −1] [3, 3, 3, . . . , 3] [1, 1, 1, . . . , 1] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [0.5, 0.5, . . . , 0.5] [−1, −1, . . . , −1] [0, 0, . . . , 0] [4, 4, . . . , 4] [1, 1, . . . , 1] [−1, −1, . . . , −1] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [2, 2, 2, . . . , 2] [1, 1, . . . , 1] [0.1, 0.1, . . . , 0.1] [0.5, 0.5, . . . , 0.5]

6

Mathematical Problems in Engineering Table 2: Numerical results.

Number

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Dim. 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000

NI 15 24 22 22 74 81 83 88 88 83 78 49 32 24 23 24 23 23 23 23 12 12 12 12 2 2 2 2 17 19 19 19 24 22 2 2 6 6 6 3 34 47 36 43 14 21 24 24 4 4 4 4

Algorithm 1 NFG CPU 38 0.093601 66 0.358802 59 0.577204 59 1.357209 163 0.327602 176 0.702005 182 1.388409 194 3.572423 262 0.124801 277 0.234002 272 0.390002 119 0.483603 87 0.109201 65 0.171601 62 0.312002 67 0.748805 51 0.0624 51 0.093601 51 0.171601 51 0.390002 26 0.0156 26 0.0468 26 0.093601 26 0.171601 9 0.0312 9 0 9 0 9 0.0312 36 0.0624 40 0.093601 40 0.156001 40 0.405603 113 0.078001 52 0.093601 9 0.0156 9 0.0624 15 0.655204 15 2.043613 15 7.207246 8 21.590538 81 1.762811 114 6.520842 88 14.820095 102 92.492993 30 0.0468 44 0.093601 50 0.202801 50 0.592804 13 0.0312 13 0 13 0.0312 13 0.0468

NI 20 21 17 17 74 81 83 88 99 112 16 46 21 27 23 29 23 23 23 23 12 12 12 12 2 2 2 2 17 19 19 19 24 21 2 2 6 6 6 3 35 34 37 39 14 21 24 24 3 3 3 3

Algorithm 2 NFG CPU 56 0.140401 59 0.280802 46 0.405603 46 0.998406 163 0.343202 176 0.686404 182 1.341609 194 3.588023 283 0.124801 329 0.265202 66 0.078001 182 0.561604 57 0.093601 74 0.187201 63 0.265202 78 0.858005 51 0.0312 51 0.093601 51 0.156001 51 0.358802 26 0.0156 26 0.0468 26 0.0624 26 0.171601 9 0.0312 9 0 9 0.0312 9 0.0468 36 0.0468 40 0.109201 40 0.156001 40 0.405603 113 0.078 50 0.109201 9 0.0156 9 0.0468 15 0.592804 15 1.981213 15 7.503648 8 21.528138 86 1.653611 77 4.383628 93 15.241298 100 86.050152 30 0.0624 44 0.109201 50 0.218401 50 0.483603 10 0.0156 10 0 10 0.0312 10 0.0312

NI 20 21 17 17 83 81 83 89 73 112 16 46 21 27 26 25 23 23 23 23 12 12 12 12 2 2 2 2 17 19 19 19 24 22 2 2 6 6 6 3 35 36 37 39 14 21 24 24 3 3 3 3

Algorithm 3 NFG CPU 56 0.124801 59 0.249602 46 0.405603 46 1.029607 181 0.358802 176 0.655204 182 1.357209 196 3.478822 216 0.093601 329 0.249602 66 0.093601 182 0.468003 57 0.0624 74 0.187201 69 0.343202 66 0.717605 51 0.0624 51 0.0624 51 0.124801 51 0.312002 26 0.0156 26 0.0312 26 0.078 26 0.156001 9 0.0312 9 0 9 0.0312 9 0.0468 36 0.0624 40 0.078 40 0.171601 40 0.374402 113 0.078 52 0.109201 9 0.0312 9 0.078 15 0.592804 15 1.996813 15 7.394447 8 21.481338 86 1.63801 88 4.74243 93 15.100897 100 85.83175 30 0.0468 44 0.078001 50 0.218401 50 0.468003 10 0 10 0 10 0 10 0.0312


7 Table 2: Continued.

Number

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

Dim. 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000

NI 3 3 3 3 9 128 24 13 33 32 32 33 9 9 9 9 11 11 47 39 28 38 159 7 97 106 93 87 36 37 34 33 42 41 41 45 42 71 37 52 3 3 2 2 4 5 7 10 12 16 21 30

Algorithm 1 NFG 9 9 9 9 48 292 114 39 76 73 73 78 44 44 44 44 88 88 207 158 56 76 504 16 212 230 206 194 90 94 88 83 92 90 90 98 88 146 80 110 7 7 5 5 8 10 14 20 24 32 42 62

CPU 0.0312 0.0312 0.0624 0.078001 0.0468 0.312002 0.140401 0.156001 0.109201 0.187201 0.312002 0.811205 0.124801 0.280802 0.499203 1.279208 0.0624 0.093601 0.436803 0.873606 0.0312 0.078 0.717605 0.078 0.234002 0.452403 0.733205 1.700411 0.0468 0.124801 0.202801 0.436803 0.0624 0.109201 0.234002 0.514803 0.109201 0.296402 0.312002 1.029607 0 0.0312 0 0.0312 0 0.0156 0.0312 0.140401 0.0312 0.078001 0.124801 0.421203

NI 3 3 3 3 9 42 29 16 33 32 32 33 9 9 9 9 11 11 20 89 28 38 170 7 97 106 93 87 36 37 33 36 42 41 41 45 42 71 37 52 3 3 2 2 4 5 7 10 12 16 21 30

Algorithm 2 NFG CPU 9 0.0312 9 0.0312 9 0.0624 9 0.078 48 0.0312 108 0.093601 139 0.156001 45 0.171601 76 0.093601 73 0.156001 73 0.312002 78 0.780005 44 0.156001 44 0.265202 44 0.514803 44 1.232408 87 0.0312 83 0.093601 115 0.202801 248 1.63801 56 0.0468 76 0.093601 594 0.733205 16 0.0468 212 0.218401 230 0.421203 206 0.702005 194 1.54441 90 0.0468 94 0.093601 86 0.156001 92 0.405603 92 0.078 90 0.124801 90 0.187201 98 0.499203 88 0.078001 146 0.296402 80 0.265202 110 0.904806 7 0 7 0.0312 5 0 5 0.0312 8 0 10 0.0156 14 0.0312 20 0.124801 24 0.0468 32 0.0468 42 0.109201 62 0.405603

NI 3 3 3 3 9 42 29 16 42 41 43 37 9 9 9 9 11 11 20 31 28 38 66 7 97 106 93 87 36 37 33 36 42 41 41 45 42 71 37 52 3 3 2 2 4 5 7 10 12 16 21 31

Algorithm 3 NFG CPU 9 0 9 0.0312 9 0.0312 9 0.078001 48 0.0312 108 0.078 139 0.156001 45 0.156001 90 0.093601 88 0.187201 95 0.390003 86 0.889206 43 0.124801 44 0.249602 44 0.514803 44 1.232408 86 0.0312 83 0.093601 115 0.187201 134 0.733205 56 0.0312 76 0.0624 162 0.249602 16 0.078 212 0.202801 230 0.390003 206 0.686404 194 1.52881 90 0.078 94 0.093601 86 0.156001 92 0.421203 92 0.078 90 0.093601 90 0.202801 98 0.483603 88 0.078 146 0.280802 80 0.280802 110 0.920406 7 0 7 0 5 0 5 0.0624 8 0 10 0.0312 14 0.0624 20 0.124801 24 0.0468 32 0.0468 42 0.109201 62 0.405603

8

Mathematical Problems in Engineering Table 2: Continued.

Number

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

Dim. 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000

NI 18 5 5 5 9 12 10 12 26 27 29 23 49 37 29 32 4 4 4 4 22 23 24 25 38 39 41 42 66 69 72 75 85 45 58 75 33 47 49 40 23 23 23 23 12 47 68 11 21 22 23 24

Algorithm 1 NFG CPU 57 1.341609 16 1.060807 16 3.494422 16 21.138135 23 0.0312 30 0.0312 25 0.078 29 0.140401 52 0.0468 54 0.078001 58 0.124801 46 0.249602 119 0.0624 92 0.109201 75 0.124801 77 0.296402 21 0 21 0.0312 21 0.0312 21 0.093601 48 0.296402 50 0.577204 52 1.201208 54 3.026419 80 0.483603 82 0.951606 86 1.950013 88 5.054432 136 0.842405 142 1.669211 148 3.478822 154 9.001258 175 26.036567 110 57.79837 146 296.932303 201 2473.645457 76 1.279208 106 7.020045 101 26.410969 95 154.050987 48 0.124801 48 0.265202 48 0.499203 48 1.232408 36 0.0468 150 0.156001 178 0.374402 30 0.140401 44 0.483603 46 1.950013 48 7.441248 50 51.979533

NI 14 8 8 17 9 12 10 11 26 27 29 23 27 26 28 27 4 4 4 4 22 23 24 25 38 39 41 42 66 69 72 75 86 84 35 96 48 50 50 41 23 23 23 23 19 26 13 42 21 22 23 24


NI 14 8 8 17 9 12 10 11 26 27 29 23 27 26 28 27 4 4 4 4 22 23 24 25 36 37 38 39 66 69 72 75 86 84 35 96 49 50 50 41 23 23 23 23 19 26 13 42 21 22 23 24



9 Table 2: Continued.

Number

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

Dim. 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 30000 3000 6000 12000 3000 6000 30000

NI 449 449 449 449 93 73 80 90 154 151 188 215 96 65 81 89 142 159 178 212 93 73 80 90 72 119 138 174 302 398 359 437 26 27 27 27 41 40 40 42 35 36 38 39 28 47 42 569 800 728

Algorithm 1 NFG CPU 906 0.468003 906 0.748805 906 1.263608 906 3.07322 192 1.170007 152 1.825212 166 3.931225 186 11.029271 318 1.996813 308 3.775224 386 9.40686 453 26.972573 205 1.248008 151 1.669211 177 4.102826 193 11.169672 294 1.856412 328 4.024826 366 9.001258 434 26.379769 192 1.170007 152 1.840812 166 3.962425 186 11.013671 163 0.951606 250 2.979619 288 6.848444 360 21.574938 613 3.946825 805 10.124465 727 18.189717 883 54.912352 58 0.343202 60 0.686404 60 1.341609 60 3.354022 86 3.276021 84 10.670468 84 37.48704 88 250.459606 72 0.0624 74 0.093601 78 0.171601 80 0.452403 97 1.918812 164 10.608068 135 30.872598 1582 193.176038 2184 1074.254086 2025 24653.74284

NI 800 800 800 800 94 73 80 90 150 151 202 239 71 58 85 90 137 153 170 199 94 73 80 89 73 115 147 172 283 368 402 437 26 27 27 27 41 40 40 42 35 36 38 39 51 56 37 800 800 784


NI 800 800 800 800 164 204 251 323 118 134 156 198 65 81 100 91 128 162 254 800 164 206 255 333 99 123 137 170 800 800 800 800 27 27 27 28 42 41 42 44 35 36 38 39 51 56 37 800 800 784


10

Mathematical Problems in Engineering 1

algorithm is competitive to other similar methods. More experiments will be done to prove the performance of the proposed algorithm in the future.

0.9 0.8 Pp : r(p, s) ≤ t

0.7

Conflicts of Interest

0.6

The authors declare that there are no conflicts of interest regarding the publication of this paper.

0.5 0.4 0.3

Authors’ Contributions

0.2 0.1 0

1

1.5

2

2.5

3

3.5

4

4.5

5

t

Mr. Xiangrong Li and Dr. Songhua Wang wrote this paper in English. Dr. Zhongzhou Jin and Dr. Hongtruong Pham carried out the experiment. All the authors read and approved the final manuscript.

Acknowledgments


This work is supported by the National Natural Science Foundation of China (Grants nos. 11661009 and 11261006), the Guangxi Science Fund for Distinguished Young Scholars (no. 2015GXNSFGA139001), the Guangxi Natural Science Key Fund (no. 2017GXNSFDA198046), and the Basic Ability Promotion Project of Guangxi Young and Middle-Aged Teachers (no. 2017KY0019).

Figure 2: NFG performance of these methods. 1 0.9 0.8 Pp : r(p, s) ≤ t

0.7

References

0.6 0.5 0.4 0.3 0.2 0.1 0

1

1.5

2

2.5

3 t

3.5

4

4.5

5


Figure 3: CPU time performance of these methods.

5. Conclusions This paper proposes a modified HS three-term conjugate gradient algorithm for large-scale optimization problems and the given algorithm has some good features. (1) The modified HS three-term conjugate gradient possesses a trust region property, which makes the global convergence of the general functions easy to get. However, the normal HS formula including many other conjugate gradient formulas does not have this feature, which may be the crucial point for the global convergence of the general functions. (2) The largest dimension of the test problems is 30000 variables and the numerical results show that the presented

[1] G. Yuan and Z. Wei, “Convergence analysis of a modified BFGS method on convex minimizations,” Computational optimization and applications, vol. 47, no. 2, pp. 237–255, 2010. [2] R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” The Computer Journal, vol. 7, pp. 149–154, 1964. [3] M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436 (1953), 1952. [4] E. Polak and G. Ribière, “Note sur la convergence de méthodes de directions conjuguées,” Revue Française d’Informatique et de Recherche Opérationnelle, vol. 3, no. 16, pp. 35–43, 1969. [5] B. T. Polyak, “The conjugate gradient method in extremal problems,” USSR Computational Mathematics and Mathematical Physics, vol. 9, no. 4, pp. 94–112, 1969. [6] G. Yuan, Z. Meng, and Y. Li, “A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations,” Journal of Optimization Theory and Applications, vol. 168, no. 1, pp. 129–152, 2016. [7] G. Yuan and M. Zhang, “A three-terms Polak-Ribière-Polyak conjugate gradient algorithm for large-scale nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 286, pp. 186–195, 2015. [8] L. Zhang, W. Zhou, and D.-H. Li, “A descent modified PolakRibière-Polyak conjugate method and its global convergence,” IMA Journal of Numerical Analysis (IMAJNA), vol. 26, no. 4, pp. 629–640, 2006. [9] G. Yuan, Z. Wei, and X. Lu, “Global convergence of BFGS and PRP methods under a modified weak Wolfe-Powell line search,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 47, pp. 811–825, 2017.

Mathematical Problems in Engineering [10] A. S. Nemirovsky and D. B. Yudin, Problem complexity and method efficiency in optimization, John Wiley & Sons, Inc., NY, New York, USA, 1983. [11] G. Yuan, Z. Sheng, B. Wang, W. Hu, and C. Li, “The global convergence of a modified BFGS method for nonconvex functions,” Journal of Computational and Applied Mathematics, vol. 327, pp. 274–294, 2018. [12] N. Andrei, “An unconstrained optimization test functions collection,” Advanced Modeling and Optimization, vol. 10, no. 1, pp. 147–161, 2008. [13] Y. Wu, “A modified three-term PRP conjugate gradient algorithm for optimization models,” Journal of Inequalities and Applications, Paper No. 97, 14 pages, 2017. [14] E. D. Dolan and J. J. Moré, “Benchmarking optimization software with performance profiles,” Mathematical Programming, vol. 91, no. 2, pp. 201–213, 2002.

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A Conjugate Gradient Algorithm under Yuan-Wei-Lu Line Search

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