A time-specified search method for game equilibrium in resource scheduling among clusters with inter-cluster correlations.

Abstract

This invention discloses a time-defined search method for the equilibrium of intra-cluster cross-correlation group resource scheduling game, belonging to the field of multi-agent game decision-making technology. First, considering the situation where the individual interests of each agent in a multi-agent system are related to the decisions of other agents within and outside the cluster, a model of the intra-cluster cross-correlation group resource scheduling game problem under coupled equality resource constraints is constructed for the multi-agent system. Second, a communication topology structure is constructed for the multi-agent system. Finally, combining time planning methods, a time-defined convergent Nash equilibrium search method for the intra-cluster cross-correlation group resource scheduling game is designed for each agent. This invention can achieve time-defined convergent Nash equilibrium search for the intra-cluster cross-correlation group resource scheduling game problem, addressing the issues of coupled equality resource constraints and the correlation between the individual interests of agents and the decisions of other agents in all groups, providing a basis for distributed group resource scheduling with complex cooperative and competitive relationships.

Description

Technical Field

[0001] This invention belongs to the field of multi-agent game decision-making technology, and in particular relates to a method for searching equilibrium at a specified time in a game of resource scheduling for inter-cluster cross-correlation groups. Background Technology

[0002] With the development of swarm intelligence technology, distributed resource scheduling has become a hot research topic. It aims to achieve optimal scheduling decisions through local information interaction, thereby improving resource utilization efficiency and reducing scheduling costs. In networked cluster systems, the entire network system is clustered into a single-group networked system that is fully cooperative and aims to maximize group benefits. However, in resource scheduling of large-scale networked systems, the relationships between individuals are complex, with both cooperation and competition. This is reflected in the fact that each individual's local benefit depends not only on its local decision but also on the decisions of other individuals within the cluster and other groups. There are not only cooperative relationships within the group but also competitive relationships between groups. In this context, single-group distributed resource scheduling methods for fully cooperative systems are no longer applicable. Therefore, it is necessary to establish a group resource scheduling game model that can characterize intra-group cooperation and inter-group competition while satisfying the supply and demand balance of resource scheduling, and to conduct research on its Nash equilibrium search method.

[0003] In existing group game Nash equilibrium algorithms, the literature ( Ye M, Hu G, Lewis F L. Nash equilibrium seeking for N-coalition noncooperative games[J]. Automatica, 2018, 95: 266-272. This paper constructs an N-group non-cooperative game, uses a dynamic average consensus protocol to estimate the average gradient of the group objective function, and utilizes the estimated gradient information and gradient descent method to achieve exponential convergence of the Nash equilibrium search. The limitation of this scheme is that the designed equilibrium search algorithm is only applicable to unconstrained group games. (Reference […]) Meng M, Li X. On the linear convergence of distributed Nash equilibrium seeking for multi- cluster games under partial-decision information[J]. Automatica, 2023, 151: 110919. Further research was conducted on discrete-time algorithms for Nash equilibrium search in group non-cooperative games under consistency constraints. An equilibrium search algorithm for scenarios with partial decision information was designed using gradient following and projection methods. However, this algorithm can only achieve linear convergence and cannot achieve convergence within a specified time. Existing group game Nash equilibrium search algorithms are not suitable for practical applications with specified convergence time requirements. Summary of the Invention

[0004] The purpose of this invention is to provide a time-limited search method for the equilibrium of resource scheduling games involving intra-cluster cross-correlation, thereby solving the problem that the Nash equilibrium of multi-group resource scheduling games involving intra-group cooperation and inter-group competition is difficult to converge within a specified time in the existing technology.

[0005] To achieve the above objectives, this invention provides a method for searching for equilibrium in a game of resource scheduling within a cluster at a specified time, comprising the following steps:

[0006] Step 1: For the case where the individual interests of each agent in a multi-agent system are related to the decisions of other agents within the cluster and other agents outside the cluster, construct a game model for intra-cluster cross-correlation group resource scheduling under coupled equation resource constraints for the multi-agent system.

[0007] Step 2: Construct a communication topology for the multi-agent system;

[0008] Step 3: Combining time planning methods, design a Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game with a specified convergence time for each agent. The specific process is as follows:

[0009] S31. Use variable substitution to transform the original problem under coupled equality constraints into an unconstrained dual problem;

[0010] S32. Combining time planning methods, each agent estimates the state information of other agents based on leader-follower consistency, and uses the estimated state information to further estimate gradient information based on gradient tracking.

[0011] S33. In conjunction with the time planning problem, each agent uses the estimated gradient information and performs state updates of the auxiliary variables of the dual problem based on pseudo-gradient descent.

[0012] Preferably, in step 1, for the scenario where the individual interests of each agent in a multi-agent system are related to the decisions of other agents within the cluster and other agents outside the cluster, the specific steps for constructing a cluster-based cross-correlation group resource scheduling game model under coupled equation resource constraints for the multi-agent system are as follows:

[0013] ;

[0014] Among them, number Represents a group among N participating groups in a game. ,serial number Indicates inclusion A group of intelligent agents The Middle j An intelligent agent. Indicates group intelligent agent set, Represents the set of all intelligent agents. For intelligent agents state, Indicates group The state of union, This represents the union state of all groups. The total number of agents in all groups. and Representing intelligent agents respectively Local resources and groups owned Resources owned, groups The state satisfies the constraints, specifically the constraints are: Continuously differentiable convex function and respectively intelligent agents and groups The cost function, function and have Li psch it z-continuous gradient: ,have and ,in for Li psch it z constant, .

[0015] Preferably, step 2, which involves constructing the communication topology for the multi-agent system, specifically includes: The communication topology among all participating agents in the game is modeled as an undirected communication topology graph. ,in Represents a set of nodes. Let pq represent the set of edges, if agent pq receives the edge set from agent pq. The information, ,and For pq's neighbors, since the communication topology is undirected, if ,but Group The internal communication topology uses an undirected communication topology subgraph. express, Indicates group In A set of agents, and an edge set. intelligent agent Neighbor set representation in a network in-degree is intelligent agent In the group The neighbor set in the middle is represented as in-degree is Define an undirected communication topology graph. The adjacency matrix is ,in Represents the matrix A's first... Line number The elements of the column, all diagonal elements of matrix A ,like Off-diagonal elements ,otherwise Define an undirected communication topology subgraph. The adjacency matrix is ,in Representation matrix No. j Line number The elements of the column define an undirected communication topology subgraph. The Laplace matrix is ,in Representation matrix No. j Line number Column elements, diagonal elements Off-diagonal elements Define an undirected communication topology graph. The weighted adjacency matrix is ,in Represents the matrix W of the first generation. Line number The elements of the column, all diagonal elements of matrix W ,like Off-diagonal elements ,otherwise The elements of matrix W satisfy The specific selection method is as follows: ,in, It is a constant, defining an undirected communication topology subgraph. The corresponding double random matrix ,like ,but ,otherwise ,matrix The selection method for elements is: diagonal elements. Off-diagonal elements .

[0016] Preferred undirected communication topology graph And undirected communication topology subgraph All are connected, among which This represents a set of N game clusters.

[0017] Preferably, the specific expression for transforming the original problem under coupled equality constraints into an unconstrained dual problem using variable substitution in S31 is as follows:

[0018] ;

[0019] in, Represents intelligent agents exist t The state at time t, its initial value , It is an intelligent agent exist The initial value of the auxiliary variable after the state at time t is replaced. , Indicates the intelligent agent exist The auxiliary variable after the state at time t is replaced. Represents intelligent agents In the group Neighbor set in Indicates the first Each sampling time, derived from the sampling interval sequence Represented as:

[0020] ;

[0021] The sampling interval sequence is designed as follows:

[0022] ;

[0023] In the formula, For a preset specified time, It is a convergent infinite series sequence, that is It is finite. Indicates the first Each sampling interval Represents the set of positive integers. Denotes the first converging infinite series sequence j item, Denotes the first converging infinite series sequence item.

[0024] Preferably, in S32, combined with a time planning method, each agent estimates the state information of other agents based on leader-follower consistency, and further estimates gradient information based on gradient tracking using the estimated state information. The specific expression is as follows:

[0025] ;

[0026] In the formula, For intelligent agents Used to estimate agent pq in state of time Auxiliary variables, Represents intelligent agents Used to estimate agent pq in t Moment State The derivative of the auxiliary variable, Represents intelligent agents Used to estimate agent pq in Moment State Auxiliary variables, Indicates that agent pq is in The state at any given moment, Represents intelligent agents For other intelligent agents t State estimation at time 10:00 Represents a cluster Middle intelligence agent j The cost function, Represents intelligent agents state, Estimate the value of agent im at time t Auxiliary variables, for The derivative of For intelligent agents Estimate at time t The auxiliary variable, its initial value , Let W be the weighted adjacency matrix. Line number Column elements, A double random matrix No. j Line number The elements of the column.

[0027] Preferably, in S33, which combines time planning, each agent utilizes estimated gradient information and performs state updates of the auxiliary variables for the dual problem based on pseudo-gradient descent, as shown in the following expression:

[0028] ;

[0029] in, These are positive constants to be designed. It is an intelligent agent Auxiliary variable derivative of the dual problem after variable substitution The value at time, For intelligent agents estimate At sampling time Auxiliary variables, Estimate the value of agent ij at time t Auxiliary variables, For intelligent agents In the group The neighbor set in.

[0030] Preferably, the Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game designed in step 3 that converges within a specified time satisfies certain preconditions and parameter conditions.

[0031] Preferably, the prerequisite is that the pseudo-gradient function is required. It is strictly monotonic and has positive constants. Make .

[0032] Preferably, the parameter conditions are as follows:

[0033] ;

[0034] in,

[0035] ;

[0036] ;

[0037] ;

[0038] ;

[0039] In the formula, These are the coefficients of the strictly monotonic condition of the pseudo-gradient. It is a constant. and Let N be the Lipschitz constant and N be the number of groups. For groups The number of intelligent agents, for 3D identity matrix The total number of agents in all groups. For undirected communication topology subgraph The corresponding Laplace matrix, ,in Representation matrix The j OK, , for 3D identity matrix ,in Representing a topology graph The corresponding double random matrix, Indicates that all elements are 1 dimensional column vectors, formed by undirected topological subgraphs It is known that they are connected. , ,matrix It is a Schur matrix, and there exists a symmetric positive definite matrix. Make W is the weighted adjacency matrix, and the elements of matrix W satisfy... It can be seen that the matrix elements From an undirected topological graph It is connected to Gershgor i The n-disk theorem yields the matrix Given a Schur matrix, there exists a symmetric positive definite matrix. Make .

[0040] Therefore, the present invention employs the above-described time-specified search method for the equilibrium of resource scheduling game within cluster cross-correlation groups, which has the following beneficial effects:

[0041] (1) This invention addresses the complex cooperative and competitive relationships in resource scheduling problems that are clustered into multiple groups. Considering that the objective function of each individual is related to both other individuals within the cluster and individuals in other groups, a novel multi-alliance game is established under the resource constraint of satisfying the coupling equation.

[0042] (2) In the solution of Nash equilibrium, the sampling interval is designed by introducing a convergent infinite series to realize the solution of Nash equilibrium at a specified time. The convergence time does not depend on the initial action and method parameters. The convergence time can be determined in advance according to actual needs, and the communication cost is greatly reduced.

[0043] (3) During the process of solving the Nash equilibrium, the resource scheduling balance equation is satisfied in real time, which can adapt to practical applications with real-time search equilibrium requirements.

[0044] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0045] Figure 1 This is a schematic diagram illustrating the steps of the present invention for a time-specified search method for game equilibrium in resource scheduling within cluster cross-correlation groups;

[0046] Figure 2 This is a schematic diagram illustrating the specific process of the present invention for the time-specified search method of the game equilibrium of resource scheduling in cluster cross-correlation groups;

[0047] Figure 3 This is a communication topology diagram of the multi-agent system provided in the embodiments of the present invention;

[0048] Figure 4 This is an evolution diagram provided by an embodiment of the present invention, showing the convergence of the states of each agent to equilibrium at a specified time of 1 second;

[0049] Figure 5 This is the evolution diagram of the cost function of each group converging to the optimum at a specified time 1s, provided by the embodiments of the present invention;

[0050] Figure 6 This is a graph showing the change in the total number of resources in each group during the balanced search process provided in this embodiment of the invention. Detailed Implementation

[0051] The following detailed description of embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0052] Please see Figure 1-6 The method for searching the equilibrium of resource scheduling in a cluster-based game within a specified time includes the following steps:

[0053] Step 1: For multi-agent systems where the individual interests of each agent are related to the decisions of other agents within and outside the cluster, construct a game model for intra-cluster cross-correlation group resource scheduling under coupled equation resource constraints; specifically:

[0054] ;

[0055] Among them, number Represents a group among N participating groups in a game. ,serial number Indicates inclusion A group of intelligent agents The Middle j An intelligent agent. Indicates group intelligent agent set, Represents the set of all intelligent agents. For intelligent agents state, Indicates group The state of union, This represents the union state of all groups. The total number of agents in all groups. and Representing intelligent agents respectively Local resources and groups owned Resources owned, groups The state satisfies the constraints, specifically the constraints are: Continuously differentiable convex function and respectively intelligent agents and groups The cost function, function and have Li psch it z-continuous gradient: ,have and ,in for Lipsch it z constant, .

[0056] Step 2: Construct a communication topology for the multi-agent system; specifically: The communication topology among all participating agents in the game is modeled as an undirected communication topology graph. ,in Represents a set of nodes. Let pq represent the set of edges, if agent pq receives the edge set from agent pq. The information, ,and For pq's neighbors, since the communication topology is undirected, if ,but Group i The internal communication topology uses an undirected communication topology subgraph. express, Indicates group In A set of agents, and an edge set. intelligent agent Neighbor set representation in a network in-degree is intelligent agent In the group The neighbor set in the middle is represented as in-degree is Define an undirected communication topology graph. The adjacency matrix is ,in Represents the matrix A's first... Line number The elements of the column, all diagonal elements of matrix A ,like Off-diagonal elements ,otherwise Define an undirected communication topology subgraph. The adjacency matrix is ,in Representation matrix No. j Line number The elements of the column define an undirected communication topology subgraph. The Laplace matrix is ,in Representation matrix No. j Line number Column elements, diagonal elements Off-diagonal elements Define an undirected communication topology graph. The weighted adjacency matrix is ,in Represents the matrix W of the first generation. Line number The elements of the column, all diagonal elements of matrix W ,like Off-diagonal elements ,otherwise The elements of matrix W satisfy The specific selection method is as follows: ,in, It is a constant, defining an undirected communication topology subgraph. The corresponding double random matrix ,like ,but ,otherwise ,matrix The selection method for elements is: diagonal elements. Off-diagonal elements The undirected communication topology graph And undirected communication topology subgraph All are connected, among which This represents a set of N game clusters.

[0057] Step 3: Combining time planning methods, design a Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game with a specified convergence time for each agent. The specific process is as follows:

[0058] S31. By using variable substitution, the original problem under coupled equality constraints is transformed into an unconstrained dual problem; the specific expression is as follows:

[0059] ;

[0060] in, Represents intelligent agents exist t The state at time t, its initial value , It is an intelligent agent exist The initial value of the auxiliary variable after the state at time t is replaced. , Indicates the intelligent agent exist The auxiliary variable after the state at time t is replaced. Represents intelligent agents In the group Neighbor set in Indicates the first Each sampling time, derived from the sampling interval sequence Represented as:

[0061] ;

[0062] The sampling interval sequence is designed as follows:

[0063] ;

[0064] In the formula, For a preset specified time, It is a convergent infinite series sequence, that is It is finite. Indicates the first Each sampling interval Represents the set of positive integers. Denotes the first converging infinite series sequence j item, Denotes the first converging infinite series sequence item.

[0065] S32. Combining time planning methods, each agent estimates the state information of other agents based on leader-follower consistency, and further estimates gradient information based on gradient tracking using the estimated state information; the specific expression is as follows:

[0066] ;

[0067] In the formula, For intelligent agents Used to estimate agent pq in state of time Auxiliary variables, Represents intelligent agents Used to estimate agent pq in t Moment State The derivative of the auxiliary variable, Represents intelligent agents Used to estimate agent pq in Moment State Auxiliary variables, Indicates that agent pq is in The state at any given moment, Represents intelligent agents For other intelligent agents t State estimation at time 10:00 Represents a cluster Middle intelligence agent j The cost function, Represents intelligent agents state, Estimate the value of agent im at time t Auxiliary variables, for The derivative of For intelligent agents Estimate at time t The auxiliary variable, its initial value , Let W be the weighted adjacency matrix. Line number Column elements, A double random matrix No. j Line number The elements of the column.

[0068] S33. Combining the time planning problem, each agent utilizes estimated gradient information and performs state updates of auxiliary variables in the dual problem based on pseudo-gradient descent; the specific expression is as follows:

[0069] ;

[0070] in, These are positive constants to be designed. It is an intelligent agent Auxiliary variable derivative of the dual problem after variable substitution The value at time, For intelligent agents estimate At sampling time Auxiliary variables, For intelligent agents Estimate at time t Auxiliary variables, For intelligent agents In the group The neighbor set in.

[0071] Among them, the Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game that converges within a specified time satisfies certain preconditions and parameter conditions. The specific precondition is that the pseudo-gradient function must meet certain preconditions and parameter conditions. It is strictly monotonic and has positive constants. Make The specific parameter conditions are as follows: ;

[0072] in,

[0073] ;

[0074] ;

[0075] ;

[0076] ;

[0077] In the formula, These are the coefficients of the strictly monotonic condition of the pseudo-gradient. It is a constant. and Let N be the Lipschitz constant and N be the number of groups. For groups The number of intelligent agents, for 3D identity matrix The total number of agents in all groups. For undirected communication topology subgraph The corresponding Laplace matrix, ,in Representation matrix The j OK, , for 3D identity matrix ,in Representing a topology graph The corresponding double random matrix, Indicates that all elements are 1 dimensional column vectors, formed by undirected topological subgraphs It is known that they are connected. , ,matrix It is a Schur matrix, and there exists a symmetric positive definite matrix. Make W is the weighted adjacency matrix, and the elements of matrix W satisfy... It can be seen that the matrix elements From an undirected topological graph It is connected to Gershgor i The n-disk theorem yields the matrix Given a Schur matrix, there exists a symmetric positive definite matrix. Make .

[0078] Example

[0079] Step 1: Consider N=3 elements that contain... , , A group of intelligent agents, each group having the following resource amounts: , and, intelligent agent ij The cost function is Correlation coefficient: , , , , , , , , , , , , , , , , , , , , , , , , , , .

[0080] Step 2: The communication topology of this multi-agent system is as follows: Figure 3 As shown.

[0081] Step 3: For the specified-time search method of the intra-cluster cross-correlation group resource scheduling game Nash equilibrium, initialize the node state by selecting... .

[0082] Step 4: Design method parameters are as follows The states of all groups of agents converge to Nash equilibrium within 1 second, and their state convergence curves are shown below. Figure 4 As shown, the convergence curves of the cost functions for each group are as follows: Figure 5 As shown, the curves showing the change in the total number of resources for each group are as follows: Figure 6 As shown. Simulation results show that the states of each agent and the costs of each group converge to Nash equilibrium in 1 second. Furthermore, the supply and demand balance of resources in each group is always met.

[0083] Therefore, the present invention employs the above-mentioned time-specified search method for the equilibrium of intra-cluster cross-correlation group resource scheduling game, which can describe the complex interactions of intra-group cooperation and inter-group competition in resource scheduling of large-scale networked systems, and can realize the time-specified search for the Nash equilibrium of group resource scheduling game; moreover, the time-specified search method for the Nash equilibrium of intra-cluster cross-correlation group resource scheduling game can be applied to complex group resource scheduling tasks with online real-time execution requirements, and has broad practical application value.

[0084] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A time-specified search method for game equilibrium in resource scheduling among inter-cluster cross-correlation groups, characterized in that, Includes the following steps: Step 1: For the case where the individual interests of each agent in a multi-agent system are related to the decisions of other agents within the cluster and other agents outside the cluster, construct a game model for intra-cluster cross-correlation group resource scheduling under coupled equation resource constraints for the multi-agent system. Step 2: Construct a communication topology for the multi-agent system; Step 3: Combining time planning methods, design a Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game with a specified convergence time for each agent. The specific process is as follows: S31. Use variable substitution to transform the original problem under coupled equality constraints into an unconstrained dual problem; S32. Combining time planning methods, each agent estimates the state information of other agents based on leader-follower consistency, and uses the estimated state information to further estimate gradient information based on gradient tracking. S33. Combining the time planning problem, each agent uses the estimated gradient information and performs state updates of the auxiliary variables of the dual problem based on pseudo-gradient descent. Step 1 addresses the scenario where the individual interests of each agent in a multi-agent system are related to the decisions of other agents within and outside the cluster. Specifically, a game model for intra-cluster cross-correlation group resource scheduling under coupled equation resource constraints is constructed for the multi-agent system as follows: ； Among them, number Represents a group among N participating groups in a game. ,serial number Indicates inclusion A group of intelligent agents The Middle An intelligent agent. Indicates group intelligent agent set, Represents the set of all intelligent agents. For intelligent agents state, Indicates group The state of union, This represents the union state of all groups. The total number of agents in all groups. and Representing intelligent agents respectively Local resources and groups owned i Resources owned, groups The state satisfies the constraints, specifically the constraints are: Continuously differentiable convex function and respectively intelligent agents and groups The cost function, function and have Continuous gradient: ,have and ,in for constant, ; Step 2, specifically constructing the communication topology for the multi-agent system, involves: The communication topology among all participating agents in the game is modeled as an undirected communication topology graph. ,in Represents a set of nodes. Let pq represent the set of edges, if agent pq receives the edge set from agent pq. The information, ,and For pq's neighbors, since the communication topology is undirected, if ,but Group The internal communication topology uses an undirected communication topology subgraph. express, Indicates group In A set of agents, and an edge set. intelligent agent Neighbor set representation in a network in-degree is intelligent agent In the group The neighbor set in the middle is represented as in-degree is Define an undirected communication topology graph. The adjacency matrix is ,in Represents the matrix A's first... Line number The elements of the column, all diagonal elements of matrix A ,like Off-diagonal elements ,otherwise Define an undirected communication topology subgraph. The adjacency matrix is ,in Representation matrix No. Line number The elements of the column define an undirected communication topology subgraph. The Laplace matrix is ,in Representation matrix No. Line number Column elements, diagonal elements Off-diagonal elements ,in Define an undirected communication topology graph. The weighted adjacency matrix is ,in Represents the matrix W of the first generation. Line number The elements of the column, all diagonal elements of matrix W ,like Off-diagonal elements ,otherwise The elements of matrix W satisfy The specific selection method is as follows: ,in, It is a constant, defining an undirected communication topology subgraph. The corresponding double random matrix ,like ,but ,otherwise ,matrix The selection method for elements is: diagonal elements. Off-diagonal elements ; Undirected communication topology diagram And undirected communication topology subgraph All are connected, among which , Represents a set of N game groups; The specific expression for transforming the original problem under coupled equality constraints into an unconstrained dual problem using variable substitution in S31 is as follows: ； in, Represents intelligent agents exist The state at time t, its initial value , It is an intelligent agent exist The initial value of the auxiliary variable after the state at time t is replaced. , Indicates the intelligent agent exist The auxiliary variable after the state at time t is replaced. Represents intelligent agents In the group Neighbor set in Indicates the first Each sampling time, derived from the sampling interval sequence Represented as: ； The sampling interval sequence is designed as follows: ； In the formula, For a preset specified time, It is a convergent infinite series sequence, where , ,Right now It is finite. Indicates the first Each sampling interval Represents the set of positive integers. Denotes the first converging infinite series sequence item, Denotes the first converging infinite series sequence item; The specific parameter conditions are as follows: ； in, ； ； ； ； In the formula, These are the coefficients of the strictly monotonic condition of the pseudo-gradient. It is a constant. and Let N be the Lipschitz constant and N be the number of groups. For groups The number of intelligent agents, for 3D identity matrix The total number of agents in all groups. For undirected communication topology subgraph The corresponding Laplace matrix, ,in Representation matrix The OK, , for 3D identity matrix ,in Representing a topology graph The corresponding double random matrix, Indicates that all elements are 1 dimensional column vectors, formed by undirected topological subgraphs It is known that they are connected. , ,matrix It is a Schur matrix, and there exists a symmetric positive definite matrix. Make W is the weighted adjacency matrix, and the elements of matrix W satisfy... It can be seen that the matrix elements From an undirected topological graph It is connected to Gershgor i The n-disk theorem yields the matrix Given a Schur matrix, there exists a symmetric positive definite matrix. Make ; In S32, a time-planning method is used. Each agent estimates the state information of other agents based on leader-follower consistency, and further estimates the gradient information based on gradient tracking using the estimated state information. The specific expression is as follows: In the formula, For intelligent agents Used to estimate agent pq in state of time Auxiliary variables, Represents intelligent agents Used to estimate agent pq in Moment State The derivative of the auxiliary variable, Represents intelligent agents Used to estimate agent pq in Moment State Auxiliary variables, Indicates that agent pq is in The state at any given moment, Represents intelligent agents For other intelligent agents State estimation at time 10:00 Represents a cluster Middle intelligence agent The cost function, Represents intelligent agents state, Estimate the value of agent im at time t Auxiliary variables, for The derivative of For intelligent agents Estimate at time t The auxiliary variable, its initial value , Let W be the weighted adjacency matrix. Line number Column elements, A double random matrix No. Line number Column elements; In S33, which combines time planning problems, each agent uses estimated gradient information and performs state updates of auxiliary variables for the dual problem based on pseudo-gradient descent, as shown in the following expression: ； in, These are positive constants to be designed. It is an intelligent agent Auxiliary variable derivative of the dual problem after variable substitution The value at time, For intelligent agents estimate At sampling time Auxiliary variables, Estimate the value of agent ij at time t Auxiliary variables, For intelligent agents In the group Neighbor set in; In step 3, a Nash equilibrium search method for intra-cluster cross-correlation group resource scheduling game that converges within a specified time is designed to satisfy the preconditions and parameter conditions. The specific prerequisite is that a pseudo-gradient function is required. It is strictly monotonic and has positive constants. Make .

Images