Abstract

In this work, considering the advantages of spectral conjugate gradient method and quasi-Newton method, a spectral three-term conjugate gradient method with random parameter is proposed. The parameter in the search direction of the new method is determined by minimizing the Frobenius norm of difference between search direction matrix and self-scaled memoryless BFGS matrix based on modified secant equation. Then, the search direction satisfying the sufficient descent condition is obtained. The global convergence of new method is proved under appropriate assumptions. Numerical experiments show that our method has better performance by comparing with the up-to-date method. Furthermore, the new method has been successfully applied to the optimization of low-carbon supply chain.

1. Introduction

Consider the following unconstrained optimization problem where is continuous differentiable and bounded from below.

Spectral conjugate gradient (SCG) method is one of the most effective methods for solving (1). It has some advantages, such as simple iterative scheme, low memory requirement, and strong global convergence, as well as the traditional conjugate gradient (CG) method [1], and outperforms the traditional CG method in numerical performance. SCG method generates a sequence of solutions with the following formula: in which is the stepsize, and the search direction is defined by where is the gradient of at iterate point , is the spectral parameter, and is the conjugate parameter. The choices of and are crucial for the global convergence and numerical performance of the algorithm, which have been widely studied by many scholars (see [29]).

Deng et al. [8] proposed an improved spectral conjugate gradient (ISCG) method for nonconvex unconstrained optimization, where the parameters and in (3) are determined by where , , and is a small constant. The obtained search direction satisfies the sufficient descent and approaches the quasi-Newton direction. Numerical experiments showed that ISCG algorithm was effective for solving large-scale problems.

Li et al. [9] proposed a spectral three-term conjugate gradient method on three-dimensional subspace . The search direction is expressed as where and () are given by [10]. They made close to general quasi-Newton direction and obtained the expression for as in which , and

In addition, they used modified secant equation [11] where , , , and . If , then ; otherwise, . Another way to choose spectral parameters was proposed, in which and are obtained by replacing with in (7). Their methods had global convergence and were superior to the three-term conjugate gradient method proposed by Sun and Liu [10].

Neculai [6] proposed a new scaled conjugate gradient (SCALCG) algorithm by using a hybridization of the memoryless Broyden-Fletcher-Goldfarb-Shanno (MBFGS) preconditioned CG method [12] and SCG method [13] for solving large-scale unconstrained optimization. The search direction is defined by where is called search direction matrix and is determined according to a two-point approximation of the standard secant equation. Numerical experiments showed that the SCALCG algorithm outperformed several well-known CG algorithms [1315].

Babaie-Kafaki and Ghanbari [16] rewrote the search direction of Dai and Liao method [17] as

They obtained the following relation in which and analyzed the eigenvalues of the matrix to determine the parameter .

Yao and Ning [18] proposed a three-term conjugate gradient method, in which the search direction was expressed as where the optimal parameter was derived by minimizing the distance between and the self-scaled memoryless BFGS (ML-BFGS) matrix in the Frobenius norm, that is, and the parameters . The search direction was always sufficiently descent at every iteration independent of any line search strategy, and this method had global convergence for general nonconvex functions.

Based on the above work, it is shown that spectral parameter plays an important role in improving the conjugate gradient method, and modified secant equation uses more information of function value and gradient value. Therefore, in order to obtain a new algorithm with good numerical performance, especially for the objective function with sharp curvature change, we introduce the spectral parameter into (12) and construct the following search direction matrix and improve ML-BFGS matrix based on the modified secant equation. The parameter in (15) is determined by minimizing the Frobenius norm of difference between and ML-BFGS matrix based on modified secant equation, and we propose a spectral three-term conjugate gradient method with random parameter. The contributions of this article are listed as follows: (i)A random parameter is introduced to simplify the format of the parameter in the search direction, and the search direction satisfying the sufficient descent condition is obtained(ii)Under appropriate assumptions, global convergence of new method for general functions is given(iii)The new method has good numerical performance for the objective function with sharp curvature change(iv)The new method is applied to the low-carbon supply chain optimization model, which shows that the new method is effective

The rest of this paper is organized as follows: in the next section, a new random parameter is given to present spectral three-term conjugate gradient method. In Section 3, global convergence of the new method for uniformly convex functions and general functions is proved under appropriate conditions. In Section 4, some numerical experiments are implemented. In Section 5, the application of new method in low-carbon supply chain optimization is studied. Conclusions are made in the last section.

2. A Spectral Three-Term Conjugate Gradient Method with Random Parameter

In this section, our main aim is to propose a new spectral three-term conjugate gradient method based on modified secant equation. Consider the following modified secant equation: where , , and ; we design ML-BFGS matrix based on modified secant equation as follows: where .

The parameter is determined by minimizing the Frobenius norm of difference between search direction matrix and ML-BFGS matrix based on modified secant equation, that is, where is the Frobenius norm and and are determined by (15) and (17), respectively.

From (15) and (17), we have where is a constant independent of . Therefore, the minimum of problem (18) is in which , is the angle between and . Instead of the mean value to in [19], let be a random number in the interval , where Therefore, in (20) can be regarded as a random parameter. There are many possible ways to choose ; we set

Substitute (20) into (15), and let ; then, where

Based on the above analysis, a new spectral three-term conjugate gradient (STCG) algorithm can be presented as follows.

Algorithm 1. STCG algorithm.
Step 0. Given , , and . Compute and ; let and .
Step 1. If , stop; else, go to step 2.
Step 2. Compute a step length satisfying strong Wolfe line search conditions Step 3. Set ; calculate , , , , and .
Step 4. Compute by (20) and (21) and search direction by (22). Set and go to step 1.

The following lemma shows that the search direction satisfies the sufficient descent property, which plays an important role in proving the convergence of the algorithm.

Lemma 1. Let the sequence be generated by STCG algorithm; then, there exists a positive constant , such that

Proof. From the search direction (22), we have The second of the above inequalities comes from the fact, in which and . In the fourth of the above inequalities, can be ensured by the strong Wolfe line search, and . Combining (21), the proof is completed.

3. Convergence Analysis

To prove the global convergence of STCG algorithm, we give the following assumptions.

Assumption 2. The level set is bounded; namely, there exists a positive constant such that

Assumption 3. The gradient of function is Lipschitz continuous in some neighborhood of ; namely, there exists satisfying

Based on the above assumptions, we can easily have that is bounded; i.e., there exists a positive constant such that

Lemma 4. If Assumption 3 holds, then is bounded, i.e.,

The proof of Lemma 4 is similar to the proof of Lemma 2 in [20], so we omit it here.

According to Lemma 4, we can see

Lemma 5. Let the sequence be generated by STCG algorithm. If Assumption 3 holds, then

Proof. According to (25), we have , then both side to subtracte , and using Lipschitz condition, we get Since is a descent direction and , (31) follows immediately.

Lemma 6. Let the sequence be generated by STCG algorithm. If Assumption 3 holds, we have

Proof. From the first inequality (24) of strong Wolfe conditions, Assumption 3, and Lemma 5, we have Since is bounded from below, the proof is completed.

Theorem 7. Suppose that Assumption 2 and Assumption 3 hold. The sequence is generated by STCG algorithm. If is a uniformly convex function on , namely, there exists a positive constant such that then we have

Proof. From the Lipschitz condition (28), we have It follows (35) that Using Cauchy inequality and (38), we obtain , i.e., Then, from (37), (38), and (39), we have Let ; we get . From (40), we obtain Therefore, from (22), (37), (38), and (42), we have From Lemma 1 and (43), we get Combined with Lemma 6, then The proof is completed.

For general nonlinear functions, we can establish a weaker convergence result:

Lemma 8. Suppose that Assumption 2 and Assumption 3 hold. Let the sequence be generated by STCG algorithm; then, we have and whenever , in which .

Proof. Define , then . From the sufficient descent condition (26), we know for each , so is well defined. To prove global convergence, we define , where . By (22), we have namely, where Using the identity , we have Since , then From (25), we have Thus, By the definition of , (54) and (55), we get If , from Lemma 1 and Lemma 6, we have Thus, (47) holds.

Property(). Consider a method of form (2) and (22), and suppose

We call that a method has Property() if there exist constants and such that and .

Lemma 9. Suppose that Assumption 2 and Assumption 3 hold. Let the sequence be generated by STCG algorithm; then, STCG algorithm has Property().

Proof. By (25) and (26), we obtain

Using (29), (58), Assumption 2, and (59), we obtain

Let If , from (60) and (61), we obtain

In the next lemma, we show that if gradient sequence is bounded away form zero, then a fraction of the steps cannot be too small.. Let be the set of positive integers, namely, the set of integers corresponding to steps greater than . Now, we need to discuss groups of consecutive iterates. Let and denote the number of elements of .

Lemma 10. Suppose that Assumption 2 and Assumption 3 hold. Let the sequences and be generated by STCG algorithm. When (58) holds, there exists such that where , in which is any index.

Proof. Suppose on the contrary that there exists , such that for and for any .
By (54) and (55), we have According to (22), we have By induction, we have for any given index , where depends on , not on . Next, we consider where . Now, we divide factors of (66) into groups of each elements; namely, if , then (66) can be divided into or groups and a possible group where for , and for . It is clear that for ; from assumption condition, we get Thus, there are indices such that and indices such that on .
From (60), we have , i.e., . In conjunction with , we have So every item in (67) is less than or equal to 1, and so is their product. In (68), we have . Then, we get where and independent of . Furthermore, But from (26), (33), and (58), we have It leads to a contradiction. The proof is completed.

Theorem 11. Suppose that Assumption 2 and Assumption 3 hold. Let the sequence be generated by STCG algorithm; then, (46) holds.

Proof. The proof by contradiction is adopted. We can obtain the proof similarly to Theorem 4.3 in [21].

4. Numerical Results

In this section, we show the numerical performance of STCG algorithm. The test problems are unconstrained problems from CUTEr library [22] and Andrei [23], in which the dimensions vary from 2 to 100000. All codes are written on MATLAB R2015b and run on PC with 1.19 GHz CPU processor, 8.00 GB RAM memory. We list these test problems and their dimensions in Table 1.

We compare STCG algorithm against the descent Dai-Liao (DDL) method [16] and the modified Polak-Ribière-Polyak (PRP+) method [21], which have better numerical performance. When and are chosen, STCG algorithm is denoted by “New1” and “New2,” respectively.

All test methods are terminated when satisfying the following condition: where and and in Wolfe conditions (24) and (25).

The result of computational experiments from partial problems in Table 1 are listed in Table 2. In Table 2, , , , and CPU stand for number of iterations, function evaluations, gradient evaluations, and CPU time, respectively. And based on the numerical results of all the test problems, we present the performance profile (including number of iterations, function evaluations, gradient evaluations, and CPU time) introduced by Dolan and Moré [24] to show the difference in numerical effects among the four algorithms. In a performance profile plot, the horizontal axis gives the percentage of the test problems for which a method is the fastest (efficiency), while the vertical side gives the percentage of the test problems that are successfully solved by each of the methods.

From Table 2, we can see that STCG is significantly superior to DDL for 88 percent of the problems; STCG is superior to PRP+ for 58 percent of the problems. Figures 14 plot the performance profiles for the number of iterations, the number of function evaluations, the number of gradient evaluations, and the CPU time, respectively. They show that the performance of New1 and New2 is superior to DDL and PRP+ in all aspects. In the overall trend, the performance of New1 is slightly better than New2. We deem that New1 is more competitive than New2. In conclusion, STCG method is competitive.

5. Application of STCG Algorithm in Low-Carbon Supply Chain Optimization

In recent years, global warming has become increasingly serious due to the dramatic increase in carbon emissions caused by human activities. As an important means to achieve sustainable development, energy conservation and emission reduction are highly valued by the government, enterprises, and consumers. Therefore, we use STCG algorithm to study the optimal pricing, warranty decision, and carbon emission level strategy of the two low-carbon supply chain (LSCS) models under the centralized game structure in [25].

As shown in Figure 5 [25], manufacturers sell products through retailers and provide consumers with free after-sales warranty services. Manufacturers produce greenhouse gases when they produce products and provide warranty services. The government will set a certain carbon emission quota for each enterprise. When the enterprise carbon emission quota is insufficient or excessive, the enterprise can trade in the carbon emission market.

For a better description of the model, the symbols are shown in Table 3.

According to the assumptions given in [25], the object problem can be transformed into profit maximization. The warranty cost function is introduced as (, ) (see [2628]). Market demand only depends on the warranty period , the demand function is , and the function expression for the number of products repaired by the manufacturer during the warranty period is . In centralized decision-making, we regard manufacturers and retailers as subjects with identical interests, and both sides cooperate to maximize LSCS profits. Therefore, the total profit function of LSCS is

We transform (72) into the following optimization problem:

Based on (72), the carbon emission reduction level of the product is considered. The demand function of product is linear, where and are positive, inversely proportional to the retail price , proportional to the warranty period , and the carbon emission reduction level . In the production process, manufacturer needs to develop carbon emission reduction technologies to increase the carbon emission reduction level of products; the investment cost function of carbon emission reduction level is [29].

Considering the carbon reduction efficiency of a product under centralized decision, the overall profit function of LSCS is