Adolescent Identity Search Algorithm Based on Fast Search and Balance Optimization for Numerical and Engineering Design Problems

This paper proposed a fast convergence and balanced adolescent identity search algorithm (FCBAISA) for numerical and engineering design problems. The main contributions are as follows. Firstly, a hierarchical optimization strategy is proposed to balance the exploration and exploitation better. Secondly, a fast search strategy is proposed to avoid the local optimization and improve the accuracy of the algorithm; that is, the current optimal solution combines with the random disturbance of Brownian motion to guide other adolescents. Thirdly, the Chebyshev functional-link network (CFLN) is improved by recursive least squares estimation (RSLE), so as to find the optimal solution more effectively. Fourthly, the terminal bounce strategy is designed to avoid the algorithm falling into local optimization in the later stage of iteration. Fifthly, FCBAISA and comparison algorithms are tested by CEC2017 and CEC2022 benchmark functions, and the practical engineering problems are solved by algorithms above. The results show that FCBAISA is superior to other algorithms in all aspects and has high precision, fast convergence speed, and excellent performance.


Introduction
Optimization is an important part to find better solutions when solving many scientific problems [1]. Many practical problems ultimately boil down to a set of decision variables that make the objective function to obtain the most optimal value. Researchers have found that meta-heuristic algorithms can solve many practical problems in the specified error range, which greatly improves the efficiency. erefore, a variety of meta-heuristic algorithms are widely proposed by researchers, which are used to find approximate solutions of many complex problems. Based on studies from many researchers, meta-heuristic approaches can be divided into four main categories [2], and its details are shown in Figure 1.
A mature global optimization method with good stability and wide applicability is named evolutionary computation. EAs are inspired by the evolutionary operation of organisms in nature. ey have characteristics of organization, adaptive, and learning, which can be applied to solve complex problems effectively, which is difficult to be solved by traditional optimization algorithms. Some of the renowned algorithms are genetic algorithm (GA) [3], differential evolution (DE) [4], estimation of distribution algorithms (EDA) [5], etc.
Swarm intelligence mainly simulates a group behavior of insects, herds, birds, and fish. Each member of the population constantly changes direction by learning its own experience and other members' experience. is phenomenon stimulates design algorithms and distributed problem solution.
Optimization is applied to various real-life applications to reduce the waste of resources, save costs, reduce expenses, and maximize benefits. Researchers develop a large number of new algorithms or hybrid algorithms to solve real-life problems. In the process of product development, the newly developed political optimization algorithm (POA) was used and minimized the product cost, which is a new idea for industrial companies to fill the gap in their product design stage [34]. e new optimizer based on the ecogeographybased optimization algorithm (EBO) was applied to vehicle design for the first time, and better design results are obtained [35]. A new optimization algorithm based on grasshopper optimization algorithm and Nerdler-Mead algorithm (HGOANM) was developed to explore robot design of the robot gripper mechanism. e results showed that this algorithm can solve practical engineering problems quickly in Reference [36]. A new hybrid Taguchi salp swarm algorithm (HTSSA) was designed and used to speed up the optimization process of industrial structure design. e results reflected that the ability of HTSSA was superiority to optimize the product design process [37]. e new optimizer was developed, which is based on Seagull optimization (SOA), and its performance was verified by large-scale industrial engineering problems [38].
With the research and development of algorithms, the continuous development of optimization algorithm diversity is encouraged. A novel meta-heuristic approach based on human behavior for solving various complex optimization problems was introduced and called adolescent identity search algorithm (AISA) [39], which are first proposed by Esref Bogar and Selami Beyhan in 2020. is paper makes a series of improvements to AISA, which can make it performance better. e main contributions can be summarized as follows: (i) is work divides the iteration into three layers and makes full use of the update mechanism of each layer to obtain the best adolescent identity, which can enrich population diversity, and balance the capabilities of exploration and exploitation better. (ii) e current optimal solution guides other adolescents to combine Brownian motion, which can accelerate the convergence speed of the algorithm and prevent the algorithm from local optimization. (iii) Recursive least squares estimation (RLSE) is proposed to estimate the weight factor better. Optimizing the improved CFLN can improve the ability of exploration and exploitation, which makes the optimal solution and can be found more effectively by the algorithm. (iv) To prevent AISA into local optimum at the late iteration, a terminal bounce strategy is proposed. e structure of this paper is listed as follows. In Section 2, the adolescent identity search algorithm (AISA) is introduced. Section 3 describes FCBAISA in detail. e experimental comparison among FCBAISA and other algorithms is presented and discussed in Section 4. e practical engineering problems are solved by FCBAISA and comparison algorithms in Section 5. In Section 6, the summaries of this paper and the future work based on FCBAISA are listed.

The Canonical AISA
An optimization algorithm constructed on human behavior was called AISA by Esref Bogar and Selami Beyhan in 2020.
rough observing the formation process of adolescent identity and modeling it mathematically, a creative algorithm has been formed. is section briefly describes AISA, the details in Reference [39].

Population Random Initialization.
In AISA, a random initial population is generated by where x i j is the j th identity feature of the i th adolescent and U(0, 1) is an random number distributed uniformly in the range [0, 1]. lb is the lower boundary vectors of search space, and ub represents the upper.

Creating a New Identity.
According to the characteristics of adolescent identity exploration, it is assumed that a situation is randomly selected during the iterative update. e three cases of adolescent identity feature selection in this algorithm are as follows: Case 1. Teenagers form their identities by observing the surrounding society, judging social values, and choosing the correct beliefs and attitudes. Specifically, the Chebyshev functional-link network (CFLN) [40] approximation model is introduced to find the best adolescent identity, and the modeling process is as follows.
Chebyshev polynomials are shown in the following equation: where s is the degree of Chebyshev polynomials. Normalizing input samples (population) for the CFLN model in [−1, 1] by using the following equation: where x i j is normalized value of the j th identity feature of the i th adolescent. lb and ub are the lower and upper boundary vectors of search space. e identity is represented by the following normalized input matrix: en, according to (2), the matrix Ψ of each input element is obtained by (5), and Ψ is the regression matrix.
Weighting factors are estimated by using the least square estimation (LSE) in approximate model as follows: where ω j represents the weight vector of the j th input. All elements in (4) after normalization, the fitness values are calculated by (7) and stored in the matrix F.
Finally, the fitness values of the random initialization matrix elements are calculated and find the row index of the minimum value of each column in the matrix through the approximate model to form the best vector of identity of the present population, as shown in the following equation: In Case 1, new identity of the i th adolescent is defined as where r 1 ∈ [0, 1] represents a random number, and x * represents the best identity feature created by each teenager in (8). e (10) represents a new identity that adolescents strive to acquire from their peer group with good behaviors.

Case 2.
Believing that a role model has noble quality, good style, and imitating the role model to form the new identity.
Adolescents imitate the role model to form the new identity because they believe that a role model has noble quality and good style. erefore, adolescents can choose a better individual than themselves through learning. In this case, the updating formula for generate a new identity is written by the following equation: where r 2 ∈ [0, 1] is a random number, and x rm is the role model, which the best individual. When p ≠ rm, x p is an adolescent selected in the population randomly.
Case 3. Adolescents may be negatively affected by the group and form bad identity choices such as smoking, dropping out of school, and fighting. In this case, the updating formula for obtaining the new identity of the i th adolescent is written by the following equation: where r 3 ∈ [0, 1] is an n-dimensional vector of uniformly distributed numbers in the interval [0, 1], and x q is a Computational Intelligence and Neuroscience 3 negative identity vector and is written by the following equation: where x u is negative identity feature, which is an element randomly selected from the population matrix to make the algorithm that has the exploration capability.

The Proposed FCBAISA
Different from other meta-heuristic algorithms, AISA tries to find the fitness of adolescents and uses CFLN optimization. AISA performance is good in exploration, exploitation, avoidance of local optimization, and convergence. However, there are also some problems such as unbalanced exploration and exploitation abilities, falling into local optimum, and premature convergence. Adolescent identity development is a complex concept, which can integrate different network structures. erefore, new ideas can still be injected into the algorithm.

Hierarchical Optimization Strategy.
CFLN optimization method is very novel and effective for exploration, which is used in Case 1. In order to better play the role of CFLN, the iteration is divided into three layers to execute each update mechanism separately in this paper. is strategy can increase the diversity of the population and balance the abilities of exploration and exploitation better. In addition, improved CFLN topology in Section 3.3 has the better ability of exploration, as shown in Figure 2.

Quick Search Strategy.
is paper uses the current optimal solution (Gbest) to guide other adolescents in the whole search process and uses the characteristic that Brownian motion obeys standard normal distribution to design a fast search strategy to speed up the convergence speed of the algorithm. e Gbest guides other adolescents to update. In most cases, the optimal solution can be found faster. In addition, Brownian motion [41] is introduced to form a new update mechanism, because Brownian motion can replace random disturbance and effectively accelerate the convergence speed of the algorithm. is method enables teenagers to obtain the best adolescent identity as soon as possible, as shown in Figure 3.
Based on the current optimal solution (Gbest) and Brownian motion, as shown in (14), the formula of Case 1 is changed to the following equation.
where b 1 is n-dimensional Brownian motion. r 1 ∈ [0, 1] represents a random number, and b 2 and b 3 are two random numbers generated by Brownian motion. In addition, Brownian motion is integrated into Cases 2 and 3, and the corresponding update formulates are changed as (16) and (17), respectively.

RLSE Weight Factor
Strategy. e classical least square estimator (LSE) can be written as follows: where A 0 is a N × n matrix, X 0 � [X 1 , X 2 , . . . , X n ] T is a n × 1 parameter vector, and b 0 � [b 1 , b 2 , . . . , b N ] T is an output vector. e LSE can be given from following equations: In AISA, the weight factor is estimated by (19), which is the classical LSE recursive least squares estimation (RSLE)  [42] and is used to optimize the least square estimation and estimate the weigh factor of its approximate model.
where we eliminate A 0 and b 0 variables, In AISA, CFLN uses the LSE to estimate the weight factor of the approximate model. In this paper, a dynamic way to estimate the weight factor of the approximate model based on the LSE by learning from the recursive proof of RLSE. RLSE can dynamically estimate the weight factor of the approximate model and make CFLN more efficient as shown in Figure 2. FCBAISA can find the optimal solution more efficient by modifying the approximate model to affect the algorithm update mechanism. e formula is changed as (27).

Terminal Bounce Mechanism.
In this paper, a terminal bounce mechanism is designed to avoid the algorithm falling into local optimization in the later stage of iteration. Specifically, the algorithm may fall into local optimization if the number of iterations increases, especially in the later stage of iteration, while the value of global optimization does not change within the specified number of iterations. In this paper, the value of the timer is set to 20 and adds a counter to monitor the change of the global optimum value, which is the end disturbance mechanism that will be triggered when there is no change in the global optimum value after 20 iterations, which can make the algorithm jump out of the local optimum. In order to achieve better disturbance effect, two individuals are selected randomly from the population and the Gbest is added for guidance when designing the end disturbance strategy. e pseudocode is given by Algorithm 1, and the specific design of the terminal bounce mechanism is as formula (26).
where Gbest represents the current optimal solution, r ∈ [0, 1] denotes a random number, and ind(1) and ind (2) are two indexes generated from the population randomly. In summary, a fast convergence and balanced AISA is proposed (FCBAISA), Algorithm 2, and Figure 4 gives the pseudocode and flowchart of FCBAISA, respectively.

Benchmark Function and Comparison Algorithm.
e CEC2017 benchmark functions are applied to check the performance of FCBAISA in this paper. Among the CEC2017 benchmark functions [43], f 1 , f 3 , f 4 ∼ f 10 , f 11 ∼ f 20 , and f 21 ∼ f 30 are unimodal functions, simple multimodal functions, hybrid functions, and composite functions, respectively. f 2 has not been tested, the reason is the instability in high dimensional, and the details can be found in Reference [44]. e CEC2022 benchmark function includes unimodal function, basic functions, hybrid function, and composition function. ese benchmark functions are detailed in Tables 1 and 2. For checking the effectiveness and superiority of FCBAISA, it is compared with the performance of eight evolutionary algorithms. In order to be more fair and reasonable, the comparison algorithms include the classical algorithm and the new excellent algorithm. ese are as follows: transient search algorithm (TSO) [45], the Archerfish Hunting Optimizer algorithm (AHO) [46], butterfly optimization algorithm (BOA) [47], dynamic differential annealed optimization (DDAO) [48], PSO [6], owl search (1) counters � counters + 1; (2) if counters ≥ 20 then (3) r � rand; (4) Compute the New best by equation (26) (5) Boundary constraint process; (6) EFs � EFs + 1; Computational Intelligence and Neuroscience 5 algorithm (OSA) [49], and gravitational search algorithm (GSA) [50]. e contents of these algorithms are shown in Table 3. To compare the performance of algorithms fairly, the population size (N) of all algorithms is 30, the dimension (n) is 30, and each algorithm runs 50 times independently. e maximum number of function evaluations is 30000, and the maximum number of iterations is 1000 in the CEC2017 benchmark functions. e population size (N) of all algorithms is 30, the dimension (n) is 20, and each algorithm runs 50 times independently. e maximum number of function evaluations is 100000, and the maximum number of iterations is 3334 in the CEC2022 benchmark functions.

Comparison between FCBAISA and Other Algorithms.
In order to be more fair and reasonable, the comparison algorithm includes the classical algorithm and the new excellent algorithm. e results of CEC2017 and CEC2022 benchmark functions are shown in Tables 4 and 5, respectively. Among the CEC2017 benchmark functions, FCBAISA ranks first in 21, second in 6, and first after the comprehensive comparison. For other algorithms, the comprehensive performance of PSO is better, ranking third. From the mean comparison, it is found that FCBAISA performs better on 21 benchmark functions, and GSA and PSO perform better on four and three test functions, respectively. From the comparison of standard deviation, it is found that the stability of FCBAISA is poor, but it also ranks first in 15 benchmark functions. In complex problems, the stability of FCBAISA is improved. By comparing the optimal solutions of each algorithm, FCBAISA can find a better optimal solution among 16 benchmark functions in CEC2017 benchmark functions. In CEC2022 benchmark functions, FCBAISA ranks first in 10 benchmark functions, second in 2 benchmark Input: FCBAISA population size N, the lower and upper bounds of variables respectively: lb, ub, maximum number of iterations MaxIter, maximum number of function evaluations MaxFEs, the degree of Chebyshev polynomials: k; Output: the best Optimal solution (1) while(Iter < � MaxIter) and (FEs < � MaxFEs) do (2) Form the matrix X by equation (3) (3) Form the regressor matrix Ψ and its subregressor vectors (ψ 1 1 , . . . , ψ N n ) by equation (5)  (4) Compute the weight vectors (ω 1 , . . . , ω n ) by equations (23), (24) (5) Form the matrix F by c (6) Find the best feature vector (x * ) by equation (9)  (7) for i � 1 to Ndo (8) if FEs > MaxFEs/3 then (12) if Fs > MaxFEs/3∧FEs < 2MaxFEs/3 then (13) Find the best adolescent and best group position (x rm ) (14) Randomly choose one of the adolescents p|p ≠ rm (15) x Generate the negative identity vector (x q ) by (13)  (18) x end if (20) end if (21) Boundary constraint process; (22) Apply the updating mechanism: (23) for end if (28) iffit(X i ) < fit(Gbest)then (29) Gbest � X i ; (30) fit(Gbest) � fit(X i ); (31) counters � 0; (32) else (33) Execute Terminal bounce mechanism in Algorithm 1 (34) end if (35) end for (36) end for (37) end while (38) Return the best solution found ALGORITHM 2: Pseudocode of FCBAISA. 6 Computational Intelligence and Neuroscience functions, and first after the comprehensive comparison. From the mean comparison, it is found that FCBAISA performs better on 10 benchmark functions. It is concluded that the FCBAISA algorithm can effectively solve the simple and complex problems, especially when solving complex problems, it is better than other algorithms. In general, FCBAISA performs better in all aspects and can find the optimal solution quickly and efficiently in most benchmark functions.     Computational Intelligence and Neuroscience compare the performance of the algorithm from the overall point of view, and finally give the ranking and p_value. rough these tests, the performance of the improved algorithm can be well tested.
For the Wilcoxon rank test, its criterion is when the significance level is 0.05, when p_value ≤ 0.05, if R + < R − is marked as "+," FCBAISA and other algorithms are significantly better. On the contrary, it will be marked as "+,"   Computational Intelligence and Neuroscience 9      Computational Intelligence and Neuroscience indicating that FCBAISA performs worse than other algorithms. If there is no significant difference between FCBAISA and other algorithms, it will be marked as "�." In Tables 6 and 7, the last row gives the sum of each tag to judge whether FCBAISA has significant advantages over other comparison algorithms. In comparison with other improved algorithms, FCBAISA got "+" on at least 25 benchmark functions, except for 20 "+" compared with PSO, and only two "−." To sum up, these tests show that the performance of FCBAISA is better than other comparing algorithms. For Friedman and Quade test, the significance level is 0.05, if p value ≤ 0.05, indicating that the test result is true. e results of the Friedman and Quade tests are shown in Tables 8 and 9 and indicate that the FCBAISA ranks first. After the Friedman test result in Table 8, the FCBAISA has a p value of 1.2942E-10 and the test result is 8.6552. For Quade test result in Table 9, it can be observed that FCBAISA's final ranking is number one. FCBAISA's result is 8.8229 with a p value of 2.2054E-43 in the Quade test. In summary, after three statistical tests, it can be proved that FCBAISA has significant advantages over 8 other comparative algorithms, including improved algorithms and other excellent algorithms.

Practical Engineering Problems
is section uses FCBAISA and all comparison algorithms to solve several problems of engineering design. e superiority of FCBAISA is further tested by analyzing those experimental results. Engineering design problems include pressure vessel design [51], welded beam design [52], gear train engineering design [53,54], and speed reducer design, and the details are as follows.

e Problem of Pressure Vessel Design.
Pressure vessels are designed to minimize costs. e pressure vessel, as shown in Figure 7, consists of a cylindrical center and hemispherical heads at both ends, where L (x 4 ), T s (x 1 ), T h (x 2 ), and R(x 3 ) are the length of the cylindrical part, the thickness of the shell, the thickness of head, and the inner radius, respectively.
is problem consists of four constraints, including three linear inequalities and a nonlinear inequality, and its model is shown in the following equation.
In Table 10, it can be seen that the optimal solution of each algorithm in solving the problem of pressure vessel design is 6.06E-10, and FCBAISA is better than that of other algorithms. e convergence curve of the algorithm involved in this paper on pressure vessel problem is shown in Figure 8.   linear and five nonlinear inequality, its model as in equation (28), ������ � x 2 3 x 6 4 /36/L 2 1 − x 3 /2L
In Table 11, it can be seen that the optimal solution of each algorithm in solving the problem of welded beam design is 2.0632, and FCBAISA is better than that of other algorithms. e convergence curve of the algorithm involved in this paper on welded beam design problem is shown in Figure 10.

e Problem of Gear Train Engineering
Design. e problem of gear train engineering design is to find the minimum value of gear and tooth ratio without affecting the efficiency as shown in Figure 11. e number of teeth must be an integer; thus, the design variables for this problem are discrete. Because constraints are constraints on design variables, the problem of constraints on discrete variables can increase its complexity. So, in this design problem, n A , n B , n D , and n F are decision variables, the integer variable of the upper bound is 60, and the lower is 12. Besides, the gear ratio is defined as (n B n D )/(n F n A ), this specific problem can be modelled as (29), where x � [x 1 , x 2 , x 3 , x 4 ] � [n A , n B , n D , n F ].
In Table 13, it can be seen that the optimal solution of each algorithm in solving the problem of speed reducer design is 3.00E + 03, and FCBAISA is better than that of other algorithms obviously. e convergence curve of the algorithm involved in this paper on speed reducer design problem is shown in Figure 14.

Conclusion
A fast convergence and balanced adolescent identity search algorithm (FCBAISA) is proposed in this work for numerical and engineering design problems to advance the quality of AISA. To balance the exploration and exploitation of FCBAISA better, a layered optimization strategy is proposed. A fast search strategy is proposed to make the algorithm break away from the local optimization and converge to the optimal value faster. e CFLN is improved by RSLE to obtain the optimal result effectively. A terminal disturbance strategy is designed to prevent the algorithm from local optimization in the later iteration. e CEC2017 benchmark functions, CEC2022 benchmark functions, and the design problems of engineering are applied to check the quality of FCBAISA. It is clear that FCBAISA has high precision, fast convergence speed, strong exploration, and exploitation ability, and the balance between them is better. In addition, future research can be carried out from the following aspects: (1) Further improvement of FCBAISA, including the Chebyshev approximation model and other effective alternative models. (2) Trying to apply FCBAISA to the problems of multiobjective optimization, and considering the combination of specific practical problems, including scheduling optimization and engineering problems.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors would like to submit the enclosed manuscript entitled "Adolescent identity search algorithm based on fast search and balance optimization for numerical and engineering design problems." e authors wish to be considered for publication in "Computational Intelligence and Neuroscience." No conflicts of interest exit in the submission of this manuscript, and manuscript is approved by all authors for publication. e authors would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.