Multi-strategy improved fox-monkey optimization algorithm for global optimization and application of multi-strategy improved fox-monkey optimization algorithm

Abstract

The invention discloses a multi-strategy improved fox-monkey optimization algorithm for global optimization and application thereof, and belongs to the technical field of optimization algorithms, the method comprises the following steps: firstly, using Chebyshev chaotic mapping to initialize a population so as to improve the diversity and coverage range of the initial population; secondly, a difference population evolution strategy is introduced in the iteration process, and the algorithm is prevented from falling into local optimum in the later period; and finally, introducing a crisscross strategy to enhance global search capability and maintain population diversity. 20 benchmark test functions and a part of complex functions in a cec2017 function set are adopted for a simulation experiment, and the experiment result shows that the optimization precision, the convergence speed and the stability of the improved algorithm are obviously improved. The improved algorithm is applied to the engineering optimization problem, compared with an original algorithm, the convergence precision of the algorithm in three actual engineering problems is improved by 19.56%, 19.18% and 6.7% respectively, the standard deviation is reduced by 91.44%, 27.56% and 92.19% respectively, and the feasibility of the improved algorithm is further verified.

Description

Technical Field

[0001] This invention relates to the field of optimization algorithm technology, and in particular to a multi-strategy improved lemur optimization algorithm for global optimization and its applications. Background Technology

[0002] With the rapid updates and development of science and technology, engineering problems are becoming increasingly complex, and the target objects are becoming more diversified. Currently, when facing large and complex optimization problems in practical engineering, traditional optimization methods urgently need improvement in terms of speed and accuracy in finding optimal solutions. To address these engineering problems, researchers have proposed various intelligent optimization algorithms based on the collective behavior of organisms in nature. The core idea of ​​these swarm intelligence algorithms is to simulate collective behavior to obtain the global optimum, possessing advantages such as robustness and adaptability, and can be used to solve multi-objective optimization problems. Existing swarm intelligence algorithms include the Sparrow Search Algorithm (SSA), Grey Wolf Optimizer (GWO), Whale Optimization Algorithm (WOA), and Dung Beetle Optimizer (DBO), which have been widely applied to various optimization problems.

[0003] In 2022, Ammar Kamal Abasi et al. proposed a novel bio-based heuristic algorithm: the Lemurs Optimizer (LO). Inspired by the movement behavior of lemur populations, the LO algorithm simulates this behavior and boasts advantages such as simple structure, few parameters, and ease of implementation. However, in solving numerical optimization problems, the LO algorithm still suffers from insufficient global search space in the early stages of iteration and low accuracy in later local optimization, leading to susceptibility to local optima and convergence stagnation, particularly noticeable in solving complex multimodal function problems. Currently, to address these shortcomings of the standard LO, Parul Punia et al. proposed introducing a dynamic scaling factor and Gaussian mutation strategy into the LO, enhancing its local optimization and global convergence capabilities. Experimental results show that the improved LO (Improved Lemurs Optimizer, ILO) improves the convergence speed and accuracy, but still suffers from local optima traps when solving multimodal function problems. Summary of the Invention

[0004] To address the aforementioned problems, this invention proposes a Multi-strategy Improved Lemurs Optimizer (MILO) algorithm for global optimization and its applications. The contents of this invention are as follows:

[0005] The first objective of this invention is to provide a multi-strategy improved lemur optimization algorithm for global optimization, the technical point of which includes the following steps:

[0006] Step 1: Initialize the initial positions of individual lemurs in the population using Chebyshev chaotic mapping;

[0007] Step 2: Update the free risk rate (FRR) of the entire lemur population, calculate the fitness value of each lemur individual in the population, sort them, and obtain the sorted index.

[0008] Step 3: Based on the ranking results in Step 3, perform differential evolution on the worst lemur individual and perturb the best lemur individual.

[0009] Step four: Use the free risk rate (FRR) and the individual risk rate (r) to determine whether the algorithm should perform a leap.

[0010] Step 5: Employ a cross-sectional strategy to prevent the entire algorithm from getting stuck in local optima.

[0011] To better address the aforementioned technical issues, the first step of the multi-strategy improved lemur optimization algorithm for global optimization in this invention initializes the initial positions of individuals in the lemur population using the Chebyshev chaotic mapping, specifically as follows:

[0012] x i+1 =cos(a*cos -1 (x(i)))

[0013] In the formula, x i Let be the current value of the chaotic sequence of generation i, and let a be the control parameter, which is set to a = 4.

[0014] To better address the aforementioned technical issues, the overall risk rate (FRR) of the lemur population in step two of the multi-strategy improved lemur optimization algorithm for global optimization in this invention is specifically as follows:

[0015]

[0016] In the formula, High_Risk_Rate and Low_Risk_Rate represent the maximum and minimum values ​​of the free risk rate (FRR), respectively, and Max_Iter and Current_Iter represent the maximum number of iterations and the current number of iterations, respectively.

[0017] To better address the aforementioned technical issues, in step three of the multi-strategy improved lemur optimization algorithm for global optimization of this invention, differential evolution is performed on the worst-performing lemur individual, specifically as follows:

[0018] E worst =swarm a +rand()*(swarm b -swarm c (a≠b≠c∈1,2,...,N)

[0019] In the formula, swarm a swarm b swarm c Three lemur individuals were randomly selected.

[0020] To better address the aforementioned technical issues, the optimal lemur individual is perturbed in step three of the multi-strategy improved lemur optimization algorithm for global optimization in this invention. Specifically:

[0021] N best =swarm best +sign(rand()-0.5)*exp(-FRR)*swarm best

[0022] In the formula, swarm best This is the globally optimal position.

[0023] To better address the aforementioned technical issues, in step four of the multi-strategy improved lemur optimization algorithm for global optimization of this invention, the free risk rate (FRR) and the individual risk rate (r) are used to determine whether the algorithm should perform a leap behavior. Specifically:

[0024] When r > FRR, the position is updated using a leap behavior;

[0025] When r < FRR, position update is performed using dance behavior.

[0026] To better address the aforementioned technical issues, this invention provides a position update mechanism for the leap behavior of a multi-strategy improved lemur optimization algorithm for global optimization, specifically as follows:

[0027]

[0028] In the formula, Let i be the position of the i-th lemur individual. The updated location for this individual lemur performing the dance behavior. For an individual in the elite group that is in a leader position, it is randomly selected from the three individuals with the best fitness values ​​for each lemur individual in step two. rand is a random number that follows a uniform distribution in the interval [0,1].

[0029] To better address the aforementioned technical issues, this invention provides a multi-strategy improved lemur optimization algorithm for global optimization, which updates the position based on the dance behavior, specifically as follows:

[0030]

[0031] In the formula, This refers to the lemur individual closest to oneself within a group of lemurs that are superior to oneself.

[0032] To better address the aforementioned technical issues, the cross-sectional strategy in step four of the multi-strategy improved lemur optimization algorithm for global optimization in this invention is specifically as follows:

[0033]

[0034]

[0035] In the formula, k1 and k2 are both random numbers uniformly distributed in the range [0,1], h1 and h2 are both random numbers uniformly distributed in the range [-1,1], and X(i,d) and X(j,d) represent the d-th dimension of the parent X(i) and the d-th dimension of the parent X(j), respectively. and Then they represent offspring. The dth dimension and offspring The d-th dimension of q is a random number that follows a uniform distribution in the range [0,1].

[0036] The second objective of this invention is to provide an application of a multi-strategy improved lemur optimization algorithm for global optimization in welded bridge design, robot gripper design, and rolling bearing design.

[0037] Compared with the prior art, the present invention has the following beneficial effects:

[0038] This invention presents a multi-strategy improved lemur optimization algorithm for global optimization and its application. First, to address the problem of the algorithm easily getting trapped in local optima due to poor ergonomics of randomly generated individuals in the early stages, a Chebyshev chaotic mapping is used to increase population diversity and global search capability, laying the foundation for improved optimization performance. Second, a differential population evolution strategy is introduced during algorithm iteration, guiding the evolution of the worst-performing individual and perturbing the update of the best-performing individual in the lemur population, enhancing the algorithm's ability to escape local optima, accelerating convergence and improving convergence accuracy to a certain extent. Finally, a cross-strategy is applied to improve the algorithm. Horizontal cross-strategy enhances the global search capability, while vertical cross-strategy maintains population diversity and avoids getting trapped in local optima, balancing the exploration and development of the algorithm. Comparison of MATLAB simulation results demonstrates that the MILO algorithm has the ability to balance global and local search, and improves both convergence speed and accuracy. Attached Figure Description

[0039] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this invention, illustrate exemplary embodiments of the invention and are used to explain the invention, but do not constitute an undue limitation of the invention. In the drawings:

[0040] Figure 1 Initialize the population CE value distribution map for different strategies in this study;

[0041] Figure 2 This invention improves the lemur's leaping behavior.

[0042] Figure 3 This is the MILO flowchart for this development;

[0043] Figure 4 Box plots of partial multimodal functions

[0044] Figure 5 Comparison of convergence curves for benchmark functions (x-axis: number of iterations; y-axis: fitness value of the current best function) Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below in conjunction with specific embodiments and corresponding drawings. Obviously, the described embodiments are only a part of the embodiments of this invention, and not all of them. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.

[0046] The technical solutions provided by the various embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0047] 1. Lemurian Optimization Algorithm

[0048] Lemurs are found only on the island of Madagascar and the nearby Comoros Islands near the Mozambique Channel in Africa, and are known for their distinctive long tails and highly social group characteristics. Lemur groups are led by a dominant female who can guide the entire group swiftly through the forest, earning them the nickname "gentle ghosts" of the Madagascar rainforest. Lemur groups demonstrate exceptional agility and adaptability in their locomotion strategies. Therefore, a novel evolutionary algorithm, the Lemur Optimization Algorithm, was developed, drawing inspiration from the group movement behavior of this species in the forest. This algorithm simulates two behaviors of lemur groups when searching for food and avoiding predators, which is very similar to the exploration and development phases in metaheuristic algorithms. One locomotion behavior is called leaping behavior, where lemurs propel themselves into the air from trees, landing on nearby branches to ensure individual safety while avoiding predators. The other is defined as dancing behavior; when the distance between trees is too great to cross by leaping alone, lemurs use dancing behavior to return to the ground and move horizontally towards nearby, safe lemur companions.

[0049] 1.1 Initializing the Lemur Population

[0050] In the initialization phase of the LO algorithm, a lemur population is first generated. Each lemur is considered as a separate solution and is represented by a vector containing its own position coordinates. Given N as the population size of lemurs and dim as the variable dimension, formula (1) is obtained as the position distribution of N lemurs in dim dimension space.

[0051]

[0052] In the formula, i represents the i-th individual in the lemur population, and j represents the j-th dimension of a single solution. For the i-th solution, its j-dimensional decision variable is obtained from formula (2).

[0053]

[0054] In the formula, rand represents a random number in the region [0,1] that follows a uniform distribution, and ub and lb represent the upper and lower boundaries, respectively.

[0055] 1.2 Update Free Risk Rate

[0056] To simulate the risk of lemur populations encountering predators during movement, the overall risk rate of the entire lemur population is represented by the free risk rate (FRR), and updated using formula (3).

[0057]

[0058] In the formula, High_Risk_Rate and Low_Risk_Rate represent the maximum and minimum values ​​of the free risk rate (FRR), respectively, and Max_Iter and Current_Iter represent the maximum number of iterations and the current number of iterations, respectively.

[0059] 1.3 Dance Behavior

[0060] When a lemur performs a dance, it will choose to move towards the nearest superior lemur in the superior lemur group. The formula for updating its position is as follows (4).

[0061]

[0062] In the formula, Let i be the position of the i-th lemur individual. The updated location for this individual lemur performing the dance behavior. This represents the closest lemur individual in a superior lemur group, and rand is a random number that follows a uniform distribution in the interval [0,1].

[0063] 1.4 Leap Behavior

[0064] The leaping behavior mainly simulates the guidance of the best lemur in the population to the other lemurs. When a lemur is threatened by a predator, it will leap towards the best lemur in the entire population. That is, the lemur will jump high from its position to a position closer to the best lemur in order to avoid being hunted by the predator. The individual position update formula is as follows (5).

[0065]

[0066] In the formula, This represents the best lemur individual in the entire population. All other variables have been explained and will not be repeated here.

[0067] 1.5 Lemur Behavioral Selection

[0068] Based on the free risk rate defined above, the risk rate of an individual lemur is compared with the free risk rate to obtain the behavioral choices of the individual lemur. When the risk rate of an individual lemur is less than the free risk rate, it means that the risk rate of the lemur at this time is less than the risk rate of the entire population, the probability of being hunted by predators is low, and it should perform dance behavior to explore the optimal position in the entire search space; when the risk rate of an individual lemur is greater than the free risk rate, it means that the risk rate of the lemur at this time is greater than the risk rate of the entire population, the probability of being hunted by predators is high, and it should perform leap behavior to quickly leave the original position and jump towards the global optimal position. This process is represented by formula (6).

[0069]

[0070] In the formula, r is a random number that follows a uniform distribution in the interval [0,1], and represents the risk rate of the i-th lemur in the j-th dimension of the decision variable.

[0071] 2. A Multi-Strategy Improved Lemurian Optimization Algorithm

[0072] 2.1 Chebyshev Chaotic Map Initialization

[0073] In the Loop algorithm, the initial population positions are generated by random distribution. This method may result in uneven distribution of individuals in the initial population throughout the search space, which limits the coverage of the population in the early stages of the search and leads to poor solution quality in the later development stages. This makes the optimization algorithm prone to getting trapped in local optima and has poor population diversity. Therefore, this invention improves the population initialization process of the Loop algorithm by using Chebyshev chaotic mapping initialization, which effectively avoids population uniformity. The formula is shown in equation (7).

[0074] x i+1 =cos(a*cos -1 (x(i))) (7)

[0075] The Clark-Evans nearest neighbor analysis method was used to measure the spatial distribution of the initial population, specifically by averaging the distances between the nearest adjacent individuals. The expected average distance from the random distribution The ratio is used to determine the degree of clustering of the distribution. The calculation formula is Equation (8).

[0076]

[0077] In the formula, CE represents the Clark-Evans index, and r i min Let represent the distance between the i-th lemur individual and its nearest neighbor, V be the spatial volume, and Γ(4 / 3) be the value of the Gamma function (approximately 0.8935), which is related to the three-dimensional spatial integral. The population distribution is determined based on the value of the Clark-Evans index: if CE < 1, the initial population is clustered; if CE = 1, the initial population is randomly distributed; if CE > 1, the initial population is uniformly distributed.

[0078] Fifty initial populations, each with a size of 30, were generated using both random distribution and Chebyshev chaotic mapping. The Clark-Evans index was used to analyze the initial populations, and their basic characteristics are shown in Table 1. Figure 1The distribution of Clark-Evans index values ​​for the 50 initial populations.

[0079] Table 1 Comparison of CE values ​​of populations initialized under different strategies.

[0080] CE value Maximum value Minimum value mean Standard deviation random distribution 1.3362 0.89743 1.090557 0.01212 Chebyshev Chaotic Mapping 1.4402 0.99531 1.210946 0.009908

[0081] By comparing the data in Table 1, the mean CE of the initial population generated based on random distribution is approximately 1.09. Figure 1 The distribution of CE values ​​in the initial population indicates that the initial population generated by this strategy is randomly distributed, which is consistent with reality. In contrast, the initial population generated based on the Chebyshev chaotic mapping has a mean CE value of approximately 1.21, combined with... Figure 1 It can be determined that the initial population generated by the Chebyshev chaotic mapping is uniformly distributed. Therefore, it can be considered that the individuals in the initial population generated by this strategy are more evenly distributed in the search space. Compared with the randomly initialized population, the lemur individuals have stronger traversal in the search space, thereby expanding the coverage of the population, increasing the diversity of the population, and improving the quality of the initial population.

[0082] 2.2 Differential Population Evolution

[0083] In the later stages of the Loop algorithm iteration, the probability of individual lemurs performing leap behavior increases, and the number of lemurs jumping to the global optimum increases, leading to a decrease in the diversity of the entire population. This increases the probability of the algorithm getting stuck in local optima, resulting in a decrease in convergence accuracy.

[0084] For the entire lemur population, individuals with poor fitness will move closer to individuals with better fitness to ensure their safety. The role of the better individuals is to explore the direction of optimization, while the role of the poor individuals is to search for the optimal value near the better individuals. Therefore, different evolutionary strategies should be matched for lemur individuals with different roles. Inspired by the survival of the fittest in nature, three lemur individuals in the population are randomly selected in each iteration. The differential evolution strategy is used on the lemur individual with the worst fitness, and the position of the individual is updated using the greedy selection principle. The purpose is to increase the search ability of the algorithm and improve the convergence speed of the algorithm. At the same time, the diversity of the population is maintained, and the convergence accuracy is avoided. The specific evolutionary strategy is as follows: (9) where swarm a swarm b swarm c Three lemur individuals were randomly selected.

[0085] E worst =swarm a +rand()*(swarm b -swarm c(a≠b≠c∈1,2,...,N) (9)

[0086] For the globally optimal lemur individual in the leader position, a perturbation strategy can be used based on the population aggregation level to enhance its ability to escape local optima in the later stages of iteration. Using the Free Risk Rate (FRR) mentioned earlier to determine the population aggregation level, its expression shows that at the beginning of the iteration, FRR is at its maximum, indicating a relatively dispersed and highly diverse population distribution. At this point, only a small perturbation is needed on the globally optimal position. However, as the algorithm iterates, FRR gradually decreases, increasing the likelihood of falling into a local optimum trap. Therefore, a larger perturbation is needed on the globally optimal position to guide the population out of local optima and enhance its global search capability. The perturbation strategy for the globally optimal position is given by formula (10). From formula (10), it can be seen that when FRR is large, the corresponding exponential function value is small, resulting in a small perturbation; while when FRR is small, the corresponding exponential function value is large, resulting in a large perturbation. In the formula, swarm... best This is the globally optimal position.

[0087] N best =swarm best +sign(rand()-0.5)*exp(-FRR)*swarm best (10)

[0088] In the hunting phase of the gray wolf optimization algorithm, an elite group guidance strategy is adopted, using the three best individuals in the population as guides to ensure population diversity. Simultaneously, an encirclement-style guidance strategy is employed to further improve the algorithm's search capability. Inspired by the gray wolf optimization algorithm, to avoid premature convergence in the LO algorithm and ensure that population diversity does not rapidly decrease in the later stages of iteration, thus effectively improving the convergence accuracy of the algorithm on multimodal functions, an improved lemur leap behavior is proposed. During the leap behavior, an elite group is used to guide the lemur population, such as... Figure 2 As shown, the position update of the leap behavior in the improved LO algorithm is as shown in the second part of equation (11).

[0089] In the formula, An individual is one of the elite groups that occupies a leading position, and is randomly selected from the three globally optimal individuals.

[0090] 2.3 Cross-sectional and cross-sectional strategies

[0091] During the iteration of the LO algorithm, the movement behavior guides individual lemurs toward the nearest optimal position and the global optimal position. If the individual acting as the guide has already fallen into a local optimum, the subsequent search will lead the entire population into a local optimum, causing "premature convergence". This invention introduces a cross-sectional strategy after the individual completes the movement behavior. The purpose of this strategy is to improve the global search capability for solving complex optimization problems, thereby improving the convergence accuracy.

[0092] Lateral crossover refers to arithmetic crossover operations performed on two different lemur individuals across all dimensions. This allows for interaction between individuals, ensuring a balance between exploration and development in the algorithm, preventing premature population convergence, and thus effectively improving the algorithm's convergence speed and search accuracy. First, 20 lemur individuals are randomly paired without repetition. Then, lateral crossover is performed on the paired individuals. If one parent pair is X(i) and X(j), their offspring are defined as... and It is generated by the following formulas (12) to (13).

[0093]

[0094] Where k1 and k2 are random numbers uniformly distributed in the range [0,1], h1 and h2 are random numbers uniformly distributed in the range [-1,1], and X(i,d) and X(j,d) represent the d-th dimension of the parent X(i) and the d-th dimension of the parent X(j), respectively. and Then they represent offspring. The dth dimension and offspring The d-th dimension. The generated offspring compete with their parents, eliminating individuals with poor fitness and retaining the best individuals.

[0095] Vertical crossover refers to the arithmetic crossover of all lemur individuals across two different dimensions. Each individual undergoes a vertical crossover, updating only one dimension without altering the data in other dimensions. This strategy considers the possibility that some dimensions may already be at their optimal values. While ensuring that other dimensions are not compromised, it enhances the ability of stagnant dimensions to escape local optima, thereby improving the search accuracy of the algorithm. Vertical crossover is performed on the d1-th and d2-th dimensions of X(i), and offspring individuals are obtained using the following formula (14).

[0096]

[0097] Where q is a random number uniformly distributed in the range [0,1]. The generated offspring compete with their parents, eliminating individuals with poor fitness and retaining the best individuals.

[0098] 2.4 MILO Algorithm Flow

[0099] The specific implementation process of the MILO algorithm is introduced in detail below. Figure 3 It is the flowchart of the improved algorithm.

[0100] Step1 Set the relevant parameters of the MILO algorithm: population size N, maximum number of iterations Max_Iter, search dimension dim, upper bound ub of the search interval, lower bound lb of the search interval, maximum value High-RiskRate and minimum value Low-RiskRate of the free risk rate FRR.

[0101] Step2 Generate the initial population using formula (7).

[0102] Step3 Update the value of the free risk rate FRR using formula (3). At the same time, calculate the fitness value of each lemur individual in the population, sort them, and obtain the sorted index.

[0103] Step4 According to the sorting result, perform differential evolution on the worst individual using formula (9), and perturb the best individual using formula (10).

[0104] Step5 Use the free risk rate FRR and the individual risk rate r to judge whether to execute the leap behavior. If r < FRR, execute the dance behavior, update the position using the first part of formula (11), and leave the optimal value using the greedy selection principle. Otherwise, execute the leap behavior and update the position using the second part of formula (11).

[0105] Step6 Prevent the algorithm from falling into local optimum according to the cross strategy formulas (12) - (14).

[0106] Step7 Judge whether the end condition is reached. If so, output the optimal result obtained by iteration. Otherwise, repeat steps Step3 - 6.

[0107] 2.5 Time complexity analysis

[0108] Time complexity is an important indicator for measuring the efficiency of optimization algorithms. Given a population size of N, a maximum number of iterations of T, and a problem dimension of dim, the time complexity of the Loop algorithm is O(N*D) in the initial stage and O(N*D*T) during the iteration process. For the improved MILO algorithm, the time complexity of the Chebyshev chaotic mapping initialization is O(N*D). During the iteration process, the time complexity of worst-case position evolution and best-case position perturbation is O(1), and the time complexity of position update through movement behavior is O(N*D). Using the cross-sectional strategy to process 2 / 3 of the randomly selected lemur individuals, divided into horizontal and vertical parts, the time complexities are O(0.5*(2 / 3N)*D) and O(2 / 3N*D) respectively. Therefore, the time complexity of the entire iteration process is O(2*(N*D+1)). After T iterations, the time complexity of this algorithm during the iteration process is O(2*N*D*T+2*T), which increases the computational complexity by a constant level compared to the original LO algorithm. However, it does not affect the order of the algorithm's time complexity. Therefore, the MILO algorithm described in this invention can be considered to have a time complexity comparable to the LO algorithm.

[0109] 3. Algorithm Performance Testing Experiment

[0110] 3.1 Experimental Environment

[0111] The computer used for simulation testing in this invention is configured with an AMD(R) Ryzen(TM) 74800H processor with a running frequency of 2.90GHz, 16.0GB of memory, a 64-bit Windows 11 operating system, and a Matlab2023b computing environment.

[0112] 3.2 Comparison Algorithm

[0113] The abbreviations for each method and comparison algorithm used in this invention are listed in Table 2.

[0114] Table 2 Algorithm Introduction

[0115]

[0116] 3.3 Introduction to Benchmark Test Functions

[0117] In this invention, the improved algorithm is simulated and tested using 20 benchmark test functions to evaluate the effectiveness of the MILO algorithm improvement strategy. The names, dimensions, value ranges, and theoretical optimal values ​​of the 20 standard test functions used are shown in the "Supplementary Information". Among them, unimodal functions F1-F7 are used to evaluate the algorithm's convergence speed and convergence accuracy; multimodal functions F8-F13 are used to evaluate the algorithm's global search capability; and fixed-dimensional multimodal functions F14-F20 are used to evaluate the algorithm's stability and ability to escape local optima. The population size is set to N=30, and the maximum number of iterations Max_iter=1000. To avoid randomness, each algorithm is run 30 times to determine the optimal value, standard deviation, and mean, and then compared and analyzed.

[0118] 3.4 Testing the effectiveness of the improved strategy (Experiment A)

[0119] To evaluate the effectiveness of the MILO algorithm improvement strategy, the MILO algorithm was compared with the Grey Wolf Optimization (GWO), Black-winged Kite (BKA), Sparrow Search (SSA), Improved Grey Wolf Optimization (IGWO), Lemur Optimization (LO), and Improved Lemur Optimization (ILO) under the same initial conditions. By comparing and analyzing the performance of the MILO algorithm on the first 20 functions of the CEC2005 standard test function set, the effectiveness of the MILO algorithm improvement strategy was comprehensively evaluated.

[0120] As shown in Table 3, for the single-peaked test functions F1 to F7, the MILO algorithm significantly improves accuracy and stability compared to the original LO algorithm. In particular, the average results for functions F1, F3, and F6 are consistent with the theoretical values ​​of the corresponding functions, with an optimization result of 0. The optimization results for F2 and F4 are not significantly different from the ILO algorithm, but are superior to other algorithms, and the optimization results are very close to the theoretical values ​​of the corresponding functions. This indicates that MILO has excellent optimization performance and excellent robustness. For the multi-peaked function F8, the distribution of results obtained from running each algorithm 30 times is as follows: Figure 4As shown in the box plot on the left, the MILO algorithm significantly outperforms other algorithms in terms of optimization accuracy, and the optimal value obtained by the MILO algorithm is consistent with the theoretical optimal value. This indicates that the optimization performance of the MILO algorithm is significantly improved through the improved strategy. Although it did not reach optimal convergence in F9, it has already achieved a significant improvement compared to the original LO function. In multimodal functions F10–F13, the improved MILO algorithm shows a clear advantage, especially in F12 and F13. In these two functions, the MILO algorithm has an improvement of more than 10 orders of magnitude in both mean and standard deviation compared to other algorithms, indicating that adding the cross-sectional strategy can significantly improve the algorithm's ability to escape local optima. For fixed-dimensional multimodal functions, MILO still outperforms other algorithms, and it converges to the theoretical optimal solution in F14, F16, F18, and F19.

[0121] To more intuitively compare the convergence process of each algorithm in Experiment A, a set of comparison graphs of convergence curves were plotted, such as... Figure 5 As shown in the figure, the convergence curves F1–F4, F6, and F9–F11 demonstrate the effect of the differential population evolution improvement strategy, which improves the convergence speed and accuracy of the algorithm. The convergence curves F5, F7, F10, and F12–F13 show obvious inflection points, indicating that the improved algorithm with the addition of the cross-sectional strategy is more likely to escape local optima compared to the original algorithm. The convergence curves F14–F20 show that the MILO algorithm effectively improves the global search capability of the initial population in the early stages of iteration by utilizing the Chebyshev chaotic mapping, preventing the algorithm from getting trapped in local optima too early.

[0122] Table 3 Benchmark Function Test Analysis

[0123]

[0124]

[0125] In summary, the MILO algorithm demonstrates excellent optimization performance in single-peak, multi-peak, and fixed-dimensional multi-peak test functions, and exhibits outstanding convergence speed and stability, thus verifying the effectiveness of the improved strategy.

[0126] 3.5 Introduction to CEC2017 Test Functions

[0127] This invention selects 19 functions from the CEC2017 test set to evaluate the optimization performance of the MILO algorithm. Specifically, unimodal functions F1 and F3 are used to accurately evaluate the algorithm's convergence speed and accuracy; multimodal functions F4–F10 are used to test the algorithm's ability to escape local optima by setting a large number of local optima; and hybrid functions F11–F20 are used to comprehensively evaluate the algorithm's flexibility and robustness in complex scenarios. The population size is set to N=50, dimension D=100, and maximum iterations Max_iter=1000. The algorithm is run 30 times, and its optimal value, standard deviation, and mean are analyzed.

[0128] 3.6 Comparison Test of MILO with Other Improved Algorithms (Experiment B)

[0129] To better evaluate the convergence accuracy and stability of the MILO algorithm, this invention selected three highly cited algorithms, including the Lemurian Optimization Algorithm (LO), the Whale Optimization Algorithm (WOA), and the Dung Beetle Optimization Algorithm (DBO), as well as four improved algorithms (EBKA, GSWOA, MSADBO, and ILO). To ensure the validity of the comparative experiments, each algorithm was tested under the same initial conditions.

[0130] In Table 4, the optimization indices Min., Std., and Avg. represent the optimal value, standard deviation, and mean, respectively. The test results for the 100-dimensional function in Table 4 show that the MILO algorithm significantly improves the optimization performance compared to the original LO algorithm and also outperforms other improved algorithms. For unimodal functions, the MILO algorithm performs exceptionally well in the F1 function, demonstrating superior convergence accuracy compared to other algorithms. In the F3 function, although its ranking is slightly lower than the EBKA algorithm, its performance is better than the other algorithms. For multimodal functions, the MILO algorithm shows a clear advantage in most functions, with its average optimization ability surpassing that of the comparative algorithms, demonstrating excellent optimization performance. Compared to composite functions, the MILO algorithm still possesses outstanding solution capabilities and exhibits a certain degree of flexibility and robustness, indicating that the MILO algorithm effectively avoids getting trapped in local optima, verifying the effectiveness of the cross-sectional strategy in the improved strategy.

[0131] Table 4. Analysis of 100-dimensional CEC2017 Complex Function Tests

[0132]

[0133]

[0134] In summary, when faced with relatively complex test functions, MILO still possesses good global search and local exploitation capabilities, and can effectively escape local optima, thus enhancing the algorithm's solution capability while maintaining good robustness.

[0135] 3.7 CEC2017 Wilcoxon Rank-Sum Test

[0136] By conducting benchmark function tests and CEC2017 complex function tests, we verified the effectiveness of the improvement strategy and analyzed the convergence accuracy and stability of the MILO algorithm. However, we have not yet demonstrated a significant difference in optimization ability between the MILO algorithm and the original LO algorithm and other improved algorithms through quantitative indicators. The Wilcoxon rank-sum test can effectively handle complex data distributions and fairly compare data from different algorithms. Therefore, we introduced the Wilcoxon rank-sum test to evaluate the superiority of the MILO algorithm. The test results are shown in Table 5. The significance level of the Wilcoxon rank-sum test is usually set to 5%. That is, if the p-value is less than 5%, it is generally considered that there is a significant difference between the two; otherwise, it is considered that there is no significant difference. In Table 5, "+", "=", and "-" are used to indicate that the performance of the MILO algorithm is "significantly better than", "basically the same as", and "lower than" compared to the comparison algorithms, respectively. The results show that the p-values ​​of the MILO algorithm relative to other algorithms are mostly less than 5%, which demonstrates the superiority of the overall performance of the MILO algorithm and verifies the performance improvement.

[0137] Table 5. Wilcoxon Test Results for 100-Dimensional CEC2017 Complex Functions

[0138] MILO-LO MILO-WOA MILO-DBO MILO-EBKA MILO-GSWOA MILO-MSADBO MILO-ILO F1 3.33839E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F3 3.01986E-11 3.01986E-11 6.01039E-08 8.15274E-11 0.428963389 0.325526587 0.070126594 F4 1.32885E-10 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 9.91863E-11 3.01986E-11 F5 5.46175E-09 6.69552E-11 4.99795E-09 5.59991E-07 3.01986E-11 1.16744E-05 3.01986E-11 F6 0.899995037 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F7 3.82016E-10 3.01986E-11 1.95678E-10 3.01986E-11 3.01986E-11 3.33839E-11 3.01986E-11 F8 9.83289E-08 3.68973E-11 1.85673E-09 5.96731E-09 3.01986E-11 3.64589E-08 3.01986E-11 F9 0.003501167 1.07018E-09 8.66343E-05 0.000691252 3.68973E-11 0.000587373 3.01986E-11 F10 3.33839E-11 0.105469947 0.000117472 0.02920541 2.83145E-08 0.428963389 7.11859E-09 F11 3.01986E-11 6.69552E-11 3.19674E-09 1.59641E-07 3.01986E-11 4.07716E-11 3.01986E-11 F12 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F13 0.899995037 3.01986E-11 3.68973E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F14 3.64589E-08 2.27802E-05 0.157975689 5.46203E-06 3.68973E-11 0.001173758 3.01986E-11 F15 0.446419436 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F16 4.97517E-11 3.01986E-11 8.88288E-06 8.48477E-09 3.01986E-11 0.000158461 3.01986E-11 F17 3.01986E-11 3.33839E-11 2.87158E-10 1.84999E-08 3.01986E-11 0.00728836 3.01986E-11 F18 4.07716E-11 0.122352926 0.662734758 1.60621E-06 6.69552E-11 3.36814E-05 3.01986E-11 F19 0.002156638 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 3.01986E-11 F20 3.01986E-11 0.000903069 0.023243447 0.001517796 3.82016E-10 0.001952677 3.33839E-11 + / ＝ / - 16 / 0 / 3 17 / 0 / 2 17 / 0 / 2 19 / 0 / 0 18 / 0 / 1 17 / 0 / 2 18 / 0 / 1

[0139] 4 Engineering Problem Optimization Experiment

[0140] Mechanical optimization problems are closely related to mathematical models. In models built based on real-world problems, optimizing parameters is the goal of algorithm development. This section selects three common engineering problems—welded beam design, robot gripper design, and rolling bearing design—to evaluate the optimization capability of the MILO algorithm in engineering applications. With a population size of N=50 and a maximum number of iterations T=1000, the Whale Optimization Algorithm (WOA), Black-winged Kite Algorithm (BKA), Harris Eagle Optimization Algorithm (HHO), Goose Optimization Algorithm (GO), Lemur Optimization Algorithm (LO), and the MILO algorithm were run 30 times each, and their means and standard deviations were analyzed.

[0141] 4.1 Welded Beam Design Issues

[0142] The design of welded beams is a common and complex challenge in structural engineering. The goal is to optimize four variables using an algorithm to improve structural performance and reduce design weight. The variables are weld thickness h (x1), beam length l (x2), height t (x3), and thickness b (x4). The model includes five inequality constraints, defined by the following formulas.

[0143] Mathematical Model:

[0144]

[0145] Subject to:

[0146]

[0147] With bounds:

[0148] x1∈[0.125,2]; x2∈[0.1,10]; x3∈[0.1,10]; x4[0.1,2].

[0149] Table 6 Optimization Results of Welded Beam Design Problems

[0150] WOA BKA GO HHO LO MILO <![CDATA[x1]]> 0.15336 0.19854 0.17331 0.18508 0.19881 0.19883 <![CDATA[x2]]> 4.4852 3.343 3.8905 3.6171 3.3378 3.3374 <![CDATA[x3]]> 9.1947 9.1919 9.2026 9.1936 9.192 9.192 <![CDATA[x4]]> 0.19882 0.19884 0.19878 0.19959 0.19883 0.19883 Optimal 1.707999012 1.670297488 1.703077991 1.696371753 1.676993935 1.670217771 Std. 0.383105619 0.240500051 0.121651028 0.054273441 0.322949678 0.027654149 Avg. 2.033284872 1.742288157 1.868952098 1.800048692 2.083808212 1.676164768 AE. 0.3631 0.0721 0.1988 0.1298 0.4136 0.00597

[0151] According to the mathematical model, the known optimal result for the welded beam design problem is 1.6702. Table 6 shows that the fitness determined by MILO is the highest value. Compared with the original LO algorithm, the optimization performance of the MILO algorithm is improved by 19.56% in this problem. The MILO algorithm effectively solves the optimization challenge of the gear train design problem, demonstrating strong stability and convergence, and proving its effectiveness and application value in practical engineering applications.

[0152] 4.2 Robot gripper problem

[0153] The robot gripper design problem is a mathematical model defined by the magnitude of the force generated by the gripper. The problem contains 7 variables and 7 inequality constraints, and the mathematical formula is shown below.

[0154] Mathematical Model:

[0155]

[0156] Subject to:

[0157]

[0158] With bounds:

[0159] e∈[0,50], c∈[100,200], f, a, b all∈[10,150], δ∈[1,3.14], l∈[100,300].

[0160] Table 7 Optimization Results of Robot Gripper Design Problem

[0161] WOA BKA GO HHO LO MILO <![CDATA[x1]]> 149.6336 149.9365 146.313 150 150 149.9999 <![CDATA[x2]]> 149.0779 149.8126 98.42269 149.8646 107.5619 149.8351 <![CDATA[x3]]> 198.5629 199.7238 179.5883 185.9737 198.8754 200 <![CDATA[x4]]> 0 0.004600424 43.65113 0 41.60376 0.04592961 <![CDATA[x5]]> 10.0164 150 130.9558 18.05599 145.2104 150 <![CDATA[x6]]> 122.9921 101.1442 150.2959 102.771 122.3547 101.1111 <![CDATA[x7]]> 1.674172 2.319771 3.015173 1.608183 2.837989 2.300177 Optimal 2.931944613 2.5521155 4.642351633 2.776958082 3.389875993 2.547136147 Std. 1.137998426 1.128209204 3.33917E+15 2.487446183 0.472345059 0.342184446 Avg. 5.372923913 3.246633396 6.09646E+14 5.46837215 4.015619261 3.245485016 AE. 2.8442 0.7179 6.0965E+14 2.9397 1.4869 0.7168

[0162] Table 7 shows the optimization results of different algorithms for the robot gripper design problem. The known optimal value of the mathematical model for this design problem is 2.5287. Compared with the optimization results of the original LO algorithm, the absolute error of the MILO algorithm's optimization results decreased from 1.4869 to 0.7168, and the convergence accuracy of the algorithm improved by 19.18%. The results show that the MILO algorithm has a significant advantage in convergence accuracy compared with the algorithm compared in the experiment, and can be used to improve the performance of robot gripper mechanisms in grasping objects.

[0163] 4.3 Design Issues of Rolling Element Bearings

[0164] The goal of the rolling bearing design problem is to optimize the load-carrying capacity of a rolling bearing using five design variables and five design parameters. These design variables are the pitch diameter (Dm), ball diameter (Db), curvature coefficients of the outer and inner raceways (fo and fi), and the total number of balls (Z). The design parameters e, ε, ζ, KDmax, and KDmin appear only in the constraints. This problem also includes nine nonlinear constraint equations, mathematically defined as follows.

[0165] Mathematical Model:

[0166]

[0167] Subject to:

[0168]

[0169] With bounds:

[0170] 0.5(D+d)≤D m ≤0.6(D+d); 0.15(D+d)≤D b ≤0.45(D+d).

[0171] Z∈[4,50],f i ∈[0.515,0.6],f o ∈[0.515,0.6],K Dmin ∈[0.4,0.5],

[0172] K Dmax ∈[0.6,0.7],ε∈[0.3,0.4],e∈[0.02,0.1],ζ∈[0.6,0.85].

[0173] Table 8 Optimization Results of Multi-Rolling Bearing Design Problem

[0174] WOA BKA GO HHO LO MILO Dm 125.6567 125.7191 125.7192 125.7152 125.3937 125.719 Db 21.41224 21.42559 21.42548 21.42463 21.35649 21.42557 Z 11.48853 10.52142 11.27191 11.13704 10.61434 11.10761 fi 0.6 0.6 0.6 0.6 0.6 0.6 fo 0.5164531 0.515 0.515 0.5152116 0.515 0.515 KDmin 0.4021724 0.5 0.4167107 0.4001669 0.4 0.494317 KDmax 0.7 0.7 0.6692435 0.7 0.7 0.6975549 ε 0.3019547 0.3 0.3 0.3001253 0.3005282 0.3000011 e 0.02057684 0.07353263 0.06770225 0.06094622 0.07869381 0.03915859 ζ 0.755361 0.85 0.85 0.828638 0.7656138 0.8217901 Optimal 4.49476E+04 4.64308E+04 4.64305E+04 4.64295E+04 4.60048E+04 4.64308E+04 Std. 3797.1416 3325.5414 4420.6762 846.9120 2367.5771 185.1265 Avg. 4.1085E+04 4.4767E+04 4.1778E+04 4.6085E+04 4.3319E+04 4.6232E+04

[0175] Table 8 shows the optimization results of each algorithm for the rolling bearing design problem. It can be seen that the improved algorithm significantly outperforms the others. Furthermore, it improves upon the original algorithm by 6.7%, and the standard deviation significantly increases from 2367.6 to 185.1. Experimental results demonstrate that the MILO algorithm achieves substantial improvements in convergence accuracy and stability, further showcasing its excellent optimization performance and ability to escape local optima.

[0176] 5. Conclusion

[0177] To further improve the optimization performance of the lemur optimization algorithm, a multi-strategy improved lemur optimization algorithm is proposed. First, Chebyshev chaotic mapping is introduced during the initialization phase to avoid homogenization of the initial population and ensure its diversity. Second, the evolution of differential populations is combined with population movement behavior, which helps the algorithm escape local optima to some extent, while improving the convergence speed and accuracy. Finally, a cross-cutting strategy is added during the population update process, effectively expanding the search range and enhancing the algorithm's ability to escape local optima, while balancing global search performance and local exploitation performance. In benchmark tests, experimental results verify the effectiveness of the improved strategies in the proposed MILO algorithm, demonstrating superior optimization accuracy and stability compared to the original algorithm. In the CEC2017 function test, the competitiveness and adaptability of the MILO function in solving complex multimodal function problems were demonstrated. The Wilcoxon rank-sum test was used to compare the performance of the MILO algorithm with other comparative algorithms on various test functions, proving that the MILO algorithm still possesses the ability to effectively escape local optima when facing complex functions, and can effectively guarantee high convergence accuracy. Finally, in three engineering design simulation experiments—welded beam design, robot gripper design, and rolling bearing design—the excellent optimization performance of the MILO algorithm in solving nonlinear complex problems was demonstrated, further verifying the effectiveness of the improvement strategies in the MILO algorithm.

[0178] The above description is merely an embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principle of the present invention should be included within the scope of the claims of the present invention.

Claims

1. A multi-strategy improved lemur optimization algorithm for global optimization, characterized in that, The method comprises the following steps: Step one, initializing the initial position of the lemur population by using Chebyshev chaotic mapping; Step two, updating the free risk rate FRR of the whole lemur population, and calculating the fitness value of each lemur individual in the population, sorting it and obtaining the sorted index; Step three, according to the sorting result of step three, the worst lemur individual is subjected to differential evolution, and the optimal lemur individual is subjected to disturbance; Step four, using the free risk rate FRR and the individual risk rate r to determine whether the algorithm performs the leap behavior; Step five, using the vertical and horizontal crossover strategy to prevent the whole algorithm from falling into local optimum.

2. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, In step one, the Chebyshev chaotic mapping initializes the initial position of the lemur population, specifically: x i+1 = cos(a * cos -1 (x(i))) In the formula, x i is the current value of the i-th chaotic sequence, and a is a control parameter, and a = 4 is taken.

3. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, In step two, the total risk rate FRR of the lemur population is specifically: In the formula, High_Risk_Rate and Low_Risk_Rate represent the maximum and minimum values of the free risk rate FRR respectively, Max_Iter and Current_Iter represent the maximum iteration number and the current iteration number respectively.

4. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, In step three, the worst lemur individual is subjected to differential evolution, specifically: E worst = swarm a + rand() * (swarm b - swarm c )(a ≠ b ≠ c ∈ 1, 2,..., N) wherein swarm a , swarm b , swarm c are three randomly selected individual lemurs.

5. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, In step three, the optimal lemur individual is subjected to disturbance, specifically: N best = swarm best + sign(rand()-0.5)*exp(-FRR)*swarm best In the formula, swarm best is the globally optimal position.

6. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, In step four, the free risk rate FRR and the individual risk rate r are used to determine whether the algorithm performs the leap behavior, specifically: When r>FRR, the leap behavior is used for position updating; When rFRR, the dance behavior is used for position updating.

7. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 6, characterized in that, The leap behavior for position updating is specifically: wherein, is the position of the i-th lemur individual, is the updated position of the lemur individual performing the dance behavior, is one individual in the elite group in the role of the leader, which is randomly selected from the three individuals with the best fitness values of each lemur individual in step two, and rand is a random number uniformly distributed in the interval [0, 1].

8. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 7, characterized in that, The dance behavior for position updating is specifically: In the formula, represents the individual of the lemur group that is closer to the individual than the individual is to the lemur group.

9. A multi-strategy improved lemur optimization algorithm for global optimization according to claim 1, characterized in that, The vertical and horizontal crossover strategy in step four is specifically: wherein k1 and k2 are random numbers in the range of [0, 1] subject to uniform distribution, h1 and h2 are random numbers in the range of [-1, 1] subject to uniform distribution, X(i, d) and X(j, d) represent the d-th dimension of the parent X(i) and the parent X(j), respectively, and X(i, d) and X(j, d) represent the d-th dimension of the parent X(i) and the parent X(j), respectively, X(i, d) and X(j, d) represent the d-th dimension of the parent X(i) and the parent X(j), respectively, X(i, d) and X(j, d) represent the d-th dimension of the parent X(i) and the parent X(j), respectively, 10. The application of the multi-strategy improved lemur optimization algorithm for global optimization in the welding bridge design, robot gripper design and rolling bearing design according to any one of claims 1-9.

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