A quasi-opposition learning and chaos local search based on walrus optimization for global optimization problems

Abstract

The Walrus Optimization (WO) algorithm, as an emerging metaheuristic algorithm, has shown excellent performance in problem-solving, however it still faces issues such as slow convergence and susceptibility to getting trapped in local optima. To this end, the study proposes a novel WO enhanced by quasi-oppositional-based learning and chaotic local search mechanisms, called QOCWO. The study aims to prevent premature convergence to local optima and enhance the diversity of the population by integrating the quasi-oppositional-based learning mechanism into the original Walrus Optimization (WO) algorithm, thereby improving the global search capability and expanding the search range. Additionally, the chaotic local search mechanism is introduced to accelerate the convergence speed of WO. To test the capabilities, the QOCWO algorithm is applied to the 23 standard functions and compared with seven other algorithms. Furthermore, the Wilcoxon rank-sum test is utilized to evaluate the significance of the results, which demonstrates the superior performance of the proposed algorithm. To assess the practicality in solving real-world problems, the QOCWO is applied to two engineering design issues, and the results indicated that QOCWO achieved lower costs compared to other algorithms.

Introduction

Optimization problems refer to the process of finding solutions that optimize certain metrics (such as cost, profit, efficiency, etc.) under given constraints. Optimization problems typically involve maximizing or minimizing an objective function by adjusting decision variables to find the optimal solution. These problems are widely applied in the fields of engineering, economics, management, and science¹. Nonetheless, these problems often appear in multidimensional forms, and the relationships between variables are complex. Traditional optimization algorithms based on mathematics cannot efficiently solve these problems. According to surveys, metaheuristic optimization algorithms have shown more effective and efficient performance in solving these problems, thus attracting the attention of many scholars².

Metaheuristic algorithms can be categorized into four main types based on their sources of inspiration: Swarm intelligence (SI), Evolutionary algorithms(EA), Physics-based algorithms (PhA), and Human-based algorithms. Among these, SI have garnered significant attention in recent years. SI algorithms are primarily inspired by the behavior of social organisms or natural phenomena, with well-known examples such as the Osprey Optimization Algorithm (OOA)³,Willow Catkin Optimization (WCO)⁴,Bedbug Meta-Heuristic Algorithm (BMHA)⁵,Nutcracker Optimizer (NO)⁶, Farmland Fertility Algorithm (FFA)⁷,Walrus Optimization (WO)⁸, Slime Mould algorithm (SMA)⁹, Sparrow Search Algorithm (SSA)¹⁰, Mountain Gazelle Optimizer (MGO)¹¹, Whale Optimization Algorithm (WOA)¹², and Pelican Optimization Algorithm (POA)¹³,Grey Wolf Optimizer (GWO)¹⁴, Moth Flame Optimization(MFO)¹⁵, and Marine Predator Algorithm (MPA)¹⁶. They have demonstrated strong performance in solving complex optimization problems by effectively balancing exploration and exploitation.However, despite their advantages, SI algorithms often face challenges related to local optima convergence and computational efficiency.

WO is a novel metaheuristic algorithm proposed by Han et al. in 2024⁸, which simulates the foraging and migration behaviors of walruses. The primary reason for choosing WO over other algorithms like CGWO¹⁷, MFO, SCA¹⁸, PSO¹⁹, and WOA lies in its unique ability to balance exploration and exploitation efficiently. Unlike PSO and WOA, which are prone to premature convergence, WO exhibits a more robust global search ability due to its walrus-inspired behaviors, making it more effective in escaping local optima. Compared to SCA and MFO, WO’s structure is simpler with fewer parameters, reducing computational complexity and making it easier to implement in large-scale problems. Additionally, WO outperforms CGWO in terms of flexibility and computational cost, providing a better trade-off between search capability and efficiency.WO demonstrates unique stability and strong competitiveness when solving high-dimensional problems, outperforming Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and White Shark Optimization (WSO) in the evaluation of benchmark functions²⁰.To date, the WO algorithm has been applied to solve various real-world problems, such as optimizing renewable energy resource allocation in power distribution systems to enhance reliability and security²¹, improving the dynamic performance of wave energy conversion systems²⁰, and precisely modeling lithium-ion batteries in industrial applications to optimize their dynamic parameters²², demonstrating its versatility and strong performance in engineering problems.

However, similar to many metaheuristic algorithms, WO may fall into local optimal solutions and cannot guarantee finding the global optimal solutions. When addressing complex problems, it may inevitably encounter slow convergence issues.To address the aforementioned issues, we propose the QOCWO algorithm. In the proposed QOCWO algorithm, the structure is designed to address two key challenges: avoiding local optima and improving convergence speed. To prevent the algorithm from getting trapped in local optima, we introduce the Quasi-Oppositional Based Learning (QOBL) mechanism. This mechanism computes a quasi-oppositional solution for the current one and compares the two, enhancing the diversity of the search process. To accelerate the convergence speed, we incorporate the Chaotic Local Search (CLS) mechanism, which exploits the ergodicity and randomness of chaotic systems, allowing the algorithm to explore the search space more effectively and accelerate the convergence speed. By alternately applying QOBL and CLS during the iteration process, the QOCWO algorithm efficiently combines the advantages of both mechanisms to improve overall performance.

To validate the efficacy of the Quantum-Optimized Cuckoo Search Algorithm (QOCWO), it was applied to 23 benchmark functions, including unimodal and multimodal functions, and compared with seven other algorithms: Gradient-Based Optimizer (GBO), Competing Leaders Grey Wolf Optimizer (CGWO), Slime Mould Algorithm (SMA), Moth Flame Optimization (MFO), Whale Optimization Algorithm (WOA), Walrus Optimization (WO), and Sine Cosine Algorithm (SCA). Furthermore, it was implemented in two engineering design problems to demonstrate its improvement over the Walrus Optimization (WO). Consequently, our contributions are as follows:

1. Proposed a novel QOCWO algorithm, which integrates Quasi-Oppositional Based Learning (QOBL) and Chaotic Local Search (CLS) into the Walrus Optimization (WO). This combination enhances search efficiency, prevents entrapment in local optima, and accelerates convergence, addressing the limitations of the original WO algorithm in terms of search scope, solution diversity, and convergence speed.

2. Demonstrated the superior performance of QOCWO on 23 benchmark functions and 2 engineering problems, showing faster and more accurate optimization compared to traditional algorithms.

The structure of this article is as follows: Section “Original WO algorithm” introduces the concept of the original WO algorithm, Section “QOCWO” describes the mechanisms and pseudocode of QOCWO, Section “Experimental results and analysis” evaluates and analyzes the algorithm through extensive experiments, and Section “Conclusion” draws a conclusion.

Original WO algorithm

The Walrus Optimization (WO) algorithm draws inspiration from the migration, roosting, fleeing, and gathering behaviors of walruses, which can more effectively address both constrained and unconstrained problems. These behaviors are abstracted into mathematical models within the algorithm to guide its search process.These behavioral characteristics are harnessed to design search strategies within the algorithm, aiding it in effectively exploring and exploiting the solution space to identify optimal or feasible solutions. In the WO algorithm, candidate solutions for optimization problems are generated by simulating the social structure and interactions of walrus populations, which includes the role differentiation among male, female, and juvenile walruses. Adult walruses comprise 90% of the population, with juveniles making up the remaining 10%, and males and females each constituting 50% of the adult population. The algorithm also introduces the concepts of “danger signals” and “safety signals” to control search behavior, mirroring the walruses’ natural behavioral patterns in their environment.

Danger signals and safety signals

When walruses are looking for food or a place to rest, one or two individuals usually patrol the area. If an emergency occurs, they immediately send out a danger signal. In the WO algorithm, the danger and safety signals are represented by the following equations:

1

2

3

4

5

where, A and R are danger factors, decreases from 1 to 0 as the number of iterations t increases, T is the maximum iteration, and r1 and r2 are random numbers within the range (0, 1).

Migration

When the danger factors are too high, the walrus herd migrates to other areas that are better suited to the survival of the population. The positions of the walrus herd are then updated accordingly, with the update formula being as follows:

6

7

8

where, is the new position of the -th walrus, is the current position of the -th walrus, the Migration step is the step size of walrus movement, and the positions of two randomly selected patrolling walruses from the herd, denoted as and . The migration step is controlled by a factor , which varies according to a smooth iterative curve, and is a random number between 0 and 1.

Roosting

Juvenile, male, and female walruses represent the classifications within a walrus herd, each with their own methods for updating their positions. The position of the male walrus, denoted as , is updated using the Halton sequence.

The position of the female walrus, denoted as , is influenced by the male walrus and the leading walrus, represented by . However, during the iteration process, the influence of the male walrus on the female walrus gradually decreases, while the influence of the leading walrus gradually increases.

9

where, is the new position of the -th female walrus, and are the current position of the -th male walrus and female walrus, respectively.

Juvenile walruses located at the edge of the population need to move and update their positions to avoid predation.

10

11

12

where, is the new position of the -th juvenile walrus, ) is determined based on its current position, a danger coefficient which is a random number within the range (0,1), a reference safe position , and a random vector based on the Lévy distribution. Here, and are two normally distributed variables, with .

13

where, and are the standard deviations, .

Fleeting

Fleeting behavior occurs when walruses are inevitably attacked by predators while searching for food underwater. In response to danger signals sent out by patrolling walruses, the foraging walruses flee the area.

14

where, represents the distance between the current walrus and the best walrus position, is a random number within the range (0,1).

Gathering

Gathering behavior is when walruses cooperate in foraging and moving based on the positions of other walruses in the population, in order to locate areas of the sea that are richer in food resources.

15

16

17

18

where, and are two weights that influence the walrus’s gathering behavior, is the position of the second walrus in the current iteration, represents the distance between the current walrus and the second walrus, a and b are gathering coefficients, and are random numbers within the range (0,1), and the value range of is from 0 to . Figure 1 is flowchart of WO.

Fig. 1.

Flow chart of WO.

QOCWO

Quasi-oppositional based learning

Tizhoosh developed a new concept in machine intelligence called Opposition-Based Learning (OBL), which is used to accelerate the convergence speed of various optimization techniques²³. The OBL considers both the current population and its opposite population to generate better candidate solutions, making it applicable to any population-based optimization technique to expand the search area and avoid being trapped in local optima. Building on the OBL method²³, Rahnamayan introduced a variant known as “Quasi-Oppositional Based Learning (QOBL)”²⁴ which has been shown to be faster and more effective in finding the global optimum.

Algorithm 1.

The pseudo-code of QOBL.

Chaotic local search (CLS)

Chaotic Local Search (CLS) is a search strategy used in optimization algorithms that makes use of certain properties of chaos theory to improve the search capability of the algorithm. Chaos theory is concerned with the sensitivity of dynamic systems to slight changes in initial conditions and the unpredictability and ergodicity of system behavior²⁵. During the search process, CLS introduces chaos variables that can effectively perturb the equilibrium state of the algorithm, increase the diversity of the population, and thus avoid falling into local optimal solutions. In addition, the randomness and ergodicity of chaotic search help the algorithm to explore the entire solution space more broadly.

The basic idea of CLS is to utilize the ergodicity and randomness of chaotic variables to perform a secondary search in the neighborhood of the current best agent, with the expectation of enhancing the convergence speed and solution quality of WO (Walrus Optimization). The definition of CLS is as follows:

19

where, represents the position of the current best agent at the nth iteration. is the newly generated agent, and denotes the chaotic variable produced by a selected chaotic system. U and L represent the upper and lower bounds of the search space, respectively. The search radius, defined as , initially set to 0.01, is then updated as follows:

20

If the fitness of the newly generated agent is higher than that of , then the newly generated agent will replace the current agent. Chaotic mappings can exhibit certain chaotic behaviors, with common chaotic mappings including the Logistic map, Tent map, Sine map, Chebyshev map, Circle map, Cubic map, and ICMIC map²⁶. The simple and effective one-dimensional chaotic map is adopted, the Logistic map, which is expressed as follows:

21

22

where, is the chaotic number at the nth iteration. Typically, CLS uses a decreasing search radius to reduce its search space. The main drawback of this CLS implementation is that the user-defined search radius is difficult to match with the search speed of WO. A search radius that is too large may cause to fall outside the search space of the current population, making it difficult to accelerate the convergence speed of the WO algorithm and improve the quality of the solution. Therefore, a modified CLS mechanism is used²⁷, the expression of which is shown as follows:

23

where, and are two individuals randomly selected from the current population.

The pseudo-code of the QOCWO algorithm is as follows.

Algorithm 2.

The pseudo-code of QOCWO.

Experimental results and analysis

To prove the advantages of QOCWO, this section will explain a series of experiments conducted on the QOCWO algorithm. In this paper, seven algorithms were selected for comparison, including Gradient-Based Optimizer (GBO), Competing Leaders Grey Wolf Optimizer (CGWO), Slime Mould Algorithm (SMA), Moth Flame Optimization (MFO), Whale Optimization Algorithm (WOA), Walrus Optimization (WO), and Sine Cosine Algorithm (SCA). During the experimental process, we set the population size to 30 and the maximum number of iterations to 1000, running each algorithm independently 50 times for each benchmark function. All experiments were performed using MATLAB R2022a, on a computer with an Intel(R) Core(TM) i5-1035G1 CPU @ 1.19 GHz–0.99 GHz and 8 GB RAM. In addition, The specific parameteison, including Genroposed algorithm and the comparative algorithms are shown in Table 1.

Table 1.

The parameters of the algorithms.

Algorithm	Parameter setting
SCA
CGWO
GBO
MFO
WOA
WO
SMA
QOCWO

Benchmark function

Unimodal functions typically have only one global optimum with a curve that has a single peak structure, while multimodal functions have multiple local optima and their curves may include multiple peaks and valleys. Therefore, testing algorithms with unimodal functions focus more on the accuracy and efficiency of global search, while testing with multimodal functions requires considering local search and avoiding trapping in local optima. Therefore, we use 23 benchmark functions with different characteristics to evaluate the properties of the algorithms: unimodal functions (F1-F7), multimodal functions (F8-13), and fixed-dimension multimodal functions (F14-F23). Table 2 provides detailed information on these 23 benchmark functions, and Fig. 2 shows the 3D diagrams of the 23 benchmark functions.

Table 2.

Benchmark function.

Function	Description	Dim	Range	fmin
Sphere		d	[− 100,100]	0
Schwefel 2.22		d	[− 10,10]	0
Schwefel 1.2		d	[− 100,100]	0
Schwefel 2.21		d	[− 100,100]	0
Rosenbrock		d	[− 30,30]	0
Step		d	[− 100,100]	0
Quartic		d	[− 1.28,1.28]	0
Schwefel 2.26		d	[− 500,500]	− 418.9829*d
Rastrigin		d	[− 5.12,5.12]	0
Ackley		d	[− 32,32]	0
Griewank		d	[− 600,600]	0
Penalized		d	[− 50,50]	0
		d	[− 50,50]	0
Foxholes		2	[− 65.536,65.536]	0.998
Kowalik		4	[-5,5]	0.0003075
Camel-Back		2	[-5,5]	-1.0316
Branin		2	[-5,10]	0.398
Goldstein-price		2	[-2,2]	3
Hartman’s family		3	[0,1]	-3.86
Shekel’s family		6	[0,1]	-3.32
		4	[0,10]	-10.1532
		4	[0,10]	-10.4028
		4	[0,10]	-10.5363

Fig. 2.

The 3D graphs of benchmark functions.

where,

Numerical analysis

The study selects the Mean and standard deviation (Std) as evaluation metrics, with the function dimension set to 30, and the specific results are shown in Table 3. (The best results for Mean and Std are highlighted in bold within the table.)

Table 3.

Comparison of results on 23 Benchmark functions among different algorithms.

Function		QOCWO	GBO	CGWO	WO	SMA	MFO	SCA	WOA
F1	Mean	0.00E + 00	1.18E-239	8.87E-59	4.12E-288	0.00E + 00	2.60E + 03	4.34E-02	2.58E-149
	Std	0.00E + 00	0.00E + 00	2.09E-58	0.00E + 00	0.00E + 00	4.87E + 03	1.80E-01	1.07E-148
F2	Mean	2.04E-233	3.43E-122	9.17E-35	2.07E-145	3.17E-208	3.28E + 01	4.53E-05	1.18E-104
	Std	0.00E + 00	0.00E + 00	1.05E-34	1.45E-144	0.00E + 00	1.98E + 01	1.42E-04	4.01E-104
F3	Mean	0.00E + 00	1.59E-203	4.12E-14	8.78E-276	0.00E + 00	185E + 04	3.07E + 03	1.67E + 04
	Std	0.00E + 00	0.00E + 00	1.83E-13	0.00E + 00	0.00E + 00	1.32E + 04	2.71E + 03	1.00E + 04
F4	Mean	1.75E-227	2.72E-111	1.89E-14	8.01E-143	3.23E-181	6.51E + 01	2.02E + 01	4.28E + 01
	Std	0.00E + 00	0.00E + 00	2.05E-14	4.47E-142	0.00E + 00	9.56E + 00	9.65E + 00	2.68E + 01
F5	Mean	2.40E-03	2.08E + 01	2.69E + 01	2.00E-02	1.43E + 00	6.43E + 06	4.36E + 02	2.72E + 01
	Std	6.50E-03	2.19E-04	8.07E-01	6.00E-02	5.24E + 00	2.19E + 07	1.61E + 03	5.40E-01
F6	Mean	7.30E-05	3.26E-09	5.99E-01	2.47E-04	9.67E-04	2.59E + 03	4.76E + 00	7.00E-02
	Std	1.38E-04	1.00E-11	3.34E-01	3.89E-04	5.03E-04	5.26E + 03	8.35E-01	1.10E-01
F7	Mean	5.96E-05	3.59E-04	8.81E-04	1.88E-04	1.06E-04	4.29E + 00	3.02E-02	1.62E-03
	Std	4.93E-05	4.91E-05	4.59E-04	1.56E-04	9.31E-05	8.91E + 00	3.02E-02	1.70E-03
F8	Mean	-1.26E + 04	-8.87E + 03	-6.11E-03	-1.26E + 04	-1.26E + 04	-8.38E + 03	-3.92E + 03	-1.13E + 04
	Std	1.00E-01	2.11E-09	7.84E-02	3.00E-01	8.42E-02	6.93E + 02	2.82E + 02	1.54E + 03
F9	Mean	0.00E + 00	0.00E + 00	3.84E-01	0.00E + 00	0.00E + 00	1.68E + 02	1.84E + 01	0.00E + 00
	Std	0.00E + 00	0.00E + 00	1.40E + 00	0.00E + 00	0.00E + 00	3.02E + 01	3.07E + 01	0.00E + 00
F10	Mean	4.44E-16	4.44E-16	1.55E-14	4.44E-16	4.44E-16	1.39E + 01	1.42E + 01	3.71E-15
	Std	0.00E + 00	0.00E + 00	2.44E-15	0.00E + 00	0.00E + 00	7.96E + 00	9.07E + 00	2.36E-15
F11	Mean	0.00E + 00	0.00E + 00	1.88E-03	0.00E + 00	0.00E + 00	1.63E + 01	2.89E-01	2.19E-03
	Std	0.00E + 00	0.00E + 00	5.14E-03	0.00E + 00	0.00E + 00	3.50E + 01	2.96E-01	1.00E-02
F12	Mean	9.77E-07	3.80E-10	3.90E-02	2.54E-06	1.25E-03	5.12E + 06	5.63E + 01	8.13E-03
	Std	1.51E-06	2.42E-12	1.79E-02	4.29E-06	2.37E-03	3.62E + 07	3.79E + 02	7.51E-03
F13	Mean	1.06E-05	2.54E-02	5.89E-01	2.02E-05	1.18E-03	4.10E-01	3.99E + 04	2.70E-01
	Std	1.92E-05	3.90E-03	2.09E-01	3.06E-05	1.69E-03	8.2E-01	2.76E + 05	1.8E-00
F14	Mean	9.98E-01	1.02E + 00	4.00E + 00	1.00E + 00	9.98E-01	2.12E + 00	1.43E + 00	2.65E + 00
	Std	2.55E-16	3.17E-17	3.84E + 00	2.07E-16	1.18E-13	1.85E + 00	8.30E-01	3.14E + 00
F15	Mean	3.11E-04	4.51E-04	3.55E-03	3.19E-04	4.58E-04	1.90E-03	8.76E-04	7.20E-04
	Std	6.36E-06	3.01E-04	7.42E-03	2.28E-05	1.86E-04	3.21E-03	3.81E-04	4.20E-04
F16	Mean	-1.03E + 00	-1.03E + 00	1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00	-1.03E + 00
	Std	1.10E-14	2.62E-16	5.74E-09	8.34E-04	2.62E-16	2.24E-16	3.28E-05	8.70E-11
F17	Mean	3.98E-01	3.98E-01	3.98E-01	3.98E01	3.98E-01	3.98E-01	3.98E-01	4.46E-01
	Std	1.46E-14	0.00E + 00	2.69E + 00	6.01E-05	1.06E-08	0.00E + 00	1.79E-03	3.38E-01
F18	Mean	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00	3.00E + 00
	Std	7.11E-14	6.65E-16	7.06E-06	8.81E-14	6.65E-16	1.75E-15	2.51E-05	2.15E-05
F19	Mean	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00	-3.86E + 00
	Std	4.03E-13	1.26E-15	2.26E-03	1.42E-11	2.89E-08	1.35E-15	2.48E-03	1.66E-02
F20	Mean	-3.29E + 00	-3.25E + 00	-3.28E + 00	-3.25E + 00	-3.24E + 00	-3.23E + 00	-2.89E + 00	-3.24E + 00
	Std	5.50E-02	5.99E-02	6.83E-02	5.92E-02	5.61E-02	6.27E-02	3.45E-01	1.00E-01
F21	Mean	-1.02E + 01	-8.22E + 00	-9.40E + 00	-1.02E + 01	-1.02E + 01	-6.09E + 00	-2.32E + 00	-8.83E + 00
	Std	1.31E-10	2.73E-15	1.91E + 00	8.81E-10	1.45E-04	3.35E + 00	1.95E + 00	2.26E + 00
F22	Mean	-1.04E + 01	-8.38E + 00	-1.03E + 01	-1.04E + 01	-1.04E + 01	-7.55E + 00	-4.14E + 00	-8.51E + 00
	Std	1.19E-10	1.54E-15	7.52E + 01	1.06E-10	8.79E-05	3.59E + 00	2.00E + 00	3.01E + 00
F23	Mean	-1.05E + 01	-8.70E + 00	-1.05E + 01	-1.05E + 01	-1.05E + 01	-8.28E + 00	-4.59E + 00	-8.63E + 00
	Std	1.67E-10	1.66E-15	2.99E-04	4.14E-10	7.48E-05	3.39E + 00	1.47E + 00	2.83E + 00

Significant values are bold.

Table 3 clearly shows that QOCWO outperforms the other algorithms on unimodal functions (F1-F5 and F7), especially on the first four function tests (F1-F4). The QOCWO accurately identifies the optimal solution, a feat that other algorithms fail to accomplish, highlighting its strong exploration capabilities.

In the six multimodal functions (F8-F13), the QOCWO is also very competitive in terms of performance. The QOCWO achieved the best performance on F9-F11 and F13-F14. Comparing the QOCWO with the original WO, it can be concluded that the introduction of the quasi-oppositional aspect allows to effectively bypass local optima.

For the multimodal functions with fixed dimensions (F14-F23), the QOCWO performs well and achieves near-perfect optimal solutions. The QOCWO achieved the best mean results on all fixed-dimension multimodal functions.

From the overall ranking results, the QOCWO performed significantly better in the comparison tests, demonstrating its strong capabilities in exploration and exploitation, thus confirming the effectiveness of the quasi-oppositional-based learning and chaotic local search mechanisms.

Convergence analysis

In this experiment, the dimensionality of the functions was set to 30. The specific average convergence curves are shown in Fig. 3. It can be observed that the QOCWO algorithm exhibits the fastest convergence rate and the highest accuracy on unimodal functions (F1-F7). Among these seven benchmark functions, QOCWO converges to the optimal solution much earlier, while other algorithms either converge slowly or fail to converge to the global optimum.

Fig. 3.

The convergence curves of the average fitness values for the 23 functions.

On multimodal functions (F8-F13), the QOCWO algorithm continues to demonstrate a relatively fast convergence rate across all functions. Compared with other algorithms, the QOCWO is capable of locating the global optimum more swiftly and does not fall into local optima.For instance, in F11, SCA and MFO frequently falls into local optima during the solution process, whereas QOCWO does not encounter such issues in its pursuit of the optimal solution. Within the fixed-dimensional multimodal functions (F14-F23), the convergence speed of the QOCWO algorithm remains relatively fast among all algorithms, and it also exhibits commendable performance in terms of convergence speed, especially on the curve functions (F15, F17, F21, F22, and F23), where it achieves nearly perfect optimization results.

In summary, the QOCWO algorithm demonstrated a remarkably swift convergence rate in the tests across 23 benchmark functions, and it perfectly managed to avoid being trapped in local optima.

Stability analysis

In this experiment, boxplots were utilized to illustrate the distribution of target values from 50 independent runs for each sample to assess the stability of QOCWO. Boxplots were drawn for unimodal, multimodal, and fixed-dimension multimodal functions, with the specific results shown in Fig. 4. The information indicates that the QOCWO has better ompared to the other algorithms for all types of functions and thus shows better performance.

Fig. 4.

Box plots of 23 functions.

High-dimensional performance analysis

Since the traditional WO may fall into local optima in high-dimensional optimization problems, the study is devoted to evaluating the performance of an improved algorithm, i.e., QOCWO, in handling high-dimensional problems. To this end, a series of experiments were designed to compare the QOCWO with seven other advanced algorithms. The experiments include benchmark functions with dimensions of 50 and 100. The specific experimental results are summarized in Tables 4 and 5, which detail the performance comparison of QOCWO under different dimensions.

Table 4.

Results and comparison of different algorithms on 13 benchmark functions with 50D.

Function		QOCWO	GBO	CGWO	WO	SMA	MFO	SCA	WOA
F1	Mean	0.00E + 00	1.50E-267	2.56E-70	6.05E-287	0.00E + 00	8.85E + 03	4.01E-03	1.06E-145
	Std	0.00E + 00	0.00E + 00	8.32E-70	0.00E + 00	0.00E + 00	9.60E + 03	1.60E-02	7.47E-145
F2	Mean	8.89E-230	6.85E-136	5.84E-41	3.32E-143	5.26E-203	6.88E + 01	5.04E-06	3.43E-103
	Std	0.00E + 00	0.00E + 00	8.47E-41	2.32E-142	0.00E + 00	3.17E + 01	9.01E-06	1.57E-102
F3	Mean	0.00E + 00	6.66E-218	7.91E-19	5.27E-267	0.00E + 00	4.94E + 04	2.16E + 03	1.25E + 05
	Std	0.00E + 00	0.00E + 00	4.77E-18	0.00E + 00	0.00E + 00	2.53E + 04	2.57E + 03	3.22E + 04
F4	Mean	1.88E-217	4.44E-123	1.43E-17	4.26E-141	1.53E-227	8.44E + 01	1.21E + 01	6.70E + 01
	Std	0.00E + 00	0.00E + 00	2.67E-17	2.81E-140	0.00E + 00	4.99E + 00	7.80E + 00	2.35E + 01
F5	Mean	7.93E-03	1.70E + 01	2.68E + 01	2.91E-02	1.97E-01	6.38E + 06	2.05E + 02	4.76E + 01
	Std	2.51E-02	3.62E-05	8.21E-01	5.38E-02	1.47E-01	2.17E + 07	6.69E + 02	5.32E-01
F6	Mean	5.91E-04	1.46E-14	3.08E-01	1.36E-03	1.46E-14	6.63E + 03	4.26E + 00	4.31E-01
	Std	1.70E-03	3.69E-14	2.60E-01	2.56E-03	3.30E-04	8.95E + 03	5.21E-01	2.41E-01
F7	Mean	1.21E-05	2.80E-04	4.95E-04	2.17E-04	6.18E-05	2.32E + 01	2.49E-02	2.04E-03
	Std	1.77E-05	3.84E-05	2.82E-04	2.26E-04	6.07E-05	2.48E + 01	1.93E-02	1.97E-03
F8	Mean	-2.09E + 04	-9.40E + 03	-6.33E + 03	-2.09E + 04	-1.26E + 04	-1.37E + 04	-4.03E + 03	-1.91E + 04
	Std	2.47E + 00	3.61E-10	9.41E + 02	3.66E + 00	3.26E-02	1.46E + 03	2.90E + 02	2.54E + 03
F9	Mean	0.00E + 00	0.00E + 00	1.19E-01	0.00E + 00	0.00E + 00	3.16E + 02	1.01E + 01	0.00E + 00
	Std	0.00E + 00	0.00E + 00	8.42E-01	0.00E + 00	0.00E + 00	5.41E + 01	1.66E + 01	0.00E + 00
F10	Mean	4.44E-16	4.44E-16	1.27E-14	4.44E-16	4.44E-16	1.96E + 01	1.28E + 01	3.64E-15
	Std	0.00E + 00	0.00E + 00	2.89E-15	0.00E + 00	0.00E + 00	4.62E-01	9.43E + 00	2.18E-15
F11	Mean	0.00E + 00	0.00E + 00	1.62E-03	0.00E + 00	0.00E + 00	1.63E + 01	1.69E-01	2.19E-03
	Std	0.00E + 00	0.00E + 00	6.92E-03	0.00E + 00	0.00E + 00	3.50E + 01	2.21E-01	1.13E-02
F12	Mean	1.15E-06	1.33E-15	2.90E-02	4.50E-06	4.75E-04	2.56E + 07	9.45E + 01	1.12E-02
	Std	1.87E-06	1.68E-14	1.31E-02	6.85E-06	4.75E-04	7.76E + 07	6.63E + 02	5.82E-03
F13	Mean	3.73E-05	1.20E-02	3.08E-01	1.44E-04	4.01E-04	4.11E + 07	3.40E + 01	5.91E-01
	Std	7.92E-05	4.61E-03	1.72E-01	3.85E-04	4.01E-04	1.49E + 08	1.78E + 02	3.12E-01

Significant values are bold.

Table 5.

Results and comparison of different algorithms on 13 benchmark functions with 100D.

Function		QOCWO	GBO	CGWO	WO	SMA	MFO	SCA	WOA
F1	Mean	0.00E + 00	6.67E-289	3.17E-85	5.13E-261	0.00E + 00	3.36E + 04	8.78E-05	6.43E-147
	Std	0.00E + 00	0.00E + 00	5.75E-85	0.00E + 00	0.00E + 00	1.42E + 04	2.68E-04	3.93E-146
F2	Mean	1.78E-219	1.21E-147	2.73E-49	1.81E-138	3.70E-240	1.70E + 02	9.74E-07	2.38E-103
	Std	0.00E + 00	0.00E + 00	2.94E-49	1.25E-137	0.00E + 00	4.75E + 01	1.98E-06	7.72E-103
F3	Mean	0.00E + 00	9.68E-238	5.10E-26	1.67E-247	0.00E + 00	2.04E + 05	1.21E + 03	8.60E + 05
	Std	0.00E + 00	0.00E + 00	2.33E-25	0.00E + 00	0.00E + 00	5.68E + 04	1.54E + 03	1.85E + 05
F4	Mean	8.40E-219	3.81E-134	1.17E-21	1.63E-134	2.68E-233	9.34E + 01	7.92E + 00	7.31E + 01
	Std	0.00E + 00	0.00E + 00	1.45E-21	1.15E-133	0.00E + 00	2.06E + 00	5.91E + 00	2.65E + 01
F5	Mean	2.88E-02	8.81E + 00	2.60E + 01	6.39E-02	9.57E-02	7.66E + 07	3.06E + 01	9.78E + 01
	Std	8.29E-02	2.88E-06	6.43E-01	1.79E-01	8.53E-02	6.32E + 7	7.27E + 00	3.65E-01
F6	Mean	5.40E-07	3.09E-25	1.80E-01	9.49E-03	3.31E-04	3.06E + 04	1.78E-04	2.07E + 00
	Std	1.33E-06	1.10E-17	2.14E-01	1.22E-02	1.78E-04	1.49E + 04	3.52E-01	7.25E-01
F7	Mean	6.73E-05	1.36E-04	2.78E-04	2.82E-04	4.08E-05	1.67E + 02	9.39E-03	2.13E-03
	Std	7.38E-05	1.44E-05	1.51E-04	1.90E-04	4.17E-05	1.16E + 02	9.19E-03	2.40E-03
F8	Mean	-4.19E + 04	-9.38E + 03	-6.56E + 03	-4.19E + 04	-1.26E + 04	-2.44E + 04	-4.19E + 03	-3.66E + 04
	Std	8.64E + 00	2.88E-10	8.29E + 02	2.85E + 01	1.22E-02	2.52E + 03	2.51E + 02	5.61E + 03
F9	Mean	0.00E + 00	0.00E + 00	1.11E-01	0.00E + 00	0.00E + 00	7.56E + 02	5.35E + 00	4.55E-15
	Std	0.00E + 00	0.00E + 00	6.54E-01	0.00E + 00	0.00E + 00	6.45E + 01	1.00E + 01	3.22E-14
F10	Mean	4.44E-16	4.44E-16	9.89E-15	4.44E-16	4.44E-16	1.98E + 01	1.23E + 01	3.64E-15
	Std	0.00E + 00	0.00E + 00	2.84E-15	0.00E + 00	0.00E + 00	2.60E-01	9.30E + 00	2.51E-15
F11	Mean	0.00E + 00	0.00E + 00	1.52E-03	0.00E + 00	0.00E + 00	2.89E + 02	1.10E-01	2.35E-03
	Std	0.00E + 00	0.00E + 00	4.99E-03	0.00E + 00	0.00E + 00	1.09E + 02	1.53E-01	1.66E-02
F12	Mean	3.67E-06	1.98E-25	1.46E-02	1.61E-05	8.87E-05	1.02E + 08	5.33E-01	1.80E-02
	Std	1.01E-05	6.10E-18	9.37E-03	3.32E-05	1.24E-04	1.29E + 08	2.52E-01	7.35E-03
F13	Mean	1.92E-04	7.01E-03	1.54E-01	3.20E-04	1.76E-04	2.70E + 08	2.34E + 00	1.64E + 00
	Std	4.06E-04	1.55E-03	1.31E-01	4.80E-04	1.37E-04	2.42E + 08	4.32E-01	6.10E-01

Significant values are bold.

The data in Tables 4 and 5 clearly show that the QOCWO algorithm performs better than the other seven algorithms on other benchmark functions except F4, F6 and F12 for the 50-dimensional problem, except F4, F6, F7 and F12 for the 100-dimensional problem. The results show that QOCWO algorithm has significant advantages in high dimensional optimization problems, and it can deal with and solve complex high dimensional optimization problems effectively.

Wilcoxon’s rank sum test analysis

Since the results of the algorithm are stochastic, statistical testing is necessary to verify the significance of the findings. The Wilcoxon rank-sum test is utilized to evaluate the statistical significance of differences between the two algorithms at a 5% significance level²⁸. We tested the two samples and used the p-value as an indicator of significance. Specifically, if the p-value is less than 0.05, it indicates that the algorithm has a significant advantage in solving the problem. With the dimensionality set to 30, the QOCWO algorithm and the other seven algorithms were compared with 23 benchmark functions. The results are shown in Table 6.

Table 6.

The results of Wilcoxon’s rank sum test for all functions.

Function		QOCWO	vs.	WOA	SCA	WO	SMA	GBO
		CGWO	MFO
F1	p-value	5.93E-18	3.33E-165	8.98E-02	3.33E-165	3.32E-165	5.85E-52	2.21E-61
F2	p-value	3.16E-05	3.33E-165	2.32E-07	3.33E-165	3.77E-72	1.69E-23	5.10E-03
F3	p-value	4.10E-122	3.33E-165	3.33E-165	3.32E-165	1.31E-82	5.61E-94	1.45E-01
F4	p-value	7.82E-28	3.33E-165	3.33E-165	2.40E-165	2.58E-90	6.73E-64	3.82E-01
F5	p-value	3.33E-165	3.33E-165	3.33E-165	3.32E-165	3.32E-165	3.32E-165	3.33E-165
F6	p-value	3.33E-165	3.33E-165	3.33E-165	3.32E-165	3.32E-165	3.32E-165	1.45E-01
F7	p-value	3.32E-165	3.33E-165	3.32E-165	3.32E-165	3.32E-165	3.32E-165	3.32E-165
F8	p-value	1.36E-165	3.33E-165	3.32E-165	2.29E-166	1.21E-165	288E-117	2.93E-165
F9	p-value	1.64E-164	3.33E-165	1.33E-17	7.50E-166	9.91E-08	9.99E-04	2.03E-27
F10	p-value	1.09E-08	3.32E-165	1.75E-05	5.80E-171	2.02E-38	6.03E-79	1.27E-48
F11	p-value	1.91E-165	3.32E-165	1.28E-10	2.51E-165	4.39E-55	1.04E-55	3.23E-27
F12	p-value	2.68E-165	3.32E-165	3.11E-165	2.50E-165	1.16E-08	1.63E-150	1.47E-09
F13	p-value	3.68E-165	3.32E-165	2.87E-165	2.66E-165	5.46E-153	2.09E-165	2.74E-01
F14	p-value	1.28E-132	3.11E-67	3.71E-167	5.28E-167	1.52E-32	2.62E-55	3.33E-66
F15	p-value	7.84E-93	1.89E-153	2.82E-162	2.3E-165	5.95E-17	1.53E-162	3.40E-119
F16	p-value	1.53E-107	9.62E-108	9.65E-97	4.89E-163	1.87E-81	3.10E-29	1.08E-150
F17	p-value	1.70E-161	2.70E-105	3.61E-109	1.58E-165	2.33E-04	8.96E-91	5.07E-135
F18	p-value	4.88E-58	1.39E-142	7.69E-02	7.84E-07	1.01E-118	7.11E-25	1.46E-165
F19	p-value	1.51E-135	5.76E-158	7.21E-01	5.23E-164	1.08E-51	5.60E-126	5.31E-164
F20	p-value	7.39E-103	4.99E-99	1.49E-100	3.90E-165	3.84E-96	3.06E-129	3.12E-165
F21	p-value	1.11E-167	1.87E-167	3.09E-165	2.07E-165	1.34E-127	1.76E-166	6.11E-131
F22	p-value	1.94E-165	7.76E-166	3.23E-162	1.32E-165	8.83E-109	3.50E-162	5.28E-56
F23	p-value	3.24E-165	2.19E-165	3.82E-163	2.51E-165	2.02E-165	1.46E-164	2.99E-06

Table 6 shows that the QOCWO algorithm outperforms the other seven algorithms, particularly excelling in functions F5and F7. When compared to CGWO, MFO, CSA, WO and SMA, the QOCWO algorithm demonstrates significant statistical differences across all functions. Compared to WOA, the QOCWO exhibits significant statistical differences in all functions except F1,F18 and F19. Similarly, when compared to GBO, noticeable statistical differences are observed in all functions except F3,F4,F6 and F13.

Application on engineering problems

To demonstrate the effectiveness of QOCWO in solving real-world problems, the algorithm was applied to two engineering design problems, that is, the design of compression springs and the design of pressure vessels.

Due to the various constraints in this section, a constraint-handling method is required. Several types of penalty functions have been proposed in the literature, including static, dynamic, annealing, adaptive, co-evolutionary, and death penalty functions. In this context, a static hard penalty function was adopted, that adds an additional penalty term to the objective function of the optimization problem. When a solution violates the constraints, the penalty term becomes very large, resulting in a significantly unfavorable objective function value for the constraint-violating solution, making it unlikely to be considered during the optimization process. This method is straightforward to implement and compute within optimization algorithms, and ensures that solutions do not significantly violate constraints. It can be combined with different optimization algorithms, which enhance the versatility of the optimization approach.

Case 1: tension/compression spring design

The objective of this test problem is to minimize the cost or weight of the spring by optimizing the wire diameter (d), the average coil diameter (D), and the effective number of coils (N) as illustrated in Fig. 5, while ensuring all constraints are satisfied. These constraints include minimum deviation (g1), shear stress (g2), impact frequency (g3), and outer diameter limit (g4). By utilizing a penalty function, the problem is converted into an unconstrained optimization problem. The formulation of the optimization problem is as follows:

Fig. 5.

Description of tension/compression spring.

Suppose

Minimize

Subject to

Variable range 0.05 ≤ x₁ ≤ 2.00,

0.25 ≤ x₂ ≤ 1.30,

2.00 ≤ x₃ ≤ 15.00.

The methods compared with the QOCWO include GA²⁹, ES³⁰, WO⁸, DE³¹, RO³², PSO³³, and WOA³⁴. Based on the optimal costs shown in Table 7, it can be seen that the QOCWO has the lowest cost and achieves the best design of the spring with a cost of 0.126633.

Table 7.

Comparison of optimization results of tension/compression spring design.

Algorithms	Optimal values for variables
	d	D	N
QOCWO	0.051985	0.36394	10.8754	0.0126633
GA	0.051480	0.351661	11.632201	0.0127048
ES	0.051989	0.363965	10.890522	0.0126810
WO	0.050000	0.311500	14.892300	0.0126650
DE	0.051609	0.354714	11.410831	0.0126702
RO	0.051370	0.349096	11.76279	0.0126788
PSO	0.051728	0.357644	11.244543	0.0126747
WOA	0.051207	0.345215	12.004032	0.0126763

Case 2: pressure vessel design

The objective of the test problem is to minimize the cost of the container by adjusting shell thickness (Ts), head thickness (Th), inner radius (R), and cylinder length (L) in Fig. 6, while ensuring that all constraints are satisfied. These constraints include the dimensional ratio (G1), the final dimensions (G2), the geometric capacity (G3), and the installation limits (G4). By using a penalty function, this problem is transformed into an unconstrained optimization problem. The optimization problem is formulated as follows:

Fig. 6.

Description of pressure vessel design.

Suppose

Minimize

Subject to

Variable range 0 ≤ x₁ ≤ 99.

0 ≤ x₂ ≤ 99.

10 ≤ x₃ ≤ 200.

10 ≤ x₄ ≤ 200.

The other optimization methods compared to the QOCWO include GA³⁵, ES³⁰, WO⁸, DE³¹, RO³², PSO³⁴, and WOA³⁴. The optimization results are presented in Table 8, and the QOCWO has shown the best optimization results. It has achieved the optimal design of the container with a cost of 3312.7752.

Table 8.

Comparison of optimization results of pressure vessel design.

Algorithms	Optimal values for variables
	T S	T h	R	L
QOCWO	0.778170	0.384650	40.319900	199.996400	5885.3431
GA	0.812500	0.437500	42.097398	176.654050	6059.9463
GSA	1.125000	0.625000	55.9886598	84.4542025	8538.8359
DE	0.812500	0.437500	42.098411	176.637690	6059.7340
PSO	0.812500	0.437500	42.091266	176.746500	6061.0777
WO	0.778190	0.384659	40.320119	199.993131	5885.3491
ES	0.812500	0.437500	42.098087	176.640518	6059.7456
WOA	0.812500	0.437500	42.0982699	176.638998	6059.7410

In addition, after conducting ten independent experiments, the best, worst, average values and standard deviations of QOCWO were determined in Table 9.

Table 9.

Statistical results of QOCWO in real-world engineering optimization problems.

Problems	Best	Mean	Worst	Std
Case1	0.0126633	0.01308	0.014748	8.22E-4
Case2	5885.3431	5954.9443	6243.3130	1.06E + 02

Conclusion

Due to the two shortcomings of the original WO, that is, slow convergence speed and susceptibility to local optimization in high-dimensional problems, the study proposes WO based on Quasi-Oppositional Based Learning and Chaos Local Search. The algorithm utilizes QOBL to generate oppositional points (Qopposed) for the current search agents’ positions and calculates the fitness of these oppositional points, which can explore areas of the solution space opposite to the current best position, thereby expanding the search space of solutions. This helps the algorithm to escape from local optima, enhances the global exploration capability of the WO algorithm, improves the quality of solutions, and enhances the robustness of the original WO algorithm. When dealing with complex problems, CLS was utilized to increase the convergence speed of WO.

In the study, the QOCWO algorithm is tested on 23 typical benchmark functions. The tests on its convergence speed, convergence accuracy, and stability show that the QOCWO is an effective algorithm for solving global optimization problems. Compared to the WO algorithm and other metaheuristic optimization algorithms, the QOCWO can bring most individuals close to the optimal value with fewer iterations, which has certain advantages. To verify the effect of the algorithm in engineering applications, two engineering design problems were also tested (tension and compression spring design and pressure vessel design), and the QOCWO algorithm still has a very strong performance, comparable to other excellent metaheuristic algorithms.

Compared to other metaheuristic algorithms, the QOCWO maintains a leading position in both optimality (i.e., performance in reaching the optimal value) and scalability (i.e., ability to handle large-scale problems). However, there are some drawbacks, such as increased computational and space complexity when dealing with large-scale optimization problems. The comparison with other algorithms is not comprehensive enough, and the evaluation of QOCWO is limited. More comparisons are needed. In future research, the internal structure of the QOCWO algorithm needs to be further optimized to make it applicable to a broader range of engineering problems, and its applications in other fields should be explored.

Author contributions

Y.L wrote the main manuscript text, L.L and Z.P provided technical support, K.Z and Y.D check the formulas and picture, all authors reviewed the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.