Optimization method, system and equipment for improving quadratic interpolation optimization algorithm based on periodic variation and storage medium

Abstract

The invention discloses an optimization method, system and equipment for improving a quadratic interpolation optimization algorithm based on periodic variation, and a storage medium, and belongs to the field of machine engineering. A mathematical model is constructed aiming at the design problem of a tension / compression spring, a quadratic interpolation optimization QIO algorithm is combined with periodic variation, Cauchy inverse cumulative distribution and a differential evolution algorithm to construct a DCVQIO algorithm, a Cauchy inverse cumulative distribution function is introduced in the exploration stage of the QIO algorithm to form a CQIO algorithm, a search mechanism of quadratic interpolation optimization is strengthened, the exploration capability is reserved, and a CQIO algorithm is constructed. A periodic variation strategy is introduced between a DE algorithm and a CQIO algorithm to improve algorithm diversity, the DE algorithm and the CQIO algorithm serve as low-level and high-level heuristic algorithms, the periodic variation strategy is fused to improve algorithm optimization precision, a DCVQIO algorithm is adopted to optimize mathematical model parameters, an optimal target solution and an optimal parameter combination are obtained, and therefore an optimal spring structure design scheme is output. According to the method, the performance of a quadratic interpolation optimization algorithm in solving a tension spring design problem is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of mechanical engineering, and in particular relates to an optimization method for improving a quadratic interpolation optimization algorithm based on periodic variation. Background Art

[0002] Against the backdrop of intensified global competition, increasing resource scarcity is driving companies to pursue cost-effectiveness maximization. This phenomenon can be transformed into a constrained extreme value solution model through mathematical modeling, a typical engineering optimization topic in modern operations research.

[0003] As a key component in mechanical systems, the constrained optimization of tension / compression springs (TCSDs) directly impacts the performance of industrial equipment. Their design quality not only affects the operational stability of equipment such as automotive suspension systems and precision instruments, but also determines energy efficiency and production economics. Traditional optimization algorithms are prone to premature convergence when dealing with such nonlinear constraints, exposing flaws such as high parameter sensitivity and low iteration efficiency. More advanced global optimization strategies are urgently needed to overcome these technical bottlenecks.

[0004] The quadratic interpolation optimization (QIO) algorithm is a new type of swarm intelligence optimization algorithm proposed by Zhao Weiguo and others in 2023. The quadratic interpolation optimization (QIO) algorithm uses the GQI method to establish a quadratic model to quickly and accurately locate extreme points. The hybrid framework combining QIO and particle swarm optimization (PSO) algorithms enhances the robustness and adaptability in applications and has been applied to photovoltaic parameter identification. The advantages of the QIO algorithm are that it effectively avoids local optimality and premature convergence, has few parameters, is simple, and is easy to implement. However, despite its superior performance, the QIO algorithm also has disadvantages, like current heuristic algorithms, namely, it quickly falls into local optimality and cannot balance local development and global exploration well. Summary of the Invention

[0005] To address the drawback of classical optimization algorithms, which are prone to falling into local optimal solutions when solving tension / compression spring design problems, resulting in insufficient solution accuracy, this paper innovatively proposes an improved quadratic interpolation optimization algorithm based on periodic variation (referred to as a super-heuristic algorithm, DCVQIO). This method introduces the Cauchy inverse cumulative distribution function into the exploration phase of the quadratic interpolation optimization algorithm, forming the Cauchy quadratic interpolation optimization (CQIO) algorithm. This enhances the search mechanism of QIO while retaining its powerful exploration capabilities. Secondly, a periodic variation strategy is introduced between the differential evolution algorithm (DE) and the CQIO algorithm to increase the diversity of DCVQIO and prevent the algorithm from falling into local optimal solutions. Finally, a hybrid strategy is proposed, which uses DE and CQIO as low-level and high-level heuristic algorithms, respectively, and integrates the periodic variation strategy to improve the algorithm's optimization accuracy. The DCVQIO algorithm proposed in this paper improves the performance of the quadratic interpolation optimization algorithm in solving tension / compression spring problems.

[0006] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a tension and compression spring design method based on a periodic variation improved quadratic interpolation optimization algorithm, the steps of which are as follows:

[0007] Step 1: Construct a mathematical model for the tension / compression spring optimization problem;

[0008] Step 2: The quadratic interpolation optimization (QIO) algorithm is combined with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct an improved quadratic interpolation optimization algorithm;

[0009] Step 3: Using the improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring optimization problem, the best target optimal solution and optimal parameter combination are obtained;

[0010] Step 4: Output the optimal structural design scheme of the spring based on the optimal decision variables of the mathematical model solved in step 3.

[0011] Furthermore, the mathematical model construction method for the above-mentioned tension / compression spring optimization problem is as follows:

[0012] The design variables are defined as a three-dimensional vector X = [x1, x2, x3], which correspond to the parameters of wire diameter d, average coil diameter D and effective coil number L respectively. The optimization objective function is set as f(X) = x1 2 x2x3+2x1 2 x2;

[0013] Identify four constraints:

[0014] Interference:

[0015]

[0016] Shear stress:

[0017]

[0018] Fluctuation frequency:

[0019]

[0020] Outer diameter:

[0021]

[0022] Among them, x1∈[0.05,2.00], x2∈[0.25,1.30], x3∈[2.00,15.0].

[0023] Furthermore, in the above step three, the three optimization parameters of the tension / compression spring are converted into population individuals; in the final stage, the position of the optimal individual is analyzed and the three optimal engineering parameters are obtained in reverse to form an optimized design scheme for the spring structure.

[0024] The steps of improving the quadratic interpolation optimization algorithm based on periodic variation are as follows:

[0025] Step 1: Convert the three optimization parameters of the tension / compression spring: wire diameter, average coil diameter, and number of effective coils into population individuals;

[0026] Step 2: Randomly initialize the population, set the population size n, dimension m, search space upper and lower limits Low and Up, and maximum number of iterations T;

[0027] Step 3: Calculate the fitness value of the population and sort it;

[0028] Step 4: Execute the Differential Evolution algorithm (DE) until the limit number T is reached DE until;

[0029] Step 5: Perform periodic mutation on all individuals;

[0030] Step 6: Improve the quadratic interpolation optimization (QIO) algorithm through the Cauchy inverse cumulative distribution, obtain the CQIO algorithm and execute it to obtain the updated individual;

[0031] Step 7: Iteration number t=t+1, determine whether the iteration number reaches the maximum iteration number T. If so, the iteration ends and the individual corresponding to the optimal fitness value, that is, the optimal solution, is output; otherwise, return to step 4.

[0032] Furthermore, the method of cyclically executing the Differential Evolution (DE) algorithm in step 4 above is:

[0033] The mutation operation of the differential evolution algorithm is performed through formula (a);

[0034] Specifically: perform the mutation operation of the differential evolution algorithm to create three random indices in [1, n], where n is the population size, and select three solution vectors X for the given index from the current solution in the search space r1 、X r2 and X r3 To generate a new solution V i .

[0035] V i =X rand1 +F*(X rand2 -X rand3 )(a)

[0036] The crossover operation of the differential evolution algorithm is performed by formula (b).

[0037] Specifically: perform the crossover operation of the differential evolution algorithm, and according to formula (b) from the new mutation solution vector V i and the original solution vector X i Generate U i :

[0038]

[0039] Where CR is the crossover rate; j rand is a random number in the range [1, m] chosen randomly from the solution vector, where m is the dimension of the problem.

[0040] The selection operation of the differential evolution algorithm is performed through formula (c).

[0041] Specifically: perform the selection operation of the differential evolution algorithm to select the best candidates based on their fitness value f(.) in X i or U i The one with better fitness value will survive to the next iteration. Considering the minimization optimization problem, the selection operation can be expressed as follows:

[0042]

[0043] From formula (c), we can observe that the differential evolution algorithm population tends to get better or remain the same, but never gets worse.

[0044] Determine whether the number of iterations T of the differential evolution algorithm has been reached DE If it is reached, execute step 5; if not, continue to execute step 4.

[0045] Furthermore, the method of period variation in step 5 above is:

[0046] Periodic mutation activation: The generalized mutation operation including the mutation strategy is described as follows:

[0047] x i,j (t) = F(g(x i,j (t)),fr)(1)

[0048] Where F is the generalized mutation function, g(xi, j(t)) is the generation vector provided by the mutation operator, and fr is the user-defined fluctuation frequency. Mutation strategies focus on studying how to apply the mutation operator during the optimization process. This operator is called the global mutation operator and is given by Equation (2).

[0049] x i,j (t) = x i,j (t)[1+A(0.5-rand)δ]

[0050] i=1,2,...,s

[0051] j=1,2,...,m

[0052]

[0053] Where A is a user-defined amplitude factor, which can be selected as a fixed number or calculated during the iteration process; rand is a random number generator that conforms to N[0, 1]. When applied, a Gaussian probability density function is used. However, other density functions can also be used in (2). Such a mutation strategy is designed to avoid premature convergence to local and obtain the correct search pattern.

[0054] Furthermore, the strategy of the CQIO algorithm in step 6 above is to introduce the Cauchy inverse cumulative distribution function into the exploration phase of the quadratic interpolation optimization (QIO) algorithm. The overall implementation steps are:

[0055] Randomly select 2 individuals from the population.

[0056] When rand>0.5(q(t)>0.5), the algorithm obtains the minimum value of the GQI function of the current individual and two randomly selected individuals through formula (3);

[0057]

[0058] w1=3n1b (4)

[0059]

[0060]

[0061] Among them, x rand1 、x rand2 and x rand3 are the positions of three different individuals randomly selected from the current population, r1, r2 and r3 are random numbers in (0, 1), and fit(.) represents the fitness value of the function. i (t), x rand1 (t), x rand2 (t), fit(x i (t)), fit(x rand1 (t)), fit(x rand2 (t)) is the GQI function, through which we can get the i , fit(x i )),(x rand2 , fit(x rand2 )) and (x rand3 ,fit(x rand3 The three randomly selected individuals in (7) can effectively expand the search range, and the GQI method helps to locate potential areas.

[0062]

[0063] The improved QIO algorithm uses formula (8) to perform exploration. Formula (8) introduces the Cauchy inverse cumulative distribution function f on the basis of the original formula (7). -1 (p; a, b).

[0064]

[0065] p=randn(1,m)(10)

[0066] In Equation (9), p is a uniformly distributed random number in the range [0, 1], calculated by Equation (10); m is the dimension of the function, the location parameter a is assigned to 0, which is consistent with the standard Cauchy distribution, and the scaling parameter b is assigned to 0.01. This setting prevents the algorithm from being too aggressive during the search process.

[0067] When rand <= 0.5 (q(t) <= 0.5), the algorithm obtains the minimum value of the GQI function of the best individual found so far and the two randomly selected individuals through formula (11), and performs development using formula (13).

[0068]

[0069] Where n2 follows the standard normal distribution, rD is a random integer from [1, m], and Lb rD and UbrD are the lower and upper bounds of the (rD)th dimension. w2 is the development weight as an adaptive coefficient. According to Equation (11), the best individual x found so far is used in the GQI method. best and two randomly selected individuals x rand1 and x rand2 To produce the minimized value x' best，rand1，rand2 , which is usually better than three individuals. From Equation (13), by using the i rD The random perturbation of x best Search for x' best，rand1，rand2 This search can help the algorithm develop effectively without stagnation.

[0070] After the exploration or development is completed, the best solution so far is found by formula (14), that is, if the fitness value of the candidate position of the i-th individual is better than the fitness value of the current position, the position of the i-th individual is replaced by its candidate position. Otherwise, it remains unchanged.

[0071]

[0072] In a second aspect, the present invention provides a computer system, characterized in that the system includes steps for implementing an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm as described above.

[0073] In a third aspect, the present invention provides a computer device comprising a memory and a processor, wherein a computer program is stored in the memory, and when the processor runs the computer program stored in the memory, the steps of the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation as described above are executed.

[0074] In a fourth aspect, the present invention provides a computer-readable storage medium, in which a plurality of computer instructions are stored, and the plurality of computer instructions are used to enable a computer to execute an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm as described above.

[0075] The beneficial effects of the present invention are:

[0076] This paper proposes an optimization method based on periodic mutation to improve the quadratic interpolation optimization algorithm (called the super-heuristic algorithm DCVQIO). First, the inverse Cauchy cumulative distribution function is introduced into the exploration phase of the quadratic interpolation optimization (QIO) algorithm to form the CQIO algorithm, enhancing the algorithm's search mechanism. Second, the differential evolution (DE) algorithm and the CQIO algorithm are used as low-level and high-level heuristic algorithms, respectively, balancing the algorithm's development and exploration capabilities. Finally, a periodic mutation strategy is adopted between the DE and CQIO algorithms to increase algorithm diversity and prevent premature convergence to local optima.

[0077] 2. Differential Evolution (DE) is considered an excellent evolutionary algorithm for solving global optimization problems, offering advantages such as simplicity, ease of implementation, and good convergence. In particular, the DE operator and DE / rand / 1 possess excellent exploration capabilities. Furthermore, the Quadratic Interpolation Optimization (QIO) algorithm utilizes the GQI method for the currently discovered best individuals, facilitating algorithm development and offering advantages such as low parameter count and simplicity. Differential evolution algorithms generate diverse solutions through mutation and crossover, but due to selection pressure, they are prone to premature convergence to suboptimal regions. The QIO algorithm utilizes the GQI method to establish a quadratic model, quickly and accurately locating extreme points, thus compensating for the local development limitations of differential evolution algorithms. The local mechanism information provided by the QIO algorithm can reverse the global search direction of the differential evolution algorithm, forming a dynamic closed-loop optimization. However, many other heuristic algorithms are too complex to implement this interaction. On the other hand, the QIO algorithm has fewer parameters, avoiding the stability issues associated with parameter sensitivity of traditional heuristic algorithms. Therefore, the present invention uses the differential evolution algorithm and the QIO algorithm (CQIO) improved by the Cauchy inverse cumulative distribution function as the low-level and high-level heuristic algorithms, respectively, to achieve a balance between development and exploration and find the global optimal solution.

[0078] 3. This paper proposes a solution using the DCVQIO algorithm for the classic structural optimization problem of tension / compression spring design (TCSD). Experimental data demonstrates that the DCVQIO algorithm successfully optimizes the spring mass to 0.012665, significantly improving the optimization accuracy compared to the QIO prototype algorithm. In comparative tests, the algorithm significantly outperformed seven high-performing optimizers, including QIO, AOA, CDO, HHO, SAO, SCSO, and PSO, in terms of convergence accuracy, validating the effectiveness and advancement of the DCVQIO algorithm under engineering constraints.

[0079] The present invention is applicable to the constrained optimization of tension / compression spring design (TCSD) in key components of mechanical systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following is a brief introduction to the drawings required for use in the specific embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0081] Figure 1 It is a flow chart of the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation proposed by the present invention;

[0082] Figure 2 It is a flow chart of the DCVQIO algorithm described in the present invention. DETAILED DESCRIPTION

[0083] The specific embodiments of the present invention are further described below in conjunction with the accompanying drawings. The following embodiments will help those skilled in the art further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that those skilled in the art may make various changes and improvements without departing from the scope of the present invention, and these are all within the scope of protection of the present invention.

[0084] Implementation method 1, see Figure 1 This embodiment describes the problem that the classical optimization algorithm is prone to fall into a local optimal solution when solving the tension / compression spring design problem, resulting in insufficient solution accuracy. Therefore, an improved quadratic interpolation optimization algorithm based on periodic variation is innovatively proposed. The method includes the following steps:

[0085] like Figure 1 As shown:

[0086] Step 1: Construct a mathematical model for the tension / compression spring optimization problem;

[0087] Step 2: The quadratic interpolation optimization (QIO) algorithm is combined with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct an improved quadratic interpolation optimization algorithm;

[0088] Step 3: Using the improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring optimization problem, the best target optimal solution and optimal parameter combination are obtained;

[0089] Step 4: Output the optimal structural design scheme of the spring based on the optimal decision variables of the mathematical model solved in step 3.

[0090] Implementation method 2, see Figure 2 This embodiment is described. This embodiment specifically describes the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation proposed in the first embodiment above.

[0091] Step 1: Construct a mathematical model for the tension / compression spring optimization problem;

[0092] Specifically:

[0093] This implementation addresses the multidimensional optimization challenges in tension / compression spring design. Its mathematical modeling requires a clear core optimization objective and multidimensional constraints. The spring's geometric configuration parameters are defined as a three-dimensional design vector consisting of the wire diameter x1 (d), average coil diameter x2 (D), and number of active coils x3 (L). The quantitative model of this system requires constructing an optimization function centered on mass minimization, supplemented by multidimensional boundary conditions such as perturbation, shear stress, fluctuation frequency, and outer diameter.

[0094] The target variable, i.e., the individual in the DCVQIO algorithm population, is: X = [x1, x2, x3] = [d, D, L];

[0095] Objective function: minf(X)=x1 2 x2x3+2x1 2 x2;

[0096] Four constraints:

[0097] Disturbance (corresponding to spring stiffness requirements):

[0098]

[0099] Shear stress (to ensure material strength):

[0100]

[0101] Fluctuation frequency (to avoid resonance effect):

[0102]

[0103] Outer diameter (meet assembly space requirements):

[0104]

[0105] The value ranges of the above design variables are:

[0106] Parameter wire diameter x1∈[0.05,2.00];

[0107] Average coil diameter x2∈[0.25,1.30];

[0108] The number of effective coils x3∈[2.00,15.0].

[0109] The mathematical model proposed in this implementation achieves optimal structural design through multi-dimensional parameter optimization, while meeting the spring's mechanical performance requirements. Constraints encompass multiple requirements, including material strength, operating characteristics, and geometric dimensions, ensuring the design is both engineering-feasible and economical.

[0110] Step 2: The quadratic interpolation optimization (QIO) algorithm is combined with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct an improved quadratic interpolation optimization algorithm;

[0111] Specifically:

[0112] Regarding the spring optimization problem solution process, the key design parameters (d, D, L) of the TCSD problem are mapped to the multidimensional position vector space of the individual in the DCVQIO algorithm, forming a three-dimensional solution vector X = (x1, x2, x3). Therefore, this embodiment combines periodic variation, Cauchy inverse cumulative distribution, and differential evolution algorithms to construct an improved quadratic interpolation optimization algorithm.

[0113] Step 3: Using the improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring optimization problem, the best target optimal solution and optimal parameter combination are obtained;

[0114] Specifically:

[0115] By integrating the differential evolution (DE) algorithm, the quadratic interpolation optimization (QIO) algorithm, the inverse Cauchy cumulative distribution function, and the periodic mutation strategy, a hyper-heuristic algorithm, DCVQIO, was constructed. The optimal individual was obtained by executing the iterative optimization process of the DCVQIO algorithm.

[0116] The main inspiration behind the Quadratic Interpolation Optimization (QIO) algorithm comes from mathematics, especially the newly proposed generalized quadratic interpolation (GQI) method. This method overcomes the limitations of traditional quadratic interpolation methods and can better minimize the quadratic function composed of any three points. QIO uses the GQI method as a promising search mechanism to handle various types of optimization problems. This search mechanism provides an exploration and exploitation strategy, where the minimum provided by the GQI method helps the QIO algorithm explore promising areas in unexplored areas and develop optimal solutions in promising areas.

[0117] The implementation process of the quadratic interpolation optimization (QIO) algorithm is as follows:

[0118] (1) Randomly initialize the population, set the population size n, dimension m, and maximum number of iterations T.

[0119] (2) Calculate the fitness value of the population and sort it.

[0120] (3) Randomly select 2 individuals from the population.

[0121] (4) When rand>0.5, the algorithm obtains the minimum value of the GQI function of the current individual and two randomly selected individuals through formula (3) and performs exploration using formula (7).

[0122]

[0123] w1=3n1b (4)

[0124]

[0125] Among them, x rand1 、x rand2 and x rand3 are the positions of three different individuals randomly selected from the current population, r1, r2 and r3 are random numbers in (0, 1), and fit(.) represents the fitness value of the function. i (t), x rand1 (t), x rand2 (t), fit(x i (t)), fit(x rand1 (t)), fit(x rand2 (t)) is the GQI function, through which we can get the i ,fit(x i )),(x rand2 ,fit(x rand2 )) and (x rand3 ,fit(x rand3 The three randomly selected individuals in formula (7) can effectively expand the search range, and the GQI method helps to locate potential areas.

[0126] (5) When rand <= 0.5, the algorithm obtains the minimum value of the GQI function of the best individual found so far and the two randomly selected individuals through formula (11), and performs development using formula (13).

[0127]

[0128] where n2 follows a standard normal distribution, rD is a random integer from [1, m], and Lb rD and Ub rD are the lower and upper bounds of the (rD)th dimension. w2 is the development weight as an adaptive coefficient.

[0129] According to formula (9), the best individual x found so far is used in the GQI method best and two randomly selected individuals xrand1 and x rand2 To produce the minimized value x' best，rand1，rand2 , which is usually better than three individuals. From Equation (11), by using the i rD The random perturbation of x best Search for x' best，rand1，rand2 This search can help the algorithm develop effectively without stagnation.

[0130] (6) When the exploration or development is completed, the best solution so far is found through formula (14), that is, if the fitness value of the candidate position of the i-th individual is better than the fitness value of the current position, the position of the i-th individual is replaced by its candidate position.

[0131] Otherwise remain unchanged.

[0132]

[0133] (7) Determine whether the set termination condition is met (the number of iterations T is reached). If not, go to step (3). Otherwise, the algorithm terminates and returns the best solution found so far.

[0134] The QIO algorithm utilizes the GQI method for the currently discovered best individuals, which facilitates algorithm development and has the advantages of few parameters and simplicity. Although the QIO algorithm has superior performance, like current heuristic algorithms, it also has the disadvantage of being easily trapped in local optimality and failing to strike a good balance between local development and global exploration.

[0135] To address this issue, this implementation proposes an optimization algorithm based on periodic variation to improve the quadratic interpolation optimization (QIO) algorithm (called a hyper-heuristic algorithm, DCVQIO). Three strategies are added to enhance the performance of the QIO algorithm: the Cauchy inverse cumulative distribution function, periodic variation, and differential evolution. Furthermore, the proposed DCVQIO algorithm is validated to be highly competitive using the CEC2017 benchmark. Finally, the DCVQIO algorithm is applied to a tension / compression spring design problem, demonstrating its practical applicability.

[0136] The process of the proposed DCVQIO algorithm is as follows: Figure 2 As shown;

[0137] (1) Randomly initialize the population, set the population size n, dimension m, search space upper and lower limits Low and Up, and maximum number of iterations T.

[0138] (2) Calculate the fitness value of the population and sort it.

[0139] (3) Perform the mutation operation of the differential evolution algorithm DE through formula (a).

[0140] V i =X rand1 +F*(X rand2 -X rand3 )(a)

[0141] (4) The crossover operation of the differential evolution algorithm DE is performed through formula (b).

[0142]

[0143] (5) The selection operation of the differential evolution algorithm DE is performed through formula (c).

[0144]

[0145] (6) Determine whether the number of iterations (T DE ), if it is reached, execute step (7), otherwise jump to step (3).

[0146] (7) Perform periodic mutation on all individuals using formula (2), which is one of the improvements of the proposed DCVQIO algorithm.

[0147] (8) Randomly select 2 individuals from the population.

[0148] (9) When rand>0.5 (q(t)>0.5), the algorithm obtains the minimum value of the GQI function of the current individual and two randomly selected individuals through formula (3), and performs exploration using formula (8). Formula (8) introduces the Cauchy inverse cumulative distribution function f- on the basis of formula (7). 1 (p; a, b).

[0149] The second improvement of the proposed DCVQIO algorithm mainly introduces the Cauchy inverse cumulative distribution function in the exploration phase of the QIO algorithm, which thoroughly searches the entire variable space to identify promising regions that may contain the global optimal solution, thereby avoiding the ability of local optimality or premature convergence.

[0150] (10) When rand <= 0.5 (q(t) <= 0.5), the algorithm obtains the minimum value of the GQI function of the best individual found so far and the two randomly selected individuals through formula (11) and performs development using formula (13).

[0151] (11) When the exploration or development is completed, the best solution so far is found by formula (14), that is, if the fitness value of the candidate position of the i-th individual is better than the fitness value of the current position, the position of the i-th individual is replaced by its candidate position. Otherwise, it remains unchanged.

[0152] (12) Determine whether the set termination iteration number T has been reached. If not, go to step (3). Otherwise, the algorithm terminates and returns the best solution found so far.

[0153] Step 4: Through the encoding mechanism adopted in step 2, the optimal group individuals are reversely transformed to obtain the key solution parameters of the TCSD problem, which are the optimal structural configuration schemes corresponding to the spring system.

[0154] The improved quadratic interpolation optimization algorithm based on periodic variation (called a super heuristic algorithm DCVQIO) proposed in this embodiment introduces the Cauchy inverse cumulative distribution function in the exploration phase of the quadratic interpolation optimization (QIO) algorithm to form the Cauchy quadratic interpolation optimization (CQIO) algorithm, which enhances the search mechanism of QIO while retaining its powerful exploration capability. Secondly, a periodic variation strategy is introduced between the differential evolution algorithm (DE) and the CQIO algorithm to increase the diversity of DCVQIO and prevent the algorithm from falling into local optimality. Finally, a hybrid strategy is proposed, which uses DE and CQIO as low-level and high-level heuristic algorithms, respectively, and integrates the periodic variation strategy to improve the optimization accuracy of the algorithm. The DCVQIO algorithm proposed in the present invention improves the performance of the quadratic interpolation optimization algorithm in solving tension / compression spring problems.

[0155] Implementation method 3: This implementation method is to verify the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation described in the above implementation method 1 or implementation method 2;

[0156] The proposed DCVQIO algorithm is compared with the basic QIO algorithm, the Arithmetic Optimization Algorithm (AOA), the Chernobyl Disaster Optimization (CDO), the Harris Hawk Algorithm (HHO), the Snow Melt Optimization Algorithm (SAO), the Sand Cat Swarm Optimization (SCSO), and the Particle Swarm Optimization (PSO) algorithm. Table 1 shows the comparative results of various algorithms for the tension / spring design problem. The data in Table 1 demonstrates that the proposed DCVQIO algorithm demonstrates significant advantages in solving the TCSD problem. The optimal structural configuration achieved by the proposed DCVQIO algorithm achieves a spring system weight index of 0.012665, achieving the highest accuracy among the comparison group compared to the basic QIO algorithm, the Arithmetic Optimization Algorithm (AOA), the Chernobyl Disaster Optimization (CDO), the Harris Hawk Algorithm (HHO), the Snow Melt Optimization (SAO), the Sand Cat Swarm Optimization (SCSO), and the Particle Swarm Optimization (PSO) algorithms. Engineering examples demonstrate that the hyper-heuristic DCVQIO algorithm, which integrates the Cauchy inverse cumulative distribution function, periodic mutation, and differential evolution algorithms, significantly outperforms the original QIO, AOA, CDO, HHO, SAO, SCSO, and PSO algorithms in terms of solution quality, confirming the effectiveness of its enhancement mechanism. This research further demonstrates the practical engineering value of DCVQIO as an optimization tool for TCSD problems, and its multi-strategy coupling framework provides a new paradigm for solving complex system design problems.

[0157] Table 1

[0158]

[0159] To further validate the performance of the DCVQIO algorithm, we selected 29 benchmark functions from the CEC2017 competition for optimization. First, to highlight the superiority of the DCVQIO algorithm, we compared the performance of QIO, AOA, LFD, HHO, SAO, SCSO, and PSO. We then compared these with various improved strategies to verify the effectiveness of the improvements in the DCVQIO algorithm.

[0160] DCVQIO was tested using 29 CEC2017 benchmark functions. The CEC2017 benchmark set can be divided into four categories: unimodal (F1, F3), multimodal (F4-F10), hybrid (F11-F20), and composite (F21-F30). Multimodal test functions are suitable for testing exploratory operations in optimization algorithms. Both composite and mixed functions are used to evaluate the avoidance of local optima by optimization algorithms, while unimodal test functions are used to evaluate exploitative operations because they only contain a single global optimum. It should be noted that F2 was removed from the benchmark set due to its unstable behavior.

[0161] The DCVQIO algorithm was compared with seven optimization algorithms: QIO, AOA, LFD, HHO, SAO, SCSO, and PSO. Each function was run 30 times independently, and the average (avg) and standard deviation (std) of the best solution from each run were considered. Detailed experimental results are shown in Table 2 (Comparison of Optimization Results of Eight Algorithms on the 30-Dimensional CEC2017 Test Set). To verify significant differences between the DCVQIO algorithm and the compared algorithms, a nonparametric Wilcoxon rank sum test was used; the relevant experimental results are shown in Table 2. The algorithms were ranked according to the average avg values in Table 2.

[0162] In Table 2, the best value for each metric is highlighted in bold. As can be seen from Table 2, for 30 dimensions, the DCVQIO algorithm achieved the best average optimization results for 25 of the 29 functions. The QIO, AOA, LFD, HHO, SAO, SCSO, and PSO algorithms achieved the best average optimization results of 0, 1, 0, 0, 3, 0, and 0, respectively. The final ranking, using the average value (avg) of the best solutions for the 29 functions, shows that DCVQIO ranked first, demonstrating superior optimization performance and outperforming the other compared algorithms in avoiding local extrema. This result is due to the fact that the DE and CQIO algorithms, as low-heuristic and high-heuristic algorithms, respectively, enable the DCVQIO algorithm to achieve a smooth transition between exploration and production. The use of periodic variation and the Cauchy inverse cumulative distribution function significantly improves the algorithm's population diversity and spatial search capabilities.

[0163] Observing the standard deviation of the best solution in Table 2, for 30 dimensions, DCVQIO, QIO, AOA, LFD, HHO, SAO, SCSO, and PSO algorithms obtain the best results of 19, 2, 0, 0, 0, 6, 0, and 2 respectively in 29 test functions, which shows that the DCVQIO algorithm has higher stability.

[0164] Table 2

[0165]

[0166]

[0167]

[0168]

[0169] Furthermore, to verify the significant differences between the DCVQIO algorithm and the comparison algorithms, a nonparametric test was performed using the Wilcoxon rank-sum test. At a 5% test level, if p ≤ 0.05, it indicates that the two algorithms have a significant difference on a function; otherwise, the difference is not significant. The test results of the DCVQIO algorithm and the comparison algorithm on the 30-dimensional CEC2017 function are shown in Table 3 (30-dimensional Wilcoxon rank-sum test results on the CEC2017 test function). Cases with p values less than or equal to 0.05 are marked in bold. From the data in Table 3, the majority of the p values are less than 0.05, indicating that the DCVQIO algorithm performs significantly better than the QIO, AOA, LFD, HHO, SAO, SCSO, and PSO algorithms.

[0170] Therefore, the Cauchy inverse cumulative distribution function, periodic mutation, and differential evolution algorithms enhance the balance between exploration and exploitation of the proposed algorithm DCVQIO, enabling it to escape from the local optimal area and achieve better performance.

[0171] To further illustrate the impact of differential evolution (DE), the Cauchy inverse cumulative distribution function (ICDF), and periodic mutation on the quadratic interpolation optimization (QIO) algorithm, ablation comparison experiments were conducted. The algorithms that improved QIO using the differential evolution (DE) algorithm, the Cauchy inverse cumulative distribution function (ICDF), periodic mutation, a fusion of DE and Cauchy ICDF, a fusion of DE and Cauchy ICDF, a fusion of DE and Cauchy ICDF and periodic mutation, and a combination of DE, Cauchy ICDF, and periodic mutation were named DEQIO, CQIO, VQIO, DCQIO, DVQIO, CVQIO, and DCVQIO, respectively. These algorithms were tested on 29 CEC2017 functions. The algorithms were run 30 times with a population size of 30 and a test dimension of 30. The mean (avg) and standard deviation (std) of the incompletely improved algorithms were compared. The specific results are shown in Table 4 (ablation experiment results). The 29 CEC2017 test functions were ranked according to their mean (avg) values, resulting in an average ranking and a final ranking.

[0172] Table 3

[0173]

[0174]

[0175]

[0176] The results are shown in Table 4 (ablation experiment results). Table 4 shows that among the DEQIO, CQIO, VQIO, DCQIO, DVQIO, CVQIO, and DCVQIO tests, DCVQIO performs best, followed by DCQIO. The results show that using the differential evolution algorithm, the Cauchy inverse cumulative distribution function, and the periodic mutation strategy simultaneously to improve the DCVQIO algorithm is the most effective.

[0177] Table 4

[0178]

[0179]

[0180]

[0181] Implementation method five: The optimization method of the quadratic interpolation optimization algorithm based on periodic variation improvement proposed in the above implementation method can all be implemented using computer software. Therefore, correspondingly, this implementation method proposes a computer system, which includes a storage device, and the storage device is used to execute the steps of the optimization method of the quadratic interpolation optimization algorithm based on periodic variation improvement as described above.

[0182] Implementation method six: This implementation method proposes a computer device including a memory and a processor, wherein a computer program is stored in the memory. When the processor runs the computer program stored in the memory, the steps of the optimization method based on the periodic variation improved quadratic interpolation optimization algorithm as described above are executed.

[0183] This embodiment provides a computer device, in which the hardware device of this part is a general model and is not shown in the form of a diagram. The system includes a processor and a memory, wherein the processor and the memory can be connected via a bus or other means. The memory, as a non-transitory computer-readable storage medium, can be used to store non-transitory software programs, non-transitory computer executable programs and modules, and corresponding program instructions / modules. The processor executes various functional applications and data processing of the processor by running the non-transitory software programs, instructions and modules stored in the memory, so as to implement the steps of the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation in the above-mentioned method embodiment.

[0184] Implementation 7. This implementation provides a computer-readable storage medium, in which a plurality of computer instructions are stored. The plurality of computer instructions are used to enable a computer to execute an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm as described above.

[0185] The foregoing description is merely an embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that the present invention is susceptible to various modifications and variations. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention shall be included within the scope of the claims.

Claims

1. An optimization method based on periodic variation to improve the quadratic interpolation optimization algorithm, characterized in that: The method is: Step 1: Construct a mathematical model for the tension / compression spring optimization problem; Step 2: The quadratic interpolation optimization algorithm is combined with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct an improved quadratic interpolation optimization algorithm; Step 3: Using the improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring optimization problem, the best target optimal solution and optimal parameter combination are obtained; Step 4: Output the optimal structural design scheme of the spring based on the optimal decision variables of the mathematical model solved in step 3.

2. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 1 is characterized in that: The mathematical model of the tension / compression spring optimization problem is constructed as follows: The design variables are defined as a three-dimensional vector X = [x1, x2, x3], which correspond to the parameters of wire diameter d, average coil diameter D and effective coil number L respectively. The optimization objective function is set as f(X) = x1 2 x2x3+2x1 2 x2; Identify four constraints: Interference: Shear stress: Fluctuation frequency: Outer diameter: Among them, x1∈[0.05,2.00], x2∈[0.25,1.30], x3∈[2.00,15.0].

3. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 1 is characterized in that: Step three is as follows: Step 1: Convert the three optimization parameters of the tension / compression spring: wire diameter, average coil diameter, and number of effective coils into population individuals; Step 2: Randomly initialize the population, set the population size n, dimension m, search space upper and lower limits Low and Up, and maximum number of iterations T; Step 3: Calculate the fitness value of the population and sort it; Step 4: Execute the differential evolution algorithm until the limit number T is reached DE until; Step 5: Perform periodic mutation on all individuals; Step 6: Improve the quadratic interpolation optimization algorithm by using the Cauchy inverse cumulative distribution to obtain the CQIO algorithm and execute it to obtain the updated individual; Step 7: Iteration number t=t+1, determine whether the iteration number reaches the maximum iteration number T. If so, the iteration ends and the individual corresponding to the optimal fitness value, that is, the optimal solution, is output; otherwise, return to step 4.

4. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 3 is characterized in that: The method of cyclically executing the differential evolution algorithm in step 4 is: The mutation operation of the differential evolution algorithm is performed through formula (a); V i =X rand1 +F*(X rand2 -X rand3 ) (a) The crossover operation of the differential evolution algorithm is performed by formula (b); The selection operation of the differential evolution algorithm is performed through formula (c); Where CR is the crossover rate; j rand is a random number in the range [1, m] chosen randomly from the solution vector, where m is the dimension of the problem.

5. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 4 is characterized in that: The mutation operation of the differential evolution algorithm is as follows: Perform the mutation operation of the differential evolution algorithm to create three random indices within [1, n], where n is the population size, and select three solution vectors X for the given index from the current solutions in the search space. r1 、X r2 and X r3 To generate a new solution V i ; The crossover operation of the differential evolution algorithm is as follows: Perform the crossover operation of the differential evolution algorithm and select the new mutation solution vector V according to formula (b) i and the original solution vector X i Generate U i ; The selection operations for executing the differential evolution algorithm are as follows: Perform the selection operation of the differential evolution algorithm to select the best candidates based on their fitness value f(.) in X i or U i Choosing between them, the one with better fitness value will survive to the next iteration.

6. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 3 is characterized in that: The periodic mutation activation in step 5: The generalized mutation operation including the mutation strategy is described as formula (1): x i,j (t)=F(g(x i,j (t)),fr) (1) Where F is the generalized variogram, g(x i,j (t)) is the offspring vector provided by the mutation operator, and fr is the user-defined fluctuation frequency.

7. The optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to claim 3 is characterized in that: The Cauchy inverse cumulative distribution function is introduced in the exploration stage of the quadratic interpolation optimization algorithm to form the CQIO algorithm.

8. A computer system, characterized in that: The system includes a storage device, which is used to execute the method and steps described in any one of claims 1 to 7.

9. A computer device, characterized in that: The invention comprises a memory and a processor, wherein a computer program is stored in the memory, and when the processor runs the computer program stored in the memory, the steps of the optimization method of the improved quadratic interpolation optimization algorithm based on periodic variation according to any one of claims 1 to 7 are executed.

10. A computer-readable storage medium, characterized in that The computer-readable storage medium stores a plurality of computer instructions, and the plurality of computer instructions are used to enable a computer to execute the steps of the optimization method based on the periodic variation improved quadratic interpolation optimization algorithm as described in any one of claims 1 to 7.

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