
I investigated soliton phenomena in a prominent nonlinear fractional partial differential equation (FPDE) namely the conformable coupled Drinfeld-Sokolov-Wilson system (CCDSWS) using a novel variant of the novel extended direct algebraic method (EDAM), namely r+mEDAM. The conformable fractional derivatives are used to generalize the model due to the memory and hereditary features that are inherent in the fractional dynamics. The model was initially transformed into a more manageable system of integer-order nonlinear ordinary differential equations (NODEs) through the implementation of complex transformation. The obtained system of NODEs is further transformed into a system of algebraic equations, which yields, by solving new plethora of soliton solutions for CCDSWS in the form of generalized trigonometrical, exponential hyperbolical, and rational functions. Moreover, we employed 2D, 3D, and contour graphics to show the behavior of acquired solitons, making it abundantly evident that the obtained solitons take the shape of kink, anti-kink, bright, dark, bright-dark, and bell-shaped kink solitons within the framework of CCDSWS. The results confirmed the efficiency of the presented approach in finding solitonic solutions, which in its turn expands knowledge of nonlinear FPDEs. The aimed to theoretical and application perspectives in fractional solitons applicable in areas such fluid mechanics, plasma physic, optical communications, etc.
Citation: Naher Mohammed A. Alsafri. Solitonic behaviors in the coupled Drinfeld-Sokolov-Wilson system with fractional dynamics[J]. AIMS Mathematics, 2025, 10(3): 4747-4774. doi: 10.3934/math.2025218
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I investigated soliton phenomena in a prominent nonlinear fractional partial differential equation (FPDE) namely the conformable coupled Drinfeld-Sokolov-Wilson system (CCDSWS) using a novel variant of the novel extended direct algebraic method (EDAM), namely r+mEDAM. The conformable fractional derivatives are used to generalize the model due to the memory and hereditary features that are inherent in the fractional dynamics. The model was initially transformed into a more manageable system of integer-order nonlinear ordinary differential equations (NODEs) through the implementation of complex transformation. The obtained system of NODEs is further transformed into a system of algebraic equations, which yields, by solving new plethora of soliton solutions for CCDSWS in the form of generalized trigonometrical, exponential hyperbolical, and rational functions. Moreover, we employed 2D, 3D, and contour graphics to show the behavior of acquired solitons, making it abundantly evident that the obtained solitons take the shape of kink, anti-kink, bright, dark, bright-dark, and bell-shaped kink solitons within the framework of CCDSWS. The results confirmed the efficiency of the presented approach in finding solitonic solutions, which in its turn expands knowledge of nonlinear FPDEs. The aimed to theoretical and application perspectives in fractional solitons applicable in areas such fluid mechanics, plasma physic, optical communications, etc.
We consider the following problem with inequality constraints:
minf(x)s.t.gi(x)≤0,i∈I, | (P) |
where I={1,2,...,m} is a finite set of integers, and f:Rn→R, gi:Rn→R, i∈I are continuously differentiable. The feasible set of (P) is denoted by X={x∈Rn|gi(x)≤0,i∈I}.
Many methods have been proposed to solve (P), such as interior-point methods [1,2], sequential quadratic programming (SQP) methods [3,4], feasible direction methods [5,6,7], and penalty function methods[8,9,10]. The penalty function method is widely utilized due to its simplicity and efficiency. Courant [8] initially proposed the exterior penalty function method, and several variants have been subsequently presented. The majority of these methods add a multiple of the violation of each constraint to the objective function, and a series of unconstrained problems have been solved to serve as replacements for the original constrained problem. Fortunately, in the exact penalty method, a single unconstrained problem (rather than a sequence) takes the place of the original constrained problem. A penalty function is exact if there is some α∗ such that an optimal solution to (P) is also an optimal solution to the penalty optimization problem for all α≥α∗. In 1967, Zangwill [9] proposed the following popular exact penalty function:
F1(x,α)=f(x)+α∑i∈Imax{gi(x),0}, |
where α>0 is a penalty parameter, and the corresponding penalty problem of (P) is given by the following:
minF1(x,α)s.t.x∈Rn. | (P1) |
In 1981, Fletcher unified the nonsmooth exact penalty function and defined it as follows:
Fp(x,α)=f(x)+α‖g+(x)‖p, |
where the corresponding penalty problem of (P) is given by the following:
minFp(x,α)s.t.x∈Rn, | (Pp) |
where α is a penalty parameter, 1≤p≤∞, and ‖g+i‖=max{gi,0}. When α≥‖λ∗‖ and g+(x∗)=0, the minimizer x∗ of (Pp) is the optimal solution of the problem (P). Especially, (Pp) is equivalent to (P1) when p=1. Moreover, when p=∞, the L∞ penalty function [10] can be written as follows:
F∞(x,α)=f(x)+α‖g+(x)‖∞=f(x)+αmax{0,g1(x),...,gm(x)}. |
The exact penalty function has been extensively discussed [11,12,13]. The majority of exact penalty functions are nonsmooth because of the presence of the max-value function. However, in most existing unconstrained optimization algorithms, the objective function needs to be smooth. Consequently, numerous smoothing techniques [14,15,16] have been devised to approximate the max-value function. In practice, the existing exact penalty function algorithm requires gradually increasing the penalty parameter to find an optimal solution. Nevertheless, excessive penalty parameters may result in ill-conditioned problems. To address these limitations, innovative forms of penalty functions have been proposed.
The objective penalty function, which is special form of the penalty function, is discussed in [17,18,19], where one kind of the objective penalty function is defined as follows:
F(x,M)=(f(x)−M)p+∑i∈Igi(x)p, | (PM) |
where p>0, and M is an objective penalty parameter. Assume that x∗ is an optimal solution and f(x∗) is the optimal value of (P). The optimal solution x(Mk) of (PM) tends to x∗ when a convergent sequence Mk tends to f(x∗). In 2011, Meng et al. [20] proposed the following objective penalty function:
FQ(x,M)=Q(f(x)−M)+∑i∈IP(gi(x)), | (PQ) |
where Q(⋅):R→R∪{∞} and P(⋅):R→R∪{∞} respectively satisfy the following conditions:
{Q(t)>0,for allt∈R∖{0},Q(0)=0,Q(t1)<Q(t2),for0≤t1<t2, | (1.1) |
and
{P(t)=0,if and only ift≤0,P(t)>0,if and only ift>0. |
If Q(t), P(t), f(x), and gi(x)(i∈I) are all differentiable, then it is obvious that FQ(x,M) is also differentiable. For a comprehensive discussion of the objective penalty function, the sufficient and necessary stability condition used to determine whether the objective penalty function is exact for a global solution is proved in [21]; different forms of the objective penalty function have been proposed in [22,23]; the smoothness property concerning the objective penalty function has been discussed in [24,25,26]; lower-order smoothed objective penalty functions based on filling properties have been discussed in [27].
In order to avoid the ill-conditioning brought about by overly large penalty parameters in the traditional penalty function method, in this study, we adopt the approach of adjusting the objective penalty parameters. Furthermore, to improve the computational efficiency of constrained optimization problems with a large number of constraints, a smooth approximation is employed using a modified flattened aggregate function.
This paper presents a new objective penalty function with the flattened aggregate function, which offers both smoothness and exactness. The proposed innovative algorithm offers a powerful solution to tackle inequality constrained optimization problems, even when faced with many constraints. The rest of this paper is organized as follows: A modified flattened aggregate function is proposed in Section 2 to smoothly approximate the max-value function, and its properties are discussed; Section 3 presents the construction of an objective penalty function using the modified flattened aggregate function, and proposes an algorithm based on this function, with its convergence thoroughly proven; and Section 4 demonstrates the efficiency of the algorithm through numerical examples.
In this section, we will construct a modified flattened aggregate function to smoothly approximate the max-value function; its specific form is as follows:
ˆfp(x)=1pln(∑i∈Iexp(pϕ(gi(x)))), |
where
ϕ(t)={0,ift≤0,t,ift>ϵ,−1ϵ2t3+2ϵt2,if0<t≤ϵ. |
Some properties of the flattened aggregate function ˆfp(x) are discussed as follows.
Proposition 2.1. For any x∈Rn, gmax(x):=max{0,g1(x),...,gm(x)}, we have the following:
(1) ˆfp(x)=1plnm, if gmax(x)≤0.
(2) 1plnm<ˆfp(x)≤1plnm+ϵ, if 0<gmax(x)≤ϵ.
(3) gmax(x)≤ˆfp(x)≤gmax(x)+1plnm, if gmax(x)>ϵ.
Proof. (1) If gmax(x)≤0, we have ϕ(gi(x))=0, then
ˆfp(x)=1pln(mexp(pϕ(gi(x))))=1plnm. |
(2) With a simple verification, we know ϕ(t) is a monotonically increasing function. For a given x, we obtain the following:
ϕ(gi(x))≤ϕ(gmax(x)). |
If 0<gmax(x)≤ϵ, we have ϕ(gi(x))≤ϵ, then we obtain the following:
ˆfp(x)=1pln(∑i∈Iexp(pϕ(gi(x))))≤1pln(mexp(pϵ))=1plnm+ϵ. |
Hence, by combining (1), we have the following:
1plnm<ˆfp(x)≤1plnm+ϵ. |
(3) If gmax(x)>ϵ, then we obtain the following:
ˆfp(x)≤1pln(mexp(pϕ(gmax(x))))=1pln(mexp(pgmax(x)))=1plnm+gmax(x). |
Obviously, gmax(x)≤ˆfp(x).
Hence,
gmax(x)≤ˆfp(x)≤gmax(x)+1plnm. |
Proposition 2.2. For any given x∈Rn, the new flattened aggregate function ˆfp(x) monotonically decreases as p increases. When p→∞ and ϵ→0, ˆfp(x) sufficiently approximates the max-value function max{0,g1(x),...,gm(x)}.
Proof. Suppose that there is a vector-valued function φ(x), whose components are as follows:
φi(x)=exp{ϕ(gi(x))},i=1,2,...,m, |
and its Lp norm can be given as follows:
‖φ(x)‖p=(m∑i=1(φi(x))p)1p=(m∑i=1exp(pϕ(gi(x))))1p. |
By Jensen's inequality, it can be seen that if q≥p≥1, then we obtain the following:
‖ϕ(x)‖p≥‖ϕ(x)‖q. | (2.1) |
Taking the natural logarithm on both sides of the inequality (2.1), we have the following:
ˆfp(x)≥ˆfq(x). |
Hence, ˆfp(x) monotonically decreases as p increases.
Next, we prove the second half of the proposition. By Proposition 2.1, if gmax(x)≤0, ˆfp(x)=1plnm. When p→∞, we obtain the following:
limp→∞ˆfp(x)=limp→∞1plnm=0. |
If 0<gmax(x)≤ϵ, then 1plnm<ˆfp(x)≤1plnm+ϵ. When p→∞ and ϵ→0, we obtain the following:
limp→∞1plnm=limϵ→0p→∞ˆfp(x)=limp→∞limϵ→0(1plnm+ϵ)=0. |
Similarly, if gmax(x)>ϵ, then gmax(x)≤ˆfp(x)≤gmax(x)+1plnm. When p→∞, we obtain the following:
limp→∞ˆfp(x)≤limp→∞(gmax(x)+1plnm)=gmax(x). |
Then, we have limp→∞ˆfp(x)=gmax(x). Hence,
limϵ→0p→∞ˆfp(x)=max{0,g1(x),...,gm(x)}. |
In this section, we propose a new objective penalty function
F(x,M,ρ)=Q(f(x)−M)+ρmax{0,g1(x),...,gm(x)}, | (P(M,ρ)) |
where Q(⋅) is defined by Eq (1.1). Due to the existence of the max-value function max{0,g1(x),...,gm(x)}, F(x,M,ρ) is not differentiable. In Section 2, we have already discussed that the flattened aggregate function smoothly approximates the max-value function. Therefore, we consider the following smooth unconstrained optimization problem:
minFp(x,M,ρ),x∈Y, | (P(p,M,ρ)) |
where Rn⊃Y⊃X={x∈Rn|gi(x)≤0} and Fp(x,M,ρ) can be rewritten as follows:
Fp(x,M,ρ)=Q(f(x)−M)+ρpln(∑i∈Iexp(pϕ(gi(x)))). | (3.1) |
If there is an M′ such that an optimal solution x∗ to (P) is also an optimal solution to (P(p,M,ρ)) for ∀M<M′, then F(x,M,ρ) is called an exact objective penalty function. If an optimal solution x∗ to (P(p,M,ρ)) is also an optimal solution to (P), then M is called an exact value of the objective penalty parameter. The assumptions and proofs of the subsequent theorems fundamentally rely on the well-established properties of the flattened aggregate function. Next, we will discuss the exactness of Fp(x,M,ρ).
Theorem 3.1. Let x∗ be an optimal solution to (P). For some ρ>0, p>0, if M=f(x∗), then x∗ is the optimal solution to (P(p,M,ρ)).
Proof. Since x∗ is the optimal solution to (P), gi(x∗)≤0. We obtain the following:
Fp(x∗,M,ρ)=Q(f(x∗)−M)+ρpln(∑i∈Iexp(pϕ(gi(x∗))))=Q(f(x∗)−M)+ρplnm. |
It follows from M=f(x∗) that Fp(x∗,M,ρ)=ρplnm. From Fp(x,M,ρ)≥ρplnm, ∀x∈Y, we conclude that x∗ is the optimal solution to (P(p,M,ρ)).
This completes the proof.
Theorem 3.2. Let x∗ be an optimal solution to (P), and x∗M∈Y be an optimal solution to (P(p,M,ρ)). Assume that the feasible set Y is connected and compact, fmin(x)=minx∈Xf(x), fmax(x)=maxx∈Xf(x), and f(x) is continuous. For a given M, ρ>0, and p is large enough, then the following three assertions hold:
(1) If Fp(x∗M,M,ρ)=ρplnm, then x∗M is a feasible solution to (P) and fmin(x)≤M≤fmax(x).
(2) If Fp(x∗M,M,ρ)>ρplnm and M≤fmax(x), then M<fmin(x).
(3) If Fp(x∗M,M,ρ)>ρplnm, M<fmin(x), and x∗M is a feasible solution to (P), then x∗M is an optimal solution to (P).
Proof. (1) From the properties of the flattened aggregate function, we know that
Fp(x∗M,M,ρ)≥ρplnm. |
It follows from (3.1) that
Fp(x∗M,M,ρ)=Q(f(x∗M)−M)+ρpln(∑i∈Iexp(pϕ(gi(x∗M))))=ρplnm. |
Since Q(⋅)≥0, we have the following:
Q(f(x∗M)−M)=0, |
and
ρpln(∑i∈Iexp(pϕ(gi(x∗M))))=ρplnm. |
Then, we obtain f(x∗M)=M and gi(x∗M)≤0. Hence, x∗M is a feasible solution to (P) and fmin(x)≤M≤fmax(x).
(2) Since Fp(x∗M,M,ρ)>ρplnm, we have the following:
ρplnm<Fp(x∗M,M,ρ)≤Fp(x,M,ρ)=Q(f(x)−M)+ρplnm,∀x∈X. |
Suppose that fmin(x)≤M; then, we have fmin(x)≤M≤fmax(x). Since f is continuous on the connected and compact set X, there is some ˆx∈X such that f(ˆx)=M. Then, we obtain Fp(ˆx,M,ρ)=ρplnm.
Furthermore, x∗M is an optimal solution to (P(p,M,ρ)) and Fp(x∗M,M,ρ)>ρplnm; then, we have the following:
ρplnm<Fp(x∗M,M,ρ)≤Fp(ˆx,M,ρ)=ρplnm, |
which leads to a contradiction. Hence, M<fmin(x).
(3) From the given conditions, we obtain Fp(x∗M,M,ρ)≤Fp(x∗,M,ρ), that is
Q(f(x∗M)−M)+ρpln(∑i∈Iexp(pϕ(gi(x∗M))))≤Q(f(x∗)−M)+ρpln(∑i∈Iexp(pϕ(gi(x∗)))). |
Both x∗ and x∗M are feasible solutions to the original problem, and we obtain the following:
Q(f(x∗M)−M)≤Q(f(x)−M),∀x∈X. | (3.2) |
Since x∗ is an optimal solution to (P), we suppose that there is some x∈X such that f(x)≤M; then, we have f(x)≤M<f(x∗), which contradicts the fact that x∗ is an optimal solution to (P). Therefore, for any x∈X, we have the following:
f(x)−M>0. | (3.3) |
Additionally, if we have f(x∗M)≥f(x∗), then
f(x∗M)−M≥f(x∗)−M>0. | (3.4) |
It follows from (3.2)–(3.4) and the properties of Q(⋅), we obtain the following:
f(x∗M)−M≤f(x)−M,∀x∈X. |
Hence, f(x∗M)≤f(x),∀x∈X. This completes the proofs.
Based on Theorems 3.1 and 3.2, it is clear that the optimal solution of (P(p,M,ρ)) is the optimal solution of (P). This solidifies the proof that the objective penalty function is both smooth and exact. After conducting the aforementioned analysis, we develop a new algorithm called the flattened aggregate objective penalty function (FAOPF) algorithm.
Definition 3.1. x is an ϵ-approximately optimal solution if
|f(x)−f(x∗)|≤ϵ, |
where f(x∗) is an optimal value to (P).
FAOPF Algorithm
Step 1. Choose x0∈X, a1<minx∈Xf(x)<b1, and a1<M1<b1. a1 and b1 are all randomly selected within a given range. M1=34b1+14a1 or 14b1+34a1. Given ρ1>0, n=2, ϵ=10−4, and p=109, let k:=1.
Step 2. Solve minx∈YFp(x,Mk,ρk). Let xk be the minimizer.
Step 3. If xk is not a feasible solution to (P), then let bk+1=bk, ak+1=Mk, Mk+1=ak+1+34(bk+1−ak+1), ρk+1=nρk, and go to Step 5; otherwise, if xk∈X, then go to Step 4.
Step 4. If Fp(xk,Mk,ρk)=ρkplnm, then let ak+1=ak, bk+1=Mk, Mk+1=ak+1+14(bk+1−ak+1), ρk+1=ρk, and go to Step 5; otherwise, if Fp(xk,Mk,ρk)>1plnm, then stop and xk is an optimal solution to (P).
Step 5. If |bk+1−ak+1|<ϵ, then stop and xk is an ϵ-approximately optimal solution to (P). Otherwise, let k:=k+1, and go to Step 2.
Remark. In the FAOPF algorithm, it is assumed that one can always get a1<minx∈Xf(x).
Next, we will prove that the FAOPF algorithm is well-defined. Assume that gi(x)(i∈I) are continuous in Rn. Let
S(L,f)={xk|L≥Q(f(xk)−Mk),k=1,2,...}, |
which is called a Q-level set. S(L,f) is called bounded if, for any given L>0 and a convergent sequence Mk→minf(x), S(L,f) is bounded.
The convergence of the FAOPF algorithm is demonstrated in the following theorem.
Theorem 3.3. Let Y=Rn or Y be an open set. Suppose that Q(⋅), f(x), and gi(x)(i∈I) are continuous on Rn, and the Q-level set is bounded. Let {xk} be the sequence generated by the FAOPF Algorithm. The following three assertions hold:
(1) If {xk}(k=1,2,...,ˆk) is a finite sequence (i.e., the FAOPF algorithm stops at the ˆk-th iteration), then xˆk is either an optimal solution or an ϵ-approximately optimal solution to (P).
(2) If {xk}(k=1,2,...,ˆk) is a infinite sequence, then {xk} is bounded and any limit point of it is either an optimal solution or an approximately optimal solution to (P).
(3) If {xk} is an infinite sequence, then for any limit point x∗ of it, there exists τi>0, such that
∇f(x∗)+∑i∈Iϵ(xk)τi∇xgi(x∗)=0. |
Proof. First, we prove that the sequence {ak} increases and {bk} decreases with
ak<Mk<bk,k=1,2,..., | (3.5) |
and
bk+1−ak+1=14(bk−ak),k=1,2,.... | (3.6) |
The proof by induction is as follows. From the setting of the initial value, it is clear that a1<M1<b1. In Step 3, we know that b2=b1, and a2=M1. We take M1=14a1+34b1; then, we have the following:
b2−a2=b1−M1=14(b1−a1). |
Similarly, in Step 4, we know that b2=M1 and a2=a1. We take M1=14b1+34a1; then, we have the following:
b2−a2=M1−a1=14(b1−a1). |
Suppose that (3.5) and (3.6) hold for some k>1, and consider the case of k+1. By Step 3, we have bk+1=bk, ak+1=Mk, and Mk+1=ak+1+34(bk+1−ak+1). Then, we obtain the following:
Mk+1=ak+1+34(bk+1−ak+1)=14ak+1+34bk+1=14Mk+34bk>Mk=ak+1, |
and
Mk+1=14Mk+34bk<bk=bk+1. |
We obtain ak<Mk=ak+1, bk+1=bk, ak+1<Mk+1<bk+1, and
bk+1−ak+1=bk−Mk=bk−(ak+34(bk−ak))=14(bk−ak). |
By Step 4, we have ak+1=ak, bk+1=Mk, and Mk+1=ak+1+14(bk+1−ak+1). Then, we have the following:
Mk+1=ak+1+14(bk+1−ak+1)=34ak+1+14bk+1=34ak+14Mk>ak=ak+1, |
and
Mk+1=34ak+14Mk<Mk=bk+1. |
We get bk>Mk=bk+1, ak+1=ak, ak+1<Mk+1<bk+1, and
bk+1−ak+1=Mk−ak=ak+14(bk−ak)−ak=14(bk−ak). |
Hence, (3.5) and (3.6) hold. It is obvious that {ak} is increasing and {bk} is decreasing; then, both {ak} and {bk} are convergent. Let ak→ˉa and bk→ˉb. By (3.6), we have ˉa=ˉb. Additionally, by (3.5), we obtain Mk→ˉa.
(1) If the FAOPF Algorithm stops at the ˆk-th iteration and the second situation of Step 4 holds, then xˆk is a feasible solution to (P), and Fp(xˆk,Mˆk,ρ)>1plnm. By Theorem 3.1(3), xˆk is an optimal solution for (P). Otherwise, the stopping condition |bˆk+1−bˆk+1|<ϵ and h(xˆk)=ρkpln(∑i∈Iexp(pϕ(gi(xˆk))))<ϵ,i=1,2,...m will occur. Then, xˆk is an ϵ-approximately optimal solution to (P).
(2) First, we demonstrate that the sequence {xk} is bounded. Since xk is an optimal solution to minx∈YFp(x,M,ρ),
Fp(xk,Mk,ρk)≤Q(f(x0)−Mk)+ρkplnm,k=1,2,.... |
Since Mk→ˉa as k→+∞, we obtain that there is some L>0 such that
L>Fp(xk,Mk,ρk)>Q(f(xk)−Mk). |
Then, we know the Q-level set is bounded. Hence, the sequence {xk} is bounded.
Next, we prove the second half of the theorem. Let M∗=minx∈Xf(x). Without a loss of generality, we assume that xk→x∗. We have shown that
ak<Mk<bk,k=1,2,..., |
and all sequences {ak}, {bk}, and {Mk} converge to ˉa. By Step 4 and Theorem 3.2(1), we know M∗≤Mk=bk+1, k=1,2,.... Additionally, by Step 3 and Theorem 3.2(2), we know ak+1=Mk<M∗, k=1,2,.... Therefore, ak+1<M∗≤bk+1. Let k→+∞, we obtain ˉa=M∗. Assume that z∗ is an optimal solution to (P), then M∗=f(z∗). By the exactness of Fp, there is some ρ such that
Fp(xk,Mk,ρ)≤Fp(z∗,Mk,ρ)=Q(f(z∗)−Mk)+ρplnm. |
Let k→+∞ in the above equation. We have
Fp(x∗,M∗,ρ)≤ρplnm, |
which implies f(z∗)=M∗=f(x∗). Hence, x∗ is an optimal solution to (P).
(3) In Theorem 3.2(2), we have shown that the sequence {xk} is bounded. For any x∈Y, set Iϵ(x)={i∈I|gi(x)>0}, and the cardinality of Iϵ(x) is denoted by |Iϵ(x)|. Then, Fp(xk,Mk,ρk) can be rewritten as follows:
Fp(xk,Mk,ρk)=Q(f(xk)−Mk)+ρkpln(∑i∈Iϵ(xk)exp(pϕ(gi(xk)))+(m−|Iϵ(xk)|)). |
From the definition of Fp(xk,Mk,ρk), k=1,2,..., we have the following:
∇Fp(xk,Mk,ρk)=Q′(f(xk)−Mk)∇f(xk)+ρk∑i∈Iϵ(xk)δip(xk)∇xgi(xk), |
where δip(xk)=exp(pϕ(gi(xk)))∗ϕ′t(gi(xk))∑i∈Iϵ(xk)exp(pϕ(gi(xk)))+m−|Iϵ(xk)|.
Set
ηk=1+ρk∑i∈Iϵ(xk)δip(xk), |
we know that ηk>1 and
1ηkQ′(f(xk)−Mk)∇f(xk)+ρkηk∑i∈Iϵ(xk)δip(xk)∇xgi(xk)=0. |
Let
μk=1ηk,λk=1ηkQ′(f(xk)−Mk),ωi,k=ρkηkδip(xk). |
Then, for any k and i∈I,
μk+∑i∈Iϵ(xk)ωi,k=1. |
It's obvious that μk→μ∈(0,1), ωi,k→ωi∈(0,1), ∀i∈I. We have shown that {Mk}→M∗ as k→+∞, and {xk} is bounded with {xk}→x∗. Since f(x) and Q(⋅) are continuous differentiable, Q(f(xk)−Mk)→Q(f(x∗)−M∗). Additionally, we know that M∗<f(x∗), and λk→λ>0 as k→+∞. Then,
∇f(x∗)+∑i∈Iϵ(xk)ωiλ∇xgi(x∗)=0. |
Let τi=ωiλ>0,
∇f(x∗)+∑i∈Iϵ(xk)τi∇xgi(x∗)=0. |
This completes the proofs.
In this section, we provide some examples to illustrate the efficiency of the FAOPF algorithm. We compare the numerical performance of FAOPF with the other three algorithms: The Two-Parameter Exact Penalty algorithm (TPEP) described in [24], the Smoothing Method of Penalty Function algorithm (SMPF) described in reference [25], and the Smoothing Objective Penalty Function algorithm (SOPF) described in reference [20]. Since each algorithm requires solving unconstrained optimization subproblems, we use either Gradient-type or Newton-type algorithms. The specific forms of penalty functions employed in the comparative experiment are as follows:
TPEP
H(x,M,ρ,ϵ)=(f(x)−M)2+ρ∑i∈IPϵ(gi(x)), |
where
Pϵ(t)={12ϵsintϵ,t≤0,t−12ϵsintϵ,t>0. |
SMPF
G(x,M,ϵ)=(f(x)−M)2+∑i∈IPϵ(gi(x)), |
where
Pϵ(t)={14ϵe2t/ϵ,t≤0,t+14ϵe−2t/ϵ,t>0. |
SOPF
Fϵ(x,M,ϵ)=Q(f(x)−M)+ρ∑i∈IPτϵ(gi(x)), |
where
Pτϵ(t)={0,t≤0,1aϵτ−aτtaτ,0≤t≤ϵ,tτ−(1−1a)ϵτ,t≤ϵ. |
In the following examples, we choose
Q(t)=t2. |
We define the constraint error at the approximate optimal solution as follows:
e(x∗)=m∑i=1max{gi(x∗),0}. |
We know that x∗ is an ϵ-feasible solution to (P), as e(x∗)<ϵ.
Our tests are performed on a PC with an Intel Core CPU (I7-12700H 2.30 GHz) and 16 GB RAM, with MATLAB, R2021b. Five test problems were selected for analysis. Problems 1–3 and Problem 5 were obtained from [28], and problem 4 was obtained from [29]. The numerical results are shown in the following tables. The symbol m represents the number of constraints. f(x∗) represents the approximate minimum and x∗ represents the approximate optimal solution to the penalty problem. e(x∗) denotes the constraint error at the approximate optimal solution. k represents the number of iterations in these algorithms. CPU represents the time required to solve the penalty problem. 'Fail' indicates a solution failure.
Example 4.1.
f(x)=∑1≤k≤n(xk−1)2,gi(x)=x1exp(−x2ti)cos(x3ti+x4)+x3x2exp(−x1ti)sin(x2ti)+x5exp(−x6ti)−2,ti=πi/m,i=1,...,m,x0=(2,2,7,0,−2,1)∈R6. |
FAOPF/TPEP/SMPF: a1=−1000, b1=1000, ϵ1=0.001, and ρ1=1.
Example 4.2.
f(x)=x2,gi(x)=−x1cos(2πti)−x2sin(2πti)−1,ti=i/m,i=1,...,m,x0=(0.8,0.5)∈R2. |
FAOPF/TPEP/SMPF: a1=−100, b1=0, ϵ1=0.001, and ρ1=1.
SOPF: a1=−100, b1=0, τ=0.5, a=4, ϵ1=0.001, and ρ1=1.
Example 4.3.
f(x)=1n∑1≤k≤n(xk−1)2,gi(x)=∏1≤k≤ncos(tixk)+ti∑1≤k≤nx3k,ti=1/2+πi/m,i=1,...,m,x0=(−2,...,−2)∈Rn. |
FAOPF/TPEP/SMPF: a1=−100, b1=100, ϵ1=0.001, ρ1=1, and n=10 (or n=100).
SOPF: a1=−100, b1=100, a=4, ϵ=0.0001, ρ1=1, and n=10 (or n=100).
Example 4.4.
f(x)=x213+x12+x22,gi(x)=(1−x21t2i)2−x1t2i−x22+x2,ti=i/m,i=1,...,m,x0=(−1,100)∈R2. |
FAOPF/TPEP/SMPF: a1=−100, b1=100, ρ1=1, and ϵ1=0.001.
SOPF: a1=−100, b1=100, τ=0.5, a=4, ϵ=0.0001, and ρ1=1.
Example 4.5.
f(x)=sin(x1−1+1.5π)+∑2≤k≤1000sin(−xk+1.5π+x2k−1),gi(x)={x1−π,i=1,x2i−1−xi−π,i=2,...,1000,−x1−π,i=1001,−x2i−1001+xi−1000−π,i=1002,...,2000,x0=(1,...,1)∈R1000. |
FAOPF/TPEP/SMPF: a1=−99910, b1=−99900, ρ1=1, and ϵ1=0.001.
The result of these problems are in following Figure 1 and Tables 1–6.
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0078 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.0975 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.0835 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
102 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0060 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.2599 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.1560 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
103 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0192 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 2.1690 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 1.0915 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
104 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0640 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 15.563 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 6.2424 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
105 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.2768 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 218.00 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 79.626 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -1.0503 | 0.00000000 | 18 | 0.0704 | (0.0029,−1.0503)′ |
TPEP | -0.6179 | 0.00257560 | 14 | 0.0192 | (0.7903,−0.6179)′ | |
SMPF | -1.0513 | 0.00000000 | 14 | 0.0447 | (0.0000,−1.0513)′ | |
SOPF | -1.0515 | 0.00000000 | 14 | 0.6546 | (0.0000,−1.0515)′ | |
102 | FAOPF | -1.0000 | 0.00001262 | 14 | 0.0498 | (0.0312,−1.0000)′ |
TPEP | -0.7293 | 0.14098368 | 14 | 0.0846 | (0.7179,−0.7293)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 0.0858 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
103 | FAOPF | -1.0000 | 0.00001195 | 14 | 0.1366 | (0.0011,−1.0000)′ |
TPEP | -1.0088 | 0.25050601 | 14 | 0.6236 | (0.0000,−1.0088)′ | |
SMPF | -0.9998 | 0.00000000 | 14 | 0.6159 | (0.0000,−0.9998)′ | |
SOPF | Fail | |||||
104 | FAOPF | -0.9991 | 0.00000037 | 12 | 0.5572 | (0.0421,−0.9991)′ |
TPEP | -1.0075 | 1.93291483 | 14 | 6.5033 | (0.0000,−1.0075)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 5.9534 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
105 | FAOPF | -1.0000 | 0.00144653 | 14 | 4.7819 | (0.0000,−1.0000)′ |
TPEP | -0.9282 | 0.00000000 | 14 | 69.2135 | (0.0000,−0.9282)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 87.9373 | (0.0000,−0.9282)′ | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.0473 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.2813 | 0.05459490 | 18 | 0.0540 | (−0.5104,...,−0,5104)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.0503 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 0.1112 | (−0.5215,...,−0,5215)′∈R10 | |
102 | FAOPF | 2.3149 | 0.00000000 | 21 | 0.0742 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3002 | 0.02394616 | 18 | 0.1642 | (−0.5167,...,−0,5167)′∈R10 | |
SMPF | 2.3153 | 0.00000000 | 18 | 0.1212 | (−0.5216,...,−0,5216)′∈R10 | |
SOPF | 2.3149 | 0.00000089 | 18 | 0.5165 | (−0.5215,...,−0,5215)′∈R10 | |
103 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.2736 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3101 | 0.00825778 | 18 | 1.6351 | (−0.5199,...,−0,5199)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.6453 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 2.2040 | (−0.5215,...,−0,5215)′∈R10 | |
104 | FAOPF | 2.3149 | 0.00000000 | 23 | 1.1659 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3417 | 0.00000000 | 18 | 5.6564 | (−0.5303,...,−0,5303)′∈R10 | |
SMPF | 2.3145 | 0.00061067 | 18 | 3.7953 | (−0.5214,...,−0,5214)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 10.8069 | (−0.5215,...,−0,5215)′∈R10 | |
105 | FAOPF | 2.3149 | 0.00000000 | 23 | 12.4410 | (−0.5215,...,−0,5215)′∈R10 |
TPEP | 2.4367 | 0.00000000 | 18 | 141.000 | (−0.5610,...,−0,5610)′∈R10 | |
SMPF | 2.3149 | 0.00002634 | 18 | 49.2752 | (−0.5215,...,−0,5215)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 147.013 | (−0.5215,...,−0,5215)′∈R10 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.2432 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | 2.4059 | 0.00000000 | 15 | 0.5205 | (−0.5511,...,−0.5511)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3057 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 0.4231 | (−0.2228,...,−0.2228)′∈R100 | |
102 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.3328 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 13.4435 | 0.0000000 | 15 | 1.2428 | (−2.6665,...,−2.6665)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3422 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000089 | 15 | 0.7039 | (−0.2228,...,−0.2228)′∈R100 | |
103 | FAOPF | 1.4915 | 0.00000000 | 17 | 1.8971 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 1.3497 | 29.9112787 | 15 | 2.1453 | (−0.1618,...,−0.1618)′∈R100 | |
SMPF | 1.4920 | 0.00000000 | 15 | 1.2871 | (−0.2215,...,−0.2215)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 2.6963 | (−0.2228,...,−0.2228)′∈R100 | |
104 | FAOPF | 1.4915 | 0.00000000 | 17 | 26.9781 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4913 | 0.00038001 | 15 | 13.1280 | (−0.2212,...,−0.2212)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 33.1198 | (−0.2228,...,−0.2228)′∈R100 | |
105 | FAOPF | 1.4915 | 0.00000000 | 17 | 233.263 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4915 | 0.00000000 | 15 | 223.992 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 368.986 | (−0.2228,...,−0.2228)′∈R100 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0630 | (−0.7579,1.6185)′ |
TPEP | 2.4262 | 0.00337124 | 15 | 0.1059 | (−0.7579,1.6185)′ | |
SMPF | 2.4267 | 0.00274804 | 15 | 0.1561 | (−0.7703,1.6168)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.1051 | (−0.7579,1.6195)′ | |
102 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0831 | (−0.7579,1.6185)′ |
TPEP | 2.4327 | 0.00000000 | 15 | 0.3213 | (−0.7068,1.6185)′ | |
SMPF | 2.4300 | 0.00137691 | 15 | 0.3046 | (−0.7758,1.6178)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.2740 | (−0.7579,1.6195)′ | |
103 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.3610 | (−0.7579,1.6185)′ |
TPEP | 2.4313 | 0.00000000 | 15 | 1.5127 | (−0.7084,1.6181)′ | |
SMPF | 2.4308 | 0.00000000 | 15 | 1.7243 | (−0.7769,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 1.5236 | (−0.7579,1.6195)′ | |
104 | FAOPF | 2.4319 | 0.00000000 | 15 | 1.8437 | (−0.7579,1.6185)′ |
TPEP | 2.6163 | 0.00000000 | 15 | 14.1481 | (−0.9297,1.6713)′ | |
SMPF | 2.4308 | 0.00047849 | 15 | 17.9017 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 15.0246 | (−0.7579,1.6195)′ | |
105 | FAOPF | 2.4319 | 0.00000000 | 15 | 16.2051 | (−0.7579,1.6185)′ |
TPEP | 2.4616 | 13.8075420 | 15 | 269.000 | (−1.0260,1.6198)′ | |
SMPF | 2.4308 | 0.00020298 | 15 | 146.000 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 228.745 | (−0.7579,1.6195)′ |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -99901 | 0.00000000 | 1 | 0.0077 | (1.0000,...,1.0000)′∈R1000 |
TPEP | -99901 | 0.00000000 | 10 | 0.2173 | (1.0000,...,1.0000)′∈R1000 | |
SMPF | -99901 | 0.00000000 | 10 | 0.2759 | (1.0000,...,1.0000)′∈R1000 | |
SOPF | Fail |
The numerical results are described as follows to explain the numerical efficiency.
(1) From Figure 1, it is obvious that the FAOPF algorithm exhibits the best performance subject to the CPU time.
(2) The flattened aggregate function works similarly to the active set, with only a portion of the function being calculated in each iteration. This significantly reduces the amount of the gradient calculation. As a result, even if the FAOPF algorithm requires more iterations than others, the actual calculations are fewer, especially in problem 3. When considering the CPU time for each algorithm, the FAOPF algorithm shows a better numerical performance as the number of constraints increases.
(3) The FAOPF algorithm exhibits the most stable numerical performance compared to other methods. Regardless of the changes in the number of constraints, the approximate minimum value, the optimal solutions, and the constraint error remain relatively constant.
(4) In our numerical experiments, we discover that a moderately sized initial value for ρ is adequate, with ρ≥1 proving to be generally sufficient.
(5) The penalty function in the SOPF algorithm is a smooth approximation based on F(x,M)=Q(f(x)−M)+ρ∑i∈IPτ(gi(x)), where
Pτ(t)={0,t≤0,tτ,t≥0. |
According to the results of the numerical experiments, this penalty function cannot solve all the above problems. However, the numerical properties of the solvable problems are more stable than those of the TPEP and SMPF algorithms.
(6) When the constraint error is greater than 0, it indicates that the approximate optimal solution obtained violates the constraints of the original problem to some extent. With the same parameter settings, the TPEP and SMPF algorithms consistently produce constraint errors greater than 0 in a larger number of instances.
In this paper, the modified flattened aggregate function was used to smoothly approximate the max-value function, and combined with the objective penalty parameter, a new objective penalty function was proposed. Then, a penalty function algorithm (FAOPF) was designed to solve the approximate optimal solution of the original problem (P), and the efficiency of the FAOPF algorithm was proven by numerical experiments. Especially when faced with multiple constraints, the objective penalty function presented in this paper demonstrates a superior effectiveness.
Zhuolin Yan: Conceptualization, methodology, writing-original draft, writing-review and editing, software, validation; Xiaowei Jiang: Conceptualization, methodology, writing-original draft; Siyao Wang: Writing-review and editing, software, validation. All authors contributed equally to the manuscript. All authors have read and approved the final version of the manuscript for publication.
This work was supported by the key project of the natural science foundation joint fund of Jilin province (YDZJ202101ZYTS167, YDZJ202201ZYTS303) and the graduate innovation project of Beihua University(2023002).
All authors declare no conflict of interest in this paper.
[1] | T. Roubíček, Nonlinear partial differential equations with applications, Birkhäuser Basel, Vol. 153, 2013. https://doi.org/10.1007/978-3-0348-0513-1 |
[2] | L. Debnath, Nonlinear partial differential equations for scientists and engineers, 2 Eds., Birkhäuser Boston, 2005. https://doi.org/10.1007/b138648 |
[3] | D. Cioranescu, J. L. Lions, Nonlinear partial differential equations and their applications, College de France Seminar Volume XIV, 1 Ed., Elsevier, 2002. |
[4] | W. F. Ames, Nonlinear partial differential equations in engineering, Vol. 18, Academic press, 1965. |
[5] | J. L. Kazdan, Applications of partial differential equations to problems in geometry, Graduate Texts in Mathematics, 1983. |
[6] |
A. P. Bassom, P. A. Clarkson, A. C. Hicks, On the application of solutions of the fourth Painlevé equation to various physically motivated nonlinear partial differential equations, Adv. Differ. Equ., 1 (1996), 175–198. https://doi.org/10.57262/ade/1366896236 doi: 10.57262/ade/1366896236
![]() |
[7] |
H. Khan, R. Shah, J. F. Gómez-Aguilar, D. Baleanu, P. Kumam, Travelling waves solution for fractional-order biological population model, Math. Model. Nat. Phenom., 16 (2021), 32. https://doi.org/10.1051/mmnp/2021016 doi: 10.1051/mmnp/2021016
![]() |
[8] | B. Barnes, E. Osei-Frimpong, F. O. Boateng, J. Ackora-Prah, A two-dimensional chebyshev wavelet method for solving partial differential equations, J. Math. Theory Model., 6 (2016), 124–138. |
[9] | M. Khalid, M. Sultana, F. Zaidi, U. Arshad, An Elzaki transform decomposition algorithm applied to a class of non-linear differential equations, J. Nat. Sci. Res., 5 (2015), 48–56. |
[10] |
M. Kumar, Umesh, Recent development of Adomian decomposition method for ordinary and partial differential equations, Int. J. Appl. Comput. Math., 8 (2022), 81. https://doi.org/10.1007/s40819-022-01285-6 doi: 10.1007/s40819-022-01285-6
![]() |
[11] | H. Gündoǧdu, Ö. F. Gözükızıl, Solving nonlinear partial differential equations by using Adomian decomposition method, modified decomposition method and Laplace decomposition method, MANAS J. Eng., 5 (2017), 1–13. |
[12] |
S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365 (2007), 345–350. https://doi.org/10.1016/j.physleta.2007.01.046 doi: 10.1016/j.physleta.2007.01.046
![]() |
[13] |
H. Khan, S. Barak, P. Kumam, M. Ariff, Analytical solutions of fractional Klein-Gordon and gas dynamics equations, via the (G′/G)-expansion method, Symmetry, 11 (2019), 566. https://doi.org/10.3390/sym11040566 doi: 10.3390/sym11040566
![]() |
[14] |
R. Shah, H. Khan, P. Kumam, M. Arif, D. Baleanu, Natural transform decomposition method for solving fractional-order partial differential equations with proportional delay, Mathematics, 7 (2019), 532. https://doi.org/10.3390/math7060532 doi: 10.3390/math7060532
![]() |
[15] |
R. Ali, E. Tag-eldin, A comparative analysis of generalized and extended (G′G)-expansion methods for travelling wave solutions of fractional Maccari's system with complex structure, Alex. Eng. J., 79 (2023), 508–530. https://doi.org/10.1016/j.aej.2023.08.007 doi: 10.1016/j.aej.2023.08.007
![]() |
[16] |
M. Kamrujjaman, A. Ahmed, J. Alam, Travelling waves: interplay of low to high Reynolds number and tan-cot function method to solve Burger's equations, J. Appl. Math. Phys., 7 (2019), 861–873. https://doi.org/10.4236/jamp.2019.74058 doi: 10.4236/jamp.2019.74058
![]() |
[17] |
M. Cinar, A. Secer, M. Ozisik, M. Bayram, Derivation of optical solitons of dimensionless Fokas-Lenells equation with perturbation term using Sardar sub-equation method, Opt. Quant. Electron., 54 (2022), 402. https://doi.org/10.1007/s11082-022-03819-0 doi: 10.1007/s11082-022-03819-0
![]() |
[18] | M. S. Islam, K. Khan, M. A. Akbar, The generalized Kudrysov method to solve some coupled nonlinear evolution equations, Asian J. Math. Comput. Res., 3 (2015), 104–121. |
[19] |
A. Bekir, E. Aksoy, A. C. Cevikel, Exact solutions of nonlinear time fractional partial differential equations by sub-equation method, Math. Methods Appl. Sci., 38 (2015), 2779–2784. https://doi.org/10.1002/mma.3260 doi: 10.1002/mma.3260
![]() |
[20] |
S. Bibi, S. T. Mohyud-Din, U. Khan, N. Ahmed, Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order, Results Phys., 7 (2017), 4440–4450. https://doi.org/10.1016/j.rinp.2017.11.008 doi: 10.1016/j.rinp.2017.11.008
![]() |
[21] |
M. Dehghan, J. Manafian Heris, A. Saadatmandi, Application of the exp-function method for solving a partial differential equation arising in biology and population genetics, Int. J. Numer. Methods Heat Fluid Flow, 21 (2011), 736–753. https://doi.org/10.1108/09615531111148482 doi: 10.1108/09615531111148482
![]() |
[22] |
S. Noor, A. S. Alshehry, A. Khan, I. Khan, Analysis of soliton phenomena in (2+1)-dimensional Nizhnik-Novikov-Veselov model via a modified analytical technique, AIMS Math., 8 (2023), 28120–28142. https://doi.org/10.3934/math.20231439 doi: 10.3934/math.20231439
![]() |
[23] |
H. Yasmin, N. H. Aljahdaly, A. M. Saeed, R. Shah, Investigating symmetric soliton solutions for the fractional coupled Konno-Onno system using improved versions of a novel analytical technique, Mathematics, 11 (2023), 2686. https://doi.org/10.3390/math11122686 doi: 10.3390/math11122686
![]() |
[24] |
R. Ali, M. M. Alam, S. Barak, Exploring chaotic behavior of optical solitons in complex structured conformable perturbed Radhakrishnan-Kundu-Lakshmanan model, Phys. Scr., 99 (2024), 095209. https://doi.org/10.1088/1402-4896/ad67b1 doi: 10.1088/1402-4896/ad67b1
![]() |
[25] |
M. Iqbal, W. A. Faridi, R. Ali, A. R. Seadawy, A. A. Rajhi, A. E. Anqi, et al., Dynamical study of optical soliton structure to the nonlinear Landau-Ginzburg-Higgs equation through computational simulation, Opt. Quant. Electron., 56 (2024), 1192. https://doi.org/10.1007/s11082-024-06401-y doi: 10.1007/s11082-024-06401-y
![]() |
[26] |
R. Ali, Z. Zhang, H. Ahmad, M. M. Alam, The analytical study of soliton dynamics in fractional coupled Higgs system using the generalized Khater method, Opt. Quant. Electron., 56 (2024), 1067. https://doi.org/10.1007/s11082-024-06924-4 doi: 10.1007/s11082-024-06924-4
![]() |
[27] |
S. Noor, A. S. Alshehry, H. M. Dutt, R. Nazir, A. Khan, R. Shah, Investigating the dynamics of time-fractional Drinfeld-Sokolov-Wilson system through analytical solutions, Symmetry, 15 (2023), 703. https://doi.org/10.3390/sym15030703 doi: 10.3390/sym15030703
![]() |
[28] |
X. Z. Zhang, M. I. Asjad, W. A. Faridi, A. Jhangeer, M. İnç, The comparative report on dynamical analysis about fractional nonlinear Drinfeld-Sokolov-Wilson systemm, Fractals, 30 (2022), 2240138. https://doi.org/10.1142/S0218348X22401387 doi: 10.1142/S0218348X22401387
![]() |
[29] |
V. G. Drinfel'd, V. V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, J. Math. Sci., 30 (1985), 1975–2036. https://doi.org/10.1007/BF02105860 doi: 10.1007/BF02105860
![]() |
[30] |
G. Wilson, The affine Lie algebra C(1)2 and an equation of Hirota and Satsuma, Phys. Lett. A, 89 (1982), 332–334. https://doi.org/10.1016/0375-9601(82)90186-4 doi: 10.1016/0375-9601(82)90186-4
![]() |
[31] |
R. Arora, A. Kumar, Solution of the coupled Drinfeld's-Sokolov-Wilson (DSW) system by homotopy analysis method, Adv. Sci. Eng. Med., 5 (2013), 1105–1111. https://doi.org/10.1166/asem.2013.1399 doi: 10.1166/asem.2013.1399
![]() |
[32] |
M. Usman, A. Hussain, F. D. Zaman, S. M. Eldin, Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water wavess, Results Phys., 51 (2023), 106613. https://doi.org/10.1016/j.rinp.2023.106613 doi: 10.1016/j.rinp.2023.106613
![]() |
[33] |
J. Singh, D. Kumar, D. Baleanu, S. Rathore, An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math. Comput., 335 (2018), 12–24. https://doi.org/10.1016/j.amc.2018.04.025 doi: 10.1016/j.amc.2018.04.025
![]() |
[34] |
V. E. Tarasov, On chain rule for fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 1–4. https://doi.org/10.1016/j.cnsns.2015.06.007 doi: 10.1016/j.cnsns.2015.06.007
![]() |
[35] |
J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), 257–259. https://doi.org/10.1016/j.physleta.2011.11.030 doi: 10.1016/j.physleta.2011.11.030
![]() |
[36] | M. Z. Sarıkaya, H. Budak, H. Usta, On generalized the conformable fractional calculus, TWMS J. Appl. Eng. Math., 9 (2019), 792–799. |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0078 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.0975 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.0835 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
102 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0060 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.2599 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.1560 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
103 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0192 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 2.1690 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 1.0915 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
104 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0640 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 15.563 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 6.2424 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
105 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.2768 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 218.00 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 79.626 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -1.0503 | 0.00000000 | 18 | 0.0704 | (0.0029,−1.0503)′ |
TPEP | -0.6179 | 0.00257560 | 14 | 0.0192 | (0.7903,−0.6179)′ | |
SMPF | -1.0513 | 0.00000000 | 14 | 0.0447 | (0.0000,−1.0513)′ | |
SOPF | -1.0515 | 0.00000000 | 14 | 0.6546 | (0.0000,−1.0515)′ | |
102 | FAOPF | -1.0000 | 0.00001262 | 14 | 0.0498 | (0.0312,−1.0000)′ |
TPEP | -0.7293 | 0.14098368 | 14 | 0.0846 | (0.7179,−0.7293)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 0.0858 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
103 | FAOPF | -1.0000 | 0.00001195 | 14 | 0.1366 | (0.0011,−1.0000)′ |
TPEP | -1.0088 | 0.25050601 | 14 | 0.6236 | (0.0000,−1.0088)′ | |
SMPF | -0.9998 | 0.00000000 | 14 | 0.6159 | (0.0000,−0.9998)′ | |
SOPF | Fail | |||||
104 | FAOPF | -0.9991 | 0.00000037 | 12 | 0.5572 | (0.0421,−0.9991)′ |
TPEP | -1.0075 | 1.93291483 | 14 | 6.5033 | (0.0000,−1.0075)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 5.9534 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
105 | FAOPF | -1.0000 | 0.00144653 | 14 | 4.7819 | (0.0000,−1.0000)′ |
TPEP | -0.9282 | 0.00000000 | 14 | 69.2135 | (0.0000,−0.9282)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 87.9373 | (0.0000,−0.9282)′ | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.0473 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.2813 | 0.05459490 | 18 | 0.0540 | (−0.5104,...,−0,5104)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.0503 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 0.1112 | (−0.5215,...,−0,5215)′∈R10 | |
102 | FAOPF | 2.3149 | 0.00000000 | 21 | 0.0742 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3002 | 0.02394616 | 18 | 0.1642 | (−0.5167,...,−0,5167)′∈R10 | |
SMPF | 2.3153 | 0.00000000 | 18 | 0.1212 | (−0.5216,...,−0,5216)′∈R10 | |
SOPF | 2.3149 | 0.00000089 | 18 | 0.5165 | (−0.5215,...,−0,5215)′∈R10 | |
103 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.2736 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3101 | 0.00825778 | 18 | 1.6351 | (−0.5199,...,−0,5199)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.6453 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 2.2040 | (−0.5215,...,−0,5215)′∈R10 | |
104 | FAOPF | 2.3149 | 0.00000000 | 23 | 1.1659 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3417 | 0.00000000 | 18 | 5.6564 | (−0.5303,...,−0,5303)′∈R10 | |
SMPF | 2.3145 | 0.00061067 | 18 | 3.7953 | (−0.5214,...,−0,5214)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 10.8069 | (−0.5215,...,−0,5215)′∈R10 | |
105 | FAOPF | 2.3149 | 0.00000000 | 23 | 12.4410 | (−0.5215,...,−0,5215)′∈R10 |
TPEP | 2.4367 | 0.00000000 | 18 | 141.000 | (−0.5610,...,−0,5610)′∈R10 | |
SMPF | 2.3149 | 0.00002634 | 18 | 49.2752 | (−0.5215,...,−0,5215)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 147.013 | (−0.5215,...,−0,5215)′∈R10 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.2432 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | 2.4059 | 0.00000000 | 15 | 0.5205 | (−0.5511,...,−0.5511)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3057 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 0.4231 | (−0.2228,...,−0.2228)′∈R100 | |
102 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.3328 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 13.4435 | 0.0000000 | 15 | 1.2428 | (−2.6665,...,−2.6665)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3422 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000089 | 15 | 0.7039 | (−0.2228,...,−0.2228)′∈R100 | |
103 | FAOPF | 1.4915 | 0.00000000 | 17 | 1.8971 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 1.3497 | 29.9112787 | 15 | 2.1453 | (−0.1618,...,−0.1618)′∈R100 | |
SMPF | 1.4920 | 0.00000000 | 15 | 1.2871 | (−0.2215,...,−0.2215)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 2.6963 | (−0.2228,...,−0.2228)′∈R100 | |
104 | FAOPF | 1.4915 | 0.00000000 | 17 | 26.9781 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4913 | 0.00038001 | 15 | 13.1280 | (−0.2212,...,−0.2212)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 33.1198 | (−0.2228,...,−0.2228)′∈R100 | |
105 | FAOPF | 1.4915 | 0.00000000 | 17 | 233.263 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4915 | 0.00000000 | 15 | 223.992 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 368.986 | (−0.2228,...,−0.2228)′∈R100 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0630 | (−0.7579,1.6185)′ |
TPEP | 2.4262 | 0.00337124 | 15 | 0.1059 | (−0.7579,1.6185)′ | |
SMPF | 2.4267 | 0.00274804 | 15 | 0.1561 | (−0.7703,1.6168)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.1051 | (−0.7579,1.6195)′ | |
102 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0831 | (−0.7579,1.6185)′ |
TPEP | 2.4327 | 0.00000000 | 15 | 0.3213 | (−0.7068,1.6185)′ | |
SMPF | 2.4300 | 0.00137691 | 15 | 0.3046 | (−0.7758,1.6178)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.2740 | (−0.7579,1.6195)′ | |
103 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.3610 | (−0.7579,1.6185)′ |
TPEP | 2.4313 | 0.00000000 | 15 | 1.5127 | (−0.7084,1.6181)′ | |
SMPF | 2.4308 | 0.00000000 | 15 | 1.7243 | (−0.7769,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 1.5236 | (−0.7579,1.6195)′ | |
104 | FAOPF | 2.4319 | 0.00000000 | 15 | 1.8437 | (−0.7579,1.6185)′ |
TPEP | 2.6163 | 0.00000000 | 15 | 14.1481 | (−0.9297,1.6713)′ | |
SMPF | 2.4308 | 0.00047849 | 15 | 17.9017 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 15.0246 | (−0.7579,1.6195)′ | |
105 | FAOPF | 2.4319 | 0.00000000 | 15 | 16.2051 | (−0.7579,1.6185)′ |
TPEP | 2.4616 | 13.8075420 | 15 | 269.000 | (−1.0260,1.6198)′ | |
SMPF | 2.4308 | 0.00020298 | 15 | 146.000 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 228.745 | (−0.7579,1.6195)′ |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -99901 | 0.00000000 | 1 | 0.0077 | (1.0000,...,1.0000)′∈R1000 |
TPEP | -99901 | 0.00000000 | 10 | 0.2173 | (1.0000,...,1.0000)′∈R1000 | |
SMPF | -99901 | 0.00000000 | 10 | 0.2759 | (1.0000,...,1.0000)′∈R1000 | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0078 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.0975 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.0835 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
102 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0060 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 0.2599 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 0.1560 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
103 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0192 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 2.1690 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 1.0915 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
104 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.0640 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 15.563 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 6.2424 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail | |||||
105 | FAOPF | 0.0000 | 0.00000000 | 1 | 0.2768 | (1.0000,...,1.0000)′∈R6 |
TPEP | 0.0000 | 0.00000000 | 15 | 218.00 | (1.0000,...,1.0000)′∈R6 | |
SMPF | 0.0000 | 0.00000000 | 15 | 79.626 | (1.0000,...,1.0000)′∈R6 | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -1.0503 | 0.00000000 | 18 | 0.0704 | (0.0029,−1.0503)′ |
TPEP | -0.6179 | 0.00257560 | 14 | 0.0192 | (0.7903,−0.6179)′ | |
SMPF | -1.0513 | 0.00000000 | 14 | 0.0447 | (0.0000,−1.0513)′ | |
SOPF | -1.0515 | 0.00000000 | 14 | 0.6546 | (0.0000,−1.0515)′ | |
102 | FAOPF | -1.0000 | 0.00001262 | 14 | 0.0498 | (0.0312,−1.0000)′ |
TPEP | -0.7293 | 0.14098368 | 14 | 0.0846 | (0.7179,−0.7293)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 0.0858 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
103 | FAOPF | -1.0000 | 0.00001195 | 14 | 0.1366 | (0.0011,−1.0000)′ |
TPEP | -1.0088 | 0.25050601 | 14 | 0.6236 | (0.0000,−1.0088)′ | |
SMPF | -0.9998 | 0.00000000 | 14 | 0.6159 | (0.0000,−0.9998)′ | |
SOPF | Fail | |||||
104 | FAOPF | -0.9991 | 0.00000037 | 12 | 0.5572 | (0.0421,−0.9991)′ |
TPEP | -1.0075 | 1.93291483 | 14 | 6.5033 | (0.0000,−1.0075)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 5.9534 | (0.0000,−1.0000)′ | |
SOPF | Fail | |||||
105 | FAOPF | -1.0000 | 0.00144653 | 14 | 4.7819 | (0.0000,−1.0000)′ |
TPEP | -0.9282 | 0.00000000 | 14 | 69.2135 | (0.0000,−0.9282)′ | |
SMPF | -1.0000 | 0.00000000 | 14 | 87.9373 | (0.0000,−0.9282)′ | |
SOPF | Fail |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.0473 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.2813 | 0.05459490 | 18 | 0.0540 | (−0.5104,...,−0,5104)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.0503 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 0.1112 | (−0.5215,...,−0,5215)′∈R10 | |
102 | FAOPF | 2.3149 | 0.00000000 | 21 | 0.0742 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3002 | 0.02394616 | 18 | 0.1642 | (−0.5167,...,−0,5167)′∈R10 | |
SMPF | 2.3153 | 0.00000000 | 18 | 0.1212 | (−0.5216,...,−0,5216)′∈R10 | |
SOPF | 2.3149 | 0.00000089 | 18 | 0.5165 | (−0.5215,...,−0,5215)′∈R10 | |
103 | FAOPF | 2.3149 | 0.00000000 | 23 | 0.2736 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3101 | 0.00825778 | 18 | 1.6351 | (−0.5199,...,−0,5199)′∈R10 | |
SMPF | 2.3157 | 0.00000000 | 18 | 0.6453 | (−0.5218,...,−0,5218)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 2.2040 | (−0.5215,...,−0,5215)′∈R10 | |
104 | FAOPF | 2.3149 | 0.00000000 | 23 | 1.1659 | (−0.5225,...,−0,5225)′∈R10 |
TPEP | 2.3417 | 0.00000000 | 18 | 5.6564 | (−0.5303,...,−0,5303)′∈R10 | |
SMPF | 2.3145 | 0.00061067 | 18 | 3.7953 | (−0.5214,...,−0,5214)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 10.8069 | (−0.5215,...,−0,5215)′∈R10 | |
105 | FAOPF | 2.3149 | 0.00000000 | 23 | 12.4410 | (−0.5215,...,−0,5215)′∈R10 |
TPEP | 2.4367 | 0.00000000 | 18 | 141.000 | (−0.5610,...,−0,5610)′∈R10 | |
SMPF | 2.3149 | 0.00002634 | 18 | 49.2752 | (−0.5215,...,−0,5215)′∈R10 | |
SOPF | 2.3149 | 0.00000000 | 18 | 147.013 | (−0.5215,...,−0,5215)′∈R10 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.2432 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | 2.4059 | 0.00000000 | 15 | 0.5205 | (−0.5511,...,−0.5511)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3057 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 0.4231 | (−0.2228,...,−0.2228)′∈R100 | |
102 | FAOPF | 1.4915 | 0.00000000 | 17 | 0.3328 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 13.4435 | 0.0000000 | 15 | 1.2428 | (−2.6665,...,−2.6665)′∈R100 | |
SMPF | 1.4917 | 0.00000000 | 15 | 0.3422 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000089 | 15 | 0.7039 | (−0.2228,...,−0.2228)′∈R100 | |
103 | FAOPF | 1.4915 | 0.00000000 | 17 | 1.8971 | (−0.2213,...,−0,2213)′∈R100 |
TPEP | 1.3497 | 29.9112787 | 15 | 2.1453 | (−0.1618,...,−0.1618)′∈R100 | |
SMPF | 1.4920 | 0.00000000 | 15 | 1.2871 | (−0.2215,...,−0.2215)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 2.6963 | (−0.2228,...,−0.2228)′∈R100 | |
104 | FAOPF | 1.4915 | 0.00000000 | 17 | 26.9781 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4913 | 0.00038001 | 15 | 13.1280 | (−0.2212,...,−0.2212)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 33.1198 | (−0.2228,...,−0.2228)′∈R100 | |
105 | FAOPF | 1.4915 | 0.00000000 | 17 | 233.263 | (−0.2213,...,−0.2213)′∈R100 |
TPEP | Fail | |||||
SMPF | 1.4915 | 0.00000000 | 15 | 223.992 | (−0.2213,...,−0.2213)′∈R100 | |
SOPF | 1.4954 | 0.00000000 | 15 | 368.986 | (−0.2228,...,−0.2228)′∈R100 |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0630 | (−0.7579,1.6185)′ |
TPEP | 2.4262 | 0.00337124 | 15 | 0.1059 | (−0.7579,1.6185)′ | |
SMPF | 2.4267 | 0.00274804 | 15 | 0.1561 | (−0.7703,1.6168)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.1051 | (−0.7579,1.6195)′ | |
102 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.0831 | (−0.7579,1.6185)′ |
TPEP | 2.4327 | 0.00000000 | 15 | 0.3213 | (−0.7068,1.6185)′ | |
SMPF | 2.4300 | 0.00137691 | 15 | 0.3046 | (−0.7758,1.6178)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 0.2740 | (−0.7579,1.6195)′ | |
103 | FAOPF | 2.4319 | 0.00000000 | 15 | 0.3610 | (−0.7579,1.6185)′ |
TPEP | 2.4313 | 0.00000000 | 15 | 1.5127 | (−0.7084,1.6181)′ | |
SMPF | 2.4308 | 0.00000000 | 15 | 1.7243 | (−0.7769,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 1.5236 | (−0.7579,1.6195)′ | |
104 | FAOPF | 2.4319 | 0.00000000 | 15 | 1.8437 | (−0.7579,1.6185)′ |
TPEP | 2.6163 | 0.00000000 | 15 | 14.1481 | (−0.9297,1.6713)′ | |
SMPF | 2.4308 | 0.00047849 | 15 | 17.9017 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 15.0246 | (−0.7579,1.6195)′ | |
105 | FAOPF | 2.4319 | 0.00000000 | 15 | 16.2051 | (−0.7579,1.6185)′ |
TPEP | 2.4616 | 13.8075420 | 15 | 269.000 | (−1.0260,1.6198)′ | |
SMPF | 2.4308 | 0.00020298 | 15 | 146.000 | (−0.7770,1.6180)′ | |
SOPF | 2.4353 | 0.00000000 | 15 | 228.745 | (−0.7579,1.6195)′ |
m | Method | f(x∗) | e(x∗) | k | CPU | x∗ |
10 | FAOPF | -99901 | 0.00000000 | 1 | 0.0077 | (1.0000,...,1.0000)′∈R1000 |
TPEP | -99901 | 0.00000000 | 10 | 0.2173 | (1.0000,...,1.0000)′∈R1000 | |
SMPF | -99901 | 0.00000000 | 10 | 0.2759 | (1.0000,...,1.0000)′∈R1000 | |
SOPF | Fail |