A distributed time-varying optimal resource allocation method based on cluster neural dynamics

Abstract

The application discloses a kind of distributed time-varying optimal resource allocation methods based on cluster neural dynamics, it is related to distributed control technical field, and the specific steps of this method are as follows: define problem: to N recurrent neural network with set formation a recurrent neural network RNN set, for solving distributed optimal time-varying resource allocation problem, in the problem, non-quadratic local objective function and global equality constraint condition are changed with time, some global information is estimated using dynamic average consensus protocol, a new type of neural dynamics method with double time scale is proposed, and the time-varying optimal resource allocation of given problem is asymptotically tracked in a distributed manner, analysis shows that the proposed neural dynamics model is regarded as singular perturbation system, and converges to time-varying optimal resource configuration by singular perturbation theory, finally, the results are verified by two examples.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of distributed control, and particularly relates to a distributed time-varying optimal resource allocation method based on cluster neural dynamics. BACKGROUND

[0002] Distributed optimal resource allocation is a method of allocating limited resources among multiple agents or nodes, aiming to achieve optimal performance or maximize overall benefits of the entire system. In this allocation, each agent or node may have different needs and constraints, and the interaction between them also needs to be considered. With the continuous development of multi-agent systems, distributed optimal resource allocation has attracted more and more attention, because it is widely studied in large-scale network systems including smart grids, communication networks and robot systems. Based on optimization methods and control techniques, people have developed methods to seek optimal resource allocation in distributed methods.

[0003] Most existing researches study resource allocation problems with fixed optimal solutions, in other words, most existing works consider time-invariant cost functions and constraints. However, in actual engineering scenarios, the optimization objective or constraint conditions may change over time, such as time-varying demand in smart grid economic dispatch and tracking moving targets in robot systems. Therefore, solving the distributed optimal resource allocation problem with time-varying cost functions or (and) constraints is a technical problem that has not been solved yet. SUMMARY

[0004] The present application aims to make up for the shortcomings of the prior art, and provides a distributed time-varying optimal resource allocation method based on cluster neural dynamics, which can effectively solve 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 singular perturbation theory. The designed algorithm has a double-time-scale structure, and the dynamic consensus estimator is adjusted on the fast time scale to quickly estimate the global information. The output of each RNN is adjusted on the slow time scale to converge to the optimal solution. At the same time, the communication network connectivity design considers dynamic variability, and can still maintain good performance when the network structure changes.

[0005] To solve the above technical problems, the present application provides the following technical scheme: a distributed time-varying optimal resource allocation method based on cluster neural dynamics, and the specific steps are as follows:

[0006] S100, defining the problem: forming a recurrent neural network (RNN) set by a set of N recurrent neural networks, and a time-varying resource allocation problem is given by the following formula: where x i ∈m-dimensional real vector space and Ji (x i ,t)∈The real number set The real number set is the output and local objective function of the i-th RNN, respectively, b∈n-dimensional real vector space is the weight coefficient, the elements are positive numbers, let The i-th RNN only knows the equality constraints b and j. i (x i ,t), x=[x1,x2,…,x N ] T Represents a resource allocation vector;

[0007] S200, Constructing Network Topology: Constructing a multi-agent-based swarm RNN network structure topology;

[0008] S300, Design the communication network: Design the connectivity of the communication network so that the communication network between RNNs is an undirected connected graph with strong connectivity.

[0009] S400, Construct the Lagrange function: Based on the time-varying resource allocation problem in S100 and the structural topology graph constructed in S300, construct the Lagrange function under the conditions of 1, 2 and 3.

[0010] S500, Allocation Method: A distributed, time-varying optimal resource allocation method based on cluster neural dynamics is designed, namely... Where τ is a small positive parameter, and ξ is a vector, given a function f(x; t) with slope as... Its Hessians matrix is Furthermore, the partial derivative with respect to t is

[0011] S600, Determine the range of algorithm parameters: Design the range of algorithm parameters τ and λ;

[0012] S700, Optimized Distribution: Substitute the parameters into the algorithm to minimize the error between the solution and the optimal solution, thus completing the distributed optimization of the cluster RNN.

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

[0014] Furthermore, the method for constructing the S200 population RNN structure topology graph is as follows:

[0015] It consists of N nodes, each representing an agent, where N is an integer and N≥1. For the i-th RNN, it accesses the constrained objective function J.i (x i ,t) and x i , which will be denoted as the output of the ith RNN;

[0016] Inspired by the consensus of multi-agent systems, the distributed way is used to update x i Each RNN needs a dynamic average consensus estimator to estimate the global information;

[0017] Each RNN can communicate with its adjacent RNNs on the undirected graph, for the undirected graph, the node set corresponding to the N RNNs is denoted as V, the edge set corresponding to the communication links between the RNNs is denoted as E, and the ith RNN is denoted as N i The neighbors of the ith RNN in the graph, for the graph, L is its Laplacian matrix, and L = D - A, where D is a diagonal matrix with the diagonal elements being the degrees of the nodes in the graph, and A is the adjacency matrix of the graph; and is the estimator of b and J i (x i ,t), define and Let ξ be an auxiliary state.

[0018] Further, the construction of the S400 Lagrange function satisfies the following conditions:

[0019] Condition 1: For all t, the local objective function J i (x i ,t) is twice continuously differentiable, and is uniformly strongly convex with respect to x and for all i, the function J i (x i ,t) is continuously differentiable with respect to t;

[0020] Condition 2: Condition 1 is feasible at all times, i.e., there exists at least one solution such that for any given time, x

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

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

[0023] For the parameter τ, its value is determined by the dynamic characteristics of the system and the desired convergence speed, 0.01-0.1, τ < 0.01 will make the dynamic consistency estimator respond faster on the fast time scale, but lead to system instability, τ > 0.1 will make the estimation process slower, affecting the efficiency of the algorithm;

[0024] For the parameter λ, it is related to the constraint condition in the Lagrange function, its value range is determined according to the specific constraints and objective function of the problem, for the strict constraint condition, the value of λ is 10-100, when the sensitivity of the objective function to the variable is high, the value of λ is 1-10, to balance the optimization of the objective function and the satisfaction of the constraint condition.

[0025] Further, the S700 optimized distributed time-varying optimal resource allocation method based on cluster neural dynamics has a double time scale structure, wherein:

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

[0027] The output of each RNN is adjusted on a relatively slow time scale, by adjusting Converges to the optimal time-varying optimal elementary binary solution, and the convergence speed is affected by the objective function and the constraint condition.

[0028] Further, the S700 dynamic average consistency estimator estimates the global information by calculating the average value of the related information of its neighbor nodes j∈N i for each node i in the undirected graph, that is, define And

[0029] Further, the adaptive adjustment mechanism of the S700 double time scale structure dynamically adjusts the parameter τ according to the real-time monitored network state and convergence condition to optimize the algorithm performance, and the specific formula is: τ = τ0 + k·ΔE, wherein τ0 is the initial set τ value, k is the adaptive coefficient, ΔE is the difference between the current convergence error and the last convergence error, through this adaptive adjustment mechanism, the algorithm can better adapt to different network environments and time-varying conditions.

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

[0031] Let ξ = 0, get the boundary layer system, its dynamic equation is: The communication network between RNNs is an undirected connected graph, for all i of the boundary layer system, exponentially converges to

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

[0033] For a positive existence, the state of the ith RNN with the dynamic formula has asymptotically converges to the optimal trajectory of the problem, i.e.

[0034] Further, the connectivity design of the S300 communication network not only satisfies the undirected connectivity and strong connectivity, but also considers the dynamic variability of the network topology. When the network nodes are added and reduced, the Lagrange matrix L and the related estimator are dynamically adjusted. The calculation formula of and guarantees the effectiveness and convergence of the algorithm, and when a new node n new joins the network, the neighbor set of the node i is updated to N i ∪{n neut}, the calculation formula of is updated to The calculation formula of is updated to At the same time, the Lagrange matrix L is also updated to adapt to the new network structure.

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

[0036] 1. The application proposes a novel continuous-time cluster neural dynamics method for solving the distributed resource allocation problem of local non-quadratic objective functions and equal constraint conditions changing over time. Compared with the prior art, the Hessian matrix of the quadratic cost function is the same. In this paper, a general objective function with a non-same Hessian matrix is considered. In addition, a dynamic average estimator is introduced in the algorithm to relax the initial conditions of each index requirement.

[0037] 2. The proposed neural dynamics method with double time scale is regarded as a singular perturbation system, which is different from the analysis method of the existing distributed time-varying algorithm. Therefore, from the perspective of control system, the singular perturbation theory is easy to analyze the convergence.

[0038] Other advantages, objects, and features of the present application will be in part apparent and in part pointed out hereinafter in the specification, and in part will be observed by persons skilled in the art upon examination of the following specification, or can be learned by practice of the application. BRIEF DESCRIPTION OF DRAWINGS

[0039] In order to make the technical solutions in the embodiments of the present application or the prior art clearer, the accompanying drawings needed in the embodiments or prior art description will be briefly introduced below. Obviously, the accompanying drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without any creative effort on the basis of these drawings.

[0040] Figure 1 Flow chart of the cluster neural dynamic method for distributed time-varying optimal resource allocation of the present application;

[0041] Figure 2 Structural topology chart for the i-th RNN;

[0042] Figure 3 Communication chart for the embodiment in which 6 RNNs are used as a multi-agent system;

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

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

[0045] Figure 6 Output of the dynamic consensus estimator in the embodiment. DETAILED DESCRIPTION

[0046] The technical solutions in the embodiments of the present application will be described clearly and completely below. Obviously, the described embodiments are only some of the embodiments of the present application, but not all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without any creative effort belong to the protection scope of the present application.

[0047] Embodiment one

[0048] A distributed time-varying optimal resource allocation method based on cluster neural dynamics, as shown in FIG. 1, comprises the following steps: Figure 1 Step 1: Let N recursive neural networks form a recursive neural network (RNN) set with the set I = {1,...,N}, and a time-varying resource allocation problem is given by the following formula:

[0049]

[0050] wherein,

[0051] ( represents an m-dimensional real vector space) and f i (x i (t),t):​ ( (representing the set of real numbers) are the output and local objective function of the i-th RNN, respectively. ( Let a be the weight coefficients in an n-dimensional real vector space, where all elements are positive. 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(x1(t),…,x N (t) represents the resource allocation vector;

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

[0053] Step 2: Construct the topology of a multi-agent swarm RNN structure;

[0054] The aforementioned swarm RNN structure topology includes N nodes, each node representing an agent, where N is an integer and N≥1; for example... Figure 2 The diagram shows the topological structure of the i-th RNN. For the i-th RNN, it has access constraints. objective function f i (x i (t)) and a i b i (t). Will Let λ be the output of the i-th RNN. Driven by the consensus mechanism of multi-agent systems, in order to update λ in a distributed manner... i (t) Each RNN requires a dynamic average consistency estimator to estimate global information. Assume that each RNN can communicate with its neighboring RNNs in an undirected graph G. For an undirected graph G, the set of nodes corresponding to the N RNNs is denoted by I = {1,...,N}, and the edge set of communication connections between the RNNs is denoted by ε. i Let θ be the neighbor of the i-th RNN in graph G. For graph G, L is its Laplacian matrix. Let θ i1 θ i2 θ i3 and θ i4 for and The estimator. Define θ. i =col(θ) i1 ,θ i2 ,θ i3 ,θi4 )and make This is 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 undirected connected graph is strongly connected.

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

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

[0059] f i )hI m , Furthermore, for all i∈I, the function f i All are continuously differentiable relative to t.

[0060] Assumption 2: Problem (1) is feasible at all times, i.e., there exists at least one solution. Such that for any given time t≥0, we have

[0061] Assumption 3: ||b i (t)‖ ∞ and It has boundaries.

[0062] Based on the above assumptions and the structural topology constructed in step 3, the following Lagrange function is constructed:

[0063]

[0064] 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-bivariate solution that satisfies the optimal conditions, which are:

[0065]

[0066] Step 5: Design a distributed time-varying optimal resource allocation method based on cluster neural dynamics, as shown in the following formula:

[0067]

[0068] where α > 0 and δ is a small positive parameter, the vector sgn(x) = [sgn(x1),..., sgn(x n )] T , given the function f(x; t): Its slope is Its Hessian matrix is H(x, t, f). In addition, The partial derivative with respect to t is

[0069] Before the optimal elementary binary solution (x * (t), λ * (t)) obtained by RNNs, it is desirable to be able to quickly estimate the global information t)-b i (t)) and Then, introduce the parameter in equation (7), so that the (θ i , w i ) subsystem is in the fast time scale in order to estimate the global information more quickly, accordingly, the (x i (t), λ i (t)) subsystem is in the slow time scale in order to converge to the optimal time-varying optimal elementary binary solution (x * (t), λ * (t)).

[0070] Example Two

[0071] In order to better illustrate the present application, six RNNs are used as a multi-agent system in this embodiment, and its communication diagram is shown as Figure 3 The local objective function of the i-th RNN is

[0072] f i (x i , t) = 0.5(3 + sin(0.68t))x i 2 + (-0.5icos(0.75it))x i

[0073] + sin(0.6it), i {1, 2, 3, 4, 5, 6}.

[0074] It is assumed that the common requirement satisfied by all RNN clusters is described by b = 60.

[0075] Step 6: Set the parameters to a = 1, d = 0.04 and b = 20, Figure 4 The evolution of the outputs x i (t), i = 1, 2, 3, 4, 5, 6, is shown, where the dashed lines depict the trajectories updated by the centralized method when a = 1, and the solid lines depict the trajectories obtained by the designed swarm neural dynamics (7) when a = 1, d = 0.04 and b = 20. As Figure 4 shown, the states x i (t), i = {1, 2, 3, 4, 5, 6}, of each RNN converge to the optimal solution of the time-varying resource allocation problem. Figure 5 The evolution of the Lagrange multipliers of the 6 RNNs is shown, which indicates the estimation of the optimal Lagrange multipliers in the time-varying resource allocation problem by the proposed neural dynamics method. The outputs of the dynamic consensus estimator are shown in Figure 6 It can be seen from Figure 4 and Figure 6 that the dynamic average consensus estimator quickly estimates the global information before the outputs of the 6 RNNs reach the optimal solution x * (t). This indicates the dual time scale of the dynamic model.

[0076] It is apparent to those skilled in the art that the application is not limited to the details of the foregoing exemplary embodiments, and that the application can be implemented in other particular forms without departing from the spirit or essential characteristics of the application. The embodiments should therefore be considered in all respects as illustrative and not restrictive, the scope of the application being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. No reference signs in the claims should be considered as limiting the scope of the claims with respect to the features they indicate. The application covers simultaneous, separate and discrete applications of these features to each aspect of the application, and equivalent arrangements and equivalent parts thereof.

Claims

1. A method for distributed time-varying optimal resource allocation based on swarm neurodynamics, characterized in that, The specific steps of the method are: S100, defining the problem: N recurrent neural networks are formed into a set to form a recurrent neural network set, a time-varying resource allocation problem is given by the following formula: Wherein, x i ∈m-dimensional real vector space and J i (x i ,t)∈real set is the output of the i-th RNN and the local objective function, b∈n-dimensional real vector space is the weight coefficient, the element is positive, and let The i-th RNN only knows b in the equality constraint and J i (x i ,t), x=[x1,x2,…,x N ] T represents the resource allocation vector; S200, constructing a network topology: constructing a multi-agent-based group RNN network structure topology graph; S300, designing a communication network: designing the connectivity of the communication network so that the communication network between RNNs is an undirected connected graph, and the connectivity is strongly connected; S400, constructing a Lagrange function: constructing a Lagrange function according to the time-varying resource allocation problem in S100 and the structure topology graph constructed in S300, under the conditions 1, 2, 3; S500, design allocation method: design a distributed time-varying optimal resource allocation method based on cluster neural dynamics, that is Where τ is a small positive parameter, the vector ξ, the given function f(x; t): its slope is Its Hessian matrix is In addition, the partial derivative with respect to t is S600, determining the range of algorithm parameters: designing the range of algorithm parameters τ, λ; S700, optimizing distribution: substituting the parameters into the algorithm to minimize the error between the solution and the optimal solution, and completing the distributed optimization of the cluster RNN.

2. The method of claim 1, wherein, The S100 time-varying resource allocation problem is characterized by the fact that the optimization objective and the constraint conditions vary with time, which is specifically manifested in the local objective function J i (x i ,t) and the equality constraint condition vary with time.

3. The method of claim 1, wherein, The construction method of the S200 group RNN structure topology graph is: comprising N nodes, each node representing an agent, where N is an integer and N > 1, and for the ith RNNs, it accesses the objective function J i (x i ,t) and x i , will be denoted as the output of the ith RNN. Inspired by the consensus of multi-agent systems, we use a distributed way to update x i Each RNN needs a dynamic average consensus estimator to estimate the global information; Definition Each RNN can communicate with its neighboring RNNs on an undirected graph, for the undirected graph, the node set corresponding to the N RNNs is denoted by V, and the edge set corresponding to the communication links between the RNNs is denoted by E, denoted as N i The neighbors of the ith RNN in the graph, for the graph, L is its Laplacian matrix, such that And For b and J i (x i , the estimator of t, is defined as And Let ξ be an auxiliary state.

4. The method of claim 1, wherein, The construction of the S400 Lagrange function satisfies the conditions: Condition 1 : for all t, the local objective function J i (x i ,t) is twice continuously differentiable, and is uniformly strongly convex with respect to t, i.e., for some μ > 0, has and for all i, the function J i (x i ,t) is continuously differentiable with respect to t; Condition 2: Condition 1 is feasible at all times, i.e. there exists at least one solution such that for any given time there is Condition 3: J i (x i ,t) and x i is bounded, the constructed Lagrange function is: where λ is the Lagrange multiplier.

5. The method of claim 1, wherein, The design range of the S500 algorithm parameters τ, λ should be determined by considering the following factors to ensure the effectiveness and convergence of the algorithm: For parameter τ, its value is determined as 0.01-0.1 according to the dynamic characteristics of the system and the desired convergence speed, τ < 0.01 will make the dynamic consensus estimator respond faster on the fast time scale, but lead to system instability, τ > 0.1 will slow down the estimation process and affect the efficiency of the algorithm; For parameter λ, it is related to the constraint condition in the Lagrange function, its value range is determined according to the specific constraints and objective function of the problem, when the constraint condition is strict, the value of λ is 10-100, when the sensitivity of the objective function to the variable is high, the value of λ is 1-10, to balance the optimization of the objective function and the satisfaction of the constraint condition.

6. The method of claim 1, wherein, The S700 optimized distributed time-varying optimal resource allocation method based on cluster neural dynamics has a double-time scale structure, in which: The dynamic consensus estimator adjusts on the fast time scale to quickly estimate the global information; The output of each RNN is conditioned on a relatively slow time scale, by adjusting converges to an optimal time-varying elementary binary solution, with a convergence speed influenced by the objective function and the constraints.

7. The method of claim 6, wherein, The estimation method of the S700 dynamic average consensus estimator estimates the global information for each node i in the undirected graph by computing the average of the relevant information of its neighbor nodes j ∈ N i i.e. it defines and 8. The method of claim 6, wherein the method is based on a cluster neural dynamics for distributed time-varying optimal resource allocation. The adaptive adjustment mechanism of the S700 double-time scale structure dynamically adjusts the parameter τ according to the real-time monitoring of the network state and the convergence situation to optimize the algorithm performance, the specific formula is: τ = τ0 + k·ΔE, where τ0 is the initial set τ value, k is the adaptive coefficient, ΔE is the difference between the current convergence error and the last convergence error, through this adaptive adjustment mechanism, the algorithm can better adapt to different network environments and time-varying conditions.

9. The method of claim 6, wherein the method is based on a cluster neural dynamics for distributed time-varying optimal resource allocation. The S700 convergence analysis method is based on the following steps and principles: Letting ξ = 0, we obtain the boundary layer system whose dynamic equation is The communication network between RNNs is an undirected connected graph, and for all i, the exponential converges to When in quasi-steady state, a simplified system is obtained, whose dynamic equation is The simplified system asymptotically converges to the optimal trajectory, i.e. For a positive existence, such that for any, the state of the ith RNN with dynamic has asymptotically converged to the optimal trajectory of the problem, i.e., for any, there exists a positive integer N such that for all n > N, ||x(n) - x*(n)|| < ε 10. The method of claim 1, wherein, The connectivity design of the S300 communication network not only satisfies the undirected connectivity and strong connectivity, but also considers the dynamic variability of the network topology structure. When the network nodes are added or reduced, the Lagrange matrix L and the related estimator and the calculation formula are dynamically adjusted to ensure the effectiveness and convergence of the algorithm. When a new node n new joins the network, the neighbor set of node i is updated to N i ∪{n neut}, and the calculation formula of is updated to The calculation formula of is updated to At the same time, the Lagrange matrix L is also updated accordingly to adapt to the new network structure.

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