A distributed accelerated optimization scheduling method, device, and medium based on time-varying directed topology
By constructing a distributed accelerated optimization scheduling method based on time-varying directed topology, and utilizing the Push-Sum protocol and Nesterov momentum acceleration term, the problem of slow convergence speed of distributed optimization algorithms in time-varying directed topology networks is solved, achieving fast and robust approximation of the global optimal solution and reducing the system's computational and communication overhead.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHANGHAI UNIVERSITY OF ELECTRIC POWER
- Filing Date
- 2026-06-03
- Publication Date
- 2026-06-30
AI Technical Summary
Existing distributed optimization algorithms suffer from slow convergence speed, low computational efficiency, and convergence bias due to asymmetric communication weights when dealing with directed topological networks, especially time-varying directed topological networks. As a result, it is difficult to obtain a scheduling strategy that meets engineering accuracy requirements within a limited time.
A distributed acceleration optimization scheduling method based on time-varying directed topology is adopted. By constructing a topology model of a distributed network system, the Push-Sum protocol and Nesterov momentum acceleration term are used, combined with gradient tracking variables and weight balancing variables, to achieve distributed iterative updates. The system matrix spectral radius is verified by matrix spectral radius theory to ensure linear convergence.
While ensuring algorithm accuracy, it significantly improves convergence speed, reduces the number of iterations and system computation and communication overhead, solves the communication congestion and privacy leakage problems of centralized scheduling, and achieves fast and robust approximation of the global optimal solution.
Smart Images
Figure CN122309094A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of intelligent agent control technology, and in particular to a distributed accelerated optimization scheduling method, device and medium based on time-varying directed topology. Background Technology
[0002] Optimization algorithms approximate the optimal solution to a problem through iterative computation, and their performance mainly depends on the number of iterations and computational complexity. Improving the convergence rate of algorithms has always been a key research focus in the field of optimization. In large-scale optimization problems, fast convergence not only significantly reduces computational time and resource consumption, but is also crucial for research and application institutions that rely on large-scale cloud training. More importantly, in application scenarios with extremely high real-time requirements, the convergence speed of the algorithm directly affects the system's security and economic efficiency. For example, in weather warning systems, distributed algorithms need to process massive amounts of data within a limited time to accurately predict severe weather; in robot formation control, computational delays in real-time trajectory optimization can lead to decreased formation efficiency or even collisions. Therefore, improving the convergence speed of distributed optimization algorithms is of significant engineering importance for meeting the practical requirements of low latency and high reliability.
[0003] In distributed cooperative control theory, the communication network topology graph is the foundation for describing the rules of information interaction between intelligent agents. Most existing mainstream algorithm designs are based on the undirected graph assumption, which assumes that information transmission between nodes is bidirectional and symmetrical. While undirected graph structures theoretically guarantee convergence, in real physical systems, limitations such as transmission power differences, communication distance constraints, or channel interference often result in asymmetric communication links, thus forming directed communication networks. Furthermore, in mobile networks or scenarios with severe environmental interference, the communication topology is often not static but rather a time-varying topology that changes dynamically over time. Compared to idealized static undirected graphs, time-varying directed topologies better reflect the real constraints of the physical world, but they also present significant theoretical challenges to the stability analysis and convergence speed improvement of algorithms.
[0004] Traditional system scheduling typically employs a centralized model, where a central node collects state information from all subsystems and performs unified calculations and command issuance. However, with the expansion of system scale and the surge in the number of nodes, centralized scheduling faces problems such as communication congestion, privacy leaks, and the curse of computational dimensionality. Distributed optimization scheduling methods based on multi-agent systems effectively overcome the drawbacks of centralized scheduling by decomposing the global optimization objective into local subproblems for each node and leveraging information exchange between neighbors to achieve collaborative optimization. However, existing distributed scheduling algorithms often suffer from slow convergence speeds, sensitivity to initial conditions, or large steady-state errors when dealing with directed network environments, especially time-varying directed networks, making it difficult to obtain scheduling strategies that meet engineering accuracy requirements within a finite timeframe.
[0005] In summary, although distributed optimization has broad application prospects in many fields, current algorithm research suffers from the following shortcomings: First, there is a relative lack of accelerated algorithms designed for directed topologies, with most algorithms still relying on the assumption of bidirectional communication; second, existing algorithms for directed graphs often sacrifice computational simplicity or experience a significant decrease in convergence performance during topology switching. Therefore, designing a distributed accelerated optimization scheduling method that can adapt to time-varying directed topologies and possesses high convergence rate and low computational complexity for complex network environments with limited communication conditions is not only of significant theoretical research value but also a key technical path to solving the real-time scheduling problem of large-scale systems. Summary of the Invention
[0006] The purpose of this invention is to overcome the shortcomings of existing distributed optimization techniques in handling time-varying directed network environments, such as slow convergence rate, low computational efficiency, and convergence deviation caused by non-double random communication weights, and to provide a distributed accelerated optimization scheduling method, device, and medium based on time-varying directed topology.
[0007] The objective of this invention can be achieved through the following technical solutions: A distributed accelerated optimization scheduling method based on time-varying directed topology, the method comprising: Each scheduling unit in the distributed network system is abstracted as an agent node. A topology model and an optimization scheduling objective function for the distributed network system are constructed, and the variables of each agent are initialized. The distributed network is a time-varying directed graph. The variables of each agent include state estimation variables, weight balancing variables, gradient tracking variables, and Nesterov momentum acceleration terms. The objective of the optimization scheduling objective function is to minimize the average value of the local objective functions of all nodes. Each node in the topology model interacts with its neighboring nodes locally to perform a distributed iterative update that includes a Nesterov momentum acceleration term. The distributed iterative update process includes: updating the weight balance variables and intermediate state variables using the Push-Sum protocol, calculating and updating the biased state estimation variables, calculating the Nesterov momentum acceleration term based on historical information, and updating the gradient tracking variables. After completing one iteration, the key error is calculated based on the updated variables of each agent, the error linear system matrix is constructed and the system matrix spectral radius is verified based on the matrix spectral radius theory. It is then determined whether the distributed iterative update is linearly converged. If it is converged, the global optimal scheduling solution of the optimization scheduling objective function is output. Otherwise, the distributed iterative update containing the Nesterov momentum acceleration term is executed again. Based on the globally optimal scheduling solution, the scheduling instructions of the distributed network system are obtained, and the optimized scheduling is completed.
[0008] Furthermore, the scheduling unit in the distributed network system includes power grid units, robots, drones, or sensors. The process of constructing the topology model of the distributed network system and optimizing the scheduling objective function includes: Using graph theory, the communication relationships between agents are described. The communication network of the system is defined as a time-varying directed graph. The directed communication links between nodes are recorded. The process of the network topology of the time-varying directed graph evolving over time is verified to satisfy the B-strong connectivity hypothesis. If the verification is successful, a topology model of the distributed network system is constructed based on the time-varying directed graph and the directed communication links. Based on the actual needs of the scheduling unit, the local objective function of each node is determined, and the optimized scheduling objective function is obtained by combining the local objective functions. The local objective function satisfies strong convexity and gradient Lipschitz continuity.
[0009] Furthermore, when initializing the variables of each intelligent agent, an initial scheduling variable is set. Let be any real value, and let the weight balance variable be... This initializes the gradient tracking variable to the local gradient value at the current time step, i.e. This ensures that the sum of gradients across the entire network is zero at the initial moment, and sets initial values for intermediate variables. and Nesterov momentum acceleration term .
[0010] Furthermore, in the k-th iteration, the process of performing the distributed iterative update including the Nesterov momentum acceleration term includes: Updating the weight balance variable and intermediate state variable using the Push-Sum protocol: based on the column random weight matrix corresponding to the time-varying directed graph. Aggregated intelligent agents Weighted information from neighboring nodes, including the node itself, is used to update intermediate state variables using the Push-Sum protocol. Simultaneously update the weight balance variable ; Calculate and update the bias-free state estimate based on the updated weight balance variable. For intermediate variables Perform normalization to obtain the current state estimate after bias removal. ; Calculating the Nesterov momentum acceleration term based on historical information: Based on the Nesterov momentum mechanism, using the current state estimate. Compared with the state estimate at the previous time step The difference information, combined with the momentum coefficient Predict and calculate the Nesterov momentum acceleration term for the next interaction time step. ; Updating gradient tracking variables: A dynamic averaging consistency strategy is used to update gradient tracking variables. The gradient tracking variable is used to asymptotically track the global average gradient.
[0011] Furthermore, the key errors include consistency error, optimality gap, state difference, and gradient tracking error; The consistency error is the deviation between the node state estimate and the average state estimate. The optimality gap is the deviation between the average state estimate variable and the global optimal scheduling solution; The state difference is divided into the changes in the state estimation variables between adjacent iterations; The gradient tracking error is the deviation between the local gradient tracking variable and the global average gradient tracking variable.
[0012] Furthermore, the key error is obtained by using the vector norm inequality and the Lipschitz continuity definition of the gradient and deriving it.
[0013] Furthermore, the process of verifying the system matrix spectral radius based on matrix spectral radius theory and determining whether the distributed iterative update is linearly convergent includes: Based on the aforementioned matrix spectral radius theory, an analysis of the error linear system matrix is conducted, and the maximum step size that guarantees linear convergence of the distributed iterative update is derived. and momentum coefficient The upper bound condition is that the maximum step size is... and maximum momentum coefficient If the above upper bound condition is satisfied, then the system matrix... spectral radius The distributed iterative update converges linearly.
[0014] Furthermore, the maximum step size The upper bound condition is: in, The correlation spectral constant of the weight matrix, Let n be a network structure constant and n be the number of nodes. The time window length for B-strong connectivity. , and For auxiliary positive vector parameters, Let be the Lipschitz continuity constant for the gradient of the local function. This is an auxiliary constant; The momentum coefficient The upper bound condition is: in, For auxiliary positive vector parameters; , , and The following conditions must be met: .
[0015] An electronic device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the distributed accelerated optimization scheduling method based on time-varying directed topology as described above.
[0016] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the distributed accelerated optimization scheduling method based on time-varying directed topology as described above.
[0017] Compared with the prior art, the beneficial effects of the present invention include: 1. This invention addresses the complex network characteristics of asymmetric and dynamically changing communication links. It innovatively constructs a collaborative optimization architecture integrating the Nesterov momentum acceleration mechanism and Push-DIGing gradient tracking technology. On one hand, this method utilizes the Push-Sum protocol to handle the weight imbalance problem in directed graphs, eliminating steady-state errors by introducing auxiliary variables. On the other hand, it introduces an inertial momentum term based on historical state differences, adding "inertial force" during gradient tracking. This ensures accurate convergence of the algorithm under time-varying directed topologies while effectively suppressing iterative oscillations, breaking through the convergence speed bottleneck of traditional algorithms, and achieving fast and robust approximation of the global optimal solution.
[0018] 2. In order to solve the weight imbalance problem caused by directed graphs, this invention uses a weight balancing variable... This initializes the gradient tracking variable to the local gradient value at the current time step, i.e. To ensure that the sum of gradients across the entire network is zero at the initial moment, unbiased convergence is achieved through weight balancing and normalization, and the final output scheduling solution is a strictly global optimal solution.
[0019] 3. Compared with the traditional Push-DIGing algorithm, this invention introduces a momentum acceleration mechanism, which integrates Nesterov momentum with Push-DIGing and uses historical state difference information to form inertial acceleration, effectively suppressing iterative oscillations, increasing the convergence speed by several times, achieving linear rate convergence, and significantly reducing the number of iterations without sacrificing convergence accuracy, thereby reducing the system's computational and communication overhead.
[0020] 4. This invention asymptotically estimates the global average gradient by tracking variables through only local interactions. Compared with centralized scheduling, it eliminates the need for a central node to collect all data, thus solving the problems of communication congestion, the curse of computational dimensionality, and privacy leakage in centralized architectures. It also protects the privacy of the local objective function of each node and reduces communication volume.
[0021] 5. This invention constructs augmented state vectors for four types of errors: consistency, optimality, state difference, and gradient pursuit, transforming complex iterations into linear system analysis. Based on matrix spectral radius theory, the maximum step size is derived. and momentum coefficient The upper bound of the algorithm ensures linear convergence.
[0022] 6. In this invention, all nodes interact peer-to-peer and compute independently, with no central node, no single point of failure, nodes can be freely added or removed, the system has strong scalability, the failure of a single node does not affect the convergence of the entire network, and the system reliability is improved. Attached Figure Description
[0023] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the distributed time-varying directed network topology of the present invention; Figure 3 This is a schematic diagram comparing the convergence speed of the method of this invention and the traditional algorithm under time-varying directed graphs; Figure 4 This is a comparative diagram showing the impact of different momentum coefficients on the convergence performance of the distributed acceleration optimization algorithm proposed in this invention under the condition of fixed step size parameters. Figure 5 This is a comparative diagram showing the impact of different time length parameters on the convergence performance of the distributed acceleration optimization algorithm proposed in this invention under the condition of fixed momentum coefficient. Detailed Implementation
[0024] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.
[0025] Example 1 A distributed accelerated optimization scheduling method based on time-varying directed topology is proposed, as follows: Figure 1 As shown, the process includes steps S1-S5, each described in detail below: Step S1: Abstract each scheduling unit in the distributed network system into an intelligent agent node, and construct the topology model and optimization scheduling objective function of the distributed network system.
[0026] The distributed network is a time-varying directed graph. The variables of each agent include state estimation variables, weight balancing variables, gradient tracking variables, and Nesterov momentum acceleration terms. The objective of optimizing the scheduling objective function is to minimize the average value of the local objective functions of all nodes.
[0027] The process of constructing a topology model for a distributed network system includes: Using graph theory, we describe the communication relationships between agents, define the system's communication network as a time-varying directed graph, record the directed communication links between nodes, and verify that the evolution of the network topology of the time-varying directed graph over time satisfies the B-strong connectivity assumption. If the verification is successful, we construct a topology model of the distributed network system based on the time-varying directed graph and the directed communication links.
[0028] The process of constructing the optimization scheduling objective function includes: Based on the actual needs of the scheduling unit, the local objective function of each node is determined, and the optimized scheduling objective function is obtained by combining the local objective functions. The local objective functions satisfy strong convexity and gradient Lipshitz continuity.
[0029] The scheduling units in a distributed network system include power grid units, robots, drones, or sensors.
[0030] Correspondingly, the actual needs of the dispatching unit are such as economic dispatching of the power grid, minimum energy consumption, and optimal formation.
[0031] Specifically, in this embodiment, the process of constructing the topology model of the distributed network system is as follows: Abstracting each participating unit in a distributed system into an intelligent agent node, constructing a system composed of... A network system composed of intelligent agents. Graph theory is used to describe the communication relationships between the agents, and the communication network of the system is defined as a time-varying directed graph. .in, It represents the set of all nodes in the system; Indicates in A set of communication links that are always active. Since the network is directed, if a node... Able to send to nodes If a message is sent, then there exists a directed edge. However, this does not mean reverse link It is bound to exist. Furthermore, considering the dynamic changes in the communication environment, network topology... Over time For an evolution to occur, the B-strong connectivity assumption must be satisfied, which means that within a time window of any length B, the joint graph of all time-varying graphs is strongly connected to ensure that information can be effectively propagated throughout the network.
[0032] Specifically, in this embodiment, the process of constructing the optimized scheduling objective function is as follows: Based on the above topology model, a global objective function for network-wide collaborative optimization is defined. The core of this problem lies in finding an optimal decision variable while satisfying network constraints. This minimizes the overall cost for all agents. The global optimization objective function is defined as the average of the sums of the local objective functions of all agents, and its mathematical form is as follows: in, Let be the global scheduling decision variables to be solved. Indicates only nodes The local objective function is known. To ensure the convergence of the algorithm and the uniqueness of the solution, the following mathematical assumptions are made for the above model: a. Strong convexity assumption: Each local objective function All are strongly convex functions, meaning they have a constant. , so that for any ,satisfy ; b. Smoothness assumption: Each local objective function The gradient is Lipschitz continuous, meaning there exists a constant. , making .
[0033] Step S2: Initialize the variables of each agent.
[0034] When initializing the variables of each agent, for any node Set initial scheduling variables Let be any real value; to solve the weight imbalance problem caused by directed graphs, let the weight balancing variable be... This initializes the gradient tracking variable to the local gradient value at the current time step, i.e. This ensures that the sum of gradients across the entire network is zero at the initial moment, and sets initial values for intermediate variables. and Nesterov momentum acceleration term This prepares the data for subsequent iterations and updates.
[0035] Step S3: Enable each node in the topology model to interact with its neighboring nodes locally and perform a distributed iterative update that includes the Nesterov momentum acceleration term. The distributed iterative update process includes: updating the weight balance variables and intermediate state variables using the Push-Sum protocol, calculating and updating the biased state estimation variables, calculating the Nesterov momentum acceleration term based on historical information, and updating the gradient tracking variables.
[0036] In the k-th iteration, the process of performing a distributed iterative update including the Nesterov momentum acceleration term includes: Updating weight balance variables and intermediate state variables using the Push-Sum protocol: based on the column random weight matrix corresponding to the time-varying directed graph. Aggregated intelligent agents Weighted information from neighboring nodes, including the node itself, is used to update intermediate state variables using the Push-Sum protocol. Simultaneously update the weight balance variable ; Calculate and update the bias-free state estimate based on the updated weight balance variables. For intermediate variables Perform normalization to obtain the current state estimate after bias removal. ; Calculating the Nesterov momentum acceleration term based on historical information: Based on the Nesterov momentum mechanism, using the current state estimate. Compared with the state estimate at the previous time step The difference information, combined with the momentum coefficient Predict and calculate the Nesterov momentum acceleration term for the next interaction time step. ; Updating gradient tracking variables: A dynamic averaging consistency strategy is used to update gradient tracking variables. The gradient tracking variable is used to asymptotically track the global average gradient.
[0037] Specifically, the Push-Sum protocol is used to update the weight balance variable and intermediate state variable: the agent Based on the column random weight matrix corresponding to the time-varying directed graph It aggregates weighted information from neighboring nodes, including itself. On one hand, it updates intermediate state variables. This process incorporates a momentum auxiliary term. With gradient tracking term The impact; on the other hand, synchronously updating the weight balance variables. The specific formula is as follows: in, The communication weight matrix of a time-varying directed graph elements, The step size.
[0038] Specifically, the bias-free state estimate is calculated and updated: since the weight matrix of a directed graph is usually not double-random, directly using intermediate variables can lead to convergence bias. Therefore, the weight balance variables obtained from the above steps are used... For intermediate variables Normalization is performed to obtain an accurate estimate of the current scheduling state after bias removal. : 。
[0039] Specifically, the Nesterov momentum acceleration term is calculated based on historical information: to improve the convergence rate, a Nesterov momentum mechanism is introduced. This utilizes the current state. State compared to the previous moment The difference information, combined with the momentum coefficient Predict and calculate the momentum auxiliary variable used for interaction at the next moment. : This step effectively utilizes historical trajectory information, which can significantly suppress oscillations and accelerate convergence.
[0040] Specifically, updating gradient tracking variables: To address the issue that nodes only know the local objective function and cannot obtain the global gradient, a dynamic averaging consistency strategy is used to update gradient tracking variables. This variable aims to asymptotically track the global average gradient, and its update formula is: .
[0041] Step S4: After completing one round of iteration, calculate the key error based on the updated agent variables, construct the error linear system matrix, and verify the system matrix spectral radius based on the matrix spectral radius theory. Determine whether the distributed iterative update is linearly converged. If it is converged, output the global optimal scheduling solution of the optimization scheduling objective function; otherwise, return to step S3 and execute the distributed iterative update containing the Nesterov momentum acceleration term again.
[0042] Key errors include consistency error, optimality gap, state difference, and gradient tracking error; Consistency error The deviation between the node state estimates and the mean state estimates; Optimal gap The deviation between the average state estimate and the global optimal scheduling solution; State difference Estimating the changes in state variables for adjacent iterations; Gradient tracking error This represents the deviation between the local gradient tracking variable and the global average gradient tracking variable.
[0043] Each key error is obtained using the vector norm inequality and the Lipschitz continuity definition of the gradient, and the specific process is as follows: First, we expand using the Lipschitz continuity of the gradient. From the scaling derivation, we can obtain: in, .
[0044] Next, we expand on the consistency error. Scaling derivation. Based on the update formula in step S3, we can obtain: in, , , .
[0045] Similarly, the optimality error can be derived as follows: The state error can be derived as follows: Finally, we expand on the gradient tracking error. The scaling derivation. Based on the update formula in step S3 and the Lipschitz continuity of the gradient, we can obtain: Therefore, we construct the error linear system as follows: .
[0046] The process of verifying the system's matrix spectral radius based on matrix spectral radius theory and determining whether the distributed iterative update is linearly convergent includes: Based on the matrix spectral radius theory, this paper analyzes the matrix of the error linear system and derives the maximum step size that guarantees the linear convergence of the distributed iterative update. and momentum coefficient The upper bound condition is that the maximum step size is... and maximum momentum coefficient If the upper bound condition is satisfied, then the system matrix is... spectral radius The distributed iterative update converges linearly.
[0047] Specifically, based on the matrix spectral radius theory, the maximum step size that guarantees the linear convergence of the system is derived by analyzing the constructed error linear system. and momentum coefficient The upper bound condition.
[0048] From the constructed error linear system, we can obtain: in: The establishment of this linear system transforms the complex convergence analysis of distributed algorithms into an analysis of the system matrix. Spectral property analysis. That is, if the system matrix... spectral radius If the system converges linearly, then the maximum step size can be calculated using the theorem. and maximum momentum coefficient Specific upper bound conditions must be met; Maximum step size The upper bound condition is: in, The correlation spectral constant of the weight matrix, Here, n is a network structure constant. The time window length for B-strong connectivity. , and For auxiliary positive vector parameters, Let be the Lipschitz continuity constant for the gradient of the local function. This is an auxiliary constant.
[0049] Momentum coefficient The upper bound condition is: in, These are auxiliary positive vector parameters.
[0050] , , and The following conditions must be met: When all parameters are selected within the above range, the algorithm can achieve the following: The linear rate converges to the global optimum. This allows for optimized scheduling.
[0051] Step S5: Obtain the scheduling instructions of the distributed network system based on the globally optimal scheduling solution to complete the optimized scheduling.
[0052] If there is a sudden change in the communication topology or a change in the device status, return to step S1 to fine-tune the model, or reset the initialization in step S2 and iterate again.
[0053] Example 2 To verify the effectiveness and convergence characteristics of the distributed accelerated optimization scheduling method based on time-varying directed topology proposed in this invention, multiple sets of numerical simulation experiments were constructed for verification.
[0054] The experiment considers a distributed network system consisting of 10 agents, who interact with each other through a time-varying directed communication topology, such as... Figure 2 As shown, the network structure changes dynamically over time and satisfies the B-strong connectivity assumption, thus ensuring that information can be effectively propagated throughout the network. The local objective function of each node is selected as a strongly convex quadratic function with a continuous Lipshitz gradient to ensure the existence and uniqueness of the global optimum.
[0055] First, under the same initial conditions and communication environment, the basic convergence performance of the proposed distributed accelerated optimization algorithm in time-varying directed topology is verified. By recording the change in the optimality error between the state estimates of each agent and the global optimum with the number of iterations, it can be observed that the algorithm can stably converge to the global optimum within a finite number of iterations, indicating that the designed weight balancing mechanism and gradient tracking strategy can effectively eliminate the bias caused by the directed communication network.
[0056] Secondly, to analyze the influence of momentum parameters on the algorithm's convergence behavior, simulation experiments were conducted with different momentum coefficients under a fixed step size. Simulation results show that appropriately introducing a momentum acceleration term can significantly improve the algorithm's convergence speed. When the momentum coefficient is small, the algorithm's convergence process is relatively stable but the convergence speed is relatively slow. As the momentum coefficient increases, the initial convergence speed of the algorithm accelerates significantly, but when the momentum parameter is too large, the system may exhibit some degree of oscillation. This result verifies the necessity of constraining the range of momentum coefficient values in theoretical analysis.
[0057] Furthermore, under the condition of a fixed momentum coefficient, the influence of different step size parameters on the convergence performance of the algorithm was analyzed. Experimental results show that the step size parameter has a significant impact on the convergence speed and stability of the algorithm: a smaller step size can ensure stable convergence of the system, but more iterations are required to achieve the same accuracy; appropriately increasing the step size can effectively accelerate the convergence process and improve optimization efficiency; however, when the step size exceeds a certain threshold, the algorithm may experience a decrease in convergence speed or oscillation, and may even affect the stability of the system.
[0058] In summary, the experimental results fully verify the effectiveness and robustness of the distributed accelerated optimization scheduling method proposed in this invention under time-varying directed topology conditions. Furthermore, simulation analysis shows that the reasonable selection of step size parameters and momentum coefficients are key factors in fully leveraging the acceleration performance of the algorithm and ensuring stable convergence of the system, providing a strong basis for parameter configuration of this method in practical complex distributed scheduling scenarios.
[0059] To verify the convergence performance of the distributed accelerated optimization scheduling method based on time-varying directed topology proposed in this invention and its sensitivity to parameter changes, multiple sets of numerical simulation experiments were constructed to analyze the convergence behavior of the algorithm under different parameter configurations. The experiments considered a distributed network system composed of 10 agents, with each agent interacting with information through a time-varying directed communication topology, such as... Figure 2 As shown, the communication network changes dynamically over time and satisfies the B-strong connectivity assumption. The local objective function of each node is chosen as a strongly convex quadratic function with a continuous Lipshitz gradient to ensure the existence and uniqueness of the global optimal solution.
[0060] First, under the same communication topology, initial conditions, and algorithm parameter configuration, the distributed accelerated optimization scheduling method proposed in this invention is compared with the classic Push-DIGing algorithm through simulation. The results are as follows: Figure 3 As shown in the figure. The horizontal axis represents the number of algorithm iterations, and the vertical axis represents the change in the optimality error of the system state relative to the global optimal solution. Simulation results show that in a time-varying directed communication network environment, the classic Push-DIGing algorithm (… Figure 3 The algorithm (black dashed line in the middle) can achieve asymptotic convergence, but its convergence speed is relatively slow; in contrast, the algorithm proposed in this invention (… Figure 3 (The solid red line in the middle) By introducing a momentum acceleration mechanism, the error decay rate is significantly accelerated without affecting the convergence accuracy. It can achieve the same optimization accuracy in fewer iterations, which fully demonstrates the technical advantages of this invention in terms of convergence speed.
[0061] Secondly, at a fixed step size Analyze different momentum coefficients under the condition of 0.05. To investigate the impact of momentum coefficients on the algorithm's convergence performance, simulation experiments were conducted with momentum coefficients of 0.3, 0.5, and 0.8, respectively. The results are as follows: Figure 4 As shown. Simulation results show that: when When the value is 0.3 (red line), the algorithm converges relatively smoothly and can stably approach the global optimum in a relatively small number of iterations; when When the value is 0.5 (blue dashed line), the convergence speed of the algorithm is further improved in the early stage, but slight oscillations may occur in the middle and late stages. when When the value is 0.8 (pink dashed line), the system oscillation becomes more pronounced, and the convergence stability decreases.
[0062] This result shows that the appropriate selection of the momentum coefficient is of great significance for fully realizing the acceleration effect while ensuring the stability of the system.
[0063] Finally, with a fixed momentum coefficient Under the condition of 0.3, the influence of different step size parameters on the convergence performance of the algorithm is further investigated, and step sizes are selected respectively. Simulation experiments were conducted with values of 0.02, 0.05, and 0.1, and the results are as follows: Figure 5 As shown. Simulation results show: when When the value is 0.02 (black solid line), the algorithm can maintain stable convergence, but the overall convergence speed is slow. when When the value is 0.05 (red dashed line), the algorithm significantly accelerates the convergence speed while maintaining stability; when When the value is 0.1 (blue dashed line), the convergence speed of the algorithm is improved in the early stage, but the system is prone to oscillation, and the stability is affected to some extent.
[0064] The results above show that the step size parameter has a significant impact on the convergence speed and stability of the algorithm.
[0065] In summary, experimental results demonstrate that, under time-varying directed topology conditions, the distributed accelerated optimization scheduling method proposed in this invention can achieve fast and stable linear convergence within a reasonable parameter configuration range. Simulation analysis of the momentum coefficient and step size parameters further verifies the rationality of the parameter value range and convergence conditions discussed in the theoretical analysis above, providing a clear basis for parameter selection in practical engineering applications of this invention.
[0066] Example 3 Based on Embodiment 1, this embodiment provides an electronic device, including: one or more processors and a memory, wherein the memory stores one or more programs, and the one or more programs include instructions for executing the aforementioned distributed acceleration optimization scheduling method based on time-varying directed topology.
[0067] At the hardware level, the electronic device includes a processor, internal bus, network interface, memory, and non-volatile memory, and may also include other hardware required for business operations. The processor reads the corresponding computer program from the non-volatile memory into memory and then runs it to implement the aforementioned distributed acceleration optimization scheduling method based on time-varying directed topology. Of course, in addition to software implementation, this invention does not exclude other implementation methods, such as logic devices or a combination of hardware and software, etc. That is to say, the execution entity of the following processing flow is not limited to individual logic units, but can also be hardware or logic devices.
[0068] Memory may include non-persistent storage in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.
[0069] Computer-readable media include both permanent and non-permanent, removable and non-removable media that can store information using any method or technology. Information can be computer-readable instructions, data structures, modules of programs, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.
[0070] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A time-varying directed topology based distributed speedup optimization scheduling method, characterized in that, The method includes: Each scheduling unit in the distributed network system is abstracted as an agent node. A topology model and an optimization scheduling objective function for the distributed network system are constructed, and the variables of each agent are initialized. The distributed network is a time-varying directed graph. The variables of each agent include state estimation variables, weight balancing variables, gradient tracking variables, and Nesterov momentum acceleration terms. The objective of the optimization scheduling objective function is to minimize the average value of the local objective functions of all nodes. Each node in the topology model interacts with its neighboring nodes locally to perform a distributed iterative update that includes a Nesterov momentum acceleration term. The distributed iterative update process includes: updating the weight balance variables and intermediate state variables using the Push-Sum protocol, calculating and updating the biased state estimation variables, calculating the Nesterov momentum acceleration term based on historical information, and updating the gradient tracking variables. After completing one iteration, the key error is calculated based on the updated variables of each agent, the error linear system matrix is constructed and the system matrix spectral radius is verified based on the matrix spectral radius theory. It is then determined whether the distributed iterative update is linearly converged. If it is converged, the global optimal scheduling solution of the optimization scheduling objective function is output. Otherwise, the distributed iterative update containing the Nesterov momentum acceleration term is executed again. Based on the globally optimal scheduling solution, the scheduling instructions of the distributed network system are obtained, and the optimized scheduling is completed.
2. The method of claim 1, wherein, The scheduling unit in the distributed network system includes power grid units, robots, drones, or sensors. The process of constructing the topology model of the distributed network system and optimizing the scheduling objective function includes: Using graph theory, the communication relationships between agents are described. The communication network of the system is defined as a time-varying directed graph. The directed communication links between nodes are recorded. The process of the network topology of the time-varying directed graph evolving over time is verified to satisfy the B-strong connectivity hypothesis. If the verification is successful, a topology model of the distributed network system is constructed based on the time-varying directed graph and the directed communication links. Based on the actual needs of the scheduling unit, the local objective function of each node is determined, and the optimized scheduling objective function is obtained by combining the local objective functions. The local objective function satisfies strong convexity and gradient Lipschitz continuity.
3. The distributed acceleration optimization scheduling method based on time-varying directed topology according to claim 1, characterized in that, When initializing the variables of the agents, set the initial scheduling variable For any real value, make the weight balance variable Initialize the gradient tracking variable to the local gradient value at the current time, that is To ensure that the total network gradient sum is zero at the initial time, and set the initial value of the intermediate variable And the Nesterov momentum acceleration term .
4. The method of claim 1, wherein, In the k-th iteration, the process of performing the distributed iterative update including the Nesterov momentum acceleration term includes: updating weight balancing variables and intermediate state variables using a Push-Sum protocol: updating weight balancing variables and intermediate state variables using a Push-Sum protocol based on a column stochastic weight matrix corresponding to the time-varying directed graph , aggregating agents from neighbor nodes including itself, updating intermediate state variables using a Push-Sum protocol while updating weight balancing variables ; computing a debiased state estimate and updating: the weight balancing variable based on the updated weight balancing variable on the intermediate variable normalizing to obtain a debiased current state estimate ; Calculating the Nesterov momentum acceleration term based on historical information: Based on the Nesterov momentum mechanism, using the current state estimate. Compared with the state estimate at the previous time step The difference information, combined with the momentum coefficient Predict and calculate the Nesterov momentum acceleration term for the next interaction time step. ; Updating gradient tracking variables: A dynamic averaging consistency strategy is used to update gradient tracking variables. The gradient tracking variable is used to asymptotically track the global average gradient.
5. The method of claim 1, wherein, The key errors include consistency error, optimality gap, state difference, and gradient tracking error; The consistency error is the deviation between the node state estimate and the average state estimate. The optimality gap is the deviation between the average state estimate variable and the global optimal scheduling solution; The state difference is divided into the changes in the state estimation variables between adjacent iterations; The gradient tracking error is the deviation between the local gradient tracking variable and the global average gradient tracking variable.
6. The method of claim 1, wherein, The key error is obtained by using the vector norm inequality and the Lipschitz continuity definition of the gradient and deriving it.
7. The method of claim 1, wherein, The process of verifying the system's matrix spectral radius based on matrix spectral radius theory and determining whether the distributed iterative update is linearly convergent includes: Based on the matrix spectral radius theory, the error linear system matrix is analyzed, and the maximum step size is derived to ensure the linear convergence of the distributed iterative update and the upper bound condition of momentum coefficient If the maximum step size and the maximum momentum coefficient satisfy the upper bound condition between them, the spectral radius of the system matrix , the linear convergence of the distributed iterative update.
8. A distributed accelerated optimization scheduling method based on time-varying directed topology according to claim 7, characterized in that, the maximum step size the upper bound condition is: wherein, is a weight matrix related spectral constant, is a network structure constant, n is the number of nodes, is a length of a time window for B-strong connectivity, , and is an auxiliary forward vector parameter, is a local function gradient Lipschitz continuity constant, is an auxiliary constant; The momentum coefficient The upper bound condition is: in, For auxiliary positive vector parameters; , , and The following conditions must be met: 。 9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the distributed acceleration optimization scheduling method based on time-varying directed topology as described in any one of claims 1-8.
10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the distributed accelerated optimization scheduling method based on time-varying directed topology as described in any one of claims 1-8.