A worst-case robust optimization method based on efficient global optimization

Abstract

The present invention discloses a robust optimization method considering the worst case based on efficient global optimization. The method comprises the following steps: Step 1, determine input parameters and set a simulation model interface for obtaining the objective function value f(x c , x n ); Step 2, generate an initial population P g through an experimental design method; Step 3, evaluate the values of the objective functions corresponding to the individuals in the population; Step 4, enter the main optimization loop and iterate until the number of iterations reaches G max ; Step 5, output the global optimal solution. The present invention models the robust optimization problem based on the idea of min-max or max-min nested optimization, and solves the problem by combining a co-evolution strategy with efficient global optimization. The efficient global optimization algorithm can improve the efficiency of robust optimization, enabling the present invention to solve the robust optimization problems in engineering design where the objective function has no closed form or the computational cost is high.

Description

Technical Field

[0001] The present invention belongs to the technical field of common basic research design, and relates to a robust optimization method considering the worst case, specifically to a robust optimization method considering the worst case based on efficient global optimization. Background Art

[0002] In order to improve the quality and reliability of the designed system, the robust optimization method can be used to adjust the control factors of the system to the optimal values, while enabling the system to resist the risks brought by uncertain interference factors. Using the idea of minimax or maximin nested optimization to construct a robust optimization problem is an easy-to-solve and widely used method. However, when the worst case of the interference factors changes with the change of the control factors, the solution of this robust optimization problem becomes very complicated, which is called an asymmetric robust optimization problem, and the traditional symmetric co-evolution algorithm cannot solve such problems. On the other hand, when solving the robust optimization problem in the engineering field, the acquisition of the objective function value of the object to be optimized is often realized through simulation. Especially in the case where the closed form of the objective function cannot be obtained, numerical methods such as the finite element method are needed to obtain the objective function value under a specific design scheme. However, in engineering design, the simulation models of many design objects have high calculation costs. For example, the calculation time to obtain the objective function value corresponding to each design scheme is very long. Therefore, the robust optimization method based on a high-cost simulation model needs to focus on the efficiency of the optimization algorithm. Summary of the Invention

[0003] In order to solve the asymmetric robust optimization problem based on the idea of minimax or maximin nested optimization, the present invention provides a robust optimization method considering the worst case based on efficient global optimization. This method is mainly aimed at the optimization problem where the objective characteristics of the object to be optimized are calculated by a black-box function. The present invention models the robust optimization problem based on the idea of minimax or maximin nested optimization, and adopts a method combining the co-evolution strategy and efficient global optimization to solve the problem. The efficient global optimization algorithm can improve the efficiency of robust optimization, enabling the present invention to complete the solution of the robust optimization problem where the objective function has no closed form or the calculation cost is high in engineering design, and can be directly applied to the robust optimization design and quality improvement design process of engineering products.

[0004] The object of the present invention is achieved by the following technical solutions:

[0005] A robust optimization method considering the worst case based on efficient global optimization, comprising the following steps:

[0006] Step 1, determine the input parameters and set the simulation model interface for obtaining the objective function value f(x c , x n ).

[0007] Step 2: Generate an initial population P g , P g ={(x c,1,g , x n,1,g ),...,(x c,N,g , x n,N,g )}, where P g represents the population after the g-th iteration, and (x c,N,g , x n,N,g ) is an array representing the N-th individual in the population P g , and x c,N,g and x n,N,g represent the control factor and the interference factor of the individual, respectively;

[0008] Step 3: Evaluate the values of the objective functions corresponding to the individuals in the population;

[0009] Step 4: Enter the main optimization loop and iterate until the number of iterations reaches G max , and the specific steps executed within the loop are as follows:

[0010] Step 41: Sort the individuals according to the values of the objective functions corresponding to the individuals in the population;

[0011] Step 42: Execute a global optimization algorithm based on the Kriging method and the expected improvement method to update the interference factor population;

[0012] Step 43: Read the individual ranked first in the population as the global optimal solution;

[0013] Step 44: Execute a differential evolution algorithm to update the control factor population;

[0014] Step 5: Output the global optimal solution.

[0015] Compared with the prior art, the present invention has the following advantages:

[0016] (1) The present invention proposes a novel co-evolutionary robust optimization strategy with an asymmetric structure. This strategy creates a control factor population and an interference factor population, and updates these two populations through an asymmetric update strategy, enabling these two populations to achieve asymmetric co-evolution and solving the asymmetric robust optimization problem that cannot be solved by the general co-evolutionary robust optimization strategy with a symmetric structure.

[0017] (2) In the architecture of the proposed asymmetric co-evolutionary robust optimization strategy, the present invention uses an efficient global optimization algorithm based on the Kriging method and the expected improvement method to update the interference factor population and improve the efficiency of the optimization method, enabling it to solve the robust optimization problem where the objective function to be optimized has no closed form or requires a high-cost simulation model to solve.

[0018] (3) In the architecture of the proposed asymmetric co-evolution robust optimization strategy of the present invention, the differential evolution algorithm is used to update the population of control factors, forming an asymmetric co-evolution optimization structure with the update and evolution of the population of interference factors.

[0019] (4) The present invention can solve the robust optimization problems constructed based on the maximin or minimax optimization idea, where the objective function satisfies or does not satisfy the symmetry condition, and the error between the solution result and the benchmark result is controlled within 0.1%.

[0020] (5) The present invention has a very high solution efficiency, and the solution efficiency is increased by more than 60% compared with the calculation method of the traditional relaxation optimization method.

[0021] (6) The present invention can solve the product robust design problems where the objective function has no closed form or its calculation comes from a high-computation-cost simulation model.

[0022] (7) While optimizing the target functional characteristics of engineering products, the present invention ensures the optimal robustness of the design parameters, that is, the product control factors, and improves the robustness, reliability and quality of engineering products. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 is a flowchart of the robust optimization method considering the worst case based on efficient global optimization of the present invention;

[0024] Figure 2 is a schematic diagram of the internal structure of a certain type of three-phase AC contactor. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0025] The technical solutions of the present invention will be further described below with reference to the drawings, but are not limited thereto. Any modification or equivalent replacement of the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention shall be covered by the protection scope of the present invention.

[0026] The present invention provides a robust optimization method considering the worst case based on efficient global optimization, as Figure 1 shown, the method includes the following steps:

[0027] Step 1, determine the input parameters, and set the simulation model interface for obtaining the objective function value f(x c , x n ), where the input parameters include: the dimension D c of the control factors and the dimension D n of the interference factors, the feasible region X c of the control factors and the feasible region X n of the interference factors, and the upper limit value G max of the total number of optimization iterations., experimental design method, the number of individuals N in the population, the number of samples N in the experimental design for the stage of updating interference factors n , the upper limit t of the number of objective function calculations performed in the stage of updating interference factors max , the number of individuals α updated in the stage of updating interference factors, the optimal individual neighborhood parameter δ in the stage of updating control factors, and the number of individuals β updated in the stage of updating control factors.

[0028] Step 2: Generate an initial population P through the experimental design method g , P g = {(x c,1,g , x n,1,g ),...,(x c,N,g , x n,N,g )}, where P g represents the population after the g-th iteration, (x c,N,g , x n,N,g ) is an array representing the N-th individual in the population P g , and x c,N,g and x n,N,g represent the control factor and the interference factor of the individual respectively.

[0029] Step 3: Evaluate the values of the objective functions corresponding to the individuals in the population.

[0030] Step 4: Enter the main optimization loop and iterate until the number of iterations reaches G max , and the specific steps executed within the loop are as follows:

[0031] Step 41: Sort the individuals according to the values of the objective functions corresponding to the individuals in the population, where: the value of the objective function corresponding to the individual ranked first is the optimal value, and the value of the objective function corresponding to the individual ranked last is the worst value.

[0032] Step 42: Execute the global optimization algorithm based on the Kriging method and the expected improvement method to update the interference factor population, and the specific steps executed are as follows:

[0033] Step 421: Read the individual (x c,1st,g , x n,1st,g ) ranked first in the population and its corresponding objective function value.

[0034] Step 422: Conduct experimental design within the feasible region X n of the interference factors to generate N n new interference factors.

[0035] Step 423: Combine the generated interference factors with x c,1st,g to generate N n new individuals and calculate the corresponding objective function values.

[0036] Step 424: Based on the Kriging method and the expected improvement method, using the obtained N n new individuals and the individual (x c,1st,g , x n,1st,g ), and their corresponding objective function values, accumulate data iteratively to find (x c,1st,g , x n,更优,g ) such that f(x c,1st,g , x n,1st,g ) is better than f(x c,1st,g , x n,更优,g ) until the number of calculations of the objective function reaches t max . The specific calculation steps of this data accumulation and iteration process are as follows:

[0037] (1) Fit the Kriging model using the known data set.

[0038] (2) Use the fitted Kriging model to find the interference factor x n,更优,g with the optimal expected improvement value.

[0039] (3) Combine the obtained interference factor x n,更优,g with x c,1st,g to form a new individual (x c,1st,g , x n,更优,g ), and calculate its objective function value.

[0040] (4) Add the new individual (x c,1st,g , x n,更优,g ) and its objective function value to the data set.

[0041] Step 425: Replace the interference factors of the first α individuals in the population P n,更优,g with the found interference factor x g . The value of α is greater than 1 and less than half of the population size, usually taking 10% of the population.

[0042] Step 426: Calculate the objective function values of the updated α individuals.

[0043] Step 427: Output the updated population P g .

[0044] Step 43: Read the individual ranked first in the population as the global optimal solution.

[0045] Step 44: Execute the differential evolution algorithm to update the control factor population. The specific execution steps are as follows:

[0046] Step 441: Arrange the individuals in the population in descending order according to their corresponding objective function values.

[0047] Step 442: Read the first individual (xc,1st,g , x n,1st,g ), that is, the optimal individual.

[0048] Step 443: Randomly select β control factors from the region centered on x c,1st,g with a radius of δ. The value of β is greater than 1 and less than half of the population size, and generally 10% of the population is taken.

[0049] Step 444: Randomly select β interference factors in X n to form β new individuals together with the β newly generated control factors in Step 443, and replace the last β individuals sorted in population P g .

[0050] Step 445: Output the updated population P g .

[0051] Step 5: Output the global optimal solution.

[0052] The above method solves the following technical problems:

[0053] 1. Solved the robust optimization design problem constructed based on the min-max or max-min nested optimization idea, as shown in Equation (1) or Equation (2) respectively:

[0054]

[0055]

[0056] where x c and x n are the control factor and the interference factor respectively, X c and X n are the feasible regions of the control factor and the interference factor respectively, f(x c , x n ) is the objective function, and x c最优 is the optimal solution of the control factor.

[0057] 2. Solved the problem that when the objective function does not satisfy the symmetry condition shown in Equation (3), the general symmetric co-evolution algorithm cannot solve the robust optimization problem shown in Equation (1) or Equation (2):

[0058]

[0059] 3. Solved the robust optimization problem where the target functional characteristics of the object to be optimized need to be solved by a simulation model with high computational cost, or solved the robust optimization problem where the objective function to be optimized has no closed form.

[0060] 4. It lays a theoretical foundation for efficiently solving the robust optimization design problem, provides a new optimization design method for the robustness, reliability, and quality improvement of engineering products, and has certain reference significance for enterprises and manufacturers committed to improving the quality of engineering products.

[0061] The present invention can be verified by solving the robust optimization benchmark problem and comparing the solution results with the benchmark results. The solution error can be calculated by the mean square error (MSE) shown in Equation (4):

[0062]

[0063] where, x c,求解 is the result of the control factors solved by the method proposed in the present invention, x c,基准 is the benchmark result, D n is the dimension of the control factors of the problem to be solved, x c,i,求解 is the i-th dimension value of x c,求解 and x c,i,基准 is the i-th dimension value of x c,基准 .

[0064] The present invention can also be verified by performing robust optimization design on actual engineering products. The specific verification method is as follows:

[0065] (1) Test the target characteristics of mass engineering products and record the mean and variance of the target characteristic test results.

[0066] (2) Establish a simulation model of the engineering product, which takes a set of determined control factors and interference factors as inputs and the corresponding optimized target characteristic values as outputs.

[0067] (3) Determine the parameters required to be set in the present invention according to the actual situation of the engineering product.

[0068] (4) Execute the method proposed in the present invention to calculate the optimal control factor design scheme.

[0069] (5) Produce mass prototypes of the engineering product according to the solved control factor design scheme.

[0070] (6) Test the target characteristics of the optimized mass prototypes of the engineering product and record the mean and variance of the target characteristic test results.

[0071] (7) Compare the test results of the target characteristics of the mass engineering products before and after optimization.

[0072] Example 1. Verification of the benchmark problem

[0073] In this embodiment, the method proposed by the present invention is verified using the following commonly used benchmark test problems:

[0074] f(x c ,x n )=(x c -5) 2 -(x n -5) 2 (5)

[0075] where x c ∈[0,10], x n ∈[0,10], (x c,基准 )=(5).

[0076] f(x c ,x n )=min{3 - 0.2x c +0.3x n , 3 + 0.2x c -0.1x n}(6)

[0077] where x c ∈[0,10], x n ∈[0,10], (x c,基准 )=(0).

[0078]

[0079] where x c ∈(0,10], x n ∈(0,10], (x c,基准 )=(10).

[0080]

[0081] where x c ∈[-0.5,0.5]×[0,1], x n ∈[0,10] 2 , (x c,基准 )=(0.5,0.25).

[0082]

[0083] where x c ∈[-1,3] 2 , x n ∈[0,10] 2 , (x c,基准 )=(1,1).

[0084] Calculated benefits: The error between the calculation results of solving the above benchmark problem and the benchmark results is less than 0.1%. The number of objective function calculations consumed is less than 100 times, 150 times, 250 times, 150 times, and 150 times respectively. Compared with the calculation method of the traditional relaxation optimization method, the solution efficiency is increased by more than 60%.

[0085] Example 2. Verification of actual engineering problems

[0086] In this example, the method proposed by the present invention is used for Figure 2 robust optimization design of a certain type of three-phase AC contactor as shown. Where a is the starting coil, b is the holding coil, c is the return spring, d is the over-travel spring, e is the vacuum tube, f and g are the contacts inside the vacuum tube, h is the rotating arm, i is the rotating shaft, T represents torque, M represents moment of inertia, θ represents the angle of rotation, l represents the opening distance, k 1 represents the stiffness of the return spring, k 2 represents the stiffness of the over-travel spring, k 3 represents the stiffness of the vacuum tube.

[0087] In this robust optimization design problem, the control factors are: the number of turns of the starting coil, the number of turns of the holding coil, the resistance of the starting coil, the resistance of the holding coil, the length of the return spring, the length of the over-travel spring, the initial length of the return spring, the initial length of the over-travel spring, the stiffness of the return spring, and the stiffness of the over-travel spring. The interference factors are: the maximum angle of rotation, the opening distance, and the ambient temperature. The optimization target characteristics are: the closing time (a characteristic of expecting a smaller value), the release time (a characteristic of expecting a smaller value), and the closing temperature rise (a characteristic of expecting a smaller value).

[0088] First, establish a simulation model of the three-phase AC contactor. This simulation model takes a set of determined control factors and interference factors as inputs and the corresponding optimization target characteristic values as outputs. Then execute the method proposed by the present invention for calculation to calculate the optimal control factor design scheme.

[0089] Calculated benefits: The closing time of the batch contactor products is optimized to a mean reduction of 20% and a variance reduction of 50%. The release time is optimized to a mean reduction of 10% and a variance reduction of 50%. The closing temperature rise is optimized to a mean reduction of 30% and a variance reduction of 60%. Compared with the calculation method of the traditional relaxation optimization method in the calculation process, the solution efficiency is increased by 65%.

Claims

1. A robust optimization method considering the worst case based on efficient global optimization, characterized in that the method first establishes a simulation model of a three-phase AC contactor, which takes a set of determined control factors and interference factors as inputs, and then performs the following steps: Step 1: Determine the input parameters and set the simulation model interface for obtaining the objective function value f(x c , x n ); Step 2: Generate the initial population P by means of experimental design method g , P g = {(x c,1,g , x n,1,g ),...,(x c,N,g , x n,N,g )}, where P g represents the population after the g-th iteration, (x c,N,g , x n,N,g ) is an array representing the N-th individual in the population P g , x c,N,g and x n,N,g respectively represent the control factor and interference factor of the individual. The control factors are: number of turns of the starting coil, number of turns of the holding coil, resistance of the starting coil, resistance of the holding coil, length of the return spring, length of the overtravel spring, initial length of the return spring, initial length of the overtravel spring, stiffness of the return spring, stiffness of the overtravel spring; the interference factors are: maximum rotation angle, opening distance, ambient temperature; Step 3: Evaluate the values of the objective functions corresponding to the individuals in the population; Step 4: Enter the main optimization loop and iterate until the number of iterations reaches G max , and the specific steps executed within the loop are as follows: Step 41: Sort the individuals according to the objective function values corresponding to the individuals in the population, where: the objective function value corresponding to the individual ranked first is the optimal value, and the objective function value corresponding to the individual ranked last is the worst value; Step 42: Execute a global optimization algorithm based on the Kriging method and the expected improvement method to update the interference factor population, and the specific execution steps are as follows: Step 421: Read the individual ranked first in the population (x c,1st,g , x n,1st,g ) and its corresponding objective function value; Step 422, perform experimental design within the feasible region X of the interference factor n to generate N n new interference factors; Step 423: Combine the generated interference factors with x c,1st,g to generate N n new individuals, and calculate the corresponding objective function values; Step 424: Based on the Kriging method and the expected improvement method, using the obtained N n new individuals and the individual (x c,1st,g , x n,1st,g ), and their corresponding objective function values, iteratively accumulate data to find (x c,1st,g , x n,更优,g ) such that f(x c,1st,g , x n,1st,g ) is better than f(x c,1st,g , x n,更优,g ) until the number of calculations of the objective function reaches t max ; Step 425, use the found interference factor x n,更优,g to replace the interference factors of the individuals of the first α before sorting in the population P g ; Step 426: Calculate the objective function values of the updated α individuals; Step 427, output the updated population P g ; Step 43: Read the individual ranked first in the population as the global optimal solution; Step 44: Execute a differential evolution algorithm to update the control factor population; Step 5: Output the global optimal solution, that is, the optimal control factor design scheme.

2. The robust optimization method considering the worst case based on efficient global optimization according to claim 1, characterized in that in the step 424, the specific calculation steps of the data accumulation and iteration process are as follows: (1) Fit a Kriging model using the known data set; (2) Use the fitted Kriging model to find the interference factor x with the optimal expected improvement value n,更优,g ; (3) Combine the obtained interference factor x n,更优,g with x c,1st,g to form a new individual (x c,1st,g , x n,更优,g ), and calculate its objective function value; (4) Add the new individual (x c,1st,g , x n,更优,g ) and its objective function value to the data set.

3. The robust optimization method considering the worst case based on efficient global optimization according to claim 1, characterized in that in the step 425, the value of α is greater than 1 and less than half of the population size.

4. The robust optimization method considering the worst case based on efficient global optimization according to claim 1, characterized in that the specific execution steps of the step 44 are as follows: Step 441: Arrange the individuals in the population from the best to the worst according to their corresponding objective function values; Step 442, read the first individual in the sorting (x c,1st,g , x n,1st,g ), that is, the optimal individual; Step 443, randomly select β control factors from the region centered at x c,1st,g with a radius of δ; Step 444: Randomly select β interference factors in X n and form β new individuals by combining them with the β new control factors newly generated in Step 443, and replace the last β individuals sorted in population P g ; Step 445, output the updated population P g .

5. The robust optimization method considering the worst case based on efficient global optimization according to claim 4, characterized in that in the step 444, the value of β is greater than 1 and less than half of the population size.

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