Improved self-adaptive single-loop method based on hybrid dynamic mean value conjugate algorithm and application of improved self-adaptive single-loop method

Abstract

The invention discloses an improved self-adaptive single-loop method based on a hybrid dynamic mean value conjugation algorithm and application, according to a Karush-Kuhn-Tucker (KKT) optimality condition, a normalized sensitivity vector is calculated based on the hybrid dynamic mean value conjugation algorithm, and then a minimum function target point is deduced. The invention discloses an improved self-adaptive single-loop method, and relates to the field of reliability design optimization. The improved self-adaptive single-loop method comprises the following steps that basic parameters are set and initialized; deriving a normalized steepest descent direction; calculating a static conjugate scalar factor; calculating a dynamic factor value; calculating a dynamic conjugate gradient vector; deriving a normalized sensitivity vector; calculating random variables and random parameters; and solving the equivalent deterministic optimization model until the algorithm converges. Compared with a mainstream double-loop method, a single-loop method and a decoupling method, the improved self-adaptive single-loop method disclosed by the invention has the advantages of high efficiency, high accuracy and high convergence.

Description

Technical Field

[0001] This invention relates to the field of reliability design optimization, and in particular to an improved adaptive single-loop method based on a hybrid dynamic mean conjugate algorithm and its application. Background Technology

[0002] Due to their stable performance and simple structure, lead-acid batteries are widely used in the power systems of traditional underwater products. However, lead-acid batteries have low energy density and short endurance, making them unsuitable for deep-sea diving or long-duration missions. Lithium-ion batteries, with their advantages of high specific capacity, high single-cell output voltage, low self-discharge rate, long lifespan, and good safety performance, have become a new type of power system for underwater products.

[0003] The weight of lithium-ion battery packs significantly restricts the operational capabilities of underwater products, including diving ability, maneuverability, and stability. Reducing the weight of lithium-ion battery power systems while ensuring their reliability is of significant engineering importance for improving the performance of underwater products. Structural reliability design optimization techniques aim to find the optimal values ​​of structural parameters under the conditions of target reliability, objective function, and deterministic constraint functions. Applying reliability design optimization strategies is of significant engineering importance for improving the operational safety of lithium-ion battery power systems and reducing structural weight.

[0004] Currently, classic reliability design optimization techniques include the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). Both are nested algorithms, where the outer layer performs deterministic optimization, and the inner layer evaluates the reliability of the iterative results of the outer layer's optimization. However, nested algorithms are extremely inefficient and have high design time costs. The Sequential Optimization and Reliability Assessment (SORA) algorithm improves computational cost by decoupling deterministic optimization and reliability analysis. For large-scale problems, SORA still requires multiple calls to reliability analysis, resulting in significant time consumption. Compared to the Single-Loop Approach (SLA), SORA's computational advantage in high-dimensional problems diminishes as the dimensionality of variables increases. SLA significantly improves reliability design optimization efficiency by replacing the reliability analysis loop with approximation techniques or KKT optimality conditions. However, traditional single-loop methods suffer from poor stability and accuracy in handling highly nonlinear problems. Summary of the Invention

[0005] To address the aforementioned problems, this invention proposes an improved self-adaptive single-loop approach (MSASLA) based on the Hybrid Dynamic Mean Conjugate (HDMC) algorithm and its application. The MSASLA method, based on the KKT optimality condition, calculates the key normalized sensitivity vector using the HDMC algorithm, thereby efficiently deriving the approximate MPTP point. The MSASLA method proposed in this invention combines the efficiency of the traditional single-loop method with the high accuracy of the traditional double-loop method.

[0006] This invention is implemented as follows: an improved self-adaptive single-loop method based on a hybrid dynamic mean conjugate algorithm, comprising the following steps: (1) Set the basic parameters for reliability design optimization, including: target reliability index Convergence tolerance of deterministic design optimization Convergence tolerance of reliability analysis Initial values ​​of deterministic design variables Upper bound of deterministic design variables and lower boundary The initial value of the mean of a random variable. The upper bound of the mean of a random variable and lower boundary variance of random variables ; mean of random parameters and variance Maximum allowed number of iterations ; (2) Initialize the iteration, i.e. set the number of iteration steps. Set the mean values ​​of deterministic design variables and random variables, then calculate and record the objective function value; take the mean values ​​of random variables and random parameters; take the zero vector of the dynamic conjugate gradient vector; jump to step (5) to start the derivation process of the approximate MPTP point; (3) Input random variables and random parameters, and apply the gradient algorithm to solve the equivalent deterministic optimization model in order to obtain the optimal values ​​of the deterministic design variables and the mean of the random variables; (4) Determine whether convergence has occurred. If convergence has occurred, stop the iteration and output the mean of the deterministic design variables and random variables; otherwise, execute step (5) to start the derivation process of the approximate MPTP point. (5) Derive the normalized steepest descent direction of the function at the iteration point; (6) Calculate the static conjugate scalar factor based on gradient information; (7) When When, calculate the dynamic factor value, then jump to step (10); when The conjugate gradient vector is derived using the CGA method; (8) Evaluate the angle between the conventional conjugate gradient vector and the steepest descent direction; (9) Apply the dual criterion to accurately assess the concavity and convexity of the function and calculate the dynamic factor value; (10) Determine the dynamic conjugate gradient vector for the next iteration step; (11) Derive the normalized sensitivity vector for the next iteration step; (12) Calculate the random variables and random parameters for the next iteration step; (13) Execution Then proceed to step (3).

[0007] Furthermore, step (2) specifically involves: initializing the iteration, i.e., setting the number of iteration steps to 1. And set the deterministic design variables as The mean of the random variable is Then calculate the objective function value. And record; random variables and random parameters Take their mean; dynamic conjugate gradient vector Take the zero vector and jump to step (5) to begin the derivation process of the approximate MPTP point:

[0008] In the formula, Represents a random variable; Indicates a random parameter; Represents the mean of a random variable; This represents the mean of the random parameters; Indicates the number of iterations.

[0009] Furthermore, step (3) specifically involves: inputting random variables. and random parameters Apply the gradient algorithm to solve the following equivalent deterministic optimization model to obtain the optimal value. and ;

[0010] In the formula, Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Represents the mean of a random variable; Represent the objective function; Indicates the first One function; Indicates the number of functional functions; and These represent the upper and lower bounds of the deterministic design variables, respectively. and These represent the upper and lower bounds of the random variable, respectively. Indicates the number of iterations.

[0011] Furthermore, steps (4)-(5) are as follows: (4) Determine if convergence has occurred. If convergence has occurred, stop the iteration and output the result. and Conversely, if the condition is not met, then step (5) is executed to initiate the derivation process of the approximate MPTP point. (5) Derive the function at the iteration point The normalized steepest descent direction at the point :

[0012] In the formula, Indicates the direction of the steepest descent under normalization; and Let represent the variances of the random variable and the random parameter, respectively. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0013] Furthermore, steps (6)-(7) are as follows: (6) Calculate the static conjugate scalar factor based on gradient information. :

[0014] In the formula, Represents the static conjugate scalar factor; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; (7) When When, the dynamic factor takes the value ;when At that time, if The dynamic factor takes the value of ;like The dynamic factor takes the value of Then jump to step (10).

[0015] In the formula, Indicates a dynamic factor; Indicates the second parameter of the judgment criterion; Indicates the number of iterations; It is represented as:

[0016] In the formula, Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; when At that time, the conjugate gradient vector was derived using the traditional CGA method. :

[0017] In the formula, Represents the conjugate gradient vector; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the static conjugate scalar factor; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0018] Furthermore, step (8) specifically involves: evaluating the conventional conjugate gradient vector. Angle value between the direction of steepest descent and the direction of most rapid descent :

[0019] In the formula, Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0020] Furthermore, step (9) specifically involves: applying the dual criterion to accurately assess the concavity and convexity of the function and calculating the dynamic factor. This includes the following steps: (9-1) Calculation judgment criterion 1:

[0021] In the formula, Indicates the parameter of judgment criterion 1; , , They represent the first , , The normalized steepest descent direction of the iteration step; Indicates the number of iterations; (9-2) If If the condition is met, calculate criterion 2 and proceed to step (9-3); otherwise, proceed to step (9-4).

[0022] In the formula, Indicates the second parameter of the judgment criterion; Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; (9-3) If The dynamic factor takes the value of Otherwise, derive the following formula to calculate the dynamic factor. Then proceed to step (10);

[0023] In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Represents the number of iterations; dynamic conjugate gradient vector It is expressed as:

[0024] In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; (9-4) Analyze the dynamic conjugate gradient vector Angle value between the direction of steepest descent and the direction of most rapid descent :

[0025] In the formula, Represents the dynamic conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; In the formula, the dynamic conjugate gradient vector It is expressed as:

[0026] In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; (9-5) If Then, the dynamic factor can be derived using the following formula. Otherwise, dynamic factors Set the value to 1 and jump to step (10):

[0027] In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Represents the number of iterations; dynamic conjugate gradient vector The calculation method is shown in (9-3); Furthermore, steps (10) to (13) are as follows: (10) Determine the dynamic conjugate gradient vector for the next iteration step. :

[0028] In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations; (11) Derive the normalized sensitivity vector for the next iteration step :

[0029] In the formula, This represents the normalized sensitivity vector; Represents the dynamic conjugate gradient vector; Indicates the number of iterations; (12) Furthermore, random variables and random parameters It is represented as:

[0030] In the formula, and These represent the random variable and the random parameter, respectively. and These represent the mean of the random variable and the random parameter, respectively. and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. Indicates the target reliability index; Indicates the number of iterations; (13) Execution , and proceed to step (3); When dynamic factors When the value is 0, the dynamic conjugate gradient vector is... That is, the direction of steepest descent, the normalized sensitivity vector. It is then represented as

[0031] In the formula, and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0032] A computer-readable storage medium includes a stored computer program, wherein, when the computer program is executed, it controls the device on which the computer-readable storage medium is located to perform the above-described method.

[0033] A computer device, characterized in that the computer device includes a memory, a processor, and a program stored in the memory and executable thereon, wherein the program, when executed by the processor, implements the steps of the above-described method.

[0034] The beneficial effects of this invention are as follows: Compared with the dual-loop method and the decoupling method, the SLA method can significantly reduce design time costs. However, the traditional SLA method lacks convergence and accuracy in handling highly nonlinear optimization problems. To address this issue, this invention proposes an improved adaptive single-loop method (MSASLA) that balances design efficiency and accuracy in reliability optimization. Based on the KKT optimality condition, the MSASLA method calculates the key normalized sensitivity vector using the HDMC algorithm, thereby efficiently deriving the approximate MPTP point. Attached Figure Description

[0035] Figure 1 This is a schematic diagram of the overall process of the present invention.

[0036] Figure 2 This is a schematic diagram of the technical route of the present invention. Detailed Implementation

[0037] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0038] like Figure 1 , 2 As shown, the MSASLA method based on the HDMC algorithm of the present invention includes: (1) Set the basic parameters for reliability design optimization, including: target reliability index Convergence tolerance of deterministic design optimization Convergence tolerance of reliability analysis Initial values ​​of deterministic design variables Upper bound of deterministic design variables and lower boundary The initial value of the mean of a random variable. The upper bound of the mean of a random variable and lower boundary variance of random variables ; mean of random parameters and variance Maximum allowed number of iterations .

[0039] (2) Initialize the iteration, that is, set the number of iteration steps to 1. And set the deterministic design variables as The mean of the random variable is Then calculate the objective function value. And record; random variables and random parameters Take its mean, which is expressed by equation (1); dynamic conjugate gradient vector Take the zero vector. Jump to step (5) to start the derivation process of the approximate MPTP point.

[0040] (1) In the formula, and These represent the random variable and the random parameter, respectively. and These represent the mean of the random variable and the random parameter, respectively. Indicates the number of iterations.

[0041] (3) Input random variables and random parameters Apply the gradient algorithm to solve the following equivalent deterministic optimization model to obtain the optimal value. and ; (2) In the formula, Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Represents the mean of a random variable; Represent the objective function; Indicates the first One function; Indicates the number of functional functions; and These represent the upper and lower bounds of the deterministic design variables, respectively. and These represent the upper and lower bounds of the random variable, respectively. Indicates the number of iterations.

[0042] (4) Determine if convergence has occurred. If convergence has occurred, stop the iteration and output the result. and Conversely, if the condition is not met, then step (5) is executed to initiate the derivation process of the approximate MPTP point.

[0043] (5) Derive the function at the iteration point The normalized steepest descent direction at the point : (3) In the formula, Indicates the direction of the steepest descent under normalization; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0044] (6) Calculate the static conjugate scalar factor based on gradient information. : (4) In the formula, Represents the static conjugate scalar factor; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0045] (7) When When, the dynamic factor takes the value .when At that time, if The dynamic factor takes the value of ;like The dynamic factor takes the value of Then proceed to step (10).

[0046] (5) In the formula, Indicates a dynamic factor; Indicates the second parameter of the judgment criterion; Indicates the number of iterations. It is represented as: (6) In the formula, Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0047] when At that time, the conjugate gradient vector was derived using the traditional CGA method. : (7) In the formula, Represents the conjugate gradient vector; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the static conjugate scalar factor; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0048] (8) Evaluate the traditional conjugate gradient vector Angle value between the direction of steepest descent and the direction of most rapid descent : (8) In the formula, Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0049] (9) Apply the dual criterion to accurately assess the concavity and convexity of the function and calculate the dynamic factor. It includes the following steps: (9-1) Calculation judgment criterion 1: (9) In the formula, Indicates the parameter of judgment criterion 1; , , They represent the first , , The normalized steepest descent direction of the iteration step; Indicates the number of iterations.

[0050] (9-2) If If the condition is met, then calculate criterion 2 and proceed to step (9-3); otherwise, proceed to step (9-4).

[0051] (10) In the formula, Indicates the second parameter of the judgment criterion; Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0052] (9-3) If The dynamic factor takes the value of Otherwise, derive equation (11) to calculate the dynamic factor. Then proceed to step (10).

[0053] (11) In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the iteration step number. Dynamic conjugate gradient vector. It is expressed as: (12) In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0054] (9-4) Analyze the dynamic conjugate gradient vector Angle value between the direction of steepest descent and the direction of most rapid descent : (13) In the formula, Represents the dynamic conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0055] In the formula, the dynamic conjugate gradient vector It is expressed as: (14) In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0056] (9-5) If Then, the dynamic factor is derived using equation (15). Otherwise, dynamic factors The value is set to 1. Then proceed to step (10).

[0057] (15) In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the iteration step number. Dynamic conjugate gradient vector. The calculation method is shown in equation (12).

[0058] (10) Determine the dynamic conjugate gradient vector for the next iteration step. : (16) In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0059] (11) Derive the normalized sensitivity vector for the next iteration step : (17) In the formula, This represents the normalized sensitivity vector; Represents the dynamic conjugate gradient vector; Indicates the number of iterations.

[0060] (12) Furthermore, random variables and random parameters It is represented as: (18) In the formula, and These represent the random variable and the random parameter, respectively. and These represent the mean of the random variable and the random parameter, respectively. and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. Indicates the target reliability index; Indicates the number of iterations.

[0061] (13) Execution Then proceed to step (3).

[0062] When dynamic factors When the value is 0, the dynamic conjugate gradient vector is... That is, the direction of steepest descent, and the normalized sensitivity vector is then expressed as... (19) In the formula, and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and These represent the random variable and the random parameter, respectively. Indicates the number of iterations.

[0063] The robustness and efficiency of the MSASLA method are fully verified through two examples below. Comparison methods include the RIA method and the Performance Measurement Method (PMA). 1 / PMA 2 Sequence Optimization and Reliability Assessment Method (SORA) 1 / SORA 2 The methods include the classic SLA method and the SLS VCG method. The reliability index of the RIA method is evaluated using the classic HL-RF iterative algorithm. PMA 1 andSORA 1 Inverse reliability analysis was performed using the CGA method; PMA 2 andSORA 2 Inverse reliability analysis is performed using the HMV method. Deterministic optimization in all embodiments applies sequential quadratic programming.

[0064] The robustness and efficiency of the MSASLA method are fully verified through two examples below. Comparison methods include the RIA method and the Performance Measurement Method (PMA). 1 / PMA 2 Sequence Optimization and Reliability Assessment Method (SORA) 1 / SORA 2 The methods include the classic SLA method and the SLS VCG method. The reliability index of the RIA method is evaluated using the classic HL-RF iterative algorithm. PMA 1 andSORA 1 Inverse reliability analysis was performed using the CGA method; PMA 2 andSORA 2 Inverse reliability analysis is performed using the HMV method. Deterministic optimization in all embodiments applies sequential quadratic programming.

[0065] Example 1 includes a linear objective function. and three nonlinear function Two of the random variables , All values ​​follow a normal distribution with a standard deviation of 0.3. The target reliability index for the three probabilistic constraints... All are 3. Deterministic design variables The initial value is The mathematical model of Example 1 is described as follows: (20) In the formula, Represents deterministic design variables; Represents a random variable; Represent the objective function; Represents a nonlinear function; Indicates the target reliability index; Indicates the initial values ​​of deterministic design variables; Represents a probability operator; This represents the standard normal cumulative distribution function.

[0066] The reliability design optimization results of this weakly nonlinear embodiment are shown in Table 1. As can be seen from the table, all eight reliability design optimization algorithms converge to the same solution, i.e. ; However, their reliability design efficiency varies significantly. PMA 1 Methods, SLSVCG method and SORA 1 The number of function evaluations for the methods are 10863, 5235, and 8667 respectively, resulting in high computational costs. This is because all three methods are related to the GCA method, which has a relatively weak ability to handle convex functional functions. The RIA method and PMA using the HMV method are also relevant. 2 Law, SORA2 The computational efficiency of the methods is comparable, with 489, 492, and 426 function evaluations respectively. Compared to the double-loop method and the decoupling method, the single-loop method (SLA) only requires 210 function evaluations to complete the reliability design optimization process for this example. The improved self-adaptive single-loop method (MSASLA) proposed in this invention further improves computational efficiency, reducing the number of function evaluations to 195.

[0067] Table 1. Reliability design optimization results of Example 1

[0068] The classic Monte Carlo simulation method was applied to accurately investigate the achievement of the target reliability for each probabilistic constraint, as shown in Table 2. The number of random sampling points was 500,000. The third probabilistic constraint was invalid, and the remaining probabilistic constraints were valid. Among them, the first probabilistic constraint slightly violated the reliability requirements, while the second probabilistic constraint met the reliability requirements.

[0069] Table 2 Reliability verification of Example 1

[0070] Equation (21) describes Example 2, which includes a nonlinear objective function, a highly concave function, and two random variables that follow independent normal distributions. The target reliability index is... The initial point of this reliability design optimization problem is: The standard deviation of a random variable is The upper and lower bounds for the deterministic design variables are 10 and 0, respectively.

[0071] (twenty one) In the formula, Represents deterministic design variables; Represents a random variable; Represent the objective function; Represents a nonlinear function; Indicates the target reliability index; Indicates the initial values ​​of deterministic design variables; Represents a probability operator; This represents the standard normal cumulative distribution function.

[0072] The reliability design optimization results of nonlinear embodiment 2 are shown in Table 3. During reliability analysis, the RIA method based on the HL-RF iterative algorithm exhibits an oscillation problem with a period of 2. Applying the CGA method, PMA... 1 Law and SORA 1 The computational efficiency of this method is relatively low, with 1188 and 1556 function evaluations respectively. PMA 2Methods, SLSVCG method and SORA 2 The reliability design efficiency of the methods is comparable, with 276, 324, and 345 function evaluations respectively. The SLA method exhibits advanced computational efficiency, with only 169 function evaluations. Using the HDMC algorithm, the MSASLA method further reduces the time cost, decreasing the number of function evaluations to 154. Except for the RIA method, all probabilistic constraints meet the target reliability requirements, as shown in Table 4.

[0073] Table 3 Reliability design optimization results of Example 2

[0074] Table 4 Reliability verification of Example 2

[0075] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented, in whole or in part, as a computer program product, the computer program product includes one or more computer instructions. When the computer program instructions are loaded or executed on a computer, all or part of the processes or functions described in the embodiments of the present invention are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another website, computer, server, or data center via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium that a computer can access or a data storage device such as a server or data center that integrates one or more available media. The available medium can be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape) or an optical medium.

[0076] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An improved adaptive single-loop method based on a hybrid dynamic mean conjugate algorithm, characterized in that, Includes the following steps: (1) Set the basic parameters for reliability design optimization, including: target reliability index Convergence tolerance of deterministic design optimization Convergence tolerance of reliability analysis ; Initial values ​​of deterministic design variables ; Upper bound of deterministic design variables and lower boundary The initial value of the mean of a random variable. ; Upper bound of the mean of a random variable and lower boundary variance of random variables ; mean of random parameters and variance Maximum allowed number of iterations ; (2) Initialize the iteration, i.e. set the number of iteration steps. Set the mean values ​​of deterministic design variables and random variables, then calculate and record the objective function value; take the mean values ​​of random variables and random parameters; take the zero vector of the dynamic conjugate gradient vector; jump to step (5) to start the derivation process of the approximate MPTP point; (3) Input random variables and random parameters, and apply the gradient algorithm to solve the equivalent deterministic optimization model in order to obtain the optimal values ​​of the deterministic design variables and the mean of the random variables; (4) Determine whether convergence has occurred. If convergence has occurred, stop the iteration and output the mean of the deterministic design variables and random variables; otherwise, execute step (5) to start the derivation process of the approximate MPTP point. (5) Derive the normalized steepest descent direction of the function at the iteration point; (6) Calculate the static conjugate scalar factor based on gradient information; (7) When When, calculate the dynamic factor value, then jump to step (10); when The conjugate gradient vector is derived using the CGA method; (8) Evaluate the angle between the conventional conjugate gradient vector and the steepest descent direction; (9) Apply the dual criterion to accurately assess the concavity and convexity of the function and calculate the dynamic factor value; (10) Determine the dynamic conjugate gradient vector for the next iteration step; (11) Derive the normalized sensitivity vector for the next iteration step; (12) Calculate the random variables and random parameters for the next iteration step; (13) Execution Then proceed to step (3).

2. The improved self-adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 1, characterized in that, Step (2) specifically involves: initializing the iteration, i.e., setting the number of iteration steps to 1. And set the deterministic design variables as The mean of the random variable is Then calculate the objective function value. And record; random variables and random parameters Take their mean; dynamic conjugate gradient vector Take the zero vector and jump to step (5) to begin the derivation process of the approximate MPTP point: In the formula, Represents a random variable; Indicates a random parameter; Represents the mean of a random variable; This represents the mean of the random parameters; Indicates the number of iterations.

3. The improved self-adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 2, characterized in that, Step (3) specifically involves: inputting random variables. and random parameters Apply the gradient algorithm to solve the following equivalent deterministic optimization model to obtain the optimal value. and ; In the formula, Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Represents the mean of a random variable; Represent the objective function; Indicates the first One function; Indicates the number of functional functions; and These represent the upper and lower bounds of the deterministic design variables, respectively. and These represent the upper and lower bounds of the random variable, respectively. Indicates the number of iterations.

4. The improved adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 3, characterized in that, Steps (4) and (5) are as follows: (4) Determine if convergence has occurred. If convergence has occurred, stop the iteration and output the result. and ; Conversely, if the condition is not met, then step (5) is executed to initiate the derivation process of the approximate MPTP point; (5) Derive the function at the iteration point The normalized steepest descent direction at the point : In the formula, Indicates the direction of the steepest descent under normalization; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations.

5. The improved self-adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 4, characterized in that, Steps (6)-(7) are as follows: (6) Calculate the static conjugate scalar factor based on gradient information. : In the formula, Represents the static conjugate scalar factor; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; (7) When When, the dynamic factor takes the value ;when At that time, if The dynamic factor takes the value of ;like The dynamic factor takes the value of Then jump to step (10). In the formula, Indicates a dynamic factor; Indicates the second parameter of the judgment criterion; Indicates the number of iterations; It is represented as: In the formula, Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; when At that time, the conjugate gradient vector was derived using the traditional CGA method. : In the formula, Represents the conjugate gradient vector; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the static conjugate scalar factor; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations.

6. The improved adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 5, characterized in that, Step (8) specifically involves: evaluating the conventional conjugate gradient vector. Angle value between the direction of steepest descent and the direction of most rapid descent : In the formula, Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the conjugate gradient vector; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations.

7. The improved self-adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 6, characterized in that, Step (9) specifically involves applying the dual criterion to accurately assess the concavity and convexity of the function and calculating the dynamic factor. This includes the following steps: (9-1) Calculation judgment criterion 1: In the formula, Indicates the parameter of judgment criterion 1; , , They represent the first , , The normalized steepest descent direction of the iteration step; Indicates the number of iterations; (9-2) If If the condition is met, calculate criterion 2 and proceed to step (9-3); otherwise, proceed to step (9-4). In the formula, Indicates the second parameter of the judgment criterion; Indicates the target reliability index; Indicates the direction of the steepest descent under normalization; Represents the standard normal random space transformation; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; (9-3) If The dynamic factor takes the value of Otherwise, derive the following formula to calculate the dynamic factor. Then proceed to step (10); In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Represents the number of iterations; dynamic conjugate gradient vector It is expressed as: In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; (9-4) Analyze the dynamic conjugate gradient vector Angle value between the direction of steepest descent and the direction of most rapid descent : In the formula, Represents the dynamic conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; In the formula, the dynamic conjugate gradient vector It is expressed as: In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; (9-5) If Then, the dynamic factor can be derived using the following formula. Otherwise, dynamic factors Set the value to 1 and jump to step (10): In the formula, and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents the dynamic conjugate gradient vector; It is a constant; Represents the traditional conjugate gradient vector The angle between the direction of steepest descent and the direction of most rapid descent; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Represents the number of iterations; dynamic conjugate gradient vector The calculation method is shown in (9-3).

8. The improved adaptive single-loop method based on the hybrid dynamic mean conjugate algorithm according to claim 7, characterized in that, Steps (10) to (13) are as follows: (10) Determine the dynamic conjugate gradient vector for the next iteration step. : In the formula, and They represent the first Iteration step and the The dynamic conjugate gradient vector of the iteration step; and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Indicates a dynamic factor; Represents the static conjugate scalar factor; Represents the product of a dynamic factor and a static conjugate scalar factor; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations; (11) Derive the normalized sensitivity vector for the next iteration step : In the formula, This represents the normalized sensitivity vector; Represents the dynamic conjugate gradient vector; Indicates the number of iterations; (12) Furthermore, random variables and random parameters It is represented as: In the formula, and Let them represent the random variable and the random parameter, respectively; and These represent the mean of the random variable and the random parameter, respectively. and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. Indicates the target reliability index; Indicates the number of iterations; (13) Execution , and proceed to step (3); When dynamic factors When the value is 0, the dynamic conjugate gradient vector is... That is, the direction of steepest descent, the normalized sensitivity vector. It is then represented as In the formula, and Let represent the normalized sensitivity vectors of the random variable and the random parameter, respectively. and Let Variance and variance be represented by , respectively, for the random variable and the random parameter. The gradient vector representing the function of performance; Represents deterministic design variables; and Let them represent the random variable and the random parameter, respectively; Indicates the number of iterations.

9. A computer-readable storage medium, characterized in that, The computer-readable storage medium includes a stored computer program, wherein, when the computer program is executed, it controls the device on which the computer-readable storage medium is located to perform the method as described in any one of claims 1-8.

10. A computer device, characterized in that, The computer device includes a memory, a processor, and a program stored in and executable on the memory, the program being executed by the processor to implement the steps of the method as described in any one of claims 1-8.

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