# A Modified Nonlinear Conjugate Gradient Method for Engineering Computation

Authors: Tiefeng Zhu; Zaizai Yan; Xiuyun Peng
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

A general criterion for the global convergence of the nonlinear conjugate gradient method is established, based on which the global convergence of a new modified three-parameter nonlinear conjugate gradient method is proved under some mild conditions. A large amount of numerical experiments is executed and reported, which show that the proposed method is competitive and alternative. Finally, one engineering example has been analyzed for illustrative purposes.


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## Body

## 1. Introduction

Unconstrained optimization methods are widely used in the fields of nonlinear dynamic systems and engineering computation to obtain the numerical solution of the optimal control problem [1–4]. In this paper, we consider the unconstrained optimization problem: (1)minfx,x∈Rn,where f:Rn→R is a continuously differentiable function. The nonlinear conjugate gradient (CG) method is highly useful for solving this kind of problems because of its simplicity and its very low memory requirement [1]. The iterative formula of the CG methods is given by (2)xk+1=xk+αkdk,where αk>0 is step length which is obtained by carrying out some linear search, such as exact or inexact line search. In practical computation, exact line search is consumption time and the workload is very large, so we usually take the following inexact line search (see [5–7]). Usually, a major inexact line search is the strong Wolfe-Powell line search. The strong Wolfe-Powell line search is to find the step-length αk in (2) satisfying (3)fxk+αkdk-fxk≤δαkgkTdk,(4)σgkTdk≤gxk+αkdkTdk≤-σgkTdk,where 0<δ<1/2 and δ<σ<1. In this paper, the following modified Wolfe-Powell line search is to find the step-length αk in (2) satisfying (3) and the following: (5)σgkTdk≤gxk+αkdkTdk≤0,and dk is the search direction defined by (6)dk=-g1k=1-gk+βkdk-1k≥2,where gk denotes the gradient ∇f(xk), βk is scalar, and βk is chosen so that dk becomes the kth conjugate direction. There have been many well-known formulae for the scalar βk, for example, (7)βkFR=gk2gk-12(Fletcher-Reeves [8], 1964),(8)βkPRP=gkTyk-1gk-12(Polak-Ribiere-Polyak [9], 1969),(9)βkDY=gk2dk-1Tyk-1(Dai-Yuan [10], 1999), and other formulae (e.g., [11–13]), where · is the Euclidean norm of vectors, yk-1=gk-gk-1, and “T” stand for the transpose. These methods are generally regarded as very efficient conjugate gradient methods in practical computation.In recent decades, in order to obtain the CG method which has not only good convergence property but also excellent computation, many researchers have studied the CG method extensively and obtained some improved methods with good properties [14–20]. Li and Feng [21] gave the modified CG method which generates a sufficient descent direction and showed its global convergence property under the strong Wolfe-Powell conditions. Dai and Wen [22] gave a scaled conjugate gradient method. They proved its global convergence property under the strong Wolfe-Powell conditions. Al-Baali [23] proved that the FR method satisfies the sufficient descent condition and converges globally for general objective functions if the strong Wolfe-Powell line search is used. Dai and Yuan [24] also introduced a formula for βk: (10)βk=1-λkgk2+λkgkTyk-11-μk-ωkgk-12+μkdk-1Tyk-1-ωkdk-1Tgk-1,where 0≤λk≤1, 0≤μk≤1, and 0≤ωk≤1-μk. Because (11)gkTyk-1=gk2-gkTgk-1,gk-1Tdk-1=-gk-12+βk-1gk-1Tdk-2,we can rewrite (10) as (12)βk∗=gk2-λkgkTgk-1gk-12+μkdk-1Tgk-ωkβk-1gk-1Tdk-2.This formula includes the above three classes of CG method as an extreme case, and global convergence of three parameters of CG method was proved under strong Wolfe-Powell line search. If ωk=0, then the family reduces to the two-parameter family of conjugate gradient methods in [25]. Further, if λk=0, μk=μ, and ωk=0, then the family reduces to the one-parameter family in [26]. Therefore, the three-parameter family has the one-parameter family in [26] and the two-parameter family in [25] as its subfamilies. In addition, some hybrid methods can also be regarded as special cases of the three-parameter family [24]. Above many modified CG methods, global convergence was obtained under strong Wolfe-Powell line search; however, in this paper, we further study the CG method, and our main aim is to improve the numerical performance of the CG method while keeping its global convergence with modified Wolfe-Powell line search.This paper is organized as follows. We first present a criterion for the global convergence of CG method in the next section. In Section3, we propose a new modified three-parameter conjugate gradient method and establish global convergence results for relative algorithm under modified Wolfe-Powell line search. The preliminary numerical results are contained in Section 4. One engineering example is analyzed for illustration in Section 5. Finally, conclusions appear in Section 6.


## 2. A Criterion for the Global Convergence of CG Method

In this section, first, we adopt the following assumption used commonly in the research literatures.Assumption 1.
The functionf is LC in a neighborhood N of the level set Ω≔{x∈Rn∣f(x)≤f(x1)} and Ω is bounded. Here, by LC, we mean that the gradient ∇f(xk) is Lipschitz continuous with modulus L; that is, there exists L>0 such that (13)gx-gy≤Lx-yfor  anyx,y∈N.Lemma 2 (Zoutendijk condition [27]).
Suppose that Assumption1 holds, xk is given by (2) and (6), and αk is obtained by the modified Wolfe-Powell line search ((3), (5)), while the direction dk satisfies gkTdk<0. Then, (14)∑k=1∞gkTdk2dk2<+∞.Lemma 3 (see [28]).
Suppose thatλk(>0) and C are constants; if {ai} satisfy ∑i=1k‍ai≥λk+C, then ∑i≥1‍ai2/i=+∞ and ∑k≥1‍ak2/∑i=1k‍ai=+∞.Theorem 4.
Suppose that the objective function satisfies Assumption1 and that xk is given by (2) and (6), where αk satisfies the modified Wolfe-Powell (3) and (5), and |βk|≤gk2/gk-12; then, either gk=0 holds for certain k or (15)liminfk→∞gk=0.Proof.
Suppose, by contradiction, that the stated conclusion is not true. Then, in view ofgk>0, there exits a constant ε>0, such that (16)gk2≥ε,k=1,2,….From (6), we have (17)dk+gk=βkdk-1,k=2,3,….By multiplying dk+gk on both sides of (17), then we have (18)dk2=-gk2-2gkTdk+βk2dk-12.Let tk=dk2/gk4 and rk=-gkTdk/gk2; then, (19)tk=-1gk2-2gkTdkgk4+βk2dk-12gk4.Thus, from (19) and |βk|≤gk2/gk-12, we get (20)tk≤-1gk2+2rkgk2+gk4gk-14·dk-12gk4=tk-1-1gk2+2rkgk2.Note that t1=1/g12 and r1=1; then, it follows from (20) that (21)tk≤-∑i=1k1gi2+2∑i=1krigi2.From Assumption 1, it follows that there exists constant M(>0), such that (22)gx2≤M,∀x∈Ω,and from (16), (21), and (22), we get (23)tk≤-kM+2ε∑i=1kri.From the above, it is obvious that (24)tk≤2ε∑i=1kri.From the other side, for tk≥0 from (23), it follows that (25)∑i=1kri≥kε2M.From (24) and (25) and Lemma 3, we have (26)∑k≥1gkTdk2dk2=∑k≥1rk2tk≥ε2∑k≥1rk2∑k≥1ri=+∞,what contradicts Lemma 2. Therefore, the global convergence is proved.


## 3. The Global Convergence for the New Formula and Algorithm Frame

### 3.1. The New Formula and the Corresponding Properties

Based on formula (10), we put forward a new formula of βk: (27)βknew=max⁡0,min⁡1-λkgk2,λkgkTgk-1-dk-11-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1,where 1/2<λk≤1, 0≤μk≤1, 0≤ωk≤1, and λk≥μk+ωk. Because of possible negative values of (28)min⁡1-λkgk2,λkgkTgk-1-dk-1,we use the maximum function to truncate zero and (29)min⁡1-λkgk2,λkgkTgk-1-dk-1.Using the equality (30)gk-1Tdk-1=-gk-12+βk-1gk-1Tdk-2,we can rewrite the denominator of (27) as (31)gk-12+μkgkTdk-1-ωkβk-1gk-1Tdk-2+1-λkgk-12-1-μk+ωkβk-1gk-1Tdk-2.When (32)1-λkgk-12=1-μk+ωkβk-1gk-1Tdk-2,then the denominator of βknew given by (27) reduces to the denominator of βk∗. On the other hand, when (33)0<λkgkTgk-1-dk-1<1-λkgk2,the numerator of (27) reduces to (34)λkgkTgk-1-dk-1.When (35)dk-1=1λkgk,then (36)λkgkTgk-1-dk-1=gk2-λkgkTgk-1.Now, the numerator of βknew (27) reduces to the numerator of βk∗. From the above analysis, we can see that (27) indeed is an extension of (10). Due to the existence of the parameters λk, μk, and ωk, it would be more flexible to call methods (2), (6), and (27) by this paper of conjugate gradient methods. Numerical experiments results in Section 4 demonstrate the influence of these parameters versus formula (27).Lemma 5.
Suppose that Assumption1 holds and that xk is given by (2) and (6), where αk satisfies the modified Wolfe-Powell conditions (3) and (5), while βk is computed by (27). Then, one has (37)gkTdkgk2<0∀k≥1.Proof.
Whenk=1, we have (38)g1Td1=-g12,g1Td1g12=-1<0.Suppose gk-1Tdk-1<0 hold, βk=βknew in formula (27), and the conclusion holds.
Ifβk=βknew, we have (39)gkTdk=-gk2+βkgkTdk-1,gkTdkgk2=-1+lk1-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1·gkTdk-1gk2,where lk=max⁡{0,min⁡{(1-λk)gk2,λkgkT(gk-1-dk-1)}}; by formulas (3) and (5), we have gkTdk-1<0. Hence, (40)gkTdk-1gk2<0,μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1≥μkσgk-1Tdk-1-1-λk+μk+ωkgk-1Tdk-1=μkσ-1+λk-μk-ωkgk-1Tdk-1.When μkσ-μk<0, -1+λk≤0, and gk-1Tdk-1<0, we obtain (41)μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1>0.Due to (1-μk-ωk)gk-12>0 and lk≥0, through the above analysis, we have (42)lk1-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1·gkTdk-1gk2<0.Hence, gkTdk/gk2<0.The result shows that the search direction satisfies descent condition (gkTdk<0); this condition may be crucial for convergence analysis of any conjugate gradient method.Lemma 6.
Suppose that Assumption1 holds and that {xk} is given by (2) and (6), where αk satisfies the modified Wolfe-Powell conditions (3) and (5), while βk is computed by (27). Then, one has (43)βk≤gk2gk-12.Proof.
Letβk=βknew; when (1-λk)gk2≤λkgkT(gk-1-dk-1), then lk=(1-λk)gk2; when (1-λk)gk2>λkgkT(gk-1-dk-1), if λkgkT(gk-1-dk-1)<0, then lk=0; if λkgkT(gk-1-dk-1)>0, lk=λkgkT(gk-1-dk-1).
By Lemma5, then μkgkTdk-1-(1-λk+μk+ωk)gk-1Tdk-1>0.
To sum up,|βk|≤(1-λk)gk2/(1-μk-ωk)gk-12, and by λk>μk+ωk, we have (1-λk)/(1-μk-ωk)<1, and hence |βk|≤gk2/gk-12.Theorem 7.
Suppose the objective functionf(x) satisfies Assumption 1; consider methods (2) and (6), where βk is given by (27) and αk satisfies the modified Wolfe-Powell line search condition. Then, either gk=0 holds for certain k or (44)liminfk→∞gk=0.Proof.
By Theorem4 and Lemma 6, Theorem 7 is proved.The result shows that the proposed algorithm with the modified Wolfe-Powell line search possesses global convergence.


### 3.2. Algorithm A

Based on the discussed above, now we can describe the algorithm frame for solving the unconstrained optimization problems (1) as follows.Step 0.
Choose an initial pointx0∈Rn, given constants ε0>0, δ∈(0,1), σ∈(δ,1) and λk∈(1/2,1], μk∈[0,1], and ωk∈[0,1], subject to λk≥μk+ωk, set d0=g0, and let k≔0.Step 1.
If a stopping criteriongk<ε0 is satisfied, then stop; otherwise, go to Step 2.Step 2.
Determine a step sizeαk by line searches (3) and (5).Step 3.
Letxk+1=xk+αkdk; compute βk and dk by (27) and (6).Step 4.
Setk≔k+1 and go to Step 1.


## 3.1. The New Formula and the Corresponding Properties

Based on formula (10), we put forward a new formula of βk: (27)βknew=max⁡0,min⁡1-λkgk2,λkgkTgk-1-dk-11-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1,where 1/2<λk≤1, 0≤μk≤1, 0≤ωk≤1, and λk≥μk+ωk. Because of possible negative values of (28)min⁡1-λkgk2,λkgkTgk-1-dk-1,we use the maximum function to truncate zero and (29)min⁡1-λkgk2,λkgkTgk-1-dk-1.Using the equality (30)gk-1Tdk-1=-gk-12+βk-1gk-1Tdk-2,we can rewrite the denominator of (27) as (31)gk-12+μkgkTdk-1-ωkβk-1gk-1Tdk-2+1-λkgk-12-1-μk+ωkβk-1gk-1Tdk-2.When (32)1-λkgk-12=1-μk+ωkβk-1gk-1Tdk-2,then the denominator of βknew given by (27) reduces to the denominator of βk∗. On the other hand, when (33)0<λkgkTgk-1-dk-1<1-λkgk2,the numerator of (27) reduces to (34)λkgkTgk-1-dk-1.When (35)dk-1=1λkgk,then (36)λkgkTgk-1-dk-1=gk2-λkgkTgk-1.Now, the numerator of βknew (27) reduces to the numerator of βk∗. From the above analysis, we can see that (27) indeed is an extension of (10). Due to the existence of the parameters λk, μk, and ωk, it would be more flexible to call methods (2), (6), and (27) by this paper of conjugate gradient methods. Numerical experiments results in Section 4 demonstrate the influence of these parameters versus formula (27).Lemma 5.
Suppose that Assumption1 holds and that xk is given by (2) and (6), where αk satisfies the modified Wolfe-Powell conditions (3) and (5), while βk is computed by (27). Then, one has (37)gkTdkgk2<0∀k≥1.Proof.
Whenk=1, we have (38)g1Td1=-g12,g1Td1g12=-1<0.Suppose gk-1Tdk-1<0 hold, βk=βknew in formula (27), and the conclusion holds.
Ifβk=βknew, we have (39)gkTdk=-gk2+βkgkTdk-1,gkTdkgk2=-1+lk1-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1·gkTdk-1gk2,where lk=max⁡{0,min⁡{(1-λk)gk2,λkgkT(gk-1-dk-1)}}; by formulas (3) and (5), we have gkTdk-1<0. Hence, (40)gkTdk-1gk2<0,μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1≥μkσgk-1Tdk-1-1-λk+μk+ωkgk-1Tdk-1=μkσ-1+λk-μk-ωkgk-1Tdk-1.When μkσ-μk<0, -1+λk≤0, and gk-1Tdk-1<0, we obtain (41)μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1>0.Due to (1-μk-ωk)gk-12>0 and lk≥0, through the above analysis, we have (42)lk1-μk-ωkgk-12+μkgkTdk-1-1-λk+μk+ωkgk-1Tdk-1·gkTdk-1gk2<0.Hence, gkTdk/gk2<0.The result shows that the search direction satisfies descent condition (gkTdk<0); this condition may be crucial for convergence analysis of any conjugate gradient method.Lemma 6.
Suppose that Assumption1 holds and that {xk} is given by (2) and (6), where αk satisfies the modified Wolfe-Powell conditions (3) and (5), while βk is computed by (27). Then, one has (43)βk≤gk2gk-12.Proof.
Letβk=βknew; when (1-λk)gk2≤λkgkT(gk-1-dk-1), then lk=(1-λk)gk2; when (1-λk)gk2>λkgkT(gk-1-dk-1), if λkgkT(gk-1-dk-1)<0, then lk=0; if λkgkT(gk-1-dk-1)>0, lk=λkgkT(gk-1-dk-1).
By Lemma5, then μkgkTdk-1-(1-λk+μk+ωk)gk-1Tdk-1>0.
To sum up,|βk|≤(1-λk)gk2/(1-μk-ωk)gk-12, and by λk>μk+ωk, we have (1-λk)/(1-μk-ωk)<1, and hence |βk|≤gk2/gk-12.Theorem 7.
Suppose the objective functionf(x) satisfies Assumption 1; consider methods (2) and (6), where βk is given by (27) and αk satisfies the modified Wolfe-Powell line search condition. Then, either gk=0 holds for certain k or (44)liminfk→∞gk=0.Proof.
By Theorem4 and Lemma 6, Theorem 7 is proved.The result shows that the proposed algorithm with the modified Wolfe-Powell line search possesses global convergence.


## 3.2. Algorithm A

Based on the discussed above, now we can describe the algorithm frame for solving the unconstrained optimization problems (1) as follows.Step 0.
Choose an initial pointx0∈Rn, given constants ε0>0, δ∈(0,1), σ∈(δ,1) and λk∈(1/2,1], μk∈[0,1], and ωk∈[0,1], subject to λk≥μk+ωk, set d0=g0, and let k≔0.Step 1.
If a stopping criteriongk<ε0 is satisfied, then stop; otherwise, go to Step 2.Step 2.
Determine a step sizeαk by line searches (3) and (5).Step 3.
Letxk+1=xk+αkdk; compute βk and dk by (27) and (6).Step 4.
Setk≔k+1 and go to Step 1.


## 4. Numerical Experiments and Results

In this section, in order to show the performance of the given algorithm, we test our proposed algorithm (algorithm A) and DY algorithm (given by formula (10)) via unconstrained optimization problems from Andrei [29] as follow. These testing functions are often used in engineering field:(1)
Sphere functionfSph(x)=∑i=1100‍xi2, xi∈[-5.12,5.12], x∗=(0,0,…,0), fSph(x∗)=0.(2)
Rastrigin functionfRas(x) = 100+∑i=110‍(xi2-10cos⁡(2πxi)), xi∈[-5.12,5.12], x∗ = (0,0,…,0), fRas(x∗) = 0.(3)
Freudenstein and Roth function (Froth)fFroth(x) = (-13+x1+((5-x2)x2-2)x2)2 + (-29+x1+((x2 + 1)x2-14)x2)2, x∗=(5,4), fFroth(x∗)=0.(4)
Perturbed Quadratic diagonal function (Pqd)fPqd(x) = (∑i=15‍xi)2 + ∑i=15‍i/100xi2, x∗=(0,0,…,0), fPqd(x∗) = 0.(5)
Extended White & Holst function (Ewh)fEwh(x) = ∑i=15‍[100(x2i-x2i-13)2 + (1-x2i-1)2], x∗=(1,1,…,1), fEwh(x∗)=0.(6)
Raydan 1 functionfRay1(x) = ∑i=12‍i/10(exp(xi)-xi), x∗=(0,0), fRay1(x∗)=0.3.(7)
Raydan 2 functionfRay2(x)=∑i=1500‍(exp(xi)-xi), x∗=(0,0,…,0), fRay2(x∗)=500.(8)
Extended Trigonometric function (Etri)fEtri(x) = ∑i=110‍[(n-∑j=110‍cos⁡xj) + i(1-cos⁡xi)-sin⁡xi]2, x∗=(0,0,…,0), fEtri(x∗)=0.(9)
Extended Powell function (Epow)fEpow(x) = (x3+10x2)2+5(x3-x2)2 + (x2-2x3)4 + 10(x1-x4)4, x∗=(0,0,…,0), fEpow(x∗)=0.(10)
Wood functionfWood(x) = 100(x2-x12)2 + (1-x1)2 + (1-x3)2 + 90(x4-x32)2 + 10(x2+x4-2)2 + 0.1(x2-x42)2, x∗=(1,1,…,1), fWood(x∗)=0.(11)
Extended Wood function (Ewood)fEwood(x) = 100(x12-x2)2 + (x1-1)2 + (1-x3)2 + 90(x32-x4)2 + 10.1{(x2-1)2 + (x4-1)2} + 19.8(x2-1)(x4-1), x∗=(1,1,…,1), fEwood(x∗)=0.(12)
Perturbed Quadratic function (Perq)fPerq(x) = ∑i=13‍ixi2 + 1/100(∑i=1n‍xi)2, x∗=(0,0,…,0), fPerq(x∗)=0.(13)
Extended Tridiagonal 1 function (Etri1)fEtri1(x) = ∑i=1500‍[(x2i-1 + x2i-3)2 + (x2i-1-x2i+1)4], x∗=(1,2,1,2,…,1,2), fEtri1(x∗)=0.(14)
Extended Miele & Cantrell function (Emic)fEmic(x) = ∑i=12‍[(exp(x4i-3)-x4i-2)2 + 100(x4i-2-x4i-1)6 + {tan⁡(x4i-1-x4i)}4 + x4i-38], x∗=(0,1,1,1,0,1,1,1,…,0,1,1,1), fEmic(x∗)=0.(15)
Extended Rosenbrock function (Erosen)fErosen(x) = ∑i=110‍[100(x2i-x2i-12)2 + (1-x2i-1)2], x∗=(1,1,…,1,1), fErosen(x∗)=0.(16)
Generalized Rosenbrock function (Grosen)fGrosen(x) = ∑i=11999‍[100(xi+1-xi2)2 + (1-xi)2], x∗=(1,1,…,1,1), fGrosen(x∗)=0.(17)
QUARTC functionfQUAR(x) = ∑i=120‍(xi-1)4, x∗=(1,1,…,1,1), fQUAR(x∗)=0.(18)
LIARWHD functionfLIAR(x) = ∑i=1100‍4(xi2-x1)2+∑i=1100‍(xi-1)2, x∗=(1,1,…,1,1), fLIAR(x∗)=0.(19)
Staircase 1 functionfStai1(x) = ∑i=14‍(∑j=1i‍xj)2, x∗=(0,0,…,0,0), fStai1(x∗)=0.(20)
Staircase 2 functionfStai2(x) = ∑i=1300‍[(∑j=1i‍xj)-i]2, x∗=(1,1,…,1,1), fStai2(x∗)=0.(21)
POWER functionfPOWER(x) = ∑i=11000‍(ixi)2, x∗=(0,0,…,0,0), fPOWER(x∗)=0.(22)
Diagonal 4 functionfDia4(x) = ∑i=12‍1/2(x2i-12+100x2i2), x∗=(0,0,…,0,0), fDia4(x∗)=0.(23)
Extended BD1 function (EBD1)fEBD1(x) = ∑i=15000‍[(exp(x2i-1)-x2i)2 + (x2i-12+x2i2-2)2, x∗=(1,1,…,1,1), fEBD1(x∗)=0.(24)
CUBE functionfCUBE(x) = (x1-1)2 + ∑i=2300‍100(xi-xi-13)2, x∗=(1,1,…,1,1), fCUBE(x∗)=0.Here,x∗ and f(x∗) are the optimal solution and the function value at the optimal solution, respectively. For each algorithm, the parameters are chosen as δ=0.04 and σ=0.5. All codes were written in MATLAB 7.5 and run on Lenovo with 1.90 GHz CPU processor, 2.43 GB RAM memory, and Windows XP operating system. The stop criterion of the iteration is one of the following conditions: (1)gk≤ε0=10-4 and (2) the number of iterations Itr>5000. If condition (2) occurs, the method is deemed to fail for solving the corresponding test problem, and denote it by “F.” For the first three test problems, we present experimental results to observe the behavior of the proposed and DY (given by formula (10)) conjugate gradient algorithm for different λk, different μk, and different ωk. Details of the schemes for parameters set are given in Table 1. Numerical results of test problems are listed in Tables 2, 3, 4, 5, 6, 7, and 8, respectively. Table 9 shows numerical results of other test problems. Here, x0 denotes the initial point of the test problems and x- and f(x-) are iteration value and the function value at the final iteration, respectively.Table 1
Several schemes for the parameters set.
Parameters set Scheme number λ k = 1.0, μk=0.1, ωk=0.1 [ 1 ] λ k = 0.8, μk=0.1, ωk=0.1 [ 2 ] λ k = 0.7, μk=0.1, ωk=0.1 [ 3 ] λ k = 0.6, μk=0.1, ωk=0.1 [ 4 ] λ k = 1.0, μk=0.1, ωk=0.8 [ 5 ] λ k = 1.0, μk=0.1, ωk=0.6 [ 6 ] λ k = 1.0, μk=0.1, ωk=0.4 [ 7 ] λ k = 1.0, μk=0.1, ωk=0.2 [ 8 ] λ k = 0.9, μk=0.3, ωk=0.5 [ 9 ] λ k = 0.9, μk=0.3, ωk=0.4 [ 10 ] λ k = 0.9, μk=0.3, ωk=0.3 [ 11 ] λ k = 0.9, μk=0.3, ωk=0.1 [ 12 ] λ k = 0.7, μk=0.2, ωk=0.1 [ 13 ] λ k = 0.7, μk=0.3, ωk=0.1 [ 14 ] λ k = 0.7, μk=0.4, ωk=0.1 [ 15 ] λ k = 0.7, μk=0.5, ωk=0.1 [ 16 ]Table 2
The numerical results of sphere function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Sphere x 0 (-1,-2,-3,-4,-5,1,2,3,4,5,-1,-2,…,1,2,3,4,5) (-1,-2,-3,-4,-5,1,2,3,4,5,-1,-2,…,1,2,3,4,5) 1 x - / 2.9098 e - 11 x - / 2.9098 e - 11 2 x - / 2.9098 e - 11 x - / 2.9098 e - 11 3 x - / 2.9098 e - 11 x - / 2.9098 e - 11 4 x - / 2.9098 e - 11 x - / 2.9098 e - 11 5 x - / 2.9098 e - 11 x - / 2.9098 e - 11 6 x - / 2.9098 e - 11 x - / 2.9098 e - 11  Scheme 7 x - / 2.9098 e - 11 x - / 2.9098 e - 11 8 x - / 2.9098 e - 11 x - / 2.9098 e - 11 9 x - / 2.9098 e - 11 x - / 2.9098 e - 11 10 x - / 2.9098 e - 11 x - / 2.9098 e - 11 11 x - / 2.9098 e - 11 x - / 2.9098 e - 11 12 x - / 2.9098 e - 11 x - / 2.9098 e - 11 Note:x-=1.0e-5∗(0.0512, 0.1024, 0.1536, 0.2048, 0.2560, −0.0512, −0.1024, −0.1536, −0.2048, −0.2560, 0.0512, 0.1024, 0.1536, 0.2048, 0.2560, …, −0.0512, −0.1024, −0.1536, −0.2048, −0.2560).Table 3
The numerical results of sphere function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Sphere x 0 (3,3,…,3) (3,3,…,3) 1 1.0e-5∗(-0.1536,-0.1536,…,-0.1536)/2.5952e-11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 2 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 3 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 4 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 5 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 6 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11  Scheme 7 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 8 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 9 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 10 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 11 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11 12 1.0 e - 5 ∗ ( - 0.1536 , - 0.1536 , … , - 0.1536 ) / 2.5952 e - 11 1.0 e - 5 ∗ ( 0.2556,0.2556 , … , 0.2556 ) / 7.1845 e - 11Table 4
The numerical results of Rastrigin function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Rastrigin x 0 ( 0.5,0.5 , … , 0.5 ) ( 0.5,0.5 , … , 0.5 ) 1 1.0e-7∗(0.6520,0.6520,…,0.6520)/2.5260e-12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … ,  0.6520 ) / 2.5260 e - 12 2 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 3 1.0 e - 7 ∗ ( 0.3504,0.3504 , … , 0.3504 ) / 7.3008 e - 13 1.0 e - 7 ∗ ( 0.3504,0.3504 , … , 0.3504 ) / 7.3008 e - 13 4 - 1.0 e - 7 ∗ ( 0.6856,0.6856 , … , 0.6856 ) / 2.7978 e - 12 - 1.0 e - 7 ∗ ( 0.6856,0.6856 , … , 0.6856 ) / 2.7978 e - 12 5 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 6 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12  Scheme 7 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 8 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 1.0 e - 7 ∗ ( 0.6520,0.6520 , … , 0.6520 ) / 2.5260 e - 12 13 - 1.0 e - 7 ∗ ( 0.7804,0.7804 , … , 0.7804 ) / 3.6238 e - 12 - 1.0 e - 7 ∗ ( 0.7804,0.7804 , … , 0.7804 ) / 3.6238 e - 12 14 1.0 e - 7 ∗ ( 0.6697,0.6697 , … , 0.6697 ) / 2.6699 e - 12 1.0 e - 7 ∗ ( 0.6697,0.6697 , … , 0.6697 ) / 2.6699 e - 12 15 - 1.0 e - 7 ∗ ( 0.7222,0.7222 , … , 0.7222 ) / 3.1015 e - 12 - 1.0 e - 7 ∗ ( 0.7222,0.7222 , … , 0.7222 ) / 3.1015 e - 12 16 1.0 e - 7 ∗ ( 0.5194,0.5194 , … , 0.5194 ) / 1.6094 e - 12 1.0 e - 7 ∗ ( 0.5194,0.5194 , … , 0.5194 ) / 1.6094 e - 12Table 5
The numerical results of Rastrigin function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Rastrigin x 0 (0.5, 0.5, 0.5, 0.5, 0.5, 0.1, 0.1, 0.1, 0.1, 0.1) (0.5, 0.5, 0.5, 0.5, 0.5, 0.1, 0.1, 0.1, 0.1, 0.1) 1 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 2 1.0 e - 7 ∗ ( 0.2044 , … , 0.2044,0.5798 , …, 0.5798)/2.4514e-13 1.0 e - 7 ∗ ( 0.2044 , … , 0.2044,0.5798 , …, 0.5798)/2.4514e-13 3 1.0 e - 7 ∗ ( 0.2786 , … , 0.2786,0.8842 , …, 0.8842)/4.5830e-13 1.0 e - 7 ∗ ( 0.2786 , … , 0.2786,0.8842 , …, 0.8842)/4.5830e-13 4 1.0 e - 6 ∗ ( 0.0288 , … , 0.0288,0.1034 , …, 0.1034)/4.9027e-13 1.0 e - 6 ∗ ( 0.0288 , … , 0.0288,0.1034 , …, 0.1034)/4.9027e-13 5 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 (-0.9950,…,-0.9950,13.9269,…, 13.9269)/2.9849 6 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 (0.0000,…,0.0000,-8.9540,…, −8.9540)/4.6096e-12  Scheme 7 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 (0.0000,…,0.0000,19.8909,…, 19.8909)/1.6094e-12 8 1.0 e - 7 ∗ ( 0.0658 , … , 0.0658,0.1700 , …, 0.1700)/2.6645e-14 (0.9550,…,0.9550,-20.8843,…, −20.8843)/2.9849 13 1.0 e - 7 ∗ ( 0.2687 , … , 0.2687,0.8076 , …, 0.8076)/4.2633e-13 F/F 14 1.0 e - 7 ∗ ( 0.2558 , … , 0.2558,0.7716 , …, 0.7716)/3.8369e-13 F/F 15 1.0 e - 7 ∗ ( 0.2430 , … , 0.2430,0.7357 , …, 0.7357)/3.5172e-13 F/F 16 1.0 e - 7 ∗ ( 0.2303 , … , 0.2303,0.7000 , …, 0.7000)/3.1442e-13 F/FTable 6
The numerical results of Freudenstein and Roth function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Froth x 0 (0.5, −2) (0.5, −2) 1 (4.9992, 4)/8.6900e-7 (5.0000, 4.0000)/4.0440e-10 2 (4.9992, 4)/8.6900e-7 F/F 3 (4.9992, 4)/8.6900e-7 (5.0000, 4.0000)/4.0440e-10 4 (4.9992, 4)/8.6900e-7 (11.4128, −0.8968)/48.9843 5 (4.9992, 4)/8.6900e-7 (5.0000, 4.0000)/4.0440e-10 6 (4.9992, 4)/8.6900e-7 (5.0000, 4.0000)/4.0440e-10  Scheme 7 (4.9992, 4)/8.6900e-7 F/F 8 (4.9992, 4)/8.6900e-7 (11.4128, −0.8968)/48.9843 9 (4.9992, 4)/8.6900e-7 (11.4128, −0.8968)/48.9843 10 (4.9992, 4)/8.6900e-7 (11.4128, −0.8968)/48.9843 11 (4.9992, 4)/8.6900e-7 (11.4128, −0.8968)/48.9843 12 (4.9992, 4)/8.6900e-7 F/FTable 7
The numerical results of Freudenstein and Roth function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Froth x 0 ( - 0.5,2 ) ( - 0.5,2 ) 1 ( 4.9986,4 ) / 3.0289 e - 6 F/F 2 ( 4.9986,4 ) / 3.0289 e - 6 F/F 3 ( 4.9986,4 ) / 3.0289 e - 6 (11.4128,  −0.8968)/48.9843 4 ( 4.9986,4 ) / 3.0289 e - 6 (4.9999,  −4.0000)/1.0003e-8 5 ( 4.9986,4 ) / 3.0289 e - 6 F/F 6 ( 4.9986,4 ) / 3.0289 e - 6 F/F  Scheme 7 ( 4.9986,4 ) / 3.0289 e - 6 F/F 8 ( 4.9986,4 ) / 3.0289 e - 6 F/F 9 ( 4.9986,4 ) / 3.0289 e - 6 F/F 10 ( 4.9986,4 ) / 3.0289 e - 6 F/F 11 ( 4.9986,4 ) / 3.0289 e - 6 F/F 12 ( 4.9986,4 ) / 3.0289 e - 6 F/FTable 8
The numerical results of Freudenstein and Roth function for different scheme.
Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Froth x 0 ( - 0.5 , - 2 ) ( - 0.5 , - 2 ) 1 (4.9989,4.0000)/1.7488e-6 ( 5.0000,4.0000 ) / 1.7660 e - 10 2 ( 4.9989,4.0000 ) / 1.7488 e - 6 (11.4128,  −0.8968)/48.9843 3 ( 4.9989,4.0000 ) / 1.7488 e - 6 F/F 4 ( 4.9989,4.0000 ) / 1.7488 e - 6 F/F 5 ( 4.9989,4.0000 ) / 1.7488 e - 6 F/F 6 ( 4.9989,4.0000 ) / 1.7488 e - 6 ( 5.0000,4.0000 ) / 1.7404 e - 10  Scheme 7 ( 4.9989,4.0000 ) / 1.7488 e - 6 (11.4128,  −0.8968)/48.9843 8 ( 4.9989,4.0000 ) / 1.7488 e - 6 F/F 13 ( 4.9989,4.0000 ) / 1.7488 e - 6 F/F 14 ( 4.9989,4.0000 ) / 1.7488 e - 6 (5.0000,  4.0000)/2.6700e-10 15 ( 4.9989,4.0000 ) / 1.7488 e - 6 (11.4128,  −0.8968)/48.9843 16 ( 4.9989,4.0000 ) / 1.7488 e - 6 (11.4128,  −0.8968)/48.9843Table 9
The numerical results of different function forλk=0.9, μk=0.3, and ωk=0.1.
Functions x 0 Algorithm A DY (given by formula (10)) algorithm x - / f ( x - ) x - / f ( x - ) Pqd (0.5, −0.5, 0.5, 0.8, 0.9) ( 0.0081,0.0081 , … , - 0.0182 ) / 2.2226 e - 4 (−0.0345, −0.0345, −0.0346, −0.0347, −0.0612)/0.0401 Ewh ( 1,2 , 1,2 , … , 1,2 , 1,2 ) ( 1.0028,1.0083 , … , 1.0028, 1.0083)/2.3050e-5 ( 1,1 , … , 1,1 ) / 5.1174 e - 11 Raydan1 (1, 1) 1.0 e - 3 ∗ ( 0.9370, 0.0005)/0.3000 1.0 e - 3 ∗ ( 0.9917, 0.0008)/0.3000 Raydan2 ( 1,1 , … , 1,1 ) ( 0,0 , … , 0,0 ) / 500 ( 0,0 , … , 0,0 ) / 500 Etri ( 1,1 , … , 1,1 ) a b Epow (3, −1, 0, 1) (2.0000, 0, 0, 2.0000)/5.8875e-11 F/F Wood (−3, −1, −3, −1) (1.0002, 1.0004, 0.9998, 0.9996)/1.6386e-7 F/F Ewood (−3, 1.2, −3, 1.2) (1.0001, 1.0002, 0.9999, 0.9998)/2.0820e-8 F/F Perq (1, 2, 3) 1.0 e - 4 ∗ ( 0.0973, −0.0020, −0.1014)/4.0320e-10 1.0 e - 4 ∗ ( 0.0480, 0.0011, 0.1524)/7.2434e-10 Etri1 ( 2,2 , … , 2 ) F/F ( 1.2,1.2 , … , 1.2 ) / 6.5680 e - 11 Emic ( 2,2 , … , 2 ) c F/F Erosen ( - 1.2,1 , … , - 1.2,1 ) ( 0.9999,0.9999 , … , 0.9999 ) / 6.3141 e - 9 ( 1,1 , … , 1 ) / 3.2274 e - 12 Grosen ( 2,2 , … , 2 ) ( 1,1 , …  , 1 ) / 5.8136 e - 10 ( 1,1 , … , 1 ) / 5.8136 e - 10 QUARTC ( 2,2 , … , 2 ) ( 0.6338,0.6338 , … , 0.6338 ) / 0.0899 ( 0.5278,0.5278 , … , 0.5278 ) /0.2487 LIARWHD ( 4,4 , … , 4 ) ( 1,1 , … , 1 ) / 4.2649 e - 10 ( 1,1 , … , 1 ) / 7.5304 e - 10 Staircase1 ( 2,2 , … , 2 ) 1.0 e - 4 ∗ ( 0.0448, −0.1380, −0.0297, −0.1576)/2.7012e-10 1.0 e - 4 ∗ ( - 0.2478, 0.7435, −0.8469, 0.5991)/4.9189e-9 Staircase2 ( 0,0 , … , 0 ) ( 1,1 , …  , 1 ) / 5.1016 e - 10 ( 1,0.9999 , … , 1,0.9999 ) / 4.1476 e - 9 POWER ( 1,1 , … , 1,1 ) - 1.0 e - 4 ∗ ( 0.1081,0 , 0 , … , 0, 0.0214)/1.9507e-9 - 1.0 e - 4 ∗ ( 0.1081,0 , 0 , … , 0, 0.0214)/1.9507e-9 Diagonal4 ( 2,2 , … , 2 ) 1.0 e - 4 ∗ ( - 0.2394, 0.0062, −0.2394, 0.0062)/6.1191e-10 1.0 e - 4 ∗ ( 0.4352, 0.0055, 0.4352, 0.0055)/1.9247e-9 EBD1 ( 0,1 , … , 0,1 ) ( 1,1 , …  , 1 ) / 3.1321 e - 10 ( 1,1 , … , 1 ) / 3.1321 e - 10 CUBE ( - 1.2,1 , … , - 1.2,1 ) ( 1,1.0001 , … , 1, 1.0001)/1.4082e-9 ( 0.9693,0.9108 , … , 0.9693, 0.9108)/9.3983e-4 a = 1.0e3∗(0.0009, 0.0258, −0.0314, 0.0067, 0.0817, 1.0242, 0.1885, −0.5213, −0.9109, −0.4524)/8.0025e-11. b = 1.0e4∗(0.0522, −0.1413, 0.0641, 0.0603, −0.2218, −0.5115, 0.2249, 0.7358, −1.3182, 0.3192)/4.0124e-10. c = (0.2140, 1.2386, 1.2810, 1.3032, −0.5016, 0.6837, 0.8063, 0.7930)/0.0105.Based on Table1’s sixteen kinds of scheme (different parameters set), we compared algorithm A with DY (given by formula (10)) conjugate gradient algorithm based on different initial point for three test problems. It is easy to see that the two algorithms based on different scheme (different parameters set) are successful for the first and the second test problems listed in Tables 2, 3, and 4. From Tables 5, 6, 7, 8, and 9, we can see that algorithm A is more successful than DY (given by formula (10)) conjugate gradient algorithm. For example, for the first three test problems based on different scheme (different parameters set), algorithm A based on different initial point all achieved satisfied iteration value and the function value at the final iteration. Nevertheless, under some scheme (parameters set), DY (given by formula (10)) conjugate gradient algorithm cannot search satisfied iteration solution and the function value at the final iteration. From Tables 5–9, we can also see that DY (given by formula (10)) conjugate gradient algorithm sometimes is failed based on some scheme (parameters set); however, our algorithm is failed only one time. These indicate that the influence of parameters value’s changing in formula (27) on the algorithm is not big. We presented the Dolan and Moré [30] performance profiles for the algorithm A and DY method. Note that the performance ratio p(τ) is the probability for a solver s for the tested problems with the factor τ of the smallest cost. As we can see from Figure 1, algorithm A is superior to DY method for the absolute errors of f(x-) versus f(x∗). Hence, compared with the DY (given by formula (10)) conjugate gradient algorithm, algorithm A has higher stability and adaptability. Therefore, algorithm A yields a better numerical performance than the DY (given by formula (10)) conjugate gradient algorithm. From the above analysis, we can conclude that algorithm A is competitive for solving unconstrained optimization problems.Figure 1
Performance profile on the absolute errors off(x-) versus f(x∗) (algorithm A versus DY).


## 5. Application to Engineering

In this section, we present a real example to illustrate application of the algorithm proposed in this article. The example is the results of tests on endurance of deep groove ball bearings. For illustrating the purposes, we applied the real dataset of 23 observed failure times that was initially reported in Lieblein and Zelen [31] and later by a number of authors including Abouammoh and Alshingiti [32] and Krishna and Kumar [33]. The following dataset represents the number of millions of revolutions before failure for each of the 23 ball bearings in a life test: 17.88, 28.92, 33.0, 41.52, 42.12, 45.60, 48.40, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, and 173.4. Dey and Pradhan [34] indicated that Weibull distribution fits this dataset better than the exponential, inverted exponential, and gamma distribution. A random variable X follows the Weibull distribution with probability density function (pdf) being that (45)fx;α,λ=αλe-λxαxα-1,x>0,where α>0 and λ>0 are the shape and scale parameters, respectively. Let n denote the number of observed failures and t1,…,tn denote the complete sample; the logarithm likelihood function is (46)log⁡Lα,λ;data=nlog⁡αλ-λ∑i=1ntiα+α-1∑i=1nlog⁡ti.The corresponding differential equations are (47)∂log⁡L∂α=nα-αλ∑i=1ntiα-1+∑i=1nlog⁡ti=0,∂log⁡L∂λ=nα-∑i=1ntiα=0.From (47), a closed-form solution of α and λ does not exist, so a numerical technique (minimization -log⁡L) must be used to find the maximum likelihood estimation (MLE) of α and λ for any given dataset. By using algorithm A, We obtain α^=3.183499 and λ^=7.0363e-7. Dey and Pradhan [34] obtained the MLE of the parameters as follows: (α^,λ^)=(3.1835,1.4329e-6). From the numerical results, we can see that our algorithm is alternative for the above real unconstrained optimization problem.


## 6. Conclusion

In this article, by modifying the scalarβk, we have proposed a three-parameter family of conjugate gradient method for solving large-scale unconstrained optimization problems. Global convergence of the proposed methods under modified Wolfe-Powell line search and general criterion are established, respectively. Numerical results show that our algorithm is competitive for solving unconstrained optimization problems. So, the proposed method is an alternative method used in the reliability data.



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