Differential-Geometry-Based Distance Optimization Calculation Method and Computer-Readable Medium

Abstract

The present invention relates to a distance optimization calculation method and a computer-readable medium based on differential geometry. The method includes obtaining decision variables and objective variables corresponding to a problem to be solved; constructing a Riemannian space based on the objective variables; selecting a theoretically optimal value of the objective variables, and obtaining a decision variable point corresponding to the decision variables in the Riemannian space; and obtaining optimal decision variables based on the distance between the theoretically optimal value and the decision variable point according to the geodesic equation of the Riemannian space. The present invention constructs a curved multi-dimensional solution space (Riemannian space) with objective variables, introduces the non-linear correlation between variables into the metric tensor of the Riemannian space, and uses the geodesic length in the Riemannian space to describe the similarity degree between two objective points, so as to solve the optimization problem including non-linear correlation.

Description

Technical Field

[0001] The present invention relates to the field of target optimization, and more specifically, to a distance optimization calculation method based on differential geometry and a computer-readable medium. Background Art

[0002] In life and engineering applications, we often encounter numerous multi-objective optimization problems. For example, when choosing a travel route, we consider factors such as the shortest distance, the least cost, the shortest time, the number of transfers, and even the comfort level of the train. These objectives often conflict with each other and cannot all be optimized simultaneously. How to quickly and accurately make a choice to obtain a globally optimal result has been a widely explored issue. Pareto first proposed the concept of the Pareto solution set. Nowadays, we can already calculate the Pareto solution set of multi-objective optimization problems through many optimization algorithms. However, the Pareto solution set still contains multiple solutions, and further judgment is needed to select the best one. The determination of the solution space distance is one of the judgment methods.

[0003] However, the traditional determination of the solution space distance is to construct a linear solution space, set an ideal solution, and use the Euclidean distance from the projection of the strategy solution in the solution space to the ideal solution as the determination condition to select the optimal strategy solution. But this method will fail when there are non-linear correlations between variables, different weights of variables, and the weight of variables is related to the value of the variables themselves. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a distance optimization calculation method based on differential geometry and a computer-readable medium for the above-mentioned defects of the prior art, which can be applied to scenarios with non-linear correlation variables and variables whose weights are related to the variables themselves.

[0005] The technical solution adopted by the present invention to solve its technical problems is: constructing a distance optimization calculation method based on differential geometry, including the following steps:

[0006] S1. Obtain the decision variables and objective variables corresponding to the problem to be solved;

[0007] S2. Construct a Riemannian space based on the objective variables;

[0008] S3. Select the theoretical optimal value of the objective variables and obtain the decision variable points corresponding to the decision variables in the Riemannian space;

[0009] S4. Based on the geodesic equation of the Riemannian space, obtain the optimal decision variables according to the distance between the theoretical optimal value and the decision variable points.

[0010] In the distance optimization calculation method based on differential geometry according to the present invention, step S1 further includes the following steps:

[0011] S11. Based on the problem to be solved, obtain the corresponding n-dimensional decision variables ;

[0012] S12. Based on the problem to be solved and the n-dimensional decision variables, obtain m-dimensional objective variables , where , ; represents the functional relationship between the objective variables and the decision variables, and n and m are positive integers.

[0013] In the distance optimization calculation method based on differential geometry according to the present invention, step S2 further includes the following steps:

[0014] S21. Span the m-dimensional objective variables to form an m-dimensional parameter space ;

[0015] S22. Based on the parameter space and the m-dimensional objective variables , construct a Riemann space , where , represents the continuous objective variables extended based on ;

[0016] S23. In the Riemann space , construct a metric tensor , where X and Y respectively represent the vectors in the Riemann space , represents the weight of the continuous objective variables and the correlation function between the continuous objective variables, and are positive integers.

[0017] In the distance optimization calculation method based on differential geometry according to the present invention, in step S23:

[0018] When the continuous objective variables are linearly independent and have the same weight, the metric tensor ;

[0019] When the continuous objective variables are linearly independent and the weight is a fixed value, the metric tensor , where represents the weight of the continuous objective variables , ;

[0020] When the continuous target variables are linearly independent but the weights of the continuous target variables are related to the values of the continuous target variables, the metric tensor , where represents the weight of the continuous target variable and is a function of the continuous target variable ;

[0021] When the continuous target variables are correlated with each other and the weights of the continuous target variables are related to the values of the continuous target variables, the metric tensor , where is the weight function of the continuous target variable .

[0022] In the distance optimization calculation method based on differential geometry according to the present invention, the step S3 further includes the following steps:

[0023] S31. Select the theoretical optimal value of the target variable ;

[0024] S32. Obtain the decision variable points corresponding to each decision variable in the Riemann space.

[0025] In the distance optimization calculation method based on differential geometry according to the present invention, the step S32 further includes the following steps:

[0026] S321. Calculate the pareto solution set of the decision variable using an optimization algorithm;

[0027] S322. Substitute the constraint conditions of the target variable to obtain the pareto solution set that satisfies the constraint conditions;

[0028] S323. Obtain the decision variable points corresponding to the decision variables in the pareto solution set that satisfies the constraint conditions in the Riemann space.

[0029] In the distance optimization calculation method based on differential geometry according to the present invention, the step S4 further includes the following steps:

[0030] S41. Calculate the distance between the theoretical optimal value and the decision variable point based on the geodesic equation of the Riemann space;

[0031] S42. Take the decision variable with the minimum distance as the optimal decision variable.

[0032] In the distance optimization calculation method based on differential geometry according to the present invention, the decision variable is the computing task sequence of the edge cloud, and the target variable represents the computing resource consumption, time delay, and power consumption during the execution of the task sequence. span the Riemann space ; for each decision variable at the corresponding decision variable point in the Riemann space there is a definite geodesic connecting to the zero coordinate. Calculate the lengths of all the geodesics, and select the geodesic with the shortest length The corresponding computing task sequence is used as the optimal computing task sequence.

[0033] In the distance optimization calculation method based on differential geometry according to the present invention, when the computing resource consumption, the time delay, and the power consumption are independent of each other and have the same weight, take the metric tensor ;

[0034] When the computing resource consumption, the time delay, and the power consumption are correlated and have the same weight, take the metric tensor , where a, b, and c are respectively correlated with the computing resource consumption, the time delay, and the power consumption.

[0035] Another technical solution adopted by the present invention to solve its technical problems is to construct a computer-readable medium, characterized in that when the computer-readable medium is executed by a processor, the distance optimization calculation method based on differential geometry described above is implemented.

[0036] Implementing the distance optimization calculation method and computer-readable medium based on differential geometry of the present invention can transform the traditional multi-objective optimization problem into a geometric problem in the Riemann space. The weight and mutual correlation of the target variables are described by the metric tensor in the Riemann space, which can expand the application scope of the traditional specific distance optimization algorithm and apply it to the fields where the target variables are not completely independent of each other and the weight of the variable itself is related to the value range of the variable. Therefore, it has a wide application and beneficial effects. Further, the computing resources can be saved by pre-optimization. Furthermore, when it is applied to mobile edge computing, resource optimization configuration can be carried out to obtain good economic benefits and optimized utilization of resources. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] The present invention will be further described below in conjunction with the drawings and embodiments. In the drawings:

[0038] Figure 1 is a flowchart of steps of a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention;

[0039] Figure 2 is a flowchart of steps of constructing a Riemann space based on the target variable in a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention;

[0040] Figure 3 is a flowchart of steps of selecting the theoretical optimal value of the target variable and the decision variable point in a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention;

[0041] Figure 4 is a flowchart of steps of obtaining in a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention. Detailed implementation manners

[0042] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0043] The present invention relates to a distance optimization calculation method based on differential geometry, including obtaining decision variables and target variables corresponding to a problem to be solved; constructing a Riemann space based on the target variables; selecting a theoretical optimal value of the target variables, and obtaining decision variable points corresponding to the decision variables in the Riemann space; and obtaining optimal decision variables based on the distance between the theoretical optimal value and the decision variable points.

[0044] Based on differential geometry, the present invention constructs a curved multi-dimensional solution space (Riemann space) with target variables, introduces the non-linear correlation between variables into the metric tensor of the Riemann space, and uses the geodesic length in the Riemann space to describe the similarity degree between two target points, so as to solve the optimization problem including non-linear correlation. Therefore, the present invention transforms the traditional multi-objective optimization problem into a geometric problem of the Riemann space, and uses the metric tensor of the Riemann space to describe the weights and mutual correlations of the target variables, which can expand the application scope of the traditional specific distance optimization algorithm and apply it to the fields where the target variables are not completely independent of each other and the weights of the variables themselves are related to the value ranges of the variables. Therefore, it has a wide application and beneficial effects.

[0045] Figure 1 is a flowchart of steps of a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention. As Figure 1As shown, in step S1, decision variables and objective variables corresponding to the problem to be solved are obtained. In a preferred embodiment of the present invention, based on the problem to be solved, corresponding n-dimensional decision variables are obtained ; based on the problem to be solved and the n-dimensional decision variables, m-dimensional objective variables are obtained , where , ; represents the functional relationship between the objective variables and the decision variables, and n and m are positive integers. The specific form of can be determined according to the problem to be solved.

[0046] In a preferred embodiment of the present invention, according to the specific multi-objective optimization problem to be solved, decision variables and objective variables corresponding to the problem to be solved are obtained. For example, the specific multi-objective optimization problem to be solved can be resource optimization configuration for mobile edge computing. Since the computing resources of the edge cloud are limited, different services have different requirements for latency, and the operation of services is accompanied by the consumption of power resources, etc. Considering these comprehensive factors at the same time, how to allocate the computing resources of the edge cloud is an urgent problem to be solved. This problem can be an example of the problem to be solved in the present invention. In other preferred embodiments of the present invention, the problem to be solved can be other types of problems. And based on the problem to be solved of how to allocate the computing resources of the edge cloud, the computing task sequence of the edge cloud can be obtained and used as decision variables, where , which represents the computing task is ranked at the i th position for execution.

[0047] Correspondingly, based on the problem to be solved of how to allocate the computing resources of the edge cloud and the decision variables , the objective variables can be obtained, which respectively represent the computing resource consumption, time delay, and power consumption when the task sequence is executed. Among them represents the weighted sum of the consumption of memory resources and CPU resources when the task sequence is executed, represents the maximum value of the sum of the waiting time, transmission time, and computing time of each task in the task sequence , represents the total power consumption of all tasks in the task sequence during waiting, computing, transmission, and execution.

[0048] In step S2, a Riemann space is constructed based on the target variable. A Riemann space is a vector space that satisfies the existence of a metric tensor in the space, such that the distance between two adjacent points is determined by a positive definite quadratic form. In a preferred embodiment of the present invention, based on the m-dimensional target variable Construct a Riemann space .

[0049] Figure 2 Specifically shows a flowchart of the steps for constructing a Riemann space based on the target variable. As Figure 2 shown, in step S21, the m-dimensional target variable is spanned into an m-dimensional parameter space .

[0050] In step S22, based on the parameter space and the m-dimensional target variable a Riemann space is constructed . Specifically, if there are discrete variables in the m-dimensional target variable , they are extended into continuous variables , that is represents the continuous target variable extended based on ; is the set composed of all , so is used to construct an infinitely differentiable mapping with locally; is a smooth manifold. At this time the smooth manifold formed by is called a Riemannian manifold or a Riemann space .

[0051] In step S23, a metric tensor is constructed in the Riemann space , where X and Y respectively represent vectors in the Riemann space , represents the weight of the continuous target variable and the correlation function between the continuous target variables, and are positive integers.

[0052] Here, the metric tensor is a smooth and everywhere non-degenerate second-order symmetric tensor field. In this application, the Einstein summation rule is used, that is , and is used to represent the weight of the continuous target variable and the correlation function between the continuous target variables, and are positive integers.

[0053] In a further preferred embodiment of the present invention, when the continuous target variable is linearly independent and has the same weight, the metric tensor . At this time, it means that the Riemannian space returns to a linear space, and the optimization algorithm also returns to a traditional optimization algorithm.

[0054] When the continuous target variable is linearly independent and has a fixed weight value, the metric tensor , where represents the weight of the continuous target variable , . At this time, it means that the Riemannian space returns to a linear space with coordinate scaling, and the optimization algorithm also retreats to a traditional weighted distance optimization algorithm.

[0055] When the continuous target variable is linearly independent but the weight of the continuous target variable is related to the value of the continuous target variable, the metric tensor , where represents the weight of the continuous target variable and is a function of the continuous target variable .

[0056] When the continuous target variable is mutually correlated and the weight of the continuous target variable is related to the value of the continuous target variable, the metric tensor , where is a weight function of the continuous target variable .

[0057] Therefore, in the latter two cases, it corresponds to the scenarios where the variables are non-linearly correlated and the variable weights are related to the variables themselves, thus expanding the application scenarios of the distance optimization calculation method based on differential geometry of the present invention.

[0058] Here, taking the problem to be solved of how to allocate the computing resources of the edge cloud as an example, the spanning the Riemannian space . All possible task sequences corresponding are points in this Riemannian space .

[0059] When it is simply considered that the computing resource consumption, the time delay, and the power consumption are mutually independent and have the same weight, the metric tensor is taken;

[0060] When it is considered that there is an association and the same weight among the computing resource consumption, the time delay, and the power consumption, the metric tensor is taken , where a, b, and c are respectively associated with the computing resource consumption, the time delay, and the power consumption, and the specific values can be obtained by those skilled in the art according to experiments.

[0061] In step S3, select the theoretical optimal value of the target variable, and obtain the corresponding decision variable point of the decision variable in the Riemann space. Figure 3 is a flowchart of the step of selecting the theoretical optimal value of the target variable and the decision variable point in the preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention.

[0062] As Figure 3 shown, in step S31, select the theoretical optimal value of the target variable . Here, the theoretical optimal value of the target variable can be determined according to the problem to be solved. Similarly, taking the problem to be solved based on how to allocate the computing resources of the edge cloud as an example, the optimization goal is to minimize the computing resource consumption, the time delay, and the power consumption. Therefore, the ideal solution can be set as the zero coordinate, that is, (0, 0, 0).

[0063] In step S32, obtain each decision variable corresponding decision variable point in the Riemann space .

[0064] Since for a specific optimization problem, the value range of the involved decision variable may be very large, and calculating one by one will consume a large amount of computing resources. Therefore, an optimization algorithm can be used to calculate the pareto solution set of the decision variable . For example, a genetic algorithm, a particle swarm algorithm, etc. can be selected here to calculate the pareto solution set of the decision variable . Then substitute it into the constraint conditions of the target variable to obtain the pareto solution set that satisfies the constraint conditions. And obtain the decision variable corresponding decision variable point in the Riemann space in the pareto solution set that satisfies the constraint conditions . In this way, the number of decision variable points that need to be calculated can be reduced, and a large amount of computing resources can be saved.

[0065] Similarly, taking the problem to be solved based on how to allocate the computing resources of the edge cloud as an example, it is known that the goal is to minimize the computing resource consumption, the time delay, and the power consumption. Therefore, using a multi-objective optimization algorithm (multi-objective particle swarm algorithm, multi-objective genetic algorithm, etc.) can obtain the pareto solution set of the problem, which belongs to the Riemann space Because are not independent. For two different scenarios the dependency relationship can be expressed as

[0066] ;

[0067] then .

[0068] As previously mentioned, when simply considering that the computing resource consumption, the time delay, and the power consumption are independent of each other and have the same weight, the metric tensor is taken; when considering that the computing resource consumption, the time delay, and the power consumption are correlated and have the same weight, the metric tensor is taken.

[0069] In step S4, according to the geodesic equation of the Riemann space, the optimal decision variable is obtained based on the distance between the theoretical optimal value and the decision variable point. Figure 4 is a flowchart of the steps for obtaining a preferred embodiment of the distance optimization calculation method based on differential geometry of the present invention. As Figure 4 shown, in step S41, the distance between the theoretical optimal value and the decision variable point is calculated based on the geodesic equation of the Riemann space. In step S42, the decision variable with the minimum distance is taken as the optimal decision variable.

[0070] Specifically, since each decision variable has a corresponding decision variable point in the Riemann space , the quality of each decision variable can be judged by the distance between and the theoretical optimal value of the target variable.

[0071] Those skilled in the art know that in the Riemann space, the space interval is defined as , so the distance between two points of and the theoretical optimal value of the target variable can be expressed as is path-dependent and is the line element of the Riemann space. Therefore, this integral path is the geodesic between two points. Therefore, the distance between two points can be calculated by calculating the geodesic between two points.

[0072] The calculation of geodesics in the Riemann space is known. For example, in the Riemann space, the geodesic equation can be expressed as , where represents the affine connection and Of course, other known formulas can also be used for calculation. All kinds of geodesic calculation methods, formulas, and parameters in Riemannian space are known, and will not be elaborated here.

[0073] After calculating one by one the decision variables in the Pareto solution set that satisfy the constraint conditions The corresponding decision variable points in the Riemannian space After the distances between the decision variable points, the decision variable with the minimum distance is taken as the optimal decision variable.

[0074] In this way, the present invention constructs a curved multi-dimensional solution space (Riemannian space) with target variables, introduces the non-linear correlation between variables into the metric tensor of the Riemannian space, and uses the geodesic length in the Riemannian space to describe the similarity degree between two target points, so as to solve the optimization problem containing non-linear correlation. Therefore, the present invention transforms the traditional multi-objective optimization problem into a geometric problem of the Riemannian space, and uses the metric tensor of the Riemannian space to describe the weights and mutual correlations of target variables, which can expand the application range of traditional specific distance optimization algorithms and apply them to fields where target variables are not completely independent of each other and the weights of variables themselves are related to the value ranges of variables. Therefore, it has a wide application and beneficial effects.

[0075] Taking the problem to be solved based on how to allocate the computing resources of the edge cloud as an example, for each of the decision variables The corresponding decision variable points in the Riemannian space There is a definite geodesic Connecting to the zero coordinate, calculate the lengths of all the geodesics, and select the geodesic with the shortest length The corresponding computing task sequence is used as the optimal computing task sequence.

[0076] Implementing the distance optimization calculation method based on differential geometry of the present invention transforms the traditional multi-objective optimization problem into a geometric problem of the Riemannian space, and uses the metric tensor of the Riemannian space to describe the weights and mutual correlations of target variables, which can expand the application range of traditional specific distance optimization algorithms and apply them to fields where target variables are not completely independent of each other and the weights of variables themselves are related to the value ranges of variables. Therefore, it has a wide application and beneficial effects. Further, the computing resources can be saved by pre-optimization. Furthermore, when applied to mobile edge computing, resource optimization configuration can be carried out to obtain good economic benefits and optimized utilization of resources.

[0077] A further preferred embodiment of the present invention also relates to a computer-readable medium, which, when executed by a processor, implements the aforementioned distance optimization calculation method based on differential geometry.

[0078] Therefore, the present invention can be implemented by hardware, software, or a combination of software and hardware. The present invention can be implemented in a centralized manner in at least one computer system, or in a decentralized manner by different parts distributed in several interconnected computer systems. Any computer system or other device capable of implementing the method of the present invention is applicable. The combination of common software and hardware can be a general computer system installed with a computer program, and by installing and executing the program to control the computer system, making it run according to the method of the present invention.

[0079] The present invention can also be implemented through a computer program product. The program contains all the features capable of implementing the method of the present invention. When it is installed in a computer system, the method of the present invention can be implemented. The computer program referred to in this document means: any expression of a set of instructions written in any programming language, code, or symbol, which enables the system to have information processing capabilities to directly implement a specific function, or to implement a specific function after performing one or both of the following steps: a) converting to other languages, encodings, or symbols; b) reproducing in a different format.

[0080] Although the present invention is illustrated by specific embodiments, those skilled in the art should understand that various transformations and equivalent substitutions can be made to the present invention without departing from the scope of the present invention. In addition, various modifications can be made to the present invention for specific situations or materials without departing from the scope of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed, but should include all implementation manners falling within the scope of the claims of the present invention.

[0081] The above are only the preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A distance optimization calculation method based on differential geometry, characterized in that Including the following steps: S1. Obtain the decision variables and objective variables corresponding to the problem to be solved; S2. Construct a Riemann space based on the objective variables; S3. Select the theoretically optimal value of the objective variables and obtain the decision variable points corresponding to the decision variables in the Riemann space; S4. According to the geodesic equation of the Riemann space, obtain the optimal decision variables based on the distance between the theoretically optimal value and the decision variable points; The step S1 further includes the following steps: S11. Obtain the corresponding n-dimensional decision variable based on the problem to be solved ; S12. Obtain m-dimensional objective variables based on the problem to be solved and the n-dimensional decision variables , where , ; represents the functional relationship between the objective variables and the decision variables, and n and m are positive integers; The step S2 further includes the following steps: S21. Span an m-dimensional parameter space with the m-dimensional target variable ; ​ S22. Based on the parameter space and the m-dimensional target variable construct a Riemannian space , where , denotes the continuous target variable extended based on ; S23. In the Riemann space construct a metric tensor , where X and Y respectively represent vectors in the Riemann space ; represents the weight of the continuous target variable and the correlation function between the continuous target variables, and are positive integers; The decision variables is the computing task sequence of the edge cloud, the target variable Represents a task sequence The computing resource consumption, time delay and power consumption during execution will be Zhang Cheng Riemann Space ; For each of the decision variables The corresponding decision variable point in the Riemann space is There is a definite geodesic Connect to the zero point coordinate, calculate the length of all the geodesics, and select the geodesic with the shortest geodesic The corresponding computing task sequence is taken as the optimal computing task sequence; When the computing resource consumption, the time delay, and the power consumption are independent of each other and have the same weight, take the metric tensor ; When there is a correlation and the weights are the same among the computing resource consumption, the time delay, and the power consumption, take the metric tensor , where a, b, and c are respectively correlated with the computing resource consumption, the time delay, and the power consumption.

2. The distance optimization calculation method based on differential geometry according to claim 1, characterized in that, The step S4 further includes the following steps: S41. Calculate the distance between the theoretically optimal value and the decision variable points based on the geodesic equation of the Riemann space; S42. Take the decision variable with the minimum distance as the optimal decision variable.

3. A computer-readable medium, characterized in that, When executed by a processor, the computer-readable medium implements the distance optimization calculation method based on differential geometry according to any one of claims 1-2.

Images