A Gaussian convolutional optimization algorithm with tent chaotic mapping

Abstract

To solve the problems of the traditional convolution optimization algorithm (COA), which are its slow convergence speed and likelihood of falling into local optima, a Gaussian mutation convolution optimization algorithm based on tent chaotic mapping (TCOA) is proposed in this article. First, the tent chaotic strategy is employed for the initialization of individual positions to ensure a uniform distribution of the population across a feasible search space. Subsequently, a Gaussian convolution kernel is used for an extensive depth search within the search space to mitigate the likelihood of any individuals converging to a local optimum. The proposed approach is validated by simulation using 23 benchmark functions and six recent evolutionary algorithms. The simulation results show that the TCOA achieves superior results in low-dimensional optimization problems and solves practical, spring-related industrial design problems. This algorithm has important applications to solving optimization problems.

Introduction

With the continuous advancement of science and technology, optimization problems have evolved into increasingly difficult challenges, encompassing scenarios such as nondifferentiable objective functions and search spaces that are characterized by robust nonlinearity¹. Traditional deterministic approaches that rely on gradients and nongradients have proven not to be adept at effectively addressing optimization problems within this domain. These challenges are intrinsic to practical applications and have prompted researchers to explore intelligent evolutionary algorithms. Intelligent evolutionary algorithms iteratively build stronger individuals from the initial population through the evolutionary process of selection and through operations such as crossover and mutation, approaching the optimal solution gradually. Evolutionary algorithms are widely used to solve complex optimization problems in fields such as medicine², engineering^3,4, and computer science^5,6.

With the development of intelligent optimization algorithms in recent years, a series of new meta-heuristic algorithms have been developed that further improve the optimization of complex problems. Some examples include a template algorithm that simulates the proliferation of template cells⁷, a social behavior algorithm that modifies strawberry varieties, a template-molecule optimization algorithm that shows excellent global search capabilities in its attempt to find an optimal solution⁸, a template slime mold algorithm⁹, a moth search algorithm for moth flight patterns¹⁰, and an algorithm that simulates group predation behavior¹¹. The development of the moth evolutionary algorithm used for two-dimensional flight patterns¹² was inspired by applications required in areas such as computer vision and artificial intelligence at the cutting edge of image processing, where different rafts are used to extract different feature information from an original image. As initial individual positions in the basic raft evolution algorithm are generated via random initialization, the distribution of each initial individual position in the solution space is often unbalanced, which effect the efficiency of the algorithm. In addition, some gradient processes have adopted stochastic gradient structures, which elicit instability in the search space. Therefore, they do not solve complex optimization problems well. The optimization process of intelligent evolutionary algorithms is mainly represented by group initialization and group position update¹³. Therefore, this study improves the individual initialization method and group position update mechanism to solve these problems.

With advances in nonlinear dynamics, the chaos theory has successfully been integrated into various intelligent evolutionary algorithms and has produced excellent results. Some notable algorithms include particle swarm optimization (PSO)¹⁴, gray wolf optimization (GWO)¹⁵, grasshopper optimization algorithm¹⁶, whale optimization algorithm¹⁷, and firefly algorithm¹⁸. To further enhance the optimization performance of convolutional evolutionary algorithms, we introduce and design a new compound convolutional evolutionary algorithm that combines tent chaotic mapping and Gaussian mutation. This algorithm has important applications to solving optimization problems. The main contributions of this study are as follows:

1. Demonstration of tent chaotic mapping in population initialization to achieve enhanced population diversity.

2. Use of a Gaussian convolution kernel of size 3 × 3 to expand the search space and enhance the algorithm’s global search performance without changing original data.

3. Evaluation of the algorithm’s performance using 23 standard benchmark functions from the CEC-2005 test suite to assess the algorithm’s efficacy in solving optimization problems.

4. Comparison between the optimization results of the proposed algorithm (the Gaussian mutation convolutional optimization algorithm based on tent chaotic mapping (TCOA)) and those of a basic convolutional evolution algorithm and five other widely recognized intelligent evolution algorithms.

The subsequent sections of this paper are organized as follows: "Convolutional optimization algorithm" presents a basic convolutional evolution algorithm in detail. "TCOA" provides a detailed description of the improved convolutional evolution algorithm. "Simulation experiments and discussion" discusses simulation studies and discusses analysis results of the efficiency of the TCOA in handling real-world applications. Finally, conclusions are provided in "Conclusion and future scope".

Convolutional optimization algorithm

Convolutional process

Chen¹² proposed a convolutional evolution algorithm in 2023, which applies the idea of processing redundant image features using two-dimensional convolutional information for solving multiobjective optimization problems in nonlinear systems. The algorithm first randomly initializes the initial positions of each individual:

1

where is the initial position of each individual, is the lower limit of the individual, and is the upper limit of the individual.

Then, by updating the positions during the search process in four directions—vertical, horizontal, regional, and overall—the search process for the optimal individual is completed. The update strategy for the vertical, horizontal, and regional positions is

2

where is the current number of iterations; is an dimensional matrix, which is the position vector of the -generation population; represents the position vectors of the population after the t-generation vertical, horizontal, and regional convolutional position updates, respectively; is a vertical convolution kernel with dimensions of ; horizontal convolution kernel has dimensions of ; and the regional convolution kernel has dimensions of .

Further,

3

In the comprehensive position update stage, the individual positions of the population after the -generation vertical convolutional position update, position vector of the population after the -generation horizontal convolutional position update, and position vector of the population after the -generation regional convolutional position update are combined using random or proportional weights to form :

4

where , , and are all random numbers between and is taken in this study.

Location update strategy

Based on the convolution process in the first part, the position of each individual is obtained. Further, by comparing the fitness sizes of all individual positions in and , the optimal replacement of the individual position in is obtained

5

where is the position of the individual of the -generation population. is the position of the individual in the population updated by the -generation’s vertical, horizontal, regional, and comprehensive convolutional positions. represents the fitness function, which is the objective function.

TCOA

Chaotic system

A convolutional optimization algorithm works by generating the initial optimal solution through random initialization, and the position of a nonlinear system obtained is unevenly distributed in the solution space, which greatly influences the efficiency of the algorithm. Chaotic techniques are used to produce stochastic initial populations^19–21. Usually, this type of random initialization generates a different initial population each time, which is convenient to use. However, there are certain drawbacks, one of which is that the distribution of initial particles in the solution space is not uniform and particles in local areas are often too dense while the initial particles in other areas are too sparse. This situation is quite unfavorable for the early convergence of optimization algorithms. For group optimization algorithms that tend to fall into local optima, this situation may lead to a decrease in the convergence speed or even an inability to converge.

Chaos initialization can effectively prevent these problems. Chaos initialization introduces the characteristics of randomness, traversal, and regularity. In chaos initialization, the search space is traversed within a certain range according to its own laws and without repetition²². This generates an initial population that demonstrates high solution accuracy and convergence speed. The chaos initialization method employed in this study uses the tent chaos model, whose formula is as follows:

6

As shown in Fig. 1, tent mapping has good traversal performance, stable initialization results, and good distribution uniformity of the initial population. Therefore, this study used tent mapping to initialize the population of the TCOA to improve and enhance the distribution uniformity of the initial population in the search space and achieve enhanced global search ability.

Fig. 1.

Chaos map and random data distribution.

Gaussian convolutional kernel

As shown in Fig. 2, a standard two-dimensional convolutional algorithm was used during the optimization process. The selection of convolution kernels has a profound effect on the performance of an algorithm^23,24. The size of the shape parameters and their number as well as the number and dimensionality of convolutional kernels determine the range, accuracy, and flexibility of the convolution operation. A larger convolution kernel is able to capture a greater amount of contextual information, but it may also increase computational complexity. Smaller convolution kernels can be computed more quickly, but they may not capture enough contextual information. The original convolutional optimization algorithm randomly followed solutions in the search space to achieve individual position updating. Although this update type maintains species diversity during the early stages of iterations, the selection of convolutional kernels introduces a certain degree of blindness with respect to the algorithm’s position update method because it is not possible to determine whether the current obtained information is the current global optimal position.

Fig. 2.

2D convolution process.

Through experimentation, we discovered that when the evolutionary algorithms considered that the solution obtained after initialization had already met the optimization objectives of the given nonlinear system, the solution position was no longer updated, which made it easy for the solution to fall into local optima. To solve this problem, it was necessary to ensure that the randomly initialized solutions were no longer evenly distributed, i.e., the convolutional kernel at the algorithm update position had to gradually decrease in weight from the center to the outside so that the weight at the center was the maximum. Therefore, in this study, we enlarged the search space of the algorithm by applying Gaussian mutation to the current algorithm’s optimal solution position and dimension to reduce the probability that the optimal solution would fall into local optima. The specific steps undertaken in the algorithm were as follows:

1. The objective was to determine the optimal size of the convolution kernel. This size influenced the search space range and feature extraction capabilities of the convolutional evolution algorithm. Given the high number of pixels in the image, the aim was to obtain the optimal solution in the shortest possible time. Therefore, this article proposes a convolution kernel size of .

2. To set the standard deviation of the Gaussian function, it is necessary to convolve the weight values of each individual position within the kernel. The convolution kernel model in the convolution process of the improved evolutionary algorithm was determined as follows:

   7

   where represents the standard deviation (here, ). Indicates the height and width of the convolution kernel.

3. Normalizing the weights such that the total weight of the convolutional kernel was 1. The goal here was to avoid amplifying or attenuating the solution after multiplying the specified region of the original position by the convolution kernel. By employing normalization, the value of the original solution was preserved.

Basic process of the TCOA

Chaos initialization

In the convolutional evolution algorithm, a chaotic strategy was employed to initialize individual positions. This approach enhanced both the consistency and diversity of the initial positions, improving the overall efficacy of the initialization process. While chaotic systems display long-term behavior with stochastic characteristics, chaos is distinct from stochastic processes. Chaotic motion traverses every state within a defined region, known as a chaotic space, following an inherent regularity. In addition, each state is accessed only once, leading to a lack of exact periodicity. To ensure that tent mapping has values between (0, 1), the initial value was ultimately determined to be and.

Perform population initialization through the tent mapping shown in Eq. (6), and Eq. (2) is modified,

8

Position update stage in the TCOA

At this stage, the main approach is to use Gaussian convolution kernels for two-dimensional solution search. During vertical search, the size of the convolution kernel used was . When searching horizontally, the size of the convolution kernel used was . Finally, when searching for regions, the size of the convolution kernel used was . The position update strategy for each individual was as follows:

9

where the convolution kernels for vertical, horizontal, and regional position updates are

10

11

12

During comprehensive position update, the update strategy of the algorithm was consistent with Eq. (4) and .

Time complexity analysis of the TCOA

Time complexity is an important indicator of algorithm performance, and changes in the time complexity of the developed algorithm compared with those of traditional evolutionary algorithms was mainly affected by the initialization of chaotic maps and Gaussian mutation optimization. Given that the population size was , number of iterations performed to find the optimal solution was , and dimension of the test function was , the time complexity of the traditional evolutionary algorithm was . The improved evolutionary algorithm involved the following additional three steps:

Step 1: Initialize the population position using the chaotic mapping method with a time complexity of .

Step 2: Use a Gaussian convolution kernel and calculate the position of the updated population after mutation with a time complexity of .

The time complexity of the improved evolutionary algorithm was the sum of the time complexity of the traditional evolutionary algorithm and the aforementioned two steps, and it was at the same order as that of the traditional evolutionary algorithm.

Simulation experiments and discussion

Qualitative analysis of the TCOA

We applied the TCOA to several common single-peak and multipeak optimization problems²⁵, and the results of our qualitative analysis are shown in Fig. 3. The performance of the optimization algorithm was analyzed based on the convergence curve, which demonstrated some improvement. The TCOA exhibited excellent capability to scan a search space at the global and local levels and quickly obtained the optimal solution with high convergence speed after it determined the optimal region. This result highlights the efficiency of the algorithm during the search process.

Fig. 3.

Qualitative analysis of TCOA.

Comparison between the TCOA and recent excellent algorithms

In this subsection, we compare the performance of the proposed TCOA with six state-of-the-art algorithms and testing functions with CEC2005²⁶, including COA, WDO²⁷, SCA²⁸, PSO²⁹, TSA³⁰, and GWO³¹. The convergence curves for the algorithms are shown in Fig. 4. The simulation results show that compared with the state-of-the-art algorithms, this method undertakes greater exploration and development and shows superior balancing ability at low dimensions and superior performance. However, the TCOA exhibits competitive advantages in terms of convergence accuracy, stability, convergence speed, and optimization ability. At the same time, it increases computational complexity, which leads to longer running times. This problem remains a subject for future research and refinement. To analyze the ability of the TCOA to provide optimal solutions, four indicators were used to report the optimization results: mean, standard deviation (std), and execution time (et). Table 1 shows the average results for all the algorithms, which were run independently for 30 iterations. All of the simulation results for the competitive algorithms are shown in Table 1. The TCOA performed well in handling the functions F1–F3 and F6–F8.

Fig. 4.

Convergence curves of TCOA and some latest outstanding algorithms in the optimization.

Table 1.

Assessment results of the CEC-2005 objective functions.

		PSO	WDO	GWO	SCA	TSA	COA	TCOA
F1	Mean	1.092E−5	1.8989E−7	− 1.453.8	914.5	− 1220.9	1080	− 1245.3
	std	0.0813E−5	7.9635E−7	85.1	69.5	89	125.1	139.5
	et	0.3476	0.2785	0.5587	0.2818	0.3181	0.2911	0.2713
F2	Mean	3.3524E−09	4.8999E−09	6.8714E + 04	1.4219	4.2907E−10	2.3463E−04	1.2604E−09
	std	8.9524E−09	1.4504E−09	3.7516E + 05	1.2094	2.8865E−10	2.7615E−04	6.2267E−09
	et	0.4374	0.3579	1.1953	0.3192	0.3550	0.3245	0.2968
F3	Mean	13,717.2511	1.34808E + 5	21,780.45696	11,543.13317	2.334839E + 5	2.35825E + 5	11,981.85266
	std	6050.865871	7.29986E + 5	40,828.58155	5560.243664	3.790254E + 5	4.78335E + 5	29.68442534
	et	2.0432	2.0144	4.6870	0.4622	0.5234	0.4946	0.4654
F4	Mean	8.0897116	1.28466E−3	33.0815288	38.6005514	0.00011160	2.69739E−05	1.65744286
	st	2.6906986	3.55292E−3	4.32440373	9.99975900	4.93083E−05	3.63304E−05	0.12549598
	et	0.5395	0.4690	1.2128	0.1513	0.2081	0.1792	0.2126
F5	Mean	480.95096	260.275343	969.149842	183.668112	271.977330	286.989401	438.0.7194
	st	1826.1395	0.34797509	674.628392	42,896,164.2	0.97612791	0.02072033	1169.33601
	et	0.5927	0.5394	1.4768	0.1598	0.2188	0.1752	0.1592
F6	Mean	0.241965478	0.004157242	5.17606E−03	101.5708991	0.521417539	0.159340125	16.02263652
	st	0.0388619	0.00155955	1.68187E−06	113.558501	0.34818717	0.07081406	1.92188239
	et	0.4374	0.3479	1.1990	0.1161	0.1637	0.1266	0.0993
F7	Mean	0.043436088	0.0217946	0.792904606	0.01123324	0.003849996	0.000812167	0.029237835
	st	0.024631308	0.0120539	0.390101499	0.01890299	0.001992456	0.000494169	0.0742575199
	et	0.8512	0.7722	1.9567	0.2063	0.2463	0.2155	0.1979
F8	Mean	− 1768.446508	− 1818.85401	− 1340.89316	− 891.304266	− 1227.92941	− 1060.44438	− 1228.61117
	st	53.504232	51.9627300	76.4625132	59.5792059	103.410507	88.9108096	110.541180
	et	0.5348	0.6006	1.6590	0.1409	0.1848	0.1561	0.1277
F9	Mean	3.0553614	0	2.03895679	3.62484199	9.26475282	8.01864275	2.42051886
	st	2.2086755	0	13.0867769	282.756086	5.80317516	23.2840111	18.6597309
	et	0.3766	0.4822	1.2437	0.1313	0.1725	0.1357	0.1477
F10	Mean	20.30756	19.29627	20.2737568	20.3045734	20.8791403	0.19719213	15.3683478
	st	0.775647	0.195145	0.04304237	0.07701260	0.07722330	0.61012740	7.89237789
	et	0.5045	0.4151	1.3885	0.1888	0.1398	0.1214	0.1492
F11	Mean	0.07533234	0	0.0077571	0.76887935	0.00638905	0.03585102	0.57188237
	st	0.08180990	0	0.00276231	0.24548519	0.01042765	0.07349119	0.04400227
	ET	0.5093	0.6481	1.4596	0.1618	0.1431	0.1666	0.1983
F12	Mean	0.363752697	0.0263165	5.455883689	3,487,112.905	0.048703069	0.103382063	3.816253547
	st	0.202784123	0.000135964	1.857121107	7,028,295.419	0.02821944	0.17541855	3.270668825
	et	1.6020	1.8830	3.6962	0.3754	0.3913	0.4309	0.4225
F13	Mean	0.235434178	0.048864404	7.001251947	27,807,984	0.551308814	0.382420501	2.35608945
	st	0.142961862	0.071162774	4.198193549	102,209,520.3	0.233052388	0.791936424	0.309975409
	et	1.7181	1.7839	3.6261	0.4014	0.4982	0.4320	0.5130
F14	Mean	1.42187982	0.998102502	0.998003838	1.732562515	4.039801603	4.307357875	1.097406545
	st	2.132480213	0.00020755	0	0.967176981	3.692969303	3.309689187	0.303306063
	et	2.5982	2.7647	5.8311	1.5992	1.6122	1.6545	1.6380
F15	Mean	0.00125	0.0010	0.0010	0.0011	0.0011	0.0049	0.0011
	st	0.0315	4.1792E−05	1.69487E−06	0.0001	0.0004	0.0198	0.3779E−3
	et	0.3635	0.3808	0.6553	0.1077	0.1886	0.1977	0.1814
F16	Mean	− 1.006602115	− 1.031627734	− 1.031628453	− 1.031560237	− 1.031628442	− 1.031607497	− 1.03123321
	st	0.010333705	9.70135E−07	6.77522E−16	9.47659E−05	1.55172E−08	2.9992E−05	0.000379405
	et	0.2903	0.3116	0.4468	0.1360	0.1418	0.1810	0.1674
F17	Mean	1.27521144	0.39916485	0.39788735	0.41460115	0.39991663	0.39910152	0.39844936
	st	1.73729562	0.00133281	0	0.04089749	0.00785870	0.00198887	0.00065634
	et	0.3765	0.3893	0.6043	0.2510	0.2584	0.2224	0.2838
F18	Mean	3.0000806	3.000212916	3	3.000094169	3.000046757	3.001065	3.03859715
	st	0.000827284	0.00054386	1.24246E−7	0.000182734	4.73783E−05	0.001250499	0.04467167
	et	0.3765	0.3893	0.6043	0.2510	0.2584	0.2224	0.2838
F19	Mean	− 3.005316489	− 3.860355401	− 3.862782148	− 3.717073745	− 3.859371285	− 3.861907593	− 2.874893963
	st	1.507366174	0.003708541	2.71009E−15	0.70213413	0.005046061	0.001239905	1.019283808
	et	0.3617	0.3769	0.5974	0.1560	0.1626	0.2028	0.2903
F20	Mean	0	− 2.19272718	− 3.43539E−08	− 0.436391417	− 0.045053118	− 3.19611236	− 2.97307E−15
	st	0	1.191916563	1.88132E−07	1.045159845	0.246766091	0.133412908	1.62841E−14
	et	0.5511	0.7158	1.4392	0.3979	0.4756	0.4390	0.5002
F21	Mean	− 2.39958134	− 10.06123764	− 10.15319968	− 3.598961213	− 8.888367709	− 8.079687143	− 5.601825935
	st	2.01218978	0.064360038	7.92858E−09	1.448855778	2.368659126	2.966678245	2.287039183
	et	0.5579	0.5859	0.8462	0.3215	0.3456	0.1726	0.4264
F22	Mean	− 2.347426403	− 10.25454113	− 10.40294057	− 4.118420526	− 10.22638756	− 8.731282348	− 7.077563903
	st	1.661192709	0.079920142	1.36005E−15	1.027643842	0.96285627	2.77327844	2.388128202
	et	0.6475	0.6905	1.1951	0.3306	0.3645	0.4527	0.5500
F23	Mean	− 8.2073	− 10.405	− 10.536	− 3.5946	− 10.355	− 7.4675	− 6.8828
	st	1.6828	0.0924	5.71336E−16	1.3561	0.9872	3.5229	2.8255
	et	0.5968	0.5648	1.0993	0.3150	0.3205	0.3539	0.3416

Tension–compression spring design

As discussed in "Comparison between the TCOA and recent excellent algorithms", the TCOA has better function optimization performance compared to other intelligent algorithms and some improved algorithms. To further verify the effectiveness of the TCOA in a practical tension–compression spring, it was applied to address design problems. Tension–compression spring designing is an optimization problem in engineering sciences, with the aim of reducing the weight of the tension–compression spring, the schematic of which is reported in³¹. The mathematical formulation of this engineering design is as follows:

13

14

15

The results of the TCOA and its competitor algorithms in finding an optimal solution for the tension–compression spring design variables are listed in Table 2. The TCOA provided an optimal solution to the tension–compression spring problem, offering values of (0.0534, 0.3940, 9.7900) for the design variables and 0.0132 for the objective function. The statistical results obtained by the TCOA and its competitor algorithms for this design problem are presented in Table 3. These results indicate the superior performance of the TCOA, as reflected by better values of its statistical metrics. The TCOA convergence curve for solving the tension–compression spring design problem is presented in Fig. 5.

Table 2.

Performance of optimization algorithms on the tension/compression spring design problem.

Algorithms	Optimal variables			Optimum cost
	x1	x2	x3
TCOA	0.0534	0.3940	9.7900	0.0132
COA	0.0644	0.6966	4.5952	0.0191
TSA	0.0536	0.4021	9.0649	0.0127
PSO	0.0696	0.9541	2.000	0.0185
GWO	0.0500	0.3172	14.0675	0.0127

Table 3.

Statistical results of optimization algorithms on the tension/compression spring design problem.

Algorithms	Best	Worst	Mean	Std
TCOA	0.0129	0.0138	0.0133	0.00026
COA	0.0133	0.0327	0.0225	0.00566
TSA	0.0126	0.0131	0.0127	0.00096
PSO	0.0127	0.0185	0.0138	0.00150
GWO	0.0127	0.0173	0.0129	0.00027

Fig. 5.

TCOA’s performance convergence on the tension/compression spring.

Conclusion and future scope

To address the challenges of low convergence speed and long computation time encountered when using traditional two-dimensional convolutional evolution algorithms to solve complex nonlinear systems, this study introduces a new composite convolutional evolution algorithm. This algorithm utilizes the chaos theory and Gaussian mutation to overcome the limitations of traditional evolution algorithms and provides a balanced approach toward global exploration and local optimization while addressing common pitfalls such as local optima. However, the trade-off between enhanced performance and increased computational complexity suggests that further refinement is needed to fully realize the potential of the proposed algorithm. Future research should focus on improving the algorithm’s efficiency, scalability, and applicability to a wider range of optimization problems, ensuring that the TCOA can be effectively utilized in theoretical and practical settings.

Author contributions

Y.Q.: Conceptualization, methodology, writing-review & editing. A.J.: Funding acquisition, resources. Y.G.: Data curation.

Funding

This research was funded by “Pioneer” and “Leading Goose” R&D Program of Zhejiang (Grant Number 2023C01024).

Data availability

The datasets used and/or analyzed during this study are available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Ethical and informed consent for data used

This paper does not require ethical and informed consent for data.