Distributed time-varying optimal resource allocation method based on cluster neurodynamics

Abstract

The invention discloses a distributed time-varying optimal resource allocation method based on cluster neurodynamics, and relates to the technical field of distributed control, and the method comprises the following specific steps: defining a problem: gathering N recurrent neural networks to form a recurrent neural network (RNN) set for solving a distributed optimal time-varying resource allocation problem, a non-quadratic local objective function and a global equal constraint condition both change along with time, a dynamic average consensus protocol is utilized to estimate certain global information, a novel neurodynamics method with double time scales is provided, time-varying optimal resource allocation of a given problem is tracked asymptotically in a distributed mode, and analysis shows that the optimal resource allocation efficiency is greatly improved. The provided neurodynamic model is regarded as a singular disturbance system, convergence to time-varying optimal resource configuration is achieved through the singular disturbance theory, and finally, the obtained result is verified through two examples.

Description

Technical Field

[0001] The present invention relates to the field of distributed control technology, and in particular to a distributed time-varying optimal resource allocation method based on cluster neural dynamics. Background Art

[0002] Distributed optimal resource allocation is a method of allocating limited resources among multiple agents or nodes in order to achieve optimal performance of the entire system or maximize the overall benefit. In this allocation, each agent or node may have different requirements and constraints, and the interactions between them also need to be taken into account. With the continuous development of multi-agent systems, the distributed optimal resource allocation problem has received increasing attention because it has been widely studied in large-scale network systems including smart grids, communication networks, and robotic systems. Based on optimization methods and control techniques, methods have been developed to seek optimal resource allocation in distributed methods.

[0003] Most existing studies study the resource allocation problem with a fixed optimal solution. In other words, most existing work considers time-invariant cost functions and constraints. However, in actual engineering scenarios, the optimization objectives or constraints may change over time, such as time-varying demands in smart grid economic scheduling and tracking of mobile targets in robotic systems. Therefore, solving the distributed optimal resource allocation problem with time-varying cost functions and / or constraints is a technical challenge that has not yet been solved. Summary of the invention

[0004] The purpose of the present invention is to make up for the shortcomings of the prior art and to provide a distributed time-varying optimal resource allocation method based on cluster neural dynamics. It can effectively handle the distributed time-varying optimal resource allocation problem through cluster neural dynamics, improve the accuracy and efficiency of resource allocation, and analyze the convergence based on the singular perturbation theory. The designed algorithm has a dual time scale structure. The dynamic consistency estimator is adjusted on a fast time scale to quickly estimate global information, and the output of each RNN is adjusted on a slow time scale to converge to the optimal solution. At the same time, the communication network connectivity design takes into account dynamic variability, and can still maintain good performance when the network structure changes.

[0005] In order to solve the above technical problems, the present invention provides the following technical solutions: a distributed time-varying optimal resource allocation method based on cluster neural dynamics, the specific steps of the method are:

[0006] S100, define the problem: for N recurrent neural networks, a recurrent neural network RNN ​​set is formed as a set, and a time-varying resource allocation problem is given by the following formula: Among them, x i ∈m-dimensional real vector space and Ji (x i ,t)∈ real number set The real number set is the output and local objective function of the i-th RNN, b∈n-dimensional real number vector space is the weight coefficient, the element is a positive number, let The i-th RNN only knows the equality constraints of b and J i (x i ,t),x=[x 1 ,x 2 ,…,x N ] T represents the resource allocation vector;

[0007] S200, build network topology: construct a multi-agent-based group RNN network structure topology diagram;

[0008] S300, design communication network: design the connectivity of the communication network so that the communication network between RNNs is an undirected connected graph and the connectivity is strong;

[0009] S400, constructing a Lagrange function: according to the time-varying resource allocation problem in S100 and the structural topology graph constructed in S300, constructing a Lagrange function under conditions 1, 2, and 3;

[0010] S500, Design Allocation Method: Design a distributed time-varying optimal resource allocation method based on cluster neural dynamics, i.e. where τ is a small positive parameter, the vector ξ, given the function f(x; t): its slope is Its Hessians matrix is In addition, the partial derivative with respect to t is

[0011] S600, determining the algorithm parameter range: designing the range of the algorithm parameters τ and λ;

[0012] S700, optimize distribution: substitute parameters into the algorithm to minimize the error between the solution and the optimal solution, and complete the distributed optimization of the cluster RNN.

[0013] Furthermore, the characteristic of the S100 time-varying resource allocation problem is that the optimization objectives and constraints change over time, which is specifically manifested as the local objective function J i (x i ,t) and equality constraints Changes over time.

[0014] Furthermore, the construction method of the S200 group RNN structure topology diagram is:

[0015] It consists of N nodes, each node represents an agent, where N is an integer and N ≥ 1. For the i-th RNNs, it accesses the constraint objective function J i (x i ,t) and x i , will be recorded as the output of the i-th RNN;

[0016] Motivated by the consensus of the multi-agent system, x is updated in a distributed manner i , each RNN needs a dynamic average consistency estimator to estimate global information;

[0017] It is defined that each RNN can communicate with its adjacent RNNs on the undirected graph. For the undirected graph, the node set corresponding to N RNNs is denoted by V, and the edge set corresponding to the communication between RNNs is denoted by E and N. i The neighbors of the ith RNN in the graph, for a graph, L is its Laplacian matrix, such that and For b and J i (x i ,t) estimator, defined and Let ξ be an auxiliary state.

[0018] Furthermore, the construction of the S400Lagrange function satisfies the condition:

[0019] Condition 1: For all t, the local objective function J i (x i ,t) can be differentiated twice continuously and is uniformly strongly convex with respect to , that is, for some μ>0, we have And for all i, the function J i (x i ,t) are continuously differentiable with respect to t;

[0020] Condition 2: Condition 1 is feasible at any time, that is, there is at least one solution such that for any given time

[0021] Condition 3: J i (x i ,t) and x i is bounded, and the constructed Lagrange function is: Among them, λ is the Lagrange multiplier.

[0022] Furthermore, the design range of the S400 algorithm parameters τ and λ should be determined by comprehensively considering the following factors to ensure the effectiveness and convergence of the algorithm:

[0023] For the parameter τ, its value is determined to be 0.01-0.1 according to the dynamic characteristics of the system and the expected convergence speed. τ < 0.01 will make the dynamic consistency estimator respond faster on a fast time scale, but will cause system instability. τ > 0.1 will slow down the estimation process and affect the efficiency of the algorithm.

[0024] The parameter λ is related to the constraints in the Lagrangian function. Its value range is determined according to the specific constraints and objective function of the problem. When the constraints are strict, the value of λ is 10-100. When the objective function is highly sensitive to the variables, the value of λ is 1-10 to balance the optimization of the objective function and the satisfaction of the constraints.

[0025] Furthermore, the distributed time-varying optimal resource allocation method based on cluster neural dynamics after optimization in S700 has a dual time scale structure, wherein:

[0026] The dynamic consistency estimator adjusts on a fast time scale to quickly estimate global information;

[0027] The output of each RNN is conditioned on a relatively slow time scale by adjusting It converges to the optimal time-varying optimal elementary binary solution, and its convergence speed is affected by the objective function and constraints.

[0028] Furthermore, the estimation method of the dynamic average consistency estimator of S700 is to calculate the neighbor nodes j∈N for each node i in the undirected graph. i The global information is estimated by averaging the relevant information, that is, defining and

[0029] Furthermore, the adaptive adjustment mechanism of the S700 dual time scale structure dynamically adjusts the parameter τ according to the real-time monitored network status and convergence status to optimize the algorithm performance. The specific formula is: τ = τ 0 +k·ΔE, where τ 0 is the initially set τ value, k is the adaptive coefficient, and ΔE is the difference between the current convergence error and the previous convergence error. Through this adaptive adjustment mechanism, the algorithm can better adapt to different network environments and time-varying conditions.

[0030] Furthermore, the S700 convergence analysis method is based on the following steps and principles:

[0031] Let ξ=0, and we get the boundary layer system, whose dynamic equation is: The communication network between RNNs is an undirected connected graph, which converges exponentially to

[0032] When in a quasi-steady state, a simplified system is obtained, and its dynamic equation is: The simplified system converges asymptotically to the optimal trajectory, i.e.

[0033] For a positive existence, such that for any, the state of the i-th RNN with a dynamic form converges asymptotically to the optimal trajectory of the problem, that is, for any, there is

[0034] Furthermore, the connectivity design of the S300 communication network not only satisfies undirected connectivity and strong connectivity, but also considers the dynamic variability of the network topology. When the number of network nodes increases or decreases, the Lagrangian matrix L and the related estimator are dynamically adjusted. The calculation formula of and ensures the effectiveness and convergence of the algorithm. When the new node n new When joining the network, the neighbor set of node i is updated to N i ∪{n neut}, then the calculation formula is updated to The calculation formula is updated to At the same time, the Lagrangian matrix L is also updated accordingly to adapt to the new network structure.

[0035] Compared with the prior art, this distributed time-varying optimal resource allocation method based on cluster neural dynamics has the following beneficial effects:

[0036] 1. This paper proposes a novel continuous-time cluster neural dynamics method to solve the distributed resource allocation problem with local non-quadratic objective functions and equality constraints that change over time. In contrast, the Hessians of quadratic cost functions are the same. This paper considers general objective functions with non-identical Hessians. In addition, a dynamic average estimator is introduced into the algorithm to relax the initial conditions required for various indicators.

[0037] Second, the present invention is different from the existing analysis methods of distributed time-varying algorithms. The proposed neural dynamics method with dual time scales is regarded as a singular perturbation system. Therefore, from the perspective of control systems, the singular perturbation theory can easily analyze the convergence.

[0038] Other advantages, objectives and features of the present invention will be set forth in part in the following description and, in part, will be apparent to those skilled in the art based on an examination of the following or may be taught from the practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the prior art descriptions are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention, and for ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0040] Figure 1 A flow chart of the cluster neural dynamics method for distributed time-varying optimal resource allocation of the present invention;

[0041] Figure 2 is the structural topology diagram of the i-th RNN;

[0042] Figure 3 This is a communication diagram of a multi-agent system using 6 RNNs in the embodiment;

[0043] Figure 4 The output of 6 RNNs in the embodiment;

[0044] Figure 5 is the λ of the 6 RNNs in the embodiment i (t);

[0045] Figure 6 is the output of the dynamic consistency estimator in the embodiment. DETAILED DESCRIPTION

[0046] The technical solutions in the embodiments of the present invention are described clearly and completely below. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0047] Embodiment 1

[0048] A distributed time-varying optimal resource allocation method based on cluster neural dynamics, such as Figure 1 As shown, the following steps are included:

[0049] Step 1: Let N recurrent neural networks form a recurrent neural network (RNN) ensemble with set I = {1, ..., N}. A time-varying resource allocation problem is given by the following formula:

[0050]

[0051] in, ( represents an m-dimensional real vector space) and f i (x i (t),t): ( represents a set of real numbers) are the output and local objective function of the i-th RNN respectively. ( represents an n-dimensional real vector space) is the weight coefficient, the elements are positive numbers, let a i x i (t) = [a i1 x i1 (t),…,a im x im (t)] T The i-th RNN only knows a in the equality constraint i as well as x(t)=col(x 1 (t),…,x N (t)) represents the resource allocation vector;

[0052] The time-varying resource allocation problem is characterized in that the optimization objectives or constraints may change over time.

[0053] Step 2: Construct a multi-agent-based group RNN structure topology diagram;

[0054] The group RNN structure topology diagram includes N nodes, each node represents an agent, where N is an integer and N≥1; Figure 2 The following figure shows the topological structure of the i-th RNN. For the i-th RNNs, it can access the constraints The objective function f i (x i (t)) and a i , b i (t). Denote as the output of the i-th RNN. Inspired by the consistency of the multi-agent system, in order to update λ in a distributed manner i (t), each RNN needs a dynamic average consistency estimator to estimate global information. Assume that each RNN can communicate with its neighboring RNNs on the undirected graph G. For the undirected graph G, the node set corresponding to N RNNs is represented by I = {1,...,N}, and the edge set corresponding to the communication between RNNs is denoted by ε. N i Let θ be the neighbors of the i-th RNN in the graph G. For the graph G, L is its Laplacian matrix. i1 ,θ i2 ,θ i3 and θ i4 for and Estimator of . Define θ i =col(θ i1 ,θ i2 ,θi3 ,θ i4 )and make An auxiliary state.

[0055] Step 3: Design the connectivity of the communication network so that the communication network between RNNs is an undirected connected graph;

[0056] The connectivity of the undirected connected graph is strong connectivity.

[0057] Step 4: Before constructing the Lagrange function, the following assumptions are made for the time-varying resource allocation problem in step 1:

[0058] Assumption 1: For all t, the local objective function f i (x i (t),t) can be differentiated twice continuously and relative to x i (t) is uniformly strongly convex. That is, for some h>0H i (x i (t),t,

[0059] f i )hI m , In addition, for all i∈I, the function f i are continuously differentiable with respect to t.

[0060] Assumption 2: Problem (1) is feasible at all times, that is, there is at least one solution

[0061] So that for any given time t ≥ 0,

[0062] Assumption 3: ‖b i (t)‖ ∞ and There are boundaries.

[0063] Combining the above assumptions and the structural topology constructed in step 3, construct the Lagrange function shown below:

[0064]

[0065] The Lagrangian function L(x(t),λ(t),t) is strongly convex in x(t) and strongly concave in λ(t). Let (x * (t),λ * (t)) is the optimal elementary-binary solution that satisfies the optimality condition, which is:

[0066]

[0067] Step 5: Design a distributed time-varying optimal resource allocation method based on cluster neural dynamics, as shown in the following formula:

[0068]

[0069] Where α>0 and δ is a small positive parameter, the vector sgn(x)=[sgn(x 1 ),...,sgn(x n )] T , given the function f(x; t): Its slope is Its Hessians matrix is ​​H(x,t,f). In addition, The partial derivative with respect to t is

[0070] The best elementary binary solution (x * (t),λ * (t)) is to quickly estimate the global information

[0071] t)-b i (t)) and Then, the parameter is introduced in equation (7) so that (θ i ,w i ) subsystem is in the fast time scale in order to estimate the global information more quickly. Accordingly, (x i (t),λ i (t)) The subsystem is in a slow time scale in order to converge to the optimal time-varying optimal elementary binary solution (x * (t),λ * (t)).

[0072] Embodiment 2

[0073] In order to better illustrate the present invention, six RNNs are used as the multi-agent system in this embodiment, and the communication diagram is as follows: Figure 3 As shown. The local objective function of the i-th RNN is

[0074] f i (x i ,t)=0.5(3+sin(0.68t))x i 2 +(-0.5icos(0.75it))x i

[0075] +sin(0.6it),i{1,2,3,4,5,6}.

[0076] Assume that the common requirement satisfied by all RNN clusters is described by b=60.

[0077] Step 6: Set the parameters to α = 1, δ = 0.04 and β = 20, Figure 4 The output x of 6 RNNs is shown i (t), i = 1, 2, 3, 4, 5, 6, where the dashed line depicts the trajectory updated by the centralized method when α = 1, and the solid line depicts the trajectory obtained by designing cluster neural dynamics (7) when α = 1, δ = 0.04 and β = 20. Figure 4 As shown, the state x of each RNN i (t), i = {1, 2, 3, 4, 5, 6}, all converge to the optimal solution of the time-varying resource allocation problem. Figure 5 The evolution of the Lagrangian multipliers of the six RNNs is shown, which shows that the proposed neural dynamics method is effective in estimating the optimal Lagrangian multiplier for the time-varying resource allocation problem. The output of the dynamic consistency estimator is shown in Figure 6 As shown. Figure 4 and Figure 6 It can be seen that the output of 6 RNNs reaches the optimal solution x * Before (t), the dynamic average consistency estimator quickly estimates the global information, which indicates the dual time scale of the dynamic model.

[0078] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and range of equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A distributed time-varying optimal resource allocation method based on cluster neural dynamics, characterized in that: The specific steps of this method are: S100, define the problem: for N recurrent neural networks, a recurrent neural network RNN ​​set is formed as a set, and a time-varying resource allocation problem is given by the following formula: Among them, x i ∈m-dimensional real vector space and J i (x i ,t)∈ real number sets are the output and local objective function of the ith RNN, b∈n-dimensional real number vector space is the weight coefficient, and the elements are positive numbers. The i-th RNN only knows the equality constraints of b and J i (x i ,t),x=[x1,x2,…,x N ] T represents the resource allocation vector; S200, build network topology: construct a multi-agent-based group RNN network structure topology diagram; S300, design communication network: design the connectivity of the communication network so that the communication network between RNNs is an undirected connected graph and the connectivity is strong; S400, constructing a Lagrange function: according to the time-varying resource allocation problem in S100 and the structural topology graph constructed in S300, constructing a Lagrange function under conditions 1, 2, and 3; S500, Design Allocation Method: Design a distributed time-varying optimal resource allocation method based on cluster neural dynamics, i.e. where τ is a small positive parameter, the vector ξ, given the function f(x; t): its slope is Its Hessians matrix is In addition, the partial derivative with respect to t is S600, determining the algorithm parameter range: designing the range of the algorithm parameters τ and λ; S700, optimize distribution: substitute parameters into the algorithm to minimize the error between the solution and the optimal solution, and complete the distributed optimization of the cluster RNN.

2. According to claim 1, a distributed time-varying optimal resource allocation method based on cluster neural dynamics is characterized in that: The characteristic of the S100 time-varying resource allocation problem is that the optimization objectives and constraints will change over time, which is specifically manifested as the local objective function J i (x i ,t) and equality constraints Changes over time.

3. According to claim 1, a distributed time-varying optimal resource allocation method based on cluster neural dynamics is characterized in that: The construction method of the S200 group RNN structure topology diagram is: It consists of N nodes, each node represents an agent, where N is an integer and N ≥ 1. For the i-th RNNs, it accesses the constraint objective function J i (x i ,t) and x i , will be recorded as the output of the i-th RNN; Motivated by the consensus of the multi-agent system, x is updated in a distributed manner i , each RNN needs a dynamic average consistency estimator to estimate global information; It is defined that each RNN can communicate with its adjacent RNNs on the undirected graph. For the undirected graph, the node set corresponding to N RNNs is denoted by V, and the edge set corresponding to the communication between RNNs is denoted by E and N. i The neighbors of the ith RNN in the graph, for a graph, L is its Laplacian matrix, such that and For b and J i (x i ,t’s estimator, defined and Let ξ be an auxiliary state.

4. According to claim 1, a distributed time-varying optimal resource allocation method based on cluster neural dynamics is characterized in that: The construction of the S400Lagrange function satisfies the following conditions: Condition 1: For all t, the local objective function J i (x i ,t) can be differentiated twice continuously and is uniformly strongly convex with respect to , that is, for some μ>0, we have And for all i, the function J i (x i ,t) are continuously differentiable with respect to t; Condition 2: Condition 1 is feasible at any time, that is, there is at least one solution such that for any given time Condition 3: J i (x i ,t) and x i is bounded, and the constructed Lagrange function is: Among them, λ is the Lagrange multiplier.

5. According to claim 1, a distributed time-varying optimal resource allocation method based on cluster neural dynamics is characterized in that: The design range of the S400 algorithm parameters τ and λ should be determined by comprehensively considering the following factors to ensure the effectiveness and convergence of the algorithm: For the parameter τ, its value is determined to be 0.01-0.1 according to the dynamic characteristics of the system and the expected convergence speed. τ < 0.01 will make the dynamic consistency estimator respond faster on a fast time scale, but will cause system instability. τ > 0.1 will slow down the estimation process and affect the efficiency of the algorithm. The parameter λ is related to the constraints in the Lagrangian function. Its value range is determined according to the specific constraints and objective function of the problem. When the constraints are strict, the value of λ is 10-100. When the objective function is highly sensitive to the variables, the value of λ is 1-10 to balance the optimization of the objective function and the satisfaction of the constraints.

6. According to claim 1, a distributed time-varying optimal resource allocation method based on cluster neural dynamics is characterized in that: The distributed time-varying optimal resource allocation method based on cluster neural dynamics after S700 optimization has a dual time scale structure, wherein: The dynamic consistency estimator adjusts on a fast time scale to quickly estimate global information; The output of each RNN is conditioned on a relatively slow time scale by adjusting It converges to the optimal time-varying optimal elementary binary solution, and its convergence speed is affected by the objective function and constraints.

7. A distributed time-varying optimal resource allocation method based on cluster neural dynamics according to claim 6, characterized in that: The estimation method of the S700 dynamic average consistency estimator is to calculate the neighbor node j∈N for each node i in the undirected graph. i The global information is estimated by averaging the relevant information, that is, defining and 8. The distributed time-varying optimal resource allocation method based on cluster neural dynamics according to claim 6 is characterized in that: The adaptive adjustment mechanism of the S700 dual time scale structure dynamically adjusts the parameter τ according to the real-time monitored network status and convergence status to optimize the algorithm performance. The specific formula is: τ=τ0+k·ΔE, where τ0 is the initially set τ value, k is the adaptive coefficient, and ΔE is the difference between the current convergence error and the previous convergence error. Through this adaptive adjustment mechanism, the algorithm can better adapt to different network environments and time-varying conditions.

9. A distributed time-varying optimal resource allocation method based on cluster neural dynamics according to claim 6, characterized in that: The S700 convergence analysis method is based on the following steps and principles: Let ξ=0, and we get the boundary layer system, whose dynamic equation is: The communication network between RNNs is an undirected connected graph, which converges exponentially to When in a quasi-steady state, a simplified system is obtained, and its dynamic equation is: The simplified system converges asymptotically to the optimal trajectory, i.e. For a positive existence, such that for any, the state of the i-th RNN with a dynamic form converges asymptotically to the optimal trajectory of the problem, that is, for any, there is 10. The distributed time-varying optimal resource allocation method based on cluster neural dynamics according to claim 1, characterized in that: The connectivity design of the S300 communication network not only satisfies undirected connectivity and strong connectivity, but also considers the dynamic variability of the network topology. When the number of network nodes increases or decreases, the Lagrangian matrix L and the related estimator are dynamically adjusted. The calculation formula of and ensures the effectiveness and convergence of the algorithm. When the new node n new When joining the network, the neighbor set of node i is updated to N i ∪{n neut }, then the calculation formula is updated to The calculation formula is updated to At the same time, the Lagrangian matrix L is also updated accordingly to adapt to the new network structure.

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