# A New Spectral Three-Term Conjugate Gradient Method with Random Parameter Based on Modified Secant Equation and Its Application to Low-Carbon Supply Chain Optimization

Authors: Guoling Zhou; Yueting Yang; Mingyuan Cao
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 1. Introduction

Consider the following unconstrained optimization problem(1)minx∈ℝnfx,where f:ℝn⟶ℝ 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 xk with the following formula:
(2)xk+1=xk+αkdk,k≥0,in which αk>0 is the stepsize, and the search direction dk is defined by
(3)dk=−gk,ifk=0,−θkgk+βkdk−1,ifk≥1,where gk=∇fxk is the gradient of fx at iterate point xk, θk is the spectral parameter, and βk is the conjugate parameter. The choices of θk and βk are crucial for the global convergence and numerical performance of the algorithm, which have been widely studied by many scholars (see [2–9]).Deng et al. [8] proposed an improved spectral conjugate gradient (ISCG) method for nonconvex unconstrained optimization, where the parameters θk and βk in (3) are determined by
(4)θk=dk−1Tyk−1−gkgkTsk−1/gk2dk−1Ty¯k−1,ifdk−1Ty¯k−1>ηgk−12,dk−1Tyk−1−gkgkT/gk2gk−1−dk−1Tgk−1,otherwise,βk=gkTyk−1−sk−1dk−1Ty¯k−1,ifdk−1Ty¯k−1>ηgk−12,gkTyk−1gk−12,otherwise,where yk−1=gk−gk−1,sk−1=xk−xk−1, y¯k−1=I−gkgkT/gk2yk−1, and η is a small constant. The obtained search direction dk 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 Ωk=spangk,dk−1,yk−1. The search direction dk is expressed as
(5)dk=−gk,ifk=0,−θkgk+βkMPRPdk−1+νkyk−1,ifk≥1,where βkMPRP=gkTyk−1/μgkTdk−1+gk−12 and νk=gkTdk−1/μgkTdk−1+gk−12 (μ≥0) are given by [10]. They made dk close to general quasi-Newton direction and obtained the expression for θk as
(6)θkJCJ=θkJCJ+,ifθkJCJ+∈τ1,τ2,1,otherwise,orθkJCJ=θkJCJ−,ifθkJCJ−∈τ1,τ2,1,otherwise,in which 0<τ1<τ2, and
(7)θkJCJ+=1μgkTdk−1+gk−12yk−1Tdk−1−gkTdk−1yk−12yk−1Tgk+sk−1Tgkyk−1Tgk,θkJCJ−=1μgkTdk−1+gk−12yk−1Tdk−1−gkTdk−1yk−12yk−1Tgk.In addition, they used modified secant equation [11]
(8)Bksk−1=zk−1,where zk−1=yk−1+ρk−1maxτk−1,0/sk−1Tuk−1uk−1, τk−1=6fk−1−fk−1+3gk+gk−1Tsk−1, uk−1=1−λkyk−1+λksk−1, and ρk−1∈0,1. If sk≤1, then ρk−1=1; otherwise, ρk−1=0. Another way to choose spectral parameters was proposed, in which θkJCJ+ and θkJCJ− are obtained by replacing yk−1 with zk−1 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 dk+1 is defined by
(9)dk+1=−Dk+1gk+1=−θk+1gk+1+θk+1gk+1TskykTskyk−1+θk+1ykTykykTskgk+1TskykTsk−θk+1gk+1TykykTsksk,where Dk+1 is called search direction matrix and θk+1=skTsk/ykTsk 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 [13–15].Babaie-Kafaki and Ghanbari [16] rewrote the search direction of Dai and Liao method [17] as
(10)dk+1=−Dk+1gk+1=−I−skykTskTyk+tkskskTskTykgk+1.They obtained the following relation(11)dk+1Tgk+1=−gk+1TDk+1gk+1=−gk+1TAk+1gk+1,in which
(12)Ak+1≜Dk+1T+Dk+12=I−12skykT+ykskTskTyk+tkskskTskTyk,and analyzed the eigenvalues of the matrix Ak+1 to determine the parameter tk.Yao and Ning [18] proposed a three-term conjugate gradient method, in which the search direction was expressed as
(13)dk+1=−Dk+1gk+1=−I−tkskykT+ykskTskTyk+skskTskTykgk+1=−gk+1+βk+dk+δkyk,where the optimal parameter tk was derived by minimizing the distance between Dk+1 and the self-scaled memoryless BFGS (ML-BFGS) matrix in the Frobenius norm, that is,
(14)tk=min11+sk2yk2/skTyk2,skTykyk2,and the parameters βk+=maxtkgk+1Tyk−gk+1Tsk/dkTyk,0,δk=tkgk+1Tsk/skTyk. The search direction dk+1 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
(15)Qk+1=θk+1I−12skykT+ykskTskTyk+tkskskTskTyk,and improve ML-BFGS matrix based on the modified secant equation. The parameter tk in (15) is determined by minimizing the Frobenius norm of difference between Qk+1 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 parametertk 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 effectiveThe 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 Section3, 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:(16)Bk+1sk=zk,where zk=yk+maxτk,0/skTukuk, τk=6fk+1−fk+3gk+1+gkTsk, and μk=yk; we design ML-BFGS matrix based on modified secant equation as follows:
(17)Bk+1−1=θk+1I−θk+1skykT+ykskTskTyk+1ρk++θk+1yk2skTykskskTskTyk,where ρk+=1+maxτk,0/skTyk.The parametertk is determined by minimizing the Frobenius norm of difference between search direction matrix and ML-BFGS matrix based on modified secant equation, that is,
(18)minQk+1−Bk+1−1F2,where ⋅F is the Frobenius norm and Qk+1 and Bk+1−1 are determined by (15) and (17), respectively.From (15) and (17), we have
(19)Qk+1−Bk+1−1F2=trQk+1−Bk+1−1TQk+1−Bk+1−1=sk4skTyk2t2+22θk+1−1sk2skTyk−1ρk+sk4skTyk2−θk+1yk2sk4skTyk3t+ξ,where ξ is a constant independent of t. Therefore, the minimum of problem (18) is
(20)tk=argmintrQk+1−Bk+1−1TQk+1−Bk+1−1=1ρk++yk2θk+1−2θk+1−1mkskTyk,in which mk=cos2ηk, ηk=sk,yk is the angle between sk and yk. Instead of the mean value to cos2ηk=1/2 in [19], let mk be a random number in the interval c¯,c¯, where 0<c¯<c¯<1/2. Therefore, tk in (20) can be regarded as a random parameter. There are many possible ways to choose θk+1; we set
(21)θk+1=max1,sk2skTykorθk+1=max1,skTykyk2.Substitute (20) into (15), and let dk+1=−Qk+1gk+1; then,
(22)dk+1=−θk+1I−12skykT+ykskTskTyk+tskskTskTykgk+1≜−θk+1gk+1+aksk+bkyk,where
(23)ak=12ykTgk+1skTyk−1ρk++yk2θk+1−2θk+1−1mkskTykskTgk+1skTyk,bk=12skTgk+1skTyk.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. Givenx0∈ℝn, ε>0, 0<c¯<c¯<1/2 and 0<ω<σ<1. Compute f0=fx0 and g0=∇fx0; let d0≔−g0 and k≔0.
Step 1. Ifgk≤ε, stop; else, go to step 2.
Step 2. Compute a step lengthαk satisfying strong Wolfe line search conditions
(24)fxk+αdk−fxk≤ωαgkTdk,(25)gTxk+αdkdk≤−σgkTdk.
Step 3. Setxk+1=xk+αkdk; calculate fk+1, gk+1, sk, yk, and ρk+.
Step 4. Computetk by (20) and (21) and search direction dk+1 by (22). Set k≔k+1 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 sequencedk+1 be generated by STCG algorithm; then, there exists a positive constant c, such that
(26)gk+1Tdk+1≤−cgk+12.Proof.
From the search direction (22), we have
(27)gk+1Tdk+1=−θk+1gk+12+ykTgk+1gk+1TskskTykskTyk2−1ρk++yk2θk+1−2θk+1−1mkskTykskTgk+12skTyk≤−θk+1gk+12+12gk+1Tsk2yk2+skTyk2gk+12skTyk2−1ρk++yk2θk+1−2θk+1−1mkskTykskTgk+12skTyk=12−θk+1gk+12−skTgk+12skTyk1ρk++yk2skTykθk+1−2θk+1−1mk−12≤12−θk+1gk+12−skTgk+12skTyk1ρk++yk2skTykθk+1−2θk+1−1c¯−12≤−12gk+12.
The second of the above inequalities comes from the factuTv≤1/2u2+v2, in which u=gk+1Tskyk and v=skTykgk+1. In the fourth of the above inequalities, skTyk>0 can be ensured by the strong Wolfe line search, and ρk+=1+maxτk,0/skTyk>0. 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 setΩ=x∈ℝn:fx≤fx0 is bounded; namely, there exists a positive constant δ such that x≤δ,∀x∈Ω.Assumption 3.
The gradient of functionf is Lipschitz continuous in some neighborhood ℕ of Ω; namely, there exists L>0 satisfying
(28)gx−gy≤Lx−y,∀x,y∈ℕ.Based on the above assumptions, we can easily have thatgx is bounded; i.e., there exists a positive constant M such that
(29)gx≤M,∀x∈Ω.Lemma 4.
If Assumption3 holds, then τk is bounded, i.e.,
(30)τk≤3Lsk2.The proof of Lemma4 is similar to the proof of Lemma 2 in [20], so we omit it here.According to Lemma4, we can see 1/ρk+=skTyk/skTyk+maxτk,0<=1.Lemma 5.
Let the sequencedk be generated by STCG algorithm. If Assumption 3 holds, then
(31)αk≥1−σgkTdkLdk2.Proof.
According to (25), we have gk+1Tdk≥σgkTdk, then both side to subtracte gkTdk, and using Lipschitz condition, we get
(32)σ−1gkTdk≤gk+1−gkTdk=ykTdk≤ykdk≤αkLdk2.
Sincedk is a descent direction and 0<σ<1, (31) follows immediately.Lemma 6.
Let the sequencedk be generated by STCG algorithm. If Assumption 3 holds, we have
(33)∑k=0∞gkTdk2dk2<+∞.Proof.
From the first inequality (24) of strong Wolfe conditions, Assumption 3, and Lemma 5, we have
(34)fk−fk+1≥−ωαkgkTdk≥−ρ1−σgkTdk2Ldk2.
Sincefx is bounded from below, the proof is completed.Theorem 7.
Suppose that Assumption2 and Assumption 3 hold. The sequence xk is generated by STCG algorithm. If f is a uniformly convex function on Ω, namely, there exists a positive constant μ such that
(35)∇fx−∇fyTx−y≥μx−y2,∀x,y∈ℕ,then we have
(36)limk⟶∞gk=0.Proof.
From the Lipschitz condition (28), we have
(37)yk=gk+1−gk≤Lsk.
It follows (35) that
(38)ykTsk≥μsk2.
Using Cauchy inequality and (38), we obtain μsk2≤ykTsk≤yksk, i.e.,
(39)μsk≤yk.
Then, from (37), (38), and (39), we have
(40)1L=sk2Lsk2≤sk2skyk≤sk2skTyk≤sk2μsk2≤1μ,(41)μL2≤μsk2L2sk2≤μsk2yk2≤skTykyk2≤skykyk2≤1μ.
Letθmax=max1,1/μ; we get θk+1≤θmax. From (40), we obtain
(42)tk=1ρk++yk2θk+1−2θk+1−1mkskTyk≤1+L2μθmax−2θmax−1c¯.
Therefore, from (22), (37), (38), and (42), we have
(43)dk+1=−θk+1gk+1+aksk+bkyk≤θmaxgk+1+12skTgk+1skTykyk+12ykTgk+1skTyksk+tkskTgk+1skTyksk≤θmaxgk+1+12gk+1ykμsk+12gk+1ykμsk+1+L2μθmax−2θmax−1c¯gk+1μ≤1+L+1μ+θmax−2θmax−1c¯L2μ2gk+1≜M1gk+1.
From Lemma1 and (43), we get
(44)gk+1Tdk+12dk+12≥c2gk+12M12.
Combined with Lemma6, then
(45)∑k=0∞gk2<∞.
The proof is completed.For general nonlinear functions, we can establish a weaker convergence result:(46)liminfk⟶∞gk=0.Lemma 8.
Suppose that Assumption2 and Assumption 3 hold. Let the sequence xk be generated by STCG algorithm; then, we have dk≠0 and
(47)∑k=0∞uk+1−uk2<∞,whenever infgk:k≥0>0, in which uk=dk/dk.Proof.
Defineγ=infgk:k≥0, then gk≥γ>0. From the sufficient descent condition (26), we know dk≠0 for each k, so uk is well defined. To prove global convergence, we define ak+=maxak′,0, where ak′=1/2ykTgk+1/dkTyk−1/ρk+yk2θk+1−2θk+1−1mk/skTykskTgk+1/dkTyk. By (22), we have
(48)dk+1dk+1=−θk+1gk+1dk+1+ak+dkdk+1+bkykdk+1=−θk+1gk+1+bkykdk+1+ak+dkdk+1dkdk,namely,
(49)uk+1=ωk+δkuk,where
(50)ωk=−θk+1gk+1+bkykdk+1,δk=ak+dkdk+1≥0.
Using the identityuk+1=uk=1, we have
(51)ωk=uk+1−δkuk=δkuk+1−uk.
Sinceδk≥0, then
(52)uk+1−uk≤1+δkuk+1−1+δkuk≤uk+1−δkuk+δkuk+1−uk=2ωk.
From (25), we have
(53)σgkTdk−σgk+1Tdk≤gk+1Tdk−σgk+1Tdk−σykTdk≤1−σgk+1Tdk−σ1−σ≤gk+1TdkykTdk,gk+1Tdk+σgk+1Tdk≤−σgkTdk+σgk+1Tdkgk+1Tdk+σgk+1Tdk≤σykTdkgk+1TdkykTdk≤σ1+σ.Thus,
(54)skTgk+1skTyk=dkTgk+1dkTyk≤σ1−σ,(55)yk≤gk+1+gkgk+1gk+1≤1+Mγgk+1.
By the definition ofωk,bk, (54) and (55), we get
(56)ωk=−θk+1gk+1+bkykdk+1≤θk+1gk+1+1/2skTgk+1/skTyk⋅ykdk+1≤θmax+σ21−σ1+Mγgk+1dk+1.
Ifgk+1>γ, from Lemma 1 and Lemma 6, we have
(57)∑k=0∞c2γ2gk+12dk+12≤∑k=0∞c2gk+14dk+12≤∑k=0∞gk+1Tdk+12dk+12<+∞.Thus, (47) holds.Property(∗). Consider a method of form (2) and (22), and suppose
(58)0<γ≤gk≤γ¯,k≥0.We call that a method has Property(∗) if there exist constants b>1 and λ>0 such that ak′<b and sk≤λ⇒ak′≤1/2b.Lemma 9.
Suppose that Assumption 2 and Assumption 3 hold. Let the sequencedk be generated by STCG algorithm; then, STCG algorithm has Property(∗).Proof.
By (25) and (26), we obtain
(59)dkTyk≥σ−1gkTdk≥c1−σgk2.Using (29), (58), Assumption 2, and (59), we obtain
(60)ak′=12ykTgk+1dkTyk−1ρk++yk2θk+1−2θk+1−1mkskTykskTgk+1dkTyk≤12ykgk+1c1−σgk2+1ρk++yk2θk+1−2θk+1−1mkskTykskgk+1c1−σgk2≤12gk+1−gkgk+1c1−σgk2+1+gk+1−gk2θmaxc1−σgk2skgk+1c1−σgk2≤γ¯2c1−σγ2+1+4γ¯2θmaxc1−σγ22δγ¯c1−σγ2≔b.Let(61)λ≔c21−σ2γ42γ¯2γ¯+1+4γ¯2θmax/c1−σγ22δL/2+1+4γ¯2θmax/c1−σγ2.If sk≤λ, from (60) and (61), we obtain
(62)ak′≤12Lskgk+1c1−σγ2+1ρk++L2sk2θmaxc1−σγ2skgk+1c1−σγ2≤12Lγ¯c1−σγ2+1+4γ¯2θmaxc1−σγ2γ¯c1−σγ2sk≤12Lγ¯c1−σγ2+1+4γ¯2θmaxc1−σγ2γ¯c1−σγ2λ=12b.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.. Letℕ be the set of positive integers, Kλ≔i∈ℕ:i≥1,si>λ,forλ>0, namely, the set of integers corresponding to steps greater than λ. Now, we need to discuss groups of Δ consecutive iterates. Let Kk,Δλ≔i∈ℕ:k≤i≤k+Δ−1,si>λ, and Kk,Δλ denote the number of elements of Kk,Δλ.Lemma 10.
Suppose that Assumption2 and Assumption 3 hold. Let the sequences xk and dk be generated by STCG algorithm. When (58) holds, there exists λ>0 such that
(63)Kk,Δλ>Δ2,forΔ∈ℕ,where k≥k0, in which k0 is any index.Proof.
Suppose on the contrary that there existsλ>0, such that Kk,Δλ≤Δ/2 for Δ∈ℕ and for any k≥k0.
By (54) and (55), we have
(64)bkyk=12skTgk+1skTykyk≤σ21−σ1+γ¯γgk+1≜M2gk+1.
According to (22), we have
(65)dk+12≤ak′dk+−θk+1gk+1+bkyk2≤2ak′2dk2+2−θk+1gk+1+bkyk2≤2ak′2dk2+22θmaxgk+12+2bkyk2≤2ak′2dk2+4θmax2+M22gk+12.
By induction, we have(66)dl2≤c11+2al−1′2+2al−1′22al−2′2+⋯+2al−1′22al−2′2⋯2ak0′2,for any given index l≥k0+1, where c1 depends on dk0−1, not on l. Next, we consider 2al−1′22al−2′2⋯2ak′2, where k0≤k≤l−1. Now, we divide 2l−k factors of (66) into groups of each 2Δ elements; namely, if Λ≔l−k/Δ, then (66) can be divided into Λ or Λ+1 groups
(67)2al1′2⋯2ak1′2,⋯,2alΛ′2⋯2akΛ′2,and a possible group
(68)2alΛ+1′2⋯2ak′2,where li=l−1−i−1Δ for i=1,2,⋯,Λ+1, and ki=li+1+1 for i=1,2,⋯,Λ. It is clear that ki≥k0 for i=1,2,⋯,Λ; from assumption condition, we get pi≔Kki,Δλ≤Δ/2. Thus, there are pi indices j such that sj>λ and Δ−pi indices j such that sj≤λ on ki,ki+Δ−1.
From (60), we have b>γ¯2/c1−σγ2>1, i.e., 2b2>1. In conjunction with 2pi−Δ≤0, we have 2ali′2⋯2aki′2≤2Δb2pi1/2b2Δ−pi=2b22pi−Δ≤1. So every item in (67) is less than or equal to 1, and so is their product. In (68), we have 2alΛ+1′2⋯2ak′2≤2b2Δ. Then, we get
(69)dl2≤c2l−k0+2,where c2>0 and independent of l. Furthermore, ∑k≥01/dk2=∞. But from (26), (33), and (58), we have
(70)c2γ4∑k≥01dk2≤c2∑k≥0gk4dk2≤∑k≥0gkTdk2dk2<∞.
It leads to a contradiction. The proof is completed.Theorem 11.
Suppose that Assumption2 and Assumption 3 hold. Let the sequence xk 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.Table 1
The test functions and dimensions.
ProblemsFunctionsDimensionsProblemsFunctionsDimensions1The penalty function I100031Variable dimension10002The penalty function I500032Variable dimension100003The penalty function I1000033Variable dimension500004The penalty function I5000034Watson10005The penalty function I10000035Watson100006Boundary value function100036Watson500007Boundary value function500037Chebyquad50008Boundary value function1000038Chebyquad100009Boundary value function5000039Broyden banded100010Boundary value function10000040Broyden banded500011Broyden tridiagonal function100041Broyden banded1000012Broyden tridiagonal function500042Generalized Rosebrock100013Broyden tridiagonal function1000043Generalized Rosebrock500014Broyden tridiagonal function5000044Generalized Rosebrock1000015Separable cubic function100045Boundary value100016Separable cubic function500046Boundary value500017Separable cubic function1000047Boundary value1000018Separable cubic function5000048Integral equation100019Separable cubic function10000049Integral equation500020The variable dimension function100050Integral equation1000021The Chebyquad function100051Yang tridiagonal100022Nearly separable function100052Yang tridiagonal500023Nearly separable function500053Yang tridiagonal1000024Schittkowski function 302100054Allgower100025Extended Rosenbrock100055Freudenstein and Roth226Extended Rosenbrock1000056Powell badly scaled227Extended Rosenbrock5000057Brown badly scaled228Extended Powell singular100058Beale229Extended Powell singular1000059Helical valley330Extended Powell singular5000060Wood4We 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 θk+1=max1,sk2/skTyk and θk+1=max1,skTyk/yk2 are chosen, STCG algorithm is denoted by “New1” and “New2,” respectively.All test methods are terminated when satisfying the following condition:(71)gk≤εorfxk+1−fxk≤εmax1.0,fxk,where ε=10−6 and ω=0.1 and σ=0.6 in Wolfe conditions (24) and (25).The result of computational experiments from partial problems in Table1 are listed in Table 2. In Table 2, k, nf, ng, 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 τ of the test problems for which a method is the fastest (efficiency), while the vertical side gives the percentage ψ of the test problems that are successfully solved by each of the methods.Table 2
Partial numerical results of several methods.
P.DDLPRP+New1New2k/nf/ng/CPUk/nf/ng/CPUk/nf/ng/CPUk/nf/ng/CPU111/30/12/0.031313/39/15/0.062511/32/12/0.031312/34/15/0.0313213/35/14/0.109418/50/23/0.14069/31/10/0.031312/40/16/0.1250314/41/15/0.140612/44/17/0.140613/41/14/0.125012/39/14/0.1563437/85/51/0.953116/49/19/0.359419/60/27/0.500011/44/15/0.2344537/96/50/1.484418/55/19/0.843812/45/13/0.546914/49/15/0.6719624/33/26/0.04699/19/11/0.031319/27/21/0.015617/27/20/0.0781724/33/26/0.10949/19/11/0.031319/27/21/0.093818/27/21/0.1094824/33/26/0.26569/19/11/0.062519/28/22/0.125021/31/24/0.1875924/33/26/0.42199/19/11/0.265618/26/20/0.453120/30/23/0.32811024/33/26/0.84389/19/11/0.259419/28/22/0.781319/26/21/0.70311144/61/49/0.062537/67/47/0.046931/46/32/0.031331/46/32/0.03131231/46/32/0.093844/76/53/0.171931/46/33/0.062531/46/32/0.18751336/53/40/0.296918/42/25/0.125035/49/37/0.281334/46/36/0.23441434/51/36/0.562541/75/50/1.031335/56/38/0.921936/59/39/0.73441536/56/39/1.578149/81/58/2.156340/60/43/1.312540/62/43/1.70311612/15/14/0.17197/16/11/0.12507/12/9/0.09387/12/9/0.09381712/15/14/5.062513/22/18/6.21888/13/10/3.31258/13/10/3.29691812/15/14/14.890612/21/16/17.18759/14/11/11.39069/14/11/11.45311914/17/16/389.562514/22/18/438.812510/15/12/285.51568/13/10/234.60942013/54/13/0.015613/54/13/0.015613/54/13/0.031313/54/13/0.03132128/48/34/2.640910/21/13/0.968818/30/20/1.437515/25/17/1.14062275/144/101/1.140615/45/23/0.359433/67/44/0.531335/71/44/0.51562332/76/47/9.000059/145/98/18.703136/84/53/10.015641/88/54/10.37502421/58/26/0.015618/56/25/0.078119/55/22/0.015625/62/29/0.0469From Table2, 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 1–4 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.Figure 1
The number of iterations.Figure 2
The number of function evaluations.Figure 3
The number of gradient evaluations.Figure 4
CPU time.


## 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 Figure5 [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.Figure 5
Supply chain operation flow chart.For a better description of the model, the symbols are shown in Table3.Table 3
Symbolic explanation table.
Symbolic variablescmUnit production cost of new productsdPotential market demandkUnit carbon emissions trading priceaThe initial amount of carbon emissions allocated by the governmentτWarranty period sensitivity coefficient for quantity of repaired productsφPotential product repair quantityμCarbon emission reduction investment cost coefficienteCarbon emissions from the production of one unit of new productbWarranty period investment cost coefficientβRevenue sharing ratioαConsumer sensitivity coefficientDMarket demandRTotal number of products returned for repairΠTiProfit function for the entire LSCS in the modeiDecision variablespUnit retail priceθWarranty periodλCarbon emission reduction level of productAccording to the assumptions given in [25], the object problem can be transformed into profit maximization. The warranty cost function is introduced as bθ2 (b>0, θ>0) (see [26–28]). Market demand only depends on the warranty period θ, the demand function is D=d−αp+nθ, and the function expression for the number of products repaired by the manufacturer during the warranty period is R=φ+τθ. 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
(72)maxp,θΠT=p−cmD+ka−eD−εeR−bθ2.We transform (72) into the following optimization problem:
(73)minp,θΠT=−p−cmD−ka−eD−εeR+bθ2.Based on (72), the carbon emission reduction level of the product is considered. The demand function of product D=d−αp+nθ+δλ is linear, where n and δ are positive, inversely proportional to the retail price p, proportional to the warranty period θ, and the carbon emission reduction level λ. 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 1/2bλ2 [29].Considering the carbon reduction efficiency of a product under centralized decision, the overall profit function of LSCS is(74)maxp,λ,θΠT=p−cmD+ka−eD−εeR−12μλ2−bθ2.Similarly, (74) is transformed into the following optimization problem:
(75)minp,λ,θΠT=−p−cmD−ka−eD−εeR+12μλ2+bθ2.Then, the STCG algorithm is used to calculate the optimization problems (73) and (75), and choose different initial points pC,θC and pC0,θC0,λC0, respectively. Based on the average unit carbon emissions trading price of April 2020 in Fujian Province of China, let k=9.1. With reference to the setting of the remaining parameters in [30–32], we set cm=500, τ=0.12, n=0.84, δ=0.8, a=5000, e=10, d=300, ε=0.3, φ=50, r=0.4, μ=15, b=10, and β=0.2; these values ensure that the optimal value is meaningful, and the results are shown in Tables 4–6.Table 4
Optimal decisions under different consumer price sensitivity coefficients.
Sensitive coefficientαInitial pointpC∗θC∗ΠTC∗α=0.1(0; 0)2052.461.2220090(50; 10)2052.661.2220090(100; 20)2052.561.3220090(1000; 40)2052.761.7220090(5000; 100)2052.861.2220090α=0.3(0; 0)807.29508.978557435(50; 10)807.29548.982857435(100; 20)807.39858.869557435(1000; 40)808.86768.916257435(5000; 100)808.00768.950557435α=0.5(0; 0)595.39610.052444145(50; 10)595.3171-0.012944145(100; 20)595.21760.049544145Table 5
Optimal decisions under different consumer price sensitivity coefficients.
Sensitive coefficientαInitial pointpC∗θC∗ΠTC∗α=0.5(1000; 40)595.60010.046044145(5000; 100)595.72730.038844145α=0.8(0; 0)480.6638-4.765853692(50; 10)480.2855-4.793553692(100; 20)480.5973-4.775553692(1000; 40)480.6685-4.764753692(5000; 100)480.2986-4.844353692α=1(0; 0)442.6351-6.365165706(50; 10)442.6246-6.361165706(100; 20)442.6080-6.439265706(1000; 40)443.0113-6.411565706(5000; 100)442.9677-6.409765706Table 6
Under the carbon emission reduction level of products, the optimal decision under different consumer price sensitivity coefficients.
Sensitive coefficientαInitial pointpC0∗θC0∗λC0∗ΠTC0∗α=0.1(0; 0; 0)2562.282.6105.1281600(50; 10; 10)2562.982.6105.2281600(100; 20; 20)2561.182.6105.0281600(1000; 40; 40)2559.782.5105.0281600(5000; 100; 100)2564.882.7105.3281600α=0.3(0; 0; 0)825.34419.679712.497958522(50; 10; 10)824.79239.669312.418158522(100; 20; 20)825.35799.717612.489858522(1000; 40; 40)825.94819.708212.533658522(5000; 100; 100)825.70939.690612.518858522α=0.5(0; 0; 0)595.1347-0.01850.165544145(50; 10; 10)595.2992-0.03870.218544145(100; 20; 20)596.21030.01460.271944145(1000; 40; 40)596.40030.04730.277444145(5000; 100; 100)596.27310.05790.294944145α=0.8(0; 0; 0)477.1441-4.9476-6.097653960(50; 10; 10)477.2710-4.9305-6.039153960(100; 20; 20)477.2881-4.9724-6.017053960(1000; 40; 40)477.6123-4.9022-6.085153960(5000; 100; 100)477.8681-4.8857-5.964653960α=1(0; 0; 0)439.3399-6.5715-8.105166185(50; 10; 10)439.2198-6.4765-8.114866185(100; 20; 20)439.6549-6.5136-8.053166185(1000; 40; 40)439.8000-6.5487-8.027566185(5000; 100; 100)439.6809-6.5573-8.073066185Tables4–6 show the optimal decision of LSCS under different initial points and different consumer price sensitivity coefficients α. As far as the consumer price sensitivity coefficient α is concerned, each optimal decision variable is negatively correlated with the consumer price sensitivity coefficient α. When α is small, indicate that the price has a relatively small impact on the market demand. On the premise of ensuring after-sales service and carbon emission reduction level of products, the retail and wholesale prices of products can be appropriately raised. When α is large, the price has become an important factor affecting market demand, and each decision-making variable is reduced. With the increase of α, the optimal profit shows a trend of rising first and then falling.In terms of different initial points, on the premise of the same consumer sensitivity coefficientα, the STCG algorithm is used to solve the supply chain optimization problem. Numerical experiment results show that the solution obtained by STCG algorithm is very close to the real solution.


## 6. Conclusion

A new spectral three-term conjugate gradient method with random parameter is proposed. By minimizing the Frobenius norm distance between search directionQk+1 and ML-BFGS matrix based on the modified secant equation, the parameter tk in Qk+1 is determined, and the random parameter is introduced to simplify the scheme of tk. The search direction is sufficiently close to the quasi-Newton direction and satisfies the sufficient descent property. Global convergence of the new algorithm is proved under appropriate assumptions. Some classical test problems are selected for numerical experiments and compared with DDL and PRP+ to verify the effectiveness of the proposed algorithm. Using STCG algorithm to solve supply chain optimization has a certain practical prospect, which reflects the effectiveness of the new algorithm. There are still many deficiencies in our research; for example, the more efficient and widely used spectral parameter θk is not taken into account. Therefore, the idea of alternating direction method will be used in the future work to explore the spectral parameter θk and parameter tk in the search direction. The use of alternating direction method can better correct the approximate degree of search direction and quasi-Newton direction and obtain a more effective and robust method.



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