Optimization method, system and device based on periodic variation improved quadratic interpolation optimization algorithm and storage medium

Abstract

The optimization method, system, device and storage medium based on a periodic variation improved quadratic interpolation optimization algorithm belong to the field of machine engineering. A mathematical model is constructed for a tension / compression spring design problem. A DCVQIO algorithm is constructed by combining a quadratic interpolation optimization QIO algorithm, a periodic variation, a Cauchy inverse cumulative distribution and a differential evolution algorithm. A Cauchy inverse cumulative distribution function is introduced in the exploration stage of the QIO algorithm to form a CQIO algorithm, the search mechanism of the quadratic interpolation optimization is strengthened, and the exploration ability is retained. A periodic variation strategy is introduced between the DE and the CQIO algorithm to improve the diversity of the algorithm. The two algorithms are used as low-level and high-level heuristic algorithms, the periodic variation strategy is fused to improve the optimization accuracy of the algorithm, the DCVQIO algorithm is used to optimize the parameters of the mathematical model, and the best target solution and the optimal parameter combination are obtained, so that the best structure design scheme of the spring is output. The performance of the quadratic interpolation optimization algorithm in solving the tension spring design problem is improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of mechanical engineering, and particularly relates to an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm. BACKGROUND

[0002] Under the background of intensified globalization competition, the continuous enhancement of resource scarcity prompts enterprises to pursue cost-effectiveness maximization. This phenomenon can be transformed into a constrained extreme value solving model through mathematical modeling, which is a typical engineering optimization topic in modern operations research.

[0003] As a key component of mechanical systems, the constrained optimization of tensile / compressive springs (TCSD) directly affects the performance of industrial equipment. The design quality not only relates to the running stability of automobile suspension systems, precision instruments and other equipment, but also determines the energy utilization efficiency and production economy. Traditional optimization algorithms are prone to premature convergence when dealing with such nonlinear constraints, exposing defects such as high parameter sensitivity and low iteration efficiency, and there is an urgent need for more advanced global optimization strategies to break through technical bottlenecks.

[0004] Quadratic interpolation optimization (QIO) algorithm is a new type of swarm intelligence optimization algorithm proposed by Zhao Weiguo et al. in 2023. Quadratic interpolation optimization (QIO) algorithm uses GQI method to establish quadratic model, quickly and accurately locates extreme value point. The hybrid framework combining QIO and particle swarm optimization (PSO) algorithm enhances the robustness and adaptability in application, and has been applied to photovoltaic parameter identification. The advantages of QIO algorithm are effective avoidance of local optimum and premature convergence, less parameters, simple and easy to implement. However, although its performance is superior, like current heuristic algorithms, QIO algorithm also has disadvantages, that is, it quickly falls into local optimum and cannot well balance local development and global exploration. SUMMARY

[0005] In view of the defects that the classical optimization algorithm is easy to fall into local optimal solution when solving the design problem of the stretching / compression spring, and the solving precision is insufficient, the application innovatively proposes a kind of improved quadratic interpolation optimization algorithm based on periodic variation (referred to as a kind of super heuristic algorithm DCVQIO), the method introduces the Cauchy inverse cumulative distribution function in the exploration stage of the quadratic interpolation optimization algorithm, forms the Cauchy quadratic interpolation optimization (CQIO) algorithm, enhances the search mechanism of QIO, and meanwhile retains its strong exploration ability. Secondly, a periodic variation strategy is introduced between the differential evolution algorithm (Differential Evolution, DE) and the CQIO algorithm, aiming at increasing the diversity of DCVQIO and preventing the algorithm from falling into local optimum. Finally, a hybrid strategy is proposed, DE and CQIO are used as low-level and high-level heuristic algorithms respectively, and the periodic variation strategy is fused, so that the optimization precision of the algorithm is improved. The DCVQIO algorithm proposed in the application improves the performance of the quadratic interpolation optimization algorithm in solving the stretching / compression spring problem.

[0006] In order to achieve the above purpose, the technical scheme of the application is realized as follows: a design method of tension / compression spring based on improved quadratic interpolation optimization algorithm with periodic variation, the steps are as follows:

[0007] Step one, constructing the mathematical model of the stretching / compression spring optimization problem;

[0008] Step two, combining quadratic interpolation optimization (QIO) algorithm with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct the improved quadratic interpolation optimization algorithm;

[0009] Step three, using the improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the stretching / compression spring optimization problem constructed, and obtaining the best target optimal solution and optimal parameter combination;

[0010] Step four, outputting the best structure design scheme of the spring according to the best decision variable of the mathematical model solved in step three.

[0011] Further, the mathematical model construction method of the above stretching / compression spring optimization problem is as follows:

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

[0013] Four constraint conditions are determined:

[0014] Degree of disturbance:

[0015]

[0016] Shear stress:

[0017]

[0018] Wave frequency:

[0019]

[0020] Outer diameter:

[0021]

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

[0023] Further, in the above step three, by converting the three major optimization parameters of the tension / compression spring into population individuals; the final stage obtains the three optimal engineering parameters by analyzing the position of the optimal individual, and constitutes the optimization design scheme of the spring structure.

[0024] The steps of the quadratic interpolation optimization algorithm based on periodic mutation are:

[0025] Step 1: Convert the three major optimization parameters of the tension / compression spring: parameter wire diameter, average coil diameter and effective coil number 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 the maximum number of iterations T;

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

[0028] Step 4: Perform the differential evolution algorithm (Differential Evolution, DE) until the limited number T is reached; DE

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

[0030] Step 6: Obtain the CQIO algorithm by the Cauchy inverse cumulative distribution improved quadratic interpolation optimization (QIO) algorithm, and execute to obtain the updated individuals;

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

[0032] ​Furthermore, the method for iteratively executing the Differential Evolution (DE) algorithm in step 4 above is as follows:

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

[0034] Specifically, the mutation operation of the differential evolution algorithm is performed to create three random indices in [1, n], where n is the population size, and three solution vectors X at a given index are selected from the current solutions 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 using equation (b).

[0037] Specifically, this involves performing the crossover operation of the differential evolution algorithm, and according to equation (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 It is a random number randomly selected from the solution vector within the range [1, m], where m is the dimension of the problem.

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

[0041] Specifically, this involves performing a selection operation using a differential evolution algorithm to determine the fitness of X based on their fitness f(.). i or U i We choose between the two options, and the one with the better fitness value survives to the next iteration. Considering a minimization optimization problem, the selection operation can be represented as follows:

[0042]

[0043] From equation (c), it can be observed that the population of the differential evolution algorithm tends to improve or remain unchanged, but never deteriorates.

[0044] Determine if the number of iterations T of the differential evolution algorithm has been reached. DE If the condition is met, proceed to step 5; otherwise, continue executing step 4 in a loop.

[0045] Further, the method of periodic variation in step 5 above is:

[0046] Periodic variation activation: the generalized variation operation description including variation strategy is as formula (1):

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

[0048] Wherein, F is a generalized variation function, g(xi, j(t)) is a variation operator to provide offspring vector, and fr is a user-defined fluctuation frequency. Mutation strategy focuses on studying how to apply variation operator in the optimization process. This operator is called global variation operator and is given by formula (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] Wherein, A is an amplitude factor defined by the user, which can be selected as a fixed number or calculated in the iteration process; rand is a random number generator conforming to N[0, 1]. In application, 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 correct search pattern.

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

[0055] Randomly select 2 individuals in the population.

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

[0057]

[0058] w1=3n1b (4)

[0059]

[0060]

[0061] where 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(.) denotes the fitness value of the function. GQI(x i (t), x rand1 (t), x rand2 (t), fit(x i (t)), fit(x rand1 (t)), fit(x rand2 (t)) is the GQI function by which the minimum value of the interpolation function composed of (x i , fit(x i )), (x rand2 , fit(x rand2 )) and (x rand3 , fit(x rand3 )) can be obtained. The three randomly selected individuals in equation (7) can effectively expand the search range and the GQI method helps to locate the potential area.

[0062]

[0063] In the improved QIO algorithm, exploration is performed using equation (8) which introduces the inverse cumulative distribution function f -1 (p; a, b) into the original formula (7).

[0064]

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

[0066] In equation (9), p is a uniformly distributed random number in the range [0, 1] and is calculated by equation (10); m is the dimension of the function, the position parameter a is assigned a value of 0, which is consistent with the standard Cauchy distribution, and the scaling parameter b is assigned a value of 0.01. This setting makes the algorithm not too aggressive in 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 two randomly selected individuals by equation (11), and performs development using equation (13).

[0068]

[0069] where n2 follows a standard normal distribution, rD is a random integer from [1, m], and Lb rD and UbrD are lower and upper bounds of dimension (rD). w2 is a development weight as an adaptive coefficient. According to equation (11), the best individual x best and two randomly selected individuals x rand1 and x rand2 are used to generate a minimized value x' best，rand1，rand2 which is usually better than the three individuals. From equation (13), x' i rD is updated by searching the neighborhood of x' best with a random perturbation from x best，rand1，rand2 This search can help the algorithm to develop effectively without stagnation.

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

[0071]

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

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

[0074] In a fourth aspect, the present application provides a computer readable storage medium, wherein the computer readable storage medium stores a plurality of computer instructions, and the plurality of computer instructions are used to make a computer execute an optimization method based on a periodic mutation improved quadratic interpolation optimization algorithm as described above.

[0075] The present application has the following beneficial effects:

[0076] 1. The present application proposes an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm (referred to as hyper-heuristic algorithm DCVQIO). First, the Cauchy inverse cumulative distribution function is introduced into the exploration stage of the quadratic interpolation optimization (QIO) algorithm to form the CQIO algorithm, which enhances the search mechanism of the algorithm. Secondly, the differential evolution algorithm (Differential Evolution, DE) and the CQIO algorithm are used as the low-level and high-level heuristic algorithms respectively, balancing the development ability and exploration ability of the algorithm. Finally, a periodic variation strategy is adopted between the differential evolution algorithm and the CQIO algorithm to increase the diversity of the algorithm and avoid premature convergence to local optimum.

[0077] 2. The differential evolution algorithm (Differential Evolution, DE) is considered to be an excellent evolutionary algorithm for solving global optimization problems, with the advantages of simplicity, easy implementation, good convergence, etc. In particular, DE operator and DE / rand / 1 have excellent exploration ability. In addition, the quadratic interpolation optimization (QIO) algorithm uses the GQI method of the best individual found so far, which is beneficial to algorithm development and has the advantages of few parameters and simplicity. The differential evolution algorithm generates diverse solutions through mutation and crossover, but due to the selection pressure, it is easy to converge to a suboptimal region prematurely. The QIO algorithm uses the GQI method to establish a quadratic model, which can quickly and accurately locate the extreme point and make up for the local development defects of the differential evolution algorithm. The local mechanism information of 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 realize this interaction. On the other hand, the QIO algorithm has fewer parameters, which can avoid the stability problems caused by the parameter sensitivity of traditional heuristic algorithms. Therefore, the present application uses the differential evolution algorithm and the QIO algorithm improved by the Cauchy inverse cumulative distribution function (CQIO) as the low-level and high-level heuristic algorithms respectively, which can balance development and exploration and find the global optimal solution.

[0078] 3. For the classical structural optimization problem - tensile / compressive spring design (TCSD), the present application proposes a solution to the DCVQIO algorithm. Experimental data show that the DCVQIO algorithm successfully optimizes the spring mass to 0.012665, which significantly improves the optimization accuracy compared to the QIO prototype algorithm. In the comparative test, the algorithm significantly surpasses seven excellent optimizers such as QIO, AOA, CDO, HHO, SAO, SCSO and PSO in convergence accuracy, verifying the effectiveness and advancement of the DCVQIO algorithm under engineering constraint conditions.

[0079] The present application is suitable for the constraint optimization of tensile / compressive spring design (TCSD) in key components of mechanical systems. BRIEF DESCRIPTION OF DRAWINGS

[0080] In order to more clearly illustrate the technical solutions in the specific embodiments or prior art of the present application, the following will briefly introduce the drawings needed to be used in the specific embodiments or prior art description. Obviously, the drawings described below are some embodiments of the present application, and for those skilled in the art, other drawings can also be obtained without creative labor.

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

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

[0083] The specific embodiments of the present application will be further described in detail below with reference to the accompanying drawings. The following embodiments will help those skilled in the art to further understand the present application, but do not limit the present application in any form. It should be noted that for those skilled in the art, without departing from the concept of the present application, a number of changes and improvements can be made, which are within the scope of protection of the present application.

[0084] Embodiment one, see Figure 1 This embodiment is described, and this embodiment is aimed at the defect that the classical optimization algorithm is easy to fall into local optimal solution when solving the design problem of tensile / compression spring, resulting in insufficient solution accuracy, and thus an improved quadratic interpolation optimization algorithm based on periodic variation is innovatively proposed, which includes the following steps:

[0085] As shown in Figure 1

[0086] Step one, constructing a mathematical model of the tensile / compression spring optimization problem;

[0087] Step two, combining quadratic interpolation optimization (QIO) algorithm with periodic variation, Cauchy inverse cumulative distribution and differential evolution algorithm to construct an improved quadratic interpolation optimization algorithm;

[0088] Step three, using the improved quadratic interpolation optimization algorithm to perform parameter optimization on the mathematical model of the tensile / compression spring optimization problem constructed, to obtain the best target optimal solution and optimal parameter combination;

[0089] Step four, outputting the best structure design scheme of the spring according to the best decision variable of the mathematical model solved in step three.

[0090] Embodiment two, see Figure 2 This embodiment is described, and this embodiment is a specific description of the optimization method based on the improved quadratic interpolation optimization algorithm with periodic variation proposed in the above embodiment one;​

[0091] Step one, building a mathematical model of the tension / compression spring optimization problem;

[0092] Specifically:

[0093] This embodiment is aimed at the multi-dimensional optimization challenge in the design of tension / compression springs, and its mathematical modeling needs to clearly define the core optimization target and multi-dimensional constraint conditions. The geometric configuration parameters of the spring consist of a parameter wire diameter x1(d), an average coil diameter x2(D), and an effective coil number x3(L) to form a three-dimensional design vector. The quantitative model of this system needs to build an optimization function with mass minimization as the core, and additional multi-dimensional boundary conditions such as perturbation, shear stress, fluctuation frequency, and outer diameter.

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

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

[0096] Four constraint conditions:

[0097] Perturbation (corresponding to spring stiffness requirement):

[0098]

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

[0100]

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

[0102]

[0103] Outer diameter (to meet the assembly space requirement):

[0104]

[0105] The value range of each design variable is:

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

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

[0108] Effective coil number x3 ∈ [2.00, 15.0].

[0109] The mathematical model proposed in the embodiment is optimized by multi-dimensional parameters, and while meeting the requirements of spring mechanical properties, the optimization target of the structure design is realized. The constraint conditions cover multiple requirements such as material strength, working characteristics and geometric size, and ensure that the design scheme has both engineering feasibility and economy.

[0110] In step two, the improved quadratic interpolation optimization algorithm is constructed by combining periodic variation, inverse cumulative distribution of Cauchy and differential evolution algorithm.

[0111] Specifically,

[0112] For the solving process of the spring optimization problem, the key design parameters (d, D, L) of the TCSD problem are mapped to the multi-dimensional position vector space of the individual in the DCVQIO algorithm, forming a three-dimensional solution vector X=(x1, x2, x3). Therefore, the improved quadratic interpolation optimization algorithm is constructed by combining periodic variation, inverse cumulative distribution of Cauchy and differential evolution algorithm.

[0113] In step three, the improved quadratic interpolation optimization algorithm is used to optimize the mathematical model of the constructed tension / compression spring optimization problem, and the best target optimal solution and optimal parameter combination are obtained.

[0114] Specifically,

[0115] By fusing differential evolution algorithm (Differential Evolution, DE), quadratic interpolation optimization (QIO) algorithm, inverse cumulative distribution function of Cauchy and periodic variation strategy, a super-heuristic algorithm DCVQIO is constructed. The optimal individual is obtained by executing the iterative optimization process of the DCVQIO algorithm.

[0116] Among them, 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 find the minimum value of a 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 exploration and exploitation strategies, in which the minimum value provided by the GQI method helps the QIO algorithm to 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, the dimension m, and the 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 the two randomly selected individuals through equation (3), and performs exploration using equation (7).

[0122]

[0123] w1=3n1b (4)

[0124]

[0125] Where, x rand1 x rand2 and x rand3 This represents the positions of three distinct individuals randomly selected from the current population. r1, r2, and r3 are random numbers in the interval (0, 1), and fit(.) represents the fitness value of the function. GQI(x i (t), x rand1 (t), x rand2 (t), fit(x) i (t)), fit(x) rand1 (t)), fit(x) rand2 (t) is a GQI function, through which we can obtain the result of (x) i fit(x) i )), (x rand2 fit(x) rand2 )) and (x rand3 fit(x) rand3 The minimum value of the interpolation function formed by the three randomly selected individuals in equation (7) can effectively expand the search range, while 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 equation (11), and performs development using equation (13).

[0127]

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

[0129] According to equation (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 a minimized value x' best，rand1，rand2 which is usually better than the three individuals. From equation (11), by utilizing the random perturbation from x i rD update the i-th individual by searching the neighborhood of x' best This search can help the algorithm to develop effectively without stagnation. best，rand1，rand2

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

[0131] Otherwise, it remains unchanged.

[0132]

[0133] (7) Determine whether the set termination condition is reached (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 of the best individual currently discovered, which is conducive to the development of the algorithm and has the advantages of few parameters and simplicity. Although the QIO algorithm has superior performance, like the current heuristic algorithm, it also has the disadvantage of being easy to fall into local optimum and cannot well balance local development and global exploration.

[0135] Based on this problem, the embodiment proposes an optimization algorithm based on the periodic mutation improved quadratic interpolation optimization (QIO) algorithm (referred to as a super heuristic algorithm DCVQIO), and adds three strategies to improve the performance of the QIO algorithm: the inverse cumulative distribution function of Cauchy, periodic mutation and differential evolution algorithm. In addition, the proposed DCVQIO is verified to have strong competitiveness by using the CEC2017 benchmark function. Finally, the DCVQIO algorithm is applied to the stretch / compression spring design problem to verify its practical application ability.

[0136] The flow of the proposed DCVQIO algorithm is as follows: as shown in Figure 2

[0137] (1) Randomly initialize the population, set the population size n, the dimension m, the upper and lower limits of the search space Low and Up, and the 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 by equation (a).​​

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

[0141] (4) Perform the crossover operation of the differential evolution algorithm DE by formula (b).

[0142]

[0143] (5) Perform the selection operation of the differential evolution algorithm DE by formula (c).

[0144]

[0145] (6) Determine whether the termination iteration number (T DE ) of the differential evolution algorithm DE is reached, if yes, perform step (7), if not, jump to step (3).

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

[0147] (8) Randomly select two individuals in the population.

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

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

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

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

[0152] (12) Determine whether the set number of iterations T is reached, if not go to step (3), otherwise the algorithm terminates and returns the best solution found so far.

[0153] Step four: reverse conversion of the preferred group individuals by the encoding mechanism adopted in step two, to obtain the key solution parameters of the TCSD problem, which is the optimal structural configuration scheme corresponding to the spring system.

[0154] The method for improving the quadratic interpolation optimization algorithm based on periodic mutation (referred to as a super heuristic algorithm DCVQIO) is proposed in the embodiment. The Cauchy inverse cumulative distribution function is introduced into the exploration stage of the quadratic interpolation optimization (QIO) algorithm to form the Cauchy quadratic interpolation optimization (CQIO) algorithm, so that the search mechanism of QIO is enhanced, and the strong exploration ability is retained. Secondly, a periodic mutation strategy is introduced between the differential evolution algorithm (Differential Evolution, DE) and the CQIO algorithm, aiming at increasing the diversity of DCVQIO and preventing the algorithm from falling into local optimum. Finally, a hybrid strategy is proposed, which takes DE and CQIO as low-level and high-level heuristic algorithms respectively, and combines the periodic mutation strategy to improve the optimization accuracy of the algorithm. The DCVQIO algorithm proposed in the embodiment improves the performance of the quadratic interpolation optimization algorithm in solving the stretching / compression spring problem.

[0155] Embodiment three, the embodiment is a verification description of the optimization method of the quadratic interpolation optimization algorithm based on periodic mutation improvement in the above-mentioned embodiment one or embodiment two;

[0156] The DCVQIO algorithm proposed in the application is compared with the basic QIO algorithm, the arithmetic optimization algorithm (AOA), the Chernobyl disaster optimization algorithm (CDO), the Harris eagle algorithm (HHO), the snow ablation optimization algorithm (SAO), the sand cat swarm optimization algorithm (SCSO), and the particle swarm optimization (PSO) algorithm. The comparison results of various algorithms in the tension / spring design problem are shown in Table 1. The comparison of data in Table 1 shows that the DCVQIO algorithm proposed in the application has significant advantages in solving the TCSD problem. The optimal structure configuration obtained by the DCVQIO algorithm has a spring system weight index of 0.012665, which is the highest accuracy level in the comparison group compared with the basic QIO algorithm, the arithmetic optimization algorithm (AOA), the Chernobyl disaster optimization algorithm (CDO), the Harris eagle algorithm (HHO), the snow ablation optimization algorithm (SAO), the sand cat swarm optimization algorithm (SCSO), and the particle swarm optimization (PSO) algorithm. The engineering example verification shows that the super-heuristic DCVQIO algorithm which fuses the Cauchy inverse cumulative distribution function, the periodic variation and the differential evolution algorithm significantly surpasses the original QIO, AOA, CDO, HHO, SAO, SCSO and PSO algorithms in solution quality, and confirms the effectiveness of the enhanced mechanism. The research conclusion further proves that the DCVQIO as an optimization tool for the TCSD problem has engineering practical value, and the multi-strategy coupled framework provides a new paradigm for solving complex system design problems.

[0157] Table 1

[0158]

[0159] Further, in order to verify the performance of the DCVQIO algorithm, 29 benchmark functions on CEC2017 are selected for optimization. First, in order to highlight the superiority of the DCVQIO algorithm, QIO, AOA, LFD, HHO, SAO, SCSO and PSO are used for performance comparison. Then, different improved strategies are compared to verify the effectiveness of the improvements in the DCVQIO algorithm.

[0160] The 29 CEC2017 benchmark functions are used to test the DCVQIO. The CEC2017 benchmark set can be divided into four categories: unimodal (F1, F3), multimodal (F4-F10), hybrid (F11-F20) and composite functions (F21-F30). Multimodal test functions are suitable for testing exploration operations in optimization algorithms. Composite functions and composite functions are used to evaluate the ability to avoid local optima through optimization algorithms, while unimodal test functions are used to evaluate development operations because they only contain one global optimum. It should be noted that F2 has been removed from the benchmark test set due to its unstable behavior.

[0161] The DCVQIO algorithm is compared with seven optimization algorithms, which are QIO, AOA, LFD, HHO, SAO, SCSO and PSO, and each function is executed 30 independent runs, considering the average (avg) and standard deviation (std) of the best solution of each run. The detailed experimental results are shown in Table 2 (the optimization results of 8 algorithms on CEC2017 test set 30D). In order to verify the significant difference between the DCVQIO algorithm and the comparison algorithm, the Wilcoxon rank-sum test is used for non-parametric test; the relevant experimental results are shown in Table 2. According to the average avg in Table 2, the algorithms are ranked.

[0162] In Table 2, the best value of each index is highlighted in bold. From Table 2, it can be seen that for 30D, the DCVQIO algorithm obtains the best average optimization result in 25 functions out of 29 function results. While the QIO, AOA, LFD, HHO, SAO, SCSO, PSO algorithms obtain the best average optimization result of 0, 1, 0, 0, 3, 0, 0, respectively. According to the final ranking of the average (avg) of the best solution of 29 functions, DCVQIO ranks first, which shows that the DCVQIO algorithm has better optimization performance and is superior to other comparison algorithms in avoiding local extreme value. The reason for this result is that DE and CQIO algorithms are used as low and high heuristic algorithms, respectively, so that the DCVQIO algorithm can realize the smooth transition between exploration and exploitation. Based on the periodic mutation and the Cauchy inverse cumulative distribution function, the population diversity and spatial search ability of the algorithm can be greatly improved.

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

[0164] Table 2

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[0169] Further, to verify the significant difference between the DCVQIO algorithm and the comparison algorithms, a non-parametric test using the Wilcoxon rank-sum test is performed. If p≤0.05 indicates that there is a significant difference between the two algorithms on a function at a test level of 5%, otherwise the difference is not significant. The test results of the DCVQIO algorithm and the comparison algorithms on the CEC2017 function with 30 dimensions are shown in Table 3 (Wilcoxon rank-sum test results on 30 dimensions of CEC2017 test functions), and the cases where the p value is less than or equal to 0.05 are marked in bold. From the data in Table 3, most of the p values are less than 0.05, indicating that the performance of the DCVQIO algorithm is significantly better than that of the QIO, AOA, LFD, HHO, SAO, SCSO and PSO algorithms.

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

[0171] To further illustrate the impact of the differential evolution algorithm, the Cauchy inverse cumulative distribution function and the periodic variation on the quadratic interpolation optimization (QIO) algorithm, an ablation comparison experiment is performed. The algorithms using the differential evolution algorithm (Differential Evolution, DE), the Cauchy inverse cumulative distribution function, the periodic variation, the fusion of the differential evolution algorithm and the Cauchy inverse cumulative distribution function, the fusion of the differential evolution algorithm and the periodic variation, the fusion of the Cauchy inverse cumulative distribution function and the periodic variation, and the simultaneous use of the differential evolution algorithm, the Cauchy inverse cumulative distribution function and the periodic variation to improve the QIO algorithm will be named DEQIO, CQIO, VQIO, DCQIO, DVQIO, CVQIO and DCVQIO, and tested on 29 CEC2017 functions. The algorithm runs 30 times, the population size is 30, and the test dimension is 30. The average (avg) and standard deviation (std) of the incompletely improved algorithm are compared. The specific results are shown in Table 4 (ablation experiment results). According to the average (avg) of the 29 CEC2017 test functions, the average ranking and the final ranking are obtained.

[0172] Table 3

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[0176] The results are shown in Table 4 (ablation experiment results). As can be seen from Table 4, among the test results of DEQIO, CQIO, VQIO, DCQIO, DVQIO, CVQIO, and DCVQIO, DCVQIO performed best, followed by DCQIO. The results indicate that simultaneously using the differential evolution algorithm, the Cauchy inverse cumulative distribution function, and the periodic mutation strategy to improve the DCVQIO algorithm is the most effective approach.

[0177] Table 4

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[0181] Implementation Method 5: The optimization method based on the improved quadratic interpolation optimization algorithm based on periodic variation proposed in the above implementation methods can all be implemented using computer software. Therefore, this implementation method proposes a computer system, which includes a storage device for executing the steps of the optimization method based on the improved quadratic interpolation optimization algorithm based on periodic variation as described above.

[0182] Implementation Method Six: This implementation method proposes a computer device, including a memory and a processor. The memory stores a computer program. When the processor runs the computer program stored in the memory, it executes the steps of an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm as described above.

[0183] This embodiment provides a computer device, the hardware of which is a general-purpose model and is not shown in the figure. The system includes a processor and a memory, which 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, as well as corresponding program instructions / modules. The processor executes various functional applications and data processing by running the non-transitory software programs, instructions and modules stored in the memory, so as to realize the steps of the optimization method based on the periodic variation improved quadratic interpolation optimization algorithm in the above method embodiment.

[0184] Implementation Method Seven: This implementation method provides a computer-readable storage medium storing multiple computer instructions, which are used to cause a computer to execute an optimization method based on a periodic variation improved quadratic interpolation optimization algorithm as described above.

[0185] The above merely describes the embodiments of the present application, and is not limited to the present application. For those skilled in the art, the present application can have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the scope of claims of the present application.

Claims

1. An optimization method based on a periodic variation-based improved quadratic interpolation optimization algorithm, characterized in that, The method is as follows: Step 1: Construct a mathematical model for the optimization problem of tension / compression springs; Step 2: Combining periodic mutation, Cauchy inverse cumulative distribution, and differential evolution algorithm, an improved quadratic interpolation optimization algorithm is constructed. The process of improving the quadratic interpolation optimization algorithm is as follows: (1) Randomly initialize the population and set the population size. n dimensionality m upper and lower limits of search space Low and Up Maximum number of iterations T ; (2) Calculate and sort the fitness values ​​of the population; (3) Perform the mutation operation of the differential evolution algorithm DE using equation (a); （a） (4) Perform the crossover operation of the differential evolution algorithm DE using equation (b); （b） (5) Perform the selection operation of the differential evolution algorithm DE using equation (c); （c） (6) Determine whether the number of iterations required to terminate the differential evolution algorithm (DE) has been reached. T DE If the condition is met, proceed to step (7); otherwise, proceed to step (3). (7) Perform periodic variation on all individuals using formula (2); （2） in, A It is a user-defined amplitude factor, which can be chosen as a fixed number or calculated during the iteration process; rand It is a combination of N A random number generator for [0, 1]; (8) Randomly select 2 individuals from the population; (9) When rand >0.5 ( q(t) When the value is greater than 0.5, the algorithm obtains the minimum value of the GQI function of the current individual and the two randomly selected individuals through equation (3), and performs exploration using equation (8), which introduces the Cauchy inverse cumulative distribution function based on equation (7). ; （3） （4） （5） （6） （7） （8） in, x rand1 , x rand2 and x rand3 It refers to the positions of three different individuals randomly selected from the current group. rand, r 1. r 2 and r 3 is a random number in (0, 1). fit (.) represents the fitness value of the function; GQI (x i (t), x rand1 (t), x rand2 (t), fit (x i (t)), fit(x) rand1 (t)), fit(x) rand2 (t)) It is a GQI function, through which you can obtain... (x i fit(x) i ))， (x rand2 fit(x) rand2 )) and (x rand3 fit(x) rand3 )) The minimum value of the interpolation function formed; (10) When rand <=0.5 ( q(t) When <=0.5), the algorithm obtains the minimum value of the GQI function between the best individual found so far and the two randomly selected individuals through equation (11), and performs development using equation (13); （11） （12） （13） in, rD It comes from [1, m A random integer. Lb rD and Ub rD Is the ( rD The lower and higher boundaries of the dimension. w 2 is the development weight as the adaptive coefficient, and according to equation (11), the best individual found so far is used in the GQI method. x best and two randomly selected individuals x rand1 and x rand2 To generate the minimum value x’ best，rand1，rand2 It is generally better than three individuals, according to equation (13), by utilizing the resources from x i rD random perturbations relative to x best search x’ best，rand1，rand2 To update the neighborhood of the first i Individual; (11) After exploration or development is completed, find the best solution so far using equation (14), that is, if the first... i If the fitness value of an individual's candidate position is better than the fitness value of its current position, then the i-th... i The position of an individual is replaced by its candidate position; otherwise, it remains unchanged. （14） (12) Determine whether the set number of iterations to terminate has been reached. T If the algorithm does not proceed to step (3), it terminates and returns the best solution found so far. Step 3: Use an improved quadratic interpolation optimization algorithm to optimize the parameters of the mathematical model of the constructed tension / compression spring optimization problem, and obtain the optimal objective solution and the optimal parameter combination. Step 4: Based on the optimal decision variables of the mathematical model obtained in Step 3, output the optimal structural design scheme for the spring.

2. The optimization method based on the improved quadratic interpolation optimization algorithm according to claim 1, characterized in that, The mathematical model for the optimization problem of tension / compression springs is constructed as follows: The design variable is defined as a three-dimensional vector X = [ x 1, x 2, x 3], and corresponding to the wire diameter respectively. d Average coil diameter D and number of effective coils L The objective function for optimization is set as f(X) = x 1² x 2 x 3 + 2 x 1² x 2; Define four constraints: Detour: Shear stress: Fluctuation frequency: Outer diameter: in, x 1∈[0.05, 2.00], x 2∈[0.25, 1.30], x 3∈[2.00, 15.0].

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

4. The optimization method based on the improved quadratic interpolation optimization algorithm according to claim 1, characterized in that, The mutation operation of the differential evolution algorithm is specifically as follows: Perform the mutation operation of the differential evolution algorithm to [1, n Create three random indexes within the [include], among which n It refers to the population size, which selects three solution vectors with a given index from the current solutions in the search space. X r1 , X r2 and X r3 To generate new solutions V i ; The crossover operation of the differential evolution algorithm is specifically performed as follows: Perform the crossover operation of the differential evolution algorithm, according to equation (b), from the new mutation solution vector V i and the original solution vector X i generate U i ; The selection operation for executing the differential evolution algorithm is as follows: Perform the selection operation of the differential evolution algorithm based on their fitness values. f (.)exist X i or U i The system selects the one with the better fitness value to survive to the next iteration.

5. The optimization method based on the improved quadratic interpolation optimization algorithm according to claim 3, characterized in that, The periodic mutation activation in step 5 includes a generalized mutation operation description of the mutation strategy as shown in equation (1): （1） in, F It is a generalized mutability function. The mutation operator provides the offspring vector. fr The fluctuation frequency defined by the user.

6. The optimization method based on the improved quadratic interpolation optimization algorithm according to claim 3, characterized in that, In the exploration phase of the quadratic interpolation optimization algorithm, the Cauchy inverse cumulative distribution function was introduced to form the CQIO algorithm.

7. A computer system, characterized in that, The system includes a storage device for performing the method and steps of any one of claims 1-6.

8. A computer device, characterized in that, It includes a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, it performs the steps of the optimization method based on the improved quadratic interpolation optimization algorithm according to any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a plurality of computer instructions, which are used to cause the computer to perform the steps of the optimization method based on the improved quadratic interpolation optimization algorithm according to any one of claims 1-6.

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