Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems

Zhi Guang Li; Zhi Guang Li

doi:10.3934/math.2023923

AIMS Mathematics

2023, Volume 8, Issue 8: 18174-18184. doi: 10.3934/math.2023923

Previous Article Next Article

Research article

Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems

Zhi Guang Li ^,

Department of Mathematics, Datong University, Datong, Shanxi 037009, China

Received: 03 March 2023 Revised: 08 May 2023 Accepted: 14 May 2023 Published: 26 May 2023
MSC : 35K15, 91G15

This paper studies variation-inequality problems with fourth order non-Newtonian polytropic operators. First, the test function of the weak solution is constructed by using the difference operator. Then global regularity of the weak solution is obtained by some difference transformation and inequality amplification techniques. The weak solution is transformed into a differential inequality of the energy function. It is proved that the weak solution will blow up in finite time. Then, the upper bound and the blowup rate estimate of the blow up are given by handling some differential inequalities.

Keywords:

variation-inequality problems,
fourth order non-Newtonian polytropic operator,
regularity,
blow up

Citation: Zhi Guang Li. Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems[J]. AIMS Mathematics, 2023, 8(8): 18174-18184. doi: 10.3934/math.2023923

Related Papers:

[1]	Xiaofang Guo, Yuping Wang, Haonan Zhang . An active learning Gaussian modeling based multi-objective evolutionary algorithm using population guided weight vector evolution strategy. Mathematical Biosciences and Engineering, 2023, 20(11): 19839-19857. doi: 10.3934/mbe.2023878
[2]	Jiande Zhang, Chenrong Huang, Ying Huo, Zhan Shi, Tinghuai Ma . Multi-population cooperative evolution-based image segmentation algorithm for complex helical surface image. Mathematical Biosciences and Engineering, 2020, 17(6): 7544-7561. doi: 10.3934/mbe.2020385
[3]	Wenqiang Zhang, Chen Li, Mitsuo Gen, Weidong Yang, Zhongwei Zhang, Guohui Zhang . Multiobjective particle swarm optimization with direction search and differential evolution for distributed flow-shop scheduling problem. Mathematical Biosciences and Engineering, 2022, 19(9): 8833-8865. doi: 10.3934/mbe.2022410
[4]	Wei Li, Wenyin Gong . Differential evolution with quasi-reflection-based mutation. Mathematical Biosciences and Engineering, 2021, 18(3): 2425-2441. doi: 10.3934/mbe.2021123
[5]	Shangbo Zhou, Yuxiao Han, Long Sha, Shufang Zhu . A multi-sample particle swarm optimization algorithm based on electric field force. Mathematical Biosciences and Engineering, 2021, 18(6): 7464-7489. doi: 10.3934/mbe.2021369
[6]	Teng Fei, Hongjun Wang, Lanxue Liu, Liyi Zhang, Kangle Wu, Jianing Guo . Research on multi-strategy improved sparrow search optimization algorithm. Mathematical Biosciences and Engineering, 2023, 20(9): 17220-17241. doi: 10.3934/mbe.2023767
[7]	Daqian Zhang, Chen Huang, Jiyou Fei . Defect reconstruction from magnetic flux leakage measurements employing modified cuckoo search algorithm. Mathematical Biosciences and Engineering, 2021, 18(2): 1898-1925. doi: 10.3934/mbe.2021099
[8]	Wenqiang Zhang, Xiaoxiao Zhang, Xinchang Hao, Mitsuo Gen, Guohui Zhang, Weidong Yang . Multi-stage hybrid evolutionary algorithm for multiobjective distributed fuzzy flow-shop scheduling problem. Mathematical Biosciences and Engineering, 2023, 20(3): 4838-4864. doi: 10.3934/mbe.2023224
[9]	Lingyu Wu, Zixu Li, Wanzhen Ge, Xinchao Zhao . An adaptive differential evolution algorithm with elite gaussian mutation and bare-bones strategy. Mathematical Biosciences and Engineering, 2022, 19(8): 8537-8553. doi: 10.3934/mbe.2022396
[10]	Teng Fei, Xinxin Wu, Liyi Zhang, Yong Zhang, Lei Chen . Research on improved ant colony optimization for traveling salesman problem. Mathematical Biosciences and Engineering, 2022, 19(8): 8152-8186. doi: 10.3934/mbe.2022381

Abstract

1. Introduction

The FWA ^[1] is an optimization algorithm inspired by the phenomenon of fireworks explosion. The fireworks in FWA interchange information to dynamically control the resource allocation and the search manner.

Due to its impressive performance in black box optimization (BBO) problems, variants of FWA have been successfully applied to many kinds of real-world problems, including blind source separation in radar signals ^[2], electromagnetic optimization problem ^[3], person re-identification ^[4], traffic flow prediction ^[5], multi-object and multi-domain interference system ^[6], energy efficiency of metro railway line ^[7], Welding robot path planning ^[8], task scheduling in fog system ^[9], medical image registration ^[10], optimizing the dispatch of controllable units in microgrid ^[11].

DE ^[12] is another optimization algorithm that has been shown to be simple yet efficient. By tuning its parameters, it can show strong ability in exploration or exploitation, and thus it is widely used in hybridization with other optimization algorithms ^{[13,14,15,16]}. Specifically, several attempts have been made to cooperate DE with FWA. In ^[17], DE operators are applied to the population that has gone through selection in FWA; in ^[18] and ^[19], DE operators are applied to the population after explosion in FWA; and in ^[20], DE operators are used to generate DE explosion sparks in FWA.

In LoTFWA ^[21], a competitive loser-out tournament strategy (LoT) is proposed as an alternative manner of information interaction.

As a new baseline of FWA, a number of improved variants have been proposed based on LoTFWA to enhance its performance. According to the focus of improvement, they can be categorized into two types. The first type mainly focuses on LoT strategy. An improved LoT strategy (ILoT) based on the idea of population migration and mutation in biogeography-based optimization is proposed in ^[22]. The expected fitness improvement of each firework is divided into four degrees to follow different mechanisms to change their search pattern. In Multi-Scale Collaborative Fireworks Algorithm (MSCFWA) ^[23], LoT is adjusted to coordinate independent local searches of non-optimal fireworks. Fireworks that have not improved for many iterations are adjusted with reference to the better firework. In Adaptive Niche Radius Fireworks Algorithm (ANRFWA) ^[24], the concept of domination space is proposed and used to extend the mechanism of LoT. The fireworks in another firework's domain space are directly eliminated to save computational resource. In Last-Position Elimination-Based Fireworks Algorithm (LEFWA) ^[25], an elimination strategy is proposed to cooperate with LoT. More fireworks are initialized at the beginning, and the worst one is eliminated for every fixed number of generations. The second type mainly focuses on explosion. In Neighborhood Information Utilization Fireworks Algorithm (NiFWA) ^[5], fireworks are divided into subpopulations, and the explosion operation is improved to give further utilization of the neighborhood subpopulations' information. In Exponentially Decaying Fireworks Algorithm (EDFWA) ^[26], a mechanism called exponentially decaying explosion, which can generate series of explosion sparks whose amplitudes and numbers decay in a exponential manner, was proposed to enhance the explosion's local search ability. In Enhanced Multi-Modal Loser-Out Tournament-Based Fireworks Algorithm (ELoTFWA) ^[27], a position-based mapping rule called PMR and a self-adaptive strategy for adjusting the number of explosion sparks are proposed. The proposed strategy can better balance the global and local search abilities of the algorithm. There are also a few works that improve both operations. In a hybrid algorithm of Covariance Matrix Adaptation Evolutionary Strategies and LoTFWA (CMAFWA) ^[28], explosion is replaced by an adaptation strategy for a Gaussian distribution, a collaboration method based on search space partition is proposed to arrange the search areas of fireworks, and LoT is also improved by introducing more restart conditions.

However, it is indicated ^[29] that there is still much room for improvement. For example, the communication among fireworks is not sufficient, and the reinitialization has too much cost. At the same time, the explosion and guiding cannot deal with a local optimum efficiently enough.

Thus, a triple-spark (TS) guiding strategy is proposed in this paper. Unlike many other variants of LoTFWA, the new guiding strategy focuses on improving the guiding stage of LoTFWA and makes the following two main contributions:

1) The new strategy significantly improves the baseline algorithm, providing a new perspective of enhancing the performance of FWA. In detail, the triple-spark guiding strategy contains three different executing mechanisms to generate guiding sparks. The first guiding strategy is the same as that of LoTFWA. The barycenter guiding strategy uses the central position of several explosion sparks as guiding spark and aims at improving the ability to search in local areas of LoTFWA. The DE guiding strategy takes the advantage of DE mutation, and it aims at providing communications among fireworks at the guiding stage.

2) The new strategy does not conflict with most FWA variants, allowing it to easily cooperate with other strategies. Specifically, for variants containing guiding strategy, like LoTFWA and ILoTFWA, cooperation can be done by substituting its guiding strategy with triple-spark (TS) guiding strategy, as shown later in the article. For variants that do not generate guiding spark, like EDFWA, cooperation can be done by adding TS guiding strategy after the explosion of fireworks. Moreover, the computational cost of TS guiding strategy is low, so there is no need to worry about significant increase in running time of the hybrid algorithm.

The remainder of this paper is organized as follows. Section 2 introduces the background knowledge that provides inspirations for this paper. Section 3 presents the component guiding strategies and describes the whole strategy. Section 4 shows the experimental results to illustrate the performance of the proposed strategy. Section 5 concludes this paper.

2. Related works

In this section, the framework of LoTFWA, DE mutation, and the improved LoT mechanism of ILoTFWA are introduced in details.

2.1. LoTFWA

As a new baseline of FWA variants, LoTFWA ^[21] has shown strong competitiveness in the field of optimization and thus is chosen to be the foundation of this work.

Figure 1 provides an illustration of the framework of LoFWA by demonstrating its basic features. Local search is conducted by fireworks through explosion, guiding and selection. Global coordination is realized by exchanging information among fireworks with LoT and resource allocation. The pseudocode of LoTFWA is shown as Algorithm 1, and its operations are described in detail in the following sections.

Figure 1. Basic features of LoTFWA.

DownLoad: Full-Size Img PowerPoint

Algorithm 1 LoTFWA

Randomly initialize

$\mu$ fireworks in search space
Evaluate the fireworks' fitness
repeat
  for each firework do
    Generate and evaluate explosion sparks
    Generate and evaluate guiding sparks
    Select the best individual (including the firework, explosion sparks and guiding sparks) as the firework of next generation
  end for
  Perform LoT as described in section 2.1.4
until termination criterion is met
return the position and the fitness of the best individual

The fireworks are first randomly initialized in the search space, and then the explosion sparks are generated randomly in the explosion amplitudes of each firework. According to the explosion sparks, guiding sparks are generated for each firework. The relationship between firework and sparks is shown in Figure 2. After that, the next generation of fireworks are chosen using elitism selection. Finally, LoT is performed to provide cooperation among fireworks.

Figure 2. Illustration of fireworks explosions.

DownLoad: Full-Size Img PowerPoint

2.1.1. Explosion

The explosion sparks of each firework are random vectors uniformly distributed within the explosion amplitude of the corresponding firework. The generation process is as shown in Eq (2.1):

$\begin{equation} \boldsymbol{s}^g_{ij} = \boldsymbol{X}^g_i +\boldsymbol{\eta}^g_j \cdot A^g_i \end{equation}$

(2.1)

where $\boldsymbol{s}_{ij}^g$ is the position of the $j$ th spark generated by the $i$ th firework in generation $g$ , $\boldsymbol{X}_i^g$ is the position of the $i$ th firework in generation $g$ , $\boldsymbol{\eta}^g_j$ is a vector with each of its components independently sampled from a uniform distribution between $-1$ and $1$ , and $A_i^g$ is the explosion amplitude of the $i$ th firework in generation $g$ .

The number of explosion sparks is allocated based on the performance of the firework:

$\begin{equation} \lambda^g_r = \begin{cases} \lambda_{r}^1, &g = 1\\ \hat{\lambda}\cdot \frac{r^{-\alpha}}{\sum^\mu_{r = 1}r^{-\alpha}}, &g\ne 1 \end{cases} \end{equation}$

(2.2)

where $r$ is the fitness ranking of the current firework, $\lambda_r^g$ is the number of sparks that will be generated by the current firework in generation $g$ , and $\hat{\lambda}$ is the total number of sparks that will be generated by all fireworks. $\mu$ is the total number of the fireworks, $\alpha$ is a parameter to control the shape of the distribution, and $\lambda^1_r$ is a preset constant.

The size of explosion amplitude is controlled in a dynamic way:

$\begin{equation} A_i^g = \left\{ \begin{aligned} &A_i^1, & &g = 1\\ &C_rA_i^{g-1}, & &f(\boldsymbol{X}_i^g)\ge f(\boldsymbol{X}_i^{g-1})\\ &C_aA_i^{g-1}, & &f(\boldsymbol{X}_i^g) < f(\boldsymbol{X}_i^{g-1}) \end{aligned} \right. \end{equation}$

(2.3)

where $A^g_i$ is the explosion amplitude of the $i$ th firework in generation g. $A_i^1, C_a, C_r$ are preset constants so that $C_a > 1, C_r < 1$ .

2.1.2. Guiding spark

The generation of the guiding spark in the LoTFWA uses the difference vector between the centroids of one group of explosion sparks with excellent fitness and one with inferior fitness. It is as indicated in Algorithm 2.

Algorithm 2 Generating the Guiding Spark for the

$i$ th Firework

Require: The

$i$ th firework's position

$\boldsymbol{X}_i$ , The

$i$ th firework's sparks' positions

$\boldsymbol{s}_{ij}$ , the

$i$ th firework's sparks' fitness

$f(\boldsymbol{s}_{ij})$ , the

$i$ th firework's fitness ranking

$r_i$ , the

$i$ th firework's spark number

$\lambda_{r_i}$ and a preset constant

$\sigma$
Sort the sparks by their fitness values

$f(\boldsymbol{s}_{ij})$ in ascending order

$\Delta_i\gets \frac{1}{\sigma\lambda_{r_i}}(\sum\limits^{\sigma\lambda_{r_i}}_{j = 1}\boldsymbol{s}_{ij}-\sum\limits^{\lambda_{r_i}}_{j = \lambda_{r_i}-\sigma\lambda_{r_i}+1}\boldsymbol{s}_{ij})$

$\boldsymbol{G}_{i}\gets \boldsymbol{X}_i+\Delta_i$
return

$\boldsymbol{G}_{i}$

2.1.3. Selection

Each firework in the next generation is selected from its own candidate pool using elitism selection mechanism:

$\begin{equation} \boldsymbol{X}_i^{g+1} = \arg\min\{f(\boldsymbol{X}_i^g), f(\boldsymbol{s}_{ij}), f(\boldsymbol{G}_i)\}. \end{equation}$

(2.4)

2.1.4. LoT

The LoT operation is designed for exchanging information among fireworks and giving those fireworks who get stuck in local optimum a chance to jump out.

Define the improvement of the $i$ th firework $\boldsymbol{X}^g_i$ in generation $g$ as

$\begin{equation} \delta_i^g = f(\boldsymbol{X}_i^{g-1})-f(\boldsymbol{X}_i^g)\ge 0. \end{equation}$

(2.5)

Then the prediction of its fitness in the final generation $g_{\max}$ can be made as

$\begin{equation} \widehat{f(\boldsymbol{X}^{g_{\max}}_i)} = f(\boldsymbol{X}_i^g)-(g_{\max}-g)\delta^g_i \end{equation}$

(2.6)

For each firework, if its prediction fitness is worse than the best firework in the current generation, the firework will be reinitialized.

2.1.5. Mapping rule

In optimization problems, the sparks might fly out of the search space in certain dimensions. If so, the decision variables in those dimensions will be uniformly randomly regenerated in the search space.

2.2. DE

In order to utilize the advantage of DE, multiple attempts have been made to combine DE with FWA. With fine-tuned parameters, DE operators can provide a new way of inter-firework communication and thus improve the performance of FWA in both exploration and exploitation. Consequently, DE operators become the possible choice of generating guiding sparks for LoTFWA.

The related DE mutation operator is described as Eq (2.7), where $\boldsymbol{v}_{i, g}$ is the $i$ th mutated individual in generation $g$ , $\boldsymbol{x}_{r0, g}, \boldsymbol{x}_{r1, g}, \boldsymbol{x}_{r2, g}$ are different individuals randomly chosen in generation $g$ , $F_i$ is a preset constant parameter, and the hybrid guiding strategy is presented in Section 3.

$\begin{equation} \boldsymbol{v}_{i, g} = \boldsymbol{x}_{r0, g}+F_i\cdot(\boldsymbol{x}_{r1, g}-\boldsymbol{x}_{r2, g}) \end{equation}$

(2.7)

2.3. ILoTFWA

In ^[22], a new loser-out tournament (LoT) mechanism is introduced and enhances the performance of LoTFWA greatly. Its procedure is presented in Algorithm 3.

After making prediction as Eq (2.6), the expected fitness improvement of each firework is divided into four degrees to follow different mechanisms to change their search pattern:

1) retains to ensure stability

2) migrates to the best fireworks population and update explosion amplitude to enhance exploitation ability

3) mutates in a certain range to enhance exploration ability

4) reinitialized

Algorithm 3 ILoT

Require: the fireworks' positions

$\boldsymbol{X}_i^g$ , a preset constant

$\theta$ , the upper bound of the search space

$L$ and the lower bound of the search space

$U$
for each firework

$\boldsymbol{X}^g_i$ do

$\Delta_i = f(\boldsymbol{X}^g_i)-\min_{j}\{f(\boldsymbol{X}_j^g)\}$

$\widehat{f(\boldsymbol{X}^{g_{\max}}_i)}\gets$ make prediction according to Eq (2.6)
if

$\widehat{f(\boldsymbol{X}^{g_{\max}}_i)} > \min_j\{f(\boldsymbol{X}_j^g)\}+\frac{3\Delta_i}{4}$ then
reinitialize

$\boldsymbol{X}_i^g$ as LoT
else if

$\widehat{f(\boldsymbol{X}^{g_{\max}}_i)} > \min_j\{f(\boldsymbol{X}_j^g)\}+\frac{\Delta_i}{2}$ then

$\boldsymbol{X}_i^g = \frac{rand(-1, 1)\cdot(L-U)}{10}+\boldsymbol{X}_i^g$
else if

$\widehat{f(\boldsymbol{X}^{g_{\max}}_i)} > \min_j \{f(\boldsymbol{X}_j^g)\}$ then

$\boldsymbol{X}_i^g = \boldsymbol{X}_i^g+rand(0, 1)\cdot(\boldsymbol{X}_{best}^g-\boldsymbol{X}_i^g)$

$\{ \boldsymbol{X}_{best}^g$ is the best firework in generation

$g \}$

$A_i = (1-\theta)A_i+\theta\|\boldsymbol{X}_{best}^g-\boldsymbol{X}_i^g\|$
end if
end for

The proposed guiding strategy can also cooperate with ILoTFWA and enhance its performance as shown later in experiments.

3. Proposed guiding strategy

There are two main problems with the original guiding spark strategy.

● The guiding spark's tendency to explore or exploit is unstable.

● The utilized information is limited in one single firework and its sparks.

In this section, we will try to settle these problems by introducing three strategies of generating guiding sparks for the fireworks and merge them together to propose TS guiding strategy.

3.1. Original guiding strategy

Li et al. ^[30] indicated that the original guiding spark of LoTFWA contributes greatly to both exploration and exploitation. It can also be observed from the later experiments that the original guiding spark can improve the algorithm's performance no matter what other guiding sparks it cooperates with.

Therefore, it is decided to retain the original guiding spark strategy as a part of the proposed strategy.

3.2. Barycenter guiding strategy

The guiding spark generated by the original guiding strategy is not stable enough because it may jump out of the explosion amplitude and lose the ability of exploiting the local region. Thus, we propose the barycenter guiding strategy to solve the problem.

As shown in Algorithm 4, the barycenter guiding strategy generates guiding spark with the central position of several best explosion sparks. It can be seen later in experiments that this guiding strategy itself performs comparably with Algorithm 2 despite scarce exploration due to its strong exploitation ability.

Algorithm 4 Barycenter Guiding Strategy

Require: The

$i$ th firework's position

$\boldsymbol{X}_i$ , The

$i$ th firework's sparks' positions

$\boldsymbol{s}_{ij}$ , the

$i$ th firework's sparks' fitness

$f(\boldsymbol{s}_{ij})$ , the

$i$ th firework's fitness ranking

$r_i$ , the

$i$ th firework's spark number

$\lambda_{r_i}$ and a preset constant

$\sigma$
Sort the sparks by their fitness values

$f(\boldsymbol{s}_{ij})$ in ascending order

$\boldsymbol{G}_{i, 2}\gets \frac{1}{\sigma\lambda_{r_i}}\sum\limits^{\sigma\lambda_{r_i}}_{j = 1}\boldsymbol{s}_{ij}$
return

$\boldsymbol{G}_{i, 2}$

3.3. DE guiding strategy

DE guiding strategy generates guiding spark with DE operators as shown in Algorithm 5. It presents a way of communicating among fireworks at the guiding stage, which can enhance exploration ability of the algorithm.

In this strategy, DE mutation operator is applied to guiding sparks generated by original guiding strategy and barycenter guiding strategy. There is a little difference between the operator we use here and the one shown in Eq (2.7): We fix the first item to be the guiding spark of the current firework, so the new spark will not be too far away from the firework and its population. The other two items are randomly chosen and different from each other, but they are not necessarily different from $\boldsymbol{G}_{i, 1}$ . This is due to the fact that the number of fireworks in LoTFWA is not large. Such design helps increase diversity of information interaction among fireworks.

In the initial stage of our trial, we found that choosing crossover probability to be 1 performs no worse than other choices. At the same time, the greater the crossover rate is, the better it can explore, which matches our designing purpose of this strategy. So, the DE crossover operator is not applied in this work.

Though the same job of information exchanging and exploration has been done by LoT in LoTFWA ^[21], the expense of DE guiding strategy is much less than LoT. Random reinitialization costs much more than one evaluation only for it totally abandons the current information carried by the current firework.

Additionally, although the designing purpose of DE guiding strategy is to enhance exploration, it can actually demonstrate different abilities according to the stage of the algorithm and the properties of the objective function. In the initial stage of the algorithm, fireworks are spread randomly in the search space, and DE operator plays a role of exploration. In the convergence stage, the operator will show its ability of exploitation if the objective function is unimodal, since the fireworks are moving toward the same target, and their difference will be lessened. If the objective function is multimodal, fireworks are easily stuck in different local optimum and are still away from each other. Consequently, DE operator provides another exploring mechanism to help them get out of local optimum.

Algorithm 5 DE Guiding Strategy

Require: all fireworks' first guiding sparks' positions

$\boldsymbol{G}_{j, 1}$ and the current firework's second guiding spark's position

$\boldsymbol{G}_{i, 2}$

$\boldsymbol{G}_{i, 3}\gets \boldsymbol{G}_{i, 2}+F\cdot(\boldsymbol{G}_{r1, 1}-\boldsymbol{G}_{r2, 1})$

$\{$ apply DE mutation identical to Eq (2.7)

$\}$
return

$\boldsymbol{G}_{i, 3}$

3.4. Triple-spark loser-out tournament-based fireworks algorithm

Combining the three guiding strategies mentioned above will result in the TS guiding strategy. The three strategies will be conducted sequentially, generating three different guiding sparks. The best one among them will be chosen to serve as the role of the original guiding spark in LoTFWA.

Replacing the original guiding strategy in LoTFWA with TS guiding strategy will yield TSLoTFWA. Algorithm 6 introduces the complete pseudocode for TSLoTFWA. Similar to LoTFWA, after initialization, explosion sparks will be generated for each firework, along with the conduction of original guiding strategy and barycenter guiding strategy. After the generation of original guiding sparks for all fireworks, DE guiding strategy will be conducted. The new generation of firework will be selected among its population. Finally, LoT will be performed. Note that the DE guiding strategy requires the original guiding sparks of all fireworks, so it should be conducted after all original guiding sparks are generated.

Algorithm 6 TSLoTFWA

Randomly initialize

$\mu$ fireworks in search space
Evaluate the fireworks' fitness
repeat
for each firework

$\boldsymbol{X}_i$ do
Generate and evaluate explosion sparks

$\boldsymbol{G}_{i, 2}\gets$ generate and evaluate a GS for

$\boldsymbol{X}_i$ according to Algorithm 4

$\boldsymbol{G}_{i, 1}\gets$ generate and evaluate a GS for

$\boldsymbol{X}_i$ according to Algorithm 2
end for
for each firework

$\boldsymbol{X}_i$ do

$\boldsymbol{G}_{i, 3}\gets$ generate and evaluate a GS for

$\boldsymbol{X}_i$ according to Algorithm 5
    Select the best individual (including the firework, explosion sparks and guiding sparks) as the firework of next generation
  end for
  Perform LoT
until termination criterion is met
return the position and the fitness of the best individual

To validate that TS guiding strategy can bring promotion to other state-of-the-art FWA variants, we also apply TS guiding strategy to ILoTFWA, resulting in TSILoTFWA, which will be examined in detail in Section 4.5.

4. Experiments and analysis

4.1. Evaluate methodology

In this section, strategy validation tests and algorithm comparison tests are performed to verify the effectiveness and efficiency of the proposed strategy.

The following tests are all conducted on 28 benchmark functions from the CEC' 13 competition ^[31], which are shown in detail in Table 1.

Table 1. Test functions of CEC 2013 single objective optimization benchmark suite.

No.	Functions	$f_i^=f_i(x^)$
1	Sphere Function	-1400
2	Rotated High Conditioned Elliptic Function	-1300
3	Rotated Bent Cigar Function	-1200
4	Rotated Discus Function	-1100
5	Different Powers Function	-1000
6	Rotated Rosenbrock's Function	-900
7	Rotated Schaffers F7 Function	-800
8	Rotated Ackley's Function	-700
9	Rotated Weierstrass Function	-600
10	Rotated Griewank's Function	-500
11	Rastrigin's Function	-400
12	Rotated Rastrigin's Function	-300
13	Non-Continuous Rotated Rastrigin's Function	-200
14	Schwefel's Function	-100
15	Rotated Schwefel's Function	100
16	Rotated Katsuura Function	200
17	Lunacek Bi_Rastrigin Function	300
18	Rotated Lunacek Bi_Rastrigin Function	400
19	Expanded Griewank's plus Rosenbrock's Function	500
20	Expanded Schaffer's F6 Function	600
21	Composition Function 1 (n=5, Rotated)	700
22	Composition Function 2 (n=3, Unrotated)	800
23	Composition Function 3 (n=3, Rotated)	900
24	Composition Function 4 (n=3, Rotated)	1000
25	Composition Function 5 (n=3, Rotated)	1100
26	Composition Function 6 (n=5, Rotated)	1200
27	Composition Function 7 (n=5, Rotated)	1300
28	Composition Function 8 (n=5, Rotated)	1400
Search Range: [−100,100]^D

| Show Table

DownLoad: CSV

The benchmark dimension is D = 30, and the maximal number of function evaluations is 10000*D. For each algorithm and test function, the test runs for 51 times, and the mean errors of optimal solutions are recorded. To evaluate the performance of the proposed algorithm, the mean errors are ranked on each function, and the averaged rankings (AR) over all functions for each algorithm are calculated. Additionally, two-sided Wilcoxon rank sum tests ^[32] (with confidence level $95\%$ ) are conducted to validate the performance improvement of the proposed strategy.

4.2. Parameter setting

Parameters inheriting from LoTFWA are shown in Table 2 and set according to the suggestion in ^[21].

Table 2. Parameter settings in LoTFWA.

Parameter	Description	Value
$\mu$	Number of fireworks	5
$\hat{\lambda}$	Total number of explosion sparks in one generation	300
$C_a$	Amplification coefficient that controls the amplitude	1.2
$C_r$	Reduction coefficient that controls the amplitude	0.9
$\sigma$	Parameter that controls the proportion of adopted explosion sparks	0.2
$\alpha$	Parameter that controls the shape of the distribution of explosion sparks	0

| Show Table

DownLoad: CSV

In order to study the impact of mutation rate $F$ in Algorithm 5 and verify if the proposed strategy is sensitive to the choice of the parameter, five different choices for $F$ are conducted as the group of $F\in \{0.1, 0.3, 0.5, 0.7, 0.9\}$ . The simulations are conducted on all 28 benchmark functions, and the results are shown in Table 3. Best results are marked in bold.

Table 3. Comparison among 5 sets of parameters.

value	$F$ =0.1	$F$ =0.3	$F$ =0.5	$F$ =0.7	$F$ =0.9
AR.	3.14	2.54	2.64	2.79	3.46

| Show Table

DownLoad: CSV

indicates that assigning $F$ too close to $0$ or $1$ will lead to relatively poor performances. With $F$ close to $0$ , the guiding spark generated by DE guiding strategy will have no significant difference compared with the guiding spark generated by barycenter guiding strategy, which will waste computation resource. With $F$ close to $1$ , the guiding spark generated by DE guiding strategy will be too far from the current firework, leading to inadequate utilization of the known information. The choices of $F = 0.3, F = 0.5, F = 0.7$ do not suffer from these problems and thus have similar performances, indicating that the algorithm is relatively insensitive to $F\in (0.3, 0.7)$ . As shows, $F = 0.3$ has a small advantage over the other two choices, so it will be used in the following experiments.

4.3. Strategy validation

In order to verify that each algorithmic component of the TS guiding strategy is absolutely necessary, experiments are conducted on all combinations of the components based on LoTFWA.

The results are shown in , with $\boldsymbol{G}_1$ representing generating guiding spark according to Algorithm 2, $\boldsymbol{G}_2$ representing generating guiding spark according to Algorithm 4, $\boldsymbol{G}_3$ representing generating guiding spark according to Algorithm 5.

Table 4. Comparison among proposed strategy and other simplified versions.

func	$\boldsymbol{G}_1$	$\boldsymbol{G}_2$	$\boldsymbol{G}_3$	$\boldsymbol{G}_1$ & $\boldsymbol{G}_2$	$\boldsymbol{G}_1$ & $\boldsymbol{G}_3$	$\boldsymbol{G}_2$ & $\boldsymbol{G}_3$	$\boldsymbol{G}_1$ & $\boldsymbol{G}_2$ & $\boldsymbol{G}_3$
1	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$
2	$1.21\times 10^{6}$	$1.50\times 10^6$	$\mathit{\boldsymbol{8.81\times 10^5}}$	$1.51\times 10^6$	$9.91\times 10^5$	$1.06\times 10^6$	$1.02\times 10^6$
3	$2.39\times 10^7$	$8.80\times 10^6$	$1.30\times 10^7$	$8.58\times 10^6$	$4.13\times 10^6$	$1.88\times 10^6$	$\mathit{\boldsymbol{9.35\times 10^5}}$
4	$1.93\times 10^3$	$9.43\times 10^2$	$1.82\times 10^3$	$8.08\times 10^2$	$1.98\times 10^3$	$\mathit{\boldsymbol{3.81\times 10^2}}$	$4.47\times 10^2$
5	$3.58\times 10^{-3}$	$5.74\times 10^{-3}$	$2.09\times 10^{-3}$	$5.31\times 10^{-3}$	$\mathit{\boldsymbol{1.80\times 10^{-3}}}$	$2.52\times 10^{-3}$	$2.41\times 10^{-3}$
6	$\mathit{\boldsymbol{1.31\times 10^1}}$	$1.59\times 10^1$	$1.45\times 10^1$	$1.46\times 10^1$	$1.32\times 10^1$	$1.51\times 10^1$	$1.48\times 10^1$
7	$5.02\times 10^1$	$4.22\times 10^1$	$4.21\times 10^1$	$4.16\times 10^1$	$4.15\times 10^1$	$3.20\times 10^1$	$\mathit{\boldsymbol{3.02\times 10^1}}$
8	$2.09\times 10^1$	$2.09\times 10^1$	$2.08\times 10^1$	$2.08\times 10^1$	$2.09\times 10^1$	$2.09\times 10^1$	$\mathit{\boldsymbol{2.08\times 10^1}}$
9	$1.45\times 10^1$	$1.30\times 10^1$	$1.74\times 10^1$	$1.29\times 10^1$	$1.43\times 10^1$	$1.33\times 10^1$	$\mathit{\boldsymbol{1.22\times 10^1}}$
10	$4.04\times 10^{-2}$	$2.95\times 10^{-2}$	$5.81\times 10^{-2}$	$3.41\times 10^{-2}$	$2.98\times 10^{-2}$	$2.57\times 10^{-2}$	$\mathit{\boldsymbol{2.54\times 10^{-2}}}$
11	$6.40\times 10^1$	$4.33\times 10^1$	$5.41\times 10^1$	$3.63\times 10^1$	$3.86\times 10^1$	$3.71\times 10^1$	$\mathit{\boldsymbol{3.20\times 10^1}}$
12	$6.96\times 10^1$	$4.31\times 10^1$	$5.70\times 10^1$	$3.40\times 10^1$	$4.20\times 10^1$	$3.71\times 10^1$	$\mathit{\boldsymbol{2.75\times 10^1}}$
13	$1.31\times 10^2$	$8.88\times 10^1$	$1.13\times 10^2$	$7.72\times 10^1$	$9.59\times 10^1$	$7.84\times 10^1$	$\mathit{\boldsymbol{7.10\times 10^1}}$
14	$2.42\times 10^3$	$2.45\times 10^3$	$2.79\times 10^3$	$2.47\times 10^3$	$2.46\times 10^3$	$2.52\times 10^3$	$\mathit{\boldsymbol{2.31\times 10^3}}$
15	$2.56\times 10^3$	$2.58\times 10^3$	$2.64\times 10^3$	$2.54\times 10^3$	$2.55\times 10^3$	$2.53\times 10^3$	$\mathit{\boldsymbol{2.49\times 10^3}}$
16	$5.74\times 10^2$	$6.81\times 10^2$	$1.59\times 10^1$	$5.64\times 10^2$	$5.85\times 10^2$	$7.46\times 10^2$	$\mathit{\boldsymbol{5.57\times 10^2}}$
17	$6.31\times 10^1$	$6.77\times 10^1$	$7.56\times 10^1$	$5.91\times 10^1$	$6.10\times 10^1$	$6.27\times 10^1$	$\mathit{\boldsymbol{5.63\times 10^1}}$
18	$6.33\times 10^1$	$6.65\times 10^1$	$7.60\times 10^1$	$6.25\times 10^1$	$6.08\times 10^1$	$6.41\times 10^1$	$\mathit{\boldsymbol{5.79\times 10^1}}$
19	$3.17\times 10^0$	$2.85\times 10^0$	$3.64\times 10^0$	$2.55\times 10^0$	$2.93\times 10^0$	$2.73\times 10^0$	$\mathit{\boldsymbol{2.50\times 10^0}}$
20	$1.34\times 10^1$	$1.32\times 10^1$	$1.34\times 10^1$	$1.33\times 10^1$	$1.35\times 10^1$	$1.33\times 10^1$	$\mathit{\boldsymbol{1.31\times 10^1}}$
21	$\mathit{\boldsymbol{2.00\times 10^2}}$	$2.28\times 10^2$	$2.69\times 10^2$	$2.30\times 10^2$	$2.69\times 10^2$	$2.69\times 10^2$	$2.65\times 10^2$
22	$2.84\times 10^3$	$2.96\times 10^3$	$3.28\times 10^3$	$\mathit{\boldsymbol{2.84\times 10^3}}$	$2.96\times 10^3$	$3.02\times 10^3$	$2.93\times 10^3$
23	$3.11\times 10^3$	$3.18\times 10^3$	$3.33\times 10^3$	$3.03\times 10^3$	$3.12\times 10^3$	$\mathit{\boldsymbol{3.03\times 10^3}}$	$3.14\times 10^3$
24	$2.40\times 10^2$	$2.10\times 10^2$	$2.32\times 10^2$	$2.07\times 10^2$	$2.12\times 10^2$	$2.03\times 10^2$	$\mathit{\boldsymbol{2.01\times 10^2}}$
25	$2.76\times 10^2$	$2.57\times 10^2$	$2.75\times 10^2$	$2.52\times 10^2$	$2.69\times 10^2$	$\mathit{\boldsymbol{2.41\times 10^2}}$	$2.51\times 10^2$
26	$2.00\times 10^2$	$2.00\times 10^2$	$2.00\times 10^2$	$2.00\times 10^2$	$2.00\times 10^2$	$\mathit{\boldsymbol{2.00\times 10^2}}$	$2.00\times 10^2$
27	$6.96\times 10^2$	$4.23\times 10^2$	$6.57\times 10^2$	$3.65\times 10^2$	$4.36\times 10^2$	$3.29\times 10^2$	$\mathit{\boldsymbol{3.19\times 10^2}}$
28	$\mathit{\boldsymbol{2.69\times 10^2}}$	$2.96\times 10^2$	$2.92\times 10^2$	$2.92\times 10^2$	$2.96\times 10^2$	$2.96\times 10^2$	$2.96\times 10^2$
AR.	$4.86$	$4.68$	$5.39$	$3.04$	$4.14$	$3.21$	$\mathit{\boldsymbol{1.82}}$

| Show Table

DownLoad: CSV

As the results indicate, the more strategies we use, the better result can be achieved. All forms of cooperation among the three strategies outperform the corresponding non-cooperative ones. Compared with algorithms without barycenter guiding strategy, the algorithms that take use of barycenter guiding strategy always perform better. This means that both the original guiding strategy and DE guiding strategy are good at exploration while relatively poor in exploitation, and the barycenter guiding strategy can remarkably make up for this shortcoming. Though DE guiding strategy itself performs worse than the original guiding strategy, the performance of the combination of all three strategies is significantly better than the cooperation of original guiding strategy and barycenter guiding strategy. This indicates that DE guiding strategy and original guiding strategy can complement each other in the ability of exploration.

In conclusion, the combination of all three strategies outperforms other combinations greatly. Thus, the three components of TS guiding strategy are all indispensable.

4.4. Performance comparison

In this section, the significance of TS guiding algorithm is presented by combining it with LoTFWA to form TSLoTFWA. Comparison is conducted between TSLoTFWA, LoTFWA, ILoTFWA and DE algorithm.

The results are shown in . In the table, the $+, -, \approx$ indicate whether a given algorithm performs significantly better ( $+$ ), significantly worse ( $-$ ), or not significantly different ( $\approx$ ) compared to TSLoTFWA according to Wilcoxon rank-sum test.

Table 5. Comparison among the proposed algorithm and other optimization algorithms.

func	TSLoTFWA	DE		LoTFWA		ILoTFWA
1	$\bf{0.00\times 10^0}$	$\bf{0.00\times 10^0}$	$\approx$	$\bf{0.00\times 10^0}$	$\approx$	$\bf{0.00\times 10^0}$	$\approx$
2	$1.02\times 10^6$	$1.02\times 10^8$	-	$1.21\times 10^6$	-	$\bf{7.64\times 10^5}$	$+$
3	$9.35\times 10^5$	$\bf{5.37\times 10^5}$	$+$	$2.39\times 10^7$	-	$4.49\times 10^6$	-
4	$\bf{4.47\times 10^2}$	$5.49\times 10^4$	-	$1.93\times 10^3$	-	$4.71\times 10^2$	$\approx$
5	$2.41\times 10^{-3}$	$\bf{0.00\times 10^0}$	$+$	$3.58\times 10^{-3}$	-	$2.39\times 10^{-3}$	$\approx$
6	$1.48\times 10^1$	$1.62\times 10^1$	-	$\bf{1.31\times 10^1}$	$\approx$	$1.54\times 10^1$	$\approx$
7	$3.02\times 10^1$	$\bf{2.29\times 10^1}$	$+$	$5.02\times 10^1$	-	$3.47\times 10^1$	$\approx$
8	$2.08\times 10^1$	$2.09\times 10^1$	-	$2.09\times 10^1$	$\approx$	$\bf{2.08\times 10^1}$	$\approx$
9	$1.22\times 10^1$	$3.93\times 10^1$	-	$1.45\times 10^1$	-	$\bf{1.21\times 10^1}$	$\approx$
10	$\bf{2.54\times 10^{-2}}$	$7.70\times 10^0$	-	$4.04\times 10^{-2}$	-	$3.67\times 10^{-2}$	-
11	$\bf{3.20\times 10^1}$	$1.10\times 10^2$	-	$6.40\times 10^1$	-	$3.89\times 10^1$	-
12	$\bf{2.75\times 10^1}$	$1.96\times 10^2$	-	$6.96\times 10^1$	-	$3.51\times 10^1$	-
13	$\bf{7.10\times 10^1}$	$1.93\times 10^2$	-	$1.31\times 10^2$	-	$8.32\times 10^1$	-
14	$2.31\times 10^3$	$4.29\times 10^3$	-	$2.42\times 10^3$	$\approx$	$\bf{2.29\times 10^3}$	$\approx$
15	$2.49\times 10^3$	$7.34\times 10^3$	-	$2.56\times 10^3$	$\approx$	$\bf{2.38\times 10^3}$	$\approx$
16	$5.57\times 10^{-2}$	$2.41\times 10^0$	-	$5.74\times 10^{-2}$	$\approx$	$\bf{5.51\times 10^{-2}}$	$\approx$
17	$5.63\times 10^1$	$1.41\times 10^2$	-	$6.31\times 10^1$	-	$\bf{5.61\times 10^1}$	$\approx$
18	$\bf{5.79\times 10^1}$	$2.23\times 10^2$	-	$6.33\times 10^1$	-	$6.10\times 10^1$	$\approx$
19	$\bf{2.50\times 10^0}$	$1.37\times 10^1$	-	$3.17\times 10^0$	-	$2.70\times 10^0$	$\approx$
20	$1.31\times 10^1$	$1.28\times 10^1$	$\approx$	$1.34\times 10^1$	$\approx$	$\bf{1.25\times 10^1}$	$+$
21	$2.65\times 10^2$	$2.94\times 10^2$	-	$\bf{2.00\times 10^2}$	$+$	$2.04\times 10^2$	$+$
22	$2.93\times 10^3$	$5.26\times 10^3$	-	$2.84\times 10^3$	$\approx$	$\bf{2.77\times 10^3}$	$+$
23	$3.14\times 10^3$	$7.88\times 10^3$	-	$3.11\times 10^3$	$\approx$	$\bf{2.78\times 10^3}$	$+$
24	$\bf{2.01\times 10^2}$	$2.16\times 10^2$	-	$2.40\times 10^2$	-	$2.13\times 10^2$	-
25	$\bf{2.51\times 10^2}$	$3.00\times 10^2$	-	$2.76\times 10^2$	-	$2.65\times 10^2$	-
26	$\bf{2.00\times 10^2}$	$2.08\times 10^2$	-	$2.00\times 10^2$	-	$2.00\times 10^2$	-
27	$\bf{3.19\times 10^2}$	$1.10\times 10^3$	-	$6.96\times 10^2$	-	$5.74\times 10^2$	-
28	$2.96\times 10^2$	$3.00\times 10^2$	$\approx$	$\bf{2.69\times 10^2}$	$+$	$2.76\times 10^2$	$\approx$
AR.	$1.79$	$3.46$		$2.82$		$\bf{1.71}$
	$+$	$3$		$2$		$5$
	-	$22$		$17$		$9$
	$\approx$	$3$		$9$		$14$

| Show Table

DownLoad: CSV

As can be seen from the result, TSLoTFWA performs significantly better than LoTFWA and DE on most benchmark functions. Over the total $28$ benchmark functions, TSLoTFWA is significantly better than DE on $22$ of them and is significantly better than LoTFWA on $17$ of them. At the same time, the number of functions TSLoTFWA performs significantly worse is small, which indicates that the improvement from the TS guiding strategy has little side effect.

Furthermore, though the average ranking of TSLoTFWA is a little worse than ILoTFWA, it still holds the advantage from the Wilcoxon rank-sum test's point of view. On the whole, the result of TSLoTFWA is comparable to ILoTFWA.

In short, TS guiding strategy helps LoTFWA achieve better results with low cost, indicating that our design enhances the ability of both exploration and exploitation. The improvement it provides is similar to ILoT.

4.5. Combining with ILoTFWA

As mentioned above, the overall improvements provided by the TS guiding strategy have no obvious advantage versus ILoT. However, the main focuses of TS guiding strategy and ILoTFWA are different from each other. At the same time, ILoTFWA does not change other parts of the algorithm significantly, which makes it possible for them to combine and achieve better results.

The combination of ILoTFWA and TS guiding strategy is very simple: substituting the original strategy used for generating guiding sparks in ILoTFWA with TS guiding strategy. Related experiment results are displayed in Table 6. Notations in this table have similar meanings as Table 5.

Table 6. Comparison among TSILoTFWA and other LoTFWA.

func	TSILoTFWA	LoTFWA		TSLoTFWA		ILoTFWA
1	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\approx$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\approx$	$\mathit{\boldsymbol{0.00\times 10^0}}$	$\approx$
2	$8.91\times 10^5$	$1.21\times 10^6$	$-$	$1.02\times 10^6$	$-$	$\mathit{\boldsymbol{7.64\times 10^5}}$	$+$
3	$3.22\times 10^6$	$2.39\times 10^7$	$-$	$\mathit{\boldsymbol{9.35\times 10^5}}$	$+$	$4.49\times 10^6$	$-$
4	$\mathit{\boldsymbol{1.11\times 10^2}}$	$1.93\times 10^3$	$-$	$4.47\times 10^2$	$-$	$4.71\times 10^2$	$-$
5	$\mathit{\boldsymbol{1.99\times 10^{-3}}}$	$3.58\times 10^{-3}$	$-$	$2.41\times 10^{-3}$	$-$	$2.39\times 10^{-3}$	$-$
6	$1.45\times 10^1$	$\mathit{\boldsymbol{1.31\times 10^1}}$	$\approx$	$1.48\times 10^1$	$-$	$1.54\times 10^1$	$\approx$
7	$\mathit{\boldsymbol{2.74\times 10^1}}$	$5.02\times 10^1$	$-$	$3.02\times 10^1$	$\approx$	$3.47\times 10^1$	$-$
8	$2.08\times 10^1$	$2.09\times 10^1$	$\approx$	$2.08\times 10^1$	$\approx$	$\mathit{\boldsymbol{2.08\times 10^1}}$	$\approx$
9	$\mathit{\boldsymbol{1.03\times 10^1}}$	$1.45\times 10^1$	$-$	$1.22\times 10^1$	$-$	$1.21\times 10^1$	$-$
10	$\mathit{\boldsymbol{2.26\times 10^{-2}}}$	$4.04\times 10^{-2}$	$-$	$2.54\times 10^{-2}$	$\approx$	$3.67\times 10^{-2}$	$-$
11	$\mathit{\boldsymbol{3.09\times 10^1}}$	$6.40\times 10^1$	$-$	$3.20\times 10^1$	$\approx$	$3.89\times 10^1$	$-$
12	$2.80\times 10^1$	$6.96\times 10^1$	$-$	$\mathit{\boldsymbol{2.75\times 10^1}}$	$\approx$	$3.51\times 10^1$	$-$
13	$\mathit{\boldsymbol{6.23\times 10^1}}$	$1.31\times 10^2$	$-$	$7.10\times 10^1$	$-$	$8.32\times 10^1$	$-$
14	$2.38\times 10^3$	$2.42\times 10^3$	$\approx$	$2.31\times 10^3$	$\approx$	$\mathit{\boldsymbol{2.29\times 10^3}}$	$\approx$
15	$\mathit{\boldsymbol{2.30\times 10^3}}$	$2.56\times 10^3$	$-$	$2.49\times 10^3$	$-$	$2.38\times 10^3$	$\approx$
16	$\mathit{\boldsymbol{5.34\times 10^{-2}}}$	$5.74\times 10^{-2}$	$\approx$	$5.57\times 10^{-2}$	$\approx$	$5.51\times 10^{-2}$	$\approx$
17	$\mathit{\boldsymbol{5.34\times 10^1}}$	$6.31\times 10^1$	$-$	$5.63\times 10^1$	$-$	$5.61\times 10^1$	$-$
18	$\mathit{\boldsymbol{5.36\times 10^1}}$	$6.33\times 10^1$	$-$	$5.79\times 10^1$	$-$	$6.10\times 10^1$	$-$
19	$\mathit{\boldsymbol{2.46\times 10^0}}$	$3.17\times 10^0$	$-$	$2.50\times 10^0$	$\approx$	$2.70\times 10^0$	$\approx$
20	$\mathit{\boldsymbol{1.23\times 10^1}}$	$1.34\times 10^1$	$-$	$1.31\times 10^1$	$-$	$1.25\times 10^1$	$\approx$
21	$2.73\times 10^2$	$\mathit{\boldsymbol{2.00\times 10^2}}$	$+$	$2.65\times 10^2$	$\approx$	$2.04\times 10^2$	$+$
22	$\mathit{\boldsymbol{2.72\times 10^3}}$	$2.84\times 10^3$	$\approx$	$2.93\times 10^3$	$-$	$2.77\times 10^3$	$\approx$
23	$2.89\times 10^3$	$3.11\times 10^3$	$-$	$3.14\times 10^3$	$-$	$\mathit{\boldsymbol{2.78\times 10^3}}$	$\approx$
24	$\mathit{\boldsymbol{2.01\times 10^2}}$	$2.40\times 10^2$	$-$	$2.01\times 10^2$	$-$	$2.13\times 10^2$	$-$
25	$2.59\times 10^2$	$2.76\times 10^2$	$-$	$\mathit{\boldsymbol{2.51\times 10^2}}$	$\approx$	$2.65\times 10^2$	$-$
26	$2.00\times 10^2$	$2.00\times 10^2$	$-$	$\mathit{\boldsymbol{2.00\times 10^2}}$	$\approx$	$2.00\times 10^2$	$-$
27	$3.31\times 10^2$	$6.96\times 10^2$	$-$	$\mathit{\boldsymbol{3.19\times 10^2}}$	$+$	$5.74\times 10^2$	$-$
28	$2.96\times 10^2$	$\mathit{\boldsymbol{2.69\times 10^2}}$	$+$	$2.96\times 10^2$	$\approx$	$2.76\times 10^2$	$\approx$
AR.	$\mathit{\boldsymbol{1.57}}$	$3.50$		$2.32$		$2.36$
	$+$	$2$		$2$		$2$
	$-$	$20$		$13$		$15$
	$\approx$	$6$		$13$		$11$

| Show Table

DownLoad: CSV

By combining with ILoTFWA, TSILoTFWA shows significantly better results compared to both TSLoTFWA and ILoTFWA. At the same time, the difference between the performance of TSILoTFWA and ILoTFWA is quite similar to the difference between the performance of TSLoTFWA and LoTFWA. This means that TS guiding strategy can have similar effects on LoTFWA and ILoTFWA and provide robust performance improvements for FWA variants.

According to the average ranking values and the rank-sum test results, the newly proposed TSILoTFWA achieves very competitive performance when comparing with the most well-known competitors. It takes the performance of LoTFWA to a brand-new level.

4.6. Computational cost

With promising performance, the computational cost of TS guiding strategy is low according to the experimental results shown in . For each algorithm, the average running time on CEC' 13 benchmark functions with maximal function evaluation time equal to 300,000 is recorded as $T2$ . $T1$ stands for the average running time of 300,000 evaluations of the benchmark functions without any other operations.

Table 7. Computational cost of TS guiding strategy and other algorithms.

Algorithm	$T1$ (s)	$T2$ (s)	$(T2-T1)/T1$
DE	3.949	6.958	0.762
$\boldsymbol{G}_1$ (LoTFWA)		4.437	0.124
$\boldsymbol{G}_2$		4.420	0.119
$\boldsymbol{G}_3$		4.503	0.140
TSLoTFWA		4.504	0.140
ILoTFWA		4.384	0.110
TSILoTFWA		4.444	0.125

| Show Table

DownLoad: CSV

Among the three component strategies, barycenter guiding strategy has the lowest cost, while DE guiding strategy has the greatest cost. This is because the computation required by barycenter guiding strategy is included in the original guiding strategy, and the DE guiding strategy requires the generation of the original guiding spark and the barycenter guiding spark. At the same time, as shown in the table, the computation complexity of DE is high due to its structure, which is another reason for the relatively high cost of DE guiding strategy.

As for the complete TS guiding strategy, its computational cost is not simple summation of the three component strategies. This is because, as mentioned above, certain computation steps are exactly the same in different strategies and thus can be simplified. Specifically, all extra computations before function evaluation in TS guiding strategy are included in DE guiding strategy. As a result, the computational cost of TSLoTFWA is similar to LoTFWA with DE guiding strategy.

As the result indicates, when cooperating with LoTFWA, the TS guiding strategy results in an increase of about $12.9\%$ in the computational cost, $13.6\%$ when cooperating with ILoTFWA, which is a reasonable trade-off between cost and performance. As for absolute running time, it takes about $0.07$ seconds to perform TS guiding strategy when solving CEC' 13 benchmark functions, which is less than $2\%$ of $T1$ . Therefore, the computational cost of TS guiding strategy is low.

5. Conclusions

This paper proposes a new guiding strategy for LoTFWA which contains three different kinds of guiding spark generation methods. Experimental results on the CEC' 13 benchmark suite indicate that the proposed strategy has low computational cost and can enhance both LoTFWA and ILoTFWA. Therefore, it is believed that the proposed strategy is efficient and effective for optimization. Future works can be done to integrate the TS guiding strategy with other variants of FWA to attain possible improvements.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61973042), Beijing Natural Science Foundation (1202020) and BUPT Excellent Ph.D. Students Foundation (CX2022157).

Conflict of interest

The authors declare no conflict of interest.

References

[1]	C. H. Guan, Z. Q. Xu, F. H. Yi, A consumption-investment model with state-dependent lower bound constraint on consumption, J. Math. Anal. Appl., 516 (2022), 126511. https://doi.org/10.1016/j.jmaa.2022.126511 doi: 10.1016/j.jmaa.2022.126511
[2]	C. H. Guan, F. H. Yi, J. Chen, Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain, J. Differ. Equ., 266 (2019), 1245-1284. https://doi.org/10.1016/j.jde.2018.07.070 doi: 10.1016/j.jde.2018.07.070
[3]	X. R. Han, F. H. Yi, An irreversible investment problem with demand on a finite horizon: The optimal investment boundary analysis, Commun. Nonlinear Sci., 109 (2022), 106302. https://doi.org/10.1016/j.cnsns.2022.106302 doi: 10.1016/j.cnsns.2022.106302
[4]	X. R. Han, F. H. Yi, An irreversible investment problem with maintenance expenditure on a finite horizon: Free boundary analysis, J. Math. Anal. Appl., 440 (2016), 597-623. https://doi.org/10.1016/j.jmaa.2016.03.048 doi: 10.1016/j.jmaa.2016.03.048
[5]	T. Wu, Some results for a variation-inequality problem with fourth order p(x)-Kirchhoff operator arising from options on fresh agricultural products, AIMS Mathematics, 8 (2023), 6749-6762. https://doi.org/10.3934/math.2023343 doi: 10.3934/math.2023343
[6]	J. Li, C. C. Bi, Study of weak solutions of variational inequality systems with degenerate parabolic operators and quasilinear terms arising Americian option pricing problems, AIMS Mathematics, 7 (2022), 19758-19769. https://doi.org/10.3934/math.20221083 doi: 10.3934/math.20221083
[7]	N. Costea, Coupled systems of nonlinear variational inequalities and applications, Commun. Nonlinear Sci., 118 (2023), 107046. https://doi.org/10.1016/j.cnsns.2022.107046 doi: 10.1016/j.cnsns.2022.107046
[8]	Y. H. Wang, C. J. Zhang, Existence results of partial differential mixed variational inequalities without Lipschitz continuity, J. Math. Anal. Appl., 484 (2020), 123710. https://doi.org/10.1016/j.jmaa.2019.123710 doi: 10.1016/j.jmaa.2019.123710
[9]	F. Parise, A. Ozdaglar, A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis, Game. Econ. Behav., 114 (2019), 47-82. https://doi.org/10.1016/j.geb.2018.11.012 doi: 10.1016/j.geb.2018.11.012
[10]	Y. R. Bai, S. Migorski, S. D. Zeng, A class of generalized mixed variational Chemivariational inequalities I: Existence and uniqueness results, Comput. Math. Appl., 79 (2020), 2897-2911. https://doi.org/10.1016/j.camwa.2019.12.025 doi: 10.1016/j.camwa.2019.12.025
[11]	M. Boukrouche, D. A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, Nonlinear Anal.-Real, 12 (2011), 2211-2224. https://doi.org/10.1016/j.nonrwa.2011.01.003 doi: 10.1016/j.nonrwa.2011.01.003
[12]	S. Migorski, Y. R. Bai, S. D. Zeng, A new class of history-dependent quasi variational-hemivariational inequalities with constraints, Commun. Nonlinear Sci., 114 (2022), 106686. https://doi.org/10.1016/j.cnsns.2022.106686 doi: 10.1016/j.cnsns.2022.106686
[13]	J. L. Zheng, J. W. Chen, X. X. Ju, Fixed-time stability of projection neurodynamic network for solving pseudomonotone variational inequalities, Neurocomputing, 505 (2022), 402-412. https://doi.org/10.1016/j.neucom.2022.07.034 doi: 10.1016/j.neucom.2022.07.034
[14]	Y. J. Liu, S. Migrski, V. T. Nguyen, S. D. Zeng, Existence and convergence results for elastic frictional contact problem with nonmonotone subdifferential boundary condtions, Acta Math. Sci., 41 (2021), 1151-1168. https://doi.org/10.1007/s10473-021-0409-5 doi: 10.1007/s10473-021-0409-5
[15]	Y. J. Liu, Z. H. Liu, C. F. Wen, J. C. Yao, S. D. Zeng, Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems, Appl. Math. Optim., 84 (2021), 2037-2059. https://doi.org/10.1007/s00245-020-09703-1 doi: 10.1007/s00245-020-09703-1
[16]	Z. Q. Wu, J. X. Zhao, H. L. Li, J. N. Yin, Nonlinear Diffusion Equations, World Scientific Publishing, 2001. https://doi.org/10.1142/4782

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1371) PDF downloads(55) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems

Related Papers:

Abstract

1. Introduction

2. Related works

2.1. LoTFWA

2.1.1. Explosion

2.1.2. Guiding spark

2.1.3. Selection

2.1.4. LoT

2.1.5. Mapping rule

2.2. DE

2.3. ILoTFWA

3. Proposed guiding strategy

3.1. Original guiding strategy

3.2. Barycenter guiding strategy

3.3. DE guiding strategy

3.4. Triple-spark loser-out tournament-based fireworks algorithm

4. Experiments and analysis

4.1. Evaluate methodology

4.2. Parameter setting

4.3. Strategy validation

4.4. Performance comparison

4.5. Combining with ILoTFWA

4.6. Computational cost

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Global regularity and blowup for a class of non-Newtonian polytropic variation-inequality problem from investment-consumption problems

Related Papers:

Abstract

1. Introduction

2. Related works

2.1. LoTFWA

2.1.1. Explosion

2.1.2. Guiding spark

2.1.3. Selection

2.1.4. LoT

2.1.5. Mapping rule

2.2. DE

2.3. ILoTFWA

3. Proposed guiding strategy

3.1. Original guiding strategy

3.2. Barycenter guiding strategy

3.3. DE guiding strategy

3.4. Triple-spark loser-out tournament-based fireworks algorithm

4. Experiments and analysis

4.1. Evaluate methodology

4.2. Parameter setting

4.3. Strategy validation

4.4. Performance comparison

4.5. Combining with ILoTFWA

4.6. Computational cost

5. Conclusions

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog