Optimization algorithm based on inverse solution of objective function

Abstract

The present application relates to the technical field of optimization algorithm, and especially relates to an optimization algorithm based on inverse solution of objective function. The optimization algorithm based on inverse solution of objective function comprises the following steps: according to evaluation index and decision variable of optimization problem, the optimization problem is induced into objective function form, and the value range of each element in the decision variable is determined; the inverse matrix of Jacobian matrix between the objective function value and the decision vector is used to solve the inverse solution of the objective function; according to the property of the objective function, the rough range of the global optimal value is determined; the bisection method is used to search the global optimal solution in the range of the global optimal value until the difference between the upper limit and the lower limit is less than the specified precision, at which time the upper limit is the global optimal value, and the decision vector solved by the inverse solution method is the global optimal solution. The present application can solve the problems that the optimization algorithm is easy to fall into local optimum and has large calculation amount.

Description

Technical Field

[0001] This invention relates to the field of optimization algorithm technology, and in particular to an optimization algorithm based on the inverse solution of the objective function. Background Technology

[0002] Optimization is one of the simplest methods to improve product performance and production efficiency, such as optimizing the design of production equipment and product structures, and optimizing the layout of production lines. These optimization problems can all be analogized to a mathematical problem—finding the extremum of an objective function. The independent variable of the objective function is the decision variable that needs to be optimized, and the value of the objective function is the evaluation index of the optimization problem. The existence of an extremum in the objective function represents the best possible outcome. The decision variable corresponding to the extremum of the objective function is the optimal solution. The process of finding the optimal solution is called optimization. Obtaining the optimal solution requires an optimization method that is simple and fast in calculation, and whose optimization results are accurate and reliable.

[0003] Existing optimization methods can be broadly categorized into two main types: deterministic optimization and heuristic optimization. Classical deterministic optimization algorithms include: the simplex method, the ellipsoidal algorithm (polynomial algorithm), interior-point methods, steepest descent methods, conjugate gradient methods, Newton's method, quasi-Newton methods, trust region methods, and Lagrange multiplier methods. Deterministic optimization algorithms typically use a feasible solution as the initial value for iteration. When searching in the solution space, their search strategy is deterministic, and most search strategies are based on the derivative of the objective function. While deterministic algorithms have relatively low computational cost, they impose stringent conditions on the properties of the objective function. Furthermore, when deterministic algorithms iterate to the vicinity of a local optimum of the objective function, it is difficult for them to escape this local optimum.

[0004] Heuristic optimization algorithms mainly include: genetic algorithms, differential evolution algorithms, ant colony optimization, simulated annealing, and particle swarm optimization. Most heuristic algorithms use a set of feasible solutions as initial values ​​for iteration. Their search strategies largely simulate natural phenomena and the cooperative behavior of biological groups. Therefore, their search strategies are both structured and randomized. The cooperative search strategy of heuristic algorithms ensures the breadth and depth of the search, but the computational cost becomes extremely high when dealing with large-scale optimization problems.

[0005] Currently, both deterministic and heuristic optimization algorithms fundamentally operate on the principle of iterative searching within the solution space. Each iteration assesses whether the objective function is better and determines the direction and step size for the next search based on the derivative or value of the objective function. Iteration terminates when the maximum number of iterations is reached or the optimal solution for the objective function is found. However, the optimal solution obtained after iteration termination is not necessarily globally optimal; it may be locally optimal. This is because deterministic optimization methods largely rely on the derivative of the objective function to determine the search direction, and the derivative provides local information, making it difficult for deterministic algorithms to escape local optima. Heuristic algorithms typically use a set of initial values ​​within the solution space for iterative searching. While the search direction and step size have some randomness, there is still a tendency to converge towards the group optimum, leading to the possibility of getting trapped in local optima. To reduce the likelihood of getting trapped in local optima, it is necessary to increase the number of initial values ​​and the breadth of the search. However, as the complexity of the optimization problem increases, the computational cost also increases. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of existing technologies and propose an optimization algorithm based on the inverse solution of the objective function, which can solve the problems of optimization algorithms easily getting trapped in local optima and large computational load.

[0007] To achieve the above objectives, the present invention adopts the following specific technical solution:

[0008] The present invention provides an optimization algorithm based on the inverse solution of the objective function, comprising the following steps:

[0009] S1. Based on the evaluation indicators and decision variables of the optimization problem, the optimization problem is reduced to the form of an objective function, and the range of values ​​for each element in the decision variables is determined.

[0010] S2. Use the inverse of the Jacobian matrix between the objective function value and the decision vector to solve for the inverse of the objective function;

[0011] S3. Determine the approximate range of the global optimum based on the properties of the objective function;

[0012] S4. Within a rough range of the global optimum, use the bisection method to search for an approximate global optimum until the difference between the upper and lower bounds is less than the specified precision. The upper bound at this point is the global optimum, and the decision vector obtained by the inverse solution method is the global optimum.

[0013] Furthermore, in step S1, the objective function takes the following form:

[0014] min f(X)

[0015]

[0016] Where f(X) is the objective function, X = (x1, x2, ..., xn) n Let g(X) be an n-dimensional decision variable, g(X) and h(X) be constraints on X, and H be the solution space.

[0017] Furthermore, in step S2, the definition of the Jacobian matrix is ​​as follows:

[0018]

[0019] in, This represents the rate of change of the objective function. J represents the rate of change of the decision variable, and J represents the Jacobian matrix. + Represents the generalized inverse of the Jacobian matrix;

[0020] The inverse of the Jacobian matrix is ​​obtained iteratively, and the iterative formula is as follows:

[0021]

[0022] Among them, f e X represents the desired objective function value. n f represents the current solution. n This represents the current objective function value, and k represents the iteration coefficient.

[0023] Furthermore, in step S2, multiple initial values ​​are used for parallel calculation to simultaneously find the inverse solution, thereby improving the solution speed and accuracy.

[0024] Furthermore, in step S3, if the approximate range of the global optimum cannot be directly determined, the following method can be used:

[0025] S31. Randomly select a feasible decision variable and calculate the corresponding objective function value as the initial value for iteration;

[0026] S32. Iterate the search for the objective function value in a better direction, and use a larger step size to explore and determine the approximate range of the global optimum value of the objective function. The search step size can be several times the absolute value of the initial value of the iteration.

[0027] S33. The feasibility of the search is determined by whether the searched objective function value has a corresponding inverse solution. When the searched objective function value does not have a corresponding decision variable, this function value is taken as the lower bound f of the interval where the optimal value is located. down The function value of the corresponding decision variable from the previous step is taken as the upper bound f of the interval. up Determine the global optimum of the objective function within the interval (f down ,f up ]Inside.

[0028] The present invention can achieve the following technical effects:

[0029] This invention proposes a novel optimization method that addresses the problems of current optimization algorithms easily getting trapped in local optima and incurring high computational costs. It proposes a search strategy based on the inverse solution of the objective function value. This algorithm does not search the solution space but rather searches for the objective function value within its range, consistently moving towards a better objective function value. At each search step, the decision vector corresponding to the objective function value is calculated using the inverse solution method. The existence of a corresponding decision vector determines the next search strategy. Since the objective function is continuous, it must possess a global optimum. The algorithm proposed in this invention searches for the objective function value in the direction of optima, theoretically unaffected by local optima. Therefore, optimization using the algorithm proposed in this invention can theoretically converge to the global optimum without getting trapped in local optima. Attached Figure Description

[0030] Figure 1 This is a flowchart illustrating an optimization algorithm based on the inverse solution of the objective function provided in an embodiment of the present invention.

[0031] Figure 2 This is a schematic diagram of the binary search process provided in an embodiment of the present invention.

[0032] Figure 3 This is a result distribution diagram of test function 8 provided according to an embodiment of the present invention.

[0033] Figure 4 This is a result distribution diagram of test function 9 provided according to an embodiment of the present invention.

[0034] Figure 5 This is the optimized result of the natural frequency of the micro-vibration simulation platform provided according to an embodiment of the present invention. Detailed Implementation

[0035] In the following description, embodiments of the invention will be described with reference to the accompanying drawings. In the description below, the same modules are denoted by the same reference numerals. Where the same reference numerals are used, their names and functions are also the same. Therefore, their detailed description will not be repeated.

[0036] 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 and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and do not constitute a limitation thereof.

[0037] Figure 1 The flowchart of the optimization algorithm based on the inverse solution of the objective function provided in the embodiment of the present invention is shown.

[0038] like Figure 1 As shown, the optimization algorithm based on the inverse solution of the objective function provided in this embodiment of the invention includes the following steps:

[0039] S1. Based on the evaluation indicators and decision variables of the optimization problem, the optimization problem is reduced to the form of an objective function, and the range of values ​​for each element in the decision variables is determined.

[0040] The objective function takes the following form:

[0041] min f(X)

[0042]

[0043] Where f(X) is the objective function, X = (x1, x2, ..., xn) n Let g(X) be an n-dimensional decision variable, g(X) and h(X) be constraints on X, and H be the solution space.

[0044] S2. Use the inverse of the Jacobian matrix between the objective function value and the decision vector to solve for the inverse of the objective function.

[0045] The definition of the Jacobian matrix is ​​as follows:

[0046]

[0047] in, This represents the rate of change of the objective function. J represents the rate of change of the decision variable, and J represents the Jacobian matrix. + Represents the generalized inverse of the Jacobian matrix;

[0048] The inverse of the Jacobian matrix is ​​obtained iteratively, and the iterative formula is as follows:

[0049]

[0050] Among them, f e X represents the desired objective function value. n f represents the current solution. n This represents the current objective function value, and k represents the iteration coefficient.

[0051] To improve the solution speed and accuracy, multiple initial values ​​can be used for parallel computation to find the inverse solution simultaneously.

[0052] S3. Determine the approximate range of the global optimum based on the properties of the objective function.

[0053] If the approximate range of the global optimum cannot be determined directly, the following method can be used:

[0054] S31. Randomly select a feasible decision variable and calculate the corresponding objective function value as the initial value for iteration;

[0055] S32. Iterate the search for the objective function value in a better direction, and use a larger step size to explore and determine the approximate range of the global optimum value of the objective function. The search step size can be several times the absolute value of the initial value of the iteration.

[0056] S33. The feasibility of the search is determined by whether the searched objective function value has a corresponding inverse solution. When the searched objective function value does not have a corresponding decision variable, this function value is taken as the lower bound f of the interval where the optimal value is located. down The function value of the corresponding decision variable from the previous step is taken as the upper bound f of the interval. up Determine the global optimum of the objective function within the interval (f down ,f up ]Inside.

[0057] S4. Within a rough range of the global optimum, use a bisection search to approximate the global optimum until the difference between the upper and lower bounds is less than the specified precision. The upper bound at this point is the global optimum, and the decision vector obtained through the inverse search method is the global optimum. The bisection search process is as follows: Figure 2 As shown.

[0058] The optimization algorithm provided by this invention searches for the value of the objective function within the range of the objective function. Therefore, as long as the value of the objective function is continuous within the domain of the decision vector, the algorithm can theoretically converge to the global optimum without being affected by local optima.

[0059] To verify the correctness of this invention, ten commonly used continuous test functions were selected for experimentation, as shown in Table 1. Particle swarm optimization (PSO), differential evolution (DEE), and simulated annealing (SAM) algorithms were used for comparison. Each algorithm performed ten calculations on each test function, and the mean and standard deviation of the ten results are shown in Table 2. The table shows that the algorithms used yielded good results for most continuous test functions. However, for functions like 5, 8, and 9, where small changes in the independent variable significantly affect the function value, the results from PSO, DEE, and SAM were less stable. The algorithm proposed in this invention, however, still maintains good performance for such continuous functions. The distribution of test results for test functions 8 and 9 was analyzed. Figure 3 The distribution of the results for test function 8 is shown. Figure 4 The distribution of the results for function 9 is shown in the figure. It can be seen from the figure that the result calculated by the optimization method proposed in this invention is closer to the optimal solution and does not get trapped in local optima.

[0060] Table 1 Test Functions

[0061]

[0062] Table 2 Test Results

[0063]

[0064]

[0065] After verifying the test function, the optimization method of this patent was applied to the configuration optimization of the Stewart platform to check its engineering application capability. The first six natural frequencies ω = (ω1, ω2, ω3, ω4, ω5, ω6) of the Stewart platform are related to the platform's structural parameters. The main structural parameters of the Stewart platform are the upper hinge circle radius R. p The radius of the lower hinge circle R b Central corner of the platform Let θ be the central angle of the lower platform and H be the height of the moving coordinate system in the fixed coordinate system. When the Stewart platform is used as a micro-vibration simulation platform, the sixth natural frequency of the parallel platform should be minimized. The objective function of this optimization problem is min{ω6}, and the decision variables are...

[0066] The dynamic model of the Stewart platform is derived and can be simplified as follows:

[0067]

[0068] Where M(q) is the generalized mass matrix and K(q) is the generalized stiffness matrix. J is the matrix of Coriolis force and centrifugal force. T f is the transpose of the Jacobian matrix from the moving coordinate system to the fixed coordinate system. a The output forces of each outrigger are calculated using the undamped free vibration equation, and the mode transformation matrix is ​​then used to decouple the generalized mass matrix, generalized damping matrix, and generalized stiffness matrix modally. Without considering system damping, the system's free vibration equation can be written as:

[0069]

[0070] The necessary and sufficient condition for the existence of a non-zero solution is:

[0071] |K(q)-ω 2 M(q)|=0

[0072] In the formula, ω is the natural frequency of the system. Solving the above formula will give the natural frequency of the system.

[0073] The proposed algorithm, along with particle swarm optimization, simulated annealing, and differential evolution algorithms, were used to optimize the configuration of the micro-vibration simulation platform. The objective function of the optimization problem was min{ω6}, and the range of decision variables is shown in Table 3. The results of each algorithm performing 10 optimizations on the micro-vibration simulation platform are shown in Tables 4 and 5. Figure 5 As shown in the figure, the algorithm proposed in this invention achieves relatively good optimization results and does not get trapped in local optima, demonstrating the algorithm's engineering application capability.

[0074] Table 3. Range of configuration parameters for the micro-vibration simulation platform

[0075]

[0076] Table 4. Optimization Results of the Micro-Vibration Simulation Platform Configuration

[0077]

[0078] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0079] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention.

[0080] The specific embodiments of the present invention described above do not constitute a limitation on the scope of protection of the present invention. Any other corresponding changes and modifications made in accordance with the technical concept of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. An optimization algorithm based on inverse solution of objective function, used for configuration optimization of Stewart platform for micro-vibration simulation and reduction of the sixth order natural frequency of the Stewart platform, comprising the following steps: S1. According to the evaluation index and decision variable of the optimization problem, the optimization problem is summarized into the form of objective function, and the value range of each element in the decision variable is determined; S2. The inverse matrix of the Jacobian matrix between the objective function value and the decision vector is used to solve the inverse solution of the objective function; S3. According to the properties of the objective function, the rough range of the global optimal value is determined; in the step S3, if the rough range of the global optimal value cannot be directly determined, the following method is used: S31. A feasible decision variable is randomly taken, and the corresponding objective function value is calculated as the initial value of iteration; S32. The objective function value is iteratively searched in the direction of better value, and the search step is determined by the absolute value of the initial value of iteration multiplied by a certain number, to determine the rough range of the global optimal value of the objective function; S33, determining the feasibility of the search by judging whether the searched objective function value has a corresponding inverse solution, and taking the function value as the lower bound of the interval where the optimal value is located when the searched objective function value has no corresponding decision variable , taking the function value with the corresponding decision variable in the previous step as the upper bound of the interval , determining that the global optimal value of the objective function is in the interval . S4. The bisection method is used to search for the global optimal solution within the rough range of the global optimal value, until the difference between the upper bound and the lower bound is less than the specified precision, at which time the upper bound is the global optimal value, and the decision vector solved by the inverse solution method is the global optimal solution.

2. The optimization algorithm based on inverse solution of an objective function according to claim 1, characterized in that, In the step S1, the form of the objective function is as follows: wherein, is the objective function, is n decision variables of dimension, and is X the constraint condition, H is the solution space.

3. The optimization algorithm based on inverse solution of an objective function according to claim 1, wherein, In the step S2, the definition of the Jacobian matrix is as follows: wherein, denotes the rate of change of the objective function, denotes the rate of change of the decision variable, denotes the Jacobian matrix, denotes the generalized inverse of the Jacobian matrix; The Jacobian matrix is solved by iteration, and the iteration formula is as follows: wherein, denotes the desired objective function value, denotes the current solution, denotes the current objective function value, k denotes the iteration coefficient.

4. The optimization algorithm based on inverse solution of an objective function according to claim 1, wherein, In the step S2, multiple initial values of iteration are taken to calculate and solve the inverse solution in parallel, to improve the solving speed and accuracy.

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