Solution of joint path and destination planning problem based on distributed algorithm for solving generalized nash equilibrium

Abstract

This invention discloses a solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm. First, the joint path and destination planning problem is modeled, transforming it into a non-cooperative game model. This model includes the objective function of each electric vehicle, global coupling constraints, and local constraints. Second, pseudo-gradients are used to transform the game model into a VI problem, introducing edge-based consistency constraints and a heterogeneous step size mechanism. Based on fixed-point iteration and proximal gradient operator theory, a distributed solution algorithm under complete information is proposed. Then, a global estimate of the plans of other users is introduced, proposing a distributed solution algorithm under partial information. This invention avoids the construction of double random matrices through edge-based consistency constraints, and can maintain solution accuracy while meeting low computational requirements when more users participate in the game model.

Description

Technical Field

[0001] This invention relates to the field of information processing technology, and specifically to a solution for the planning problem of joint path and destination based on the generalized Nash equilibrium algorithm. Background Technology

[0002] With the emergence of new technologies such as various map software applications, users can access real-time information to choose routes and destinations. Research shows that users play an active role in infrastructure; therefore, if the state of the transportation network changes, users will quickly react to these changes and alter their decisions. A decision-making problem arises when users choose routes and destinations based on their preferences and data provided by online platforms (e.g., road congestion and destination crowding). This competitive nature of the decision-making problem becomes apparent when users are willing to reach the least congested destination (e.g., a charging station) in the shortest possible time. This situation occurs in issues such as population migration, supermarket selection, on-demand autonomous travel, public parking, and electric vehicle (EV) charging station selection, where congestion can affect electricity prices and waiting times. Clearly, route and destination planning are not two disjoint decision variables, as the choice of destination (e.g., charging station) constrains route selection, and vice versa. On the other hand, with the increasing number of electric vehicles and the limited number of public charging facilities with limited power resources, the operation of transportation networks and charging stations faces several challenges. In this situation, when users are willing to drive to the nearest station, the power demand at some stations will increase significantly. Due to limited resources and facilities, it is necessary to study and control for this impact. Existing research solutions cannot achieve the required accuracy when there are more electric vehicles, and also require a large amount of computation.

[0003] Game theory is a tool for studying the behavior of multiple decision-makers and has wide applications in sociology, economics, engineering, and other fields. In real-world problems, the objective functions of individuals (or decision-makers) are often mutually constrained (e.g., due to competition), and due to limitations such as network bandwidth, scarce resources, or supply and demand balance, individual decision variables are interconnected. In fact, a significant branch of game theory focuses on providing theoretical basis, effective analysis and prediction, and designing learning algorithms that can achieve equilibrium in response to various couplings and conflicts and the resulting phenomena. Therefore, introducing game theory into multi-agent systems allows us to leverage the unique theoretical framework to solve numerous real-world problems where multiple agents, in a state of mutual constraint and competition, avoid conflict and ultimately achieve relative equilibrium. This not only effectively improves system performance but also greatly expands the thinking of scholars regarding system algorithm design. The solution to a game problem (i.e., a non-cooperative game) is generally called a Nash equilibrium. A Nash equilibrium can be simply explained as all players having reached their relatively optimal strategy given the states of the other players; that is, unilaterally changing one's strategy would result in a reduction in payoff. However, in many applications, due to constraints such as network resources, the action sets of all participants are coupled together through globally shared affine constraints. That is, for each participant, the feasible action set depends on the actions of other participants. In this case, the solution to the non-cooperative game is called a generalized Nash equilibrium. Summary of the Invention

[0004] To address the aforementioned problems, this invention discloses a solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm.

[0005] The present invention adopts the following technical solution:

[0006] A solution to the joint path and destination planning problem based on a distributed algorithm for solving generalized Nash equilibrium, assuming that it includes... There are N electric vehicles = {1, ..., n}. In the hypothetical scenario, since the electric vehicles are driven by users, the term "electric vehicle" also refers to the user. road as well as One charging station A charging station refers to a parking lot equipped with charging stations, and also to the user's destination. In this hypothetical scenario, the charging pile is equivalent to the destination, and includes the following steps:

[0007] Model the planning problem of joint paths and destinations, and establish the objective function for each user. Based on the maximum traffic load of the road and the maximum power supply load of the charging station, the global coupling constraints of the planning problem of the joint path and destination are obtained. At the same time, according to probability, each user also needs to satisfy its own local constraints.

[0008] The above game theory model is transformed into a variational inequality problem. The solution to this variational inequality problem (called the variational GNE, v-GNE) has an economic interpretation without price discrimination, and simultaneously, when all local Lagrange multipliers... When consensus is reached in steady state, the solution to the variational inequality problem is also the solution to the original game model;

[0009] against To address consistency issues, an edge-based consistency constraint is introduced, and a distributed solution algorithm under complete information is designed based on fixed-point iteration and proximal gradient operator theory. Furthermore, a global estimate of the plans of other users is introduced. A distributed solution algorithm based on partial information is proposed.

[0010] As a preferred embodiment of the present invention, the problem of planning the combined path and destination specifically includes:

[0011] Model building

[0012] Each user Decide on your own plans: ,in, Indicates user The probability of choosing each path Indicates user The probability of choosing each charging station.

[0013] Each user The objective function is:

[0014]

[0015] in Indicates deviation from user The associated costs of habitual choices User Estimated travel time User Estimated service costs (such as charging fees, parking fees), It is a weighting factor representing the time term. Each user The three parts of the objective function are defined as follows:

[0016] (1) Deviating from the user The associated costs of habitual choices

[0017]

[0018] in and Indicates user Based on past experience, the probability of choosing the preferred destination and route. and These represent the weighting factors for these two preferences, respectively.

[0019] (2) Travel time

[0020]

[0021] in It indicates a road The strictly monotonically increasing function of the boarding traffic flow is defined as:

[0022]

[0023] in Indicates road Travel time under uncongested conditions With roads The length is related to the speed limit. Indicates road Traffic capacity, . Indicates road The projected traffic volume is defined as:

[0024]

[0025] in This indicates the traffic flow of non-charging vehicles.

[0026] (3) Service Costs

[0027]

[0028] in Indicates user energy demand, This indicates the parking fee. The price function of energy is defined as:

[0029]

[0030] in It is a price coefficient. It is a charging station The charging capacity, It is a charging station Total energy demand is projected. User It is a charging station Energy demand.

[0031] To ensure that the game model has a GNE (reaching Nash equilibrium), when the user... From the starting point Depart and reach a feasible destination. When, its probability is ,user The following constraints must be met:

[0032]

[0033] The above constraints can be understood as user Leave the starting point The probability of reaching the 1st is equal to 1. The probability of a charging station is equal to The probability of entering an intersection and leaving an intersection should be the same. Meanwhile, the user... You can only go to a specific set of destinations, that is Because the chargers there are compatible with their electric vehicles. Therefore, users... Need to meet In addition, users The sum of the probabilities of going to all charging stations needs to satisfy... .

[0034] Considering the maximum capacity constraints of roads and charging stations, the global coupling constraints of the joint path and destination planning problem can be modeled as follows:

[0035]

[0036] in Indicate destination The largest energy supply, Indicates road Maximum traffic flow.

[0037] As a preferred technical solution of the present invention, the game theory model is transformed into a variational inequality problem and the relationship between the solution of the variational inequality problem and the solution of the game theory model is explained, specifically including:

[0038] The game theory model described above can be summarized into another general game theory model as follows:

[0039]

[0040] in The constraints of the general game model are as follows: ,in The feasible set is defined as follows:

[0041]

[0042] in Indicates user Local constraints that need to be satisfied Define the global coupling constraints that all users must satisfy; , as well as The general game theory model can be written as:

[0043]

[0044] Furthermore, the Karush-Kuhn-Tucker conditions for the general game theory model need to satisfy:

[0045]

[0046] The pseudo-gradient mapping of the general game theory model is represented as:

[0047]

[0048] The variational inequality problem can then be expressed as:

[0049]

[0050] in The Karush-Kuhn-Tucker conditions for variational inequality problems must satisfy:

[0051]

[0052] By observing the Karush-Kuhn-Tucker conditions of general game theory models and variational inequality problems, it is inferred that when , In this case, any solution to the variational inequality problem is a solution to the game theory model.

[0053] As a preferred embodiment of the present invention, the distributed solution algorithm under complete information specifically includes:

[0054] Distributed solution algorithms under complete information

[0055] definition and for each edge Introducing operators , making ,in It is an operator that achieves consistency in local Lagrange multipliers, defined as:

[0056]

[0057] set Defined as Therefore, we can obtain when Sometimes, , .

[0058] definition As The dual variables, where .make , , as well as Therefore, the Karush-Kuhn-Tucker (KKT) conditions are as follows:

[0059]

[0060] Based on the KKT conditions, using fixed-point iteration and applying the proximal gradient operator, the intensive form of the algorithm is:

[0061]

[0062] in , , The three heterogeneous step size matrices in the above formula are represented as follows:

[0063]

[0064] Decomposing the centralized algorithm, we can obtain the distributed form of the distributed solution algorithm under complete information as follows (EDPDPG-FI):

[0065]

[0066] As a preferred embodiment of the present invention, the distributed solution algorithm based on the partial information specifically includes:

[0067] Distributed solution algorithm with partial information

[0068] Considering a more practical application, where some users cannot know the plans of all other users, this invention introduces an estimate of the plans of other users for each user: Accordingly, define Therefore, users The local objective function can be written as All users' local estimates need to reach a consensus in steady state, i.e. .

[0069] Similar to EDPDPG-FI, for each edge Operators were introduced Make

[0070]

[0071] in It is an operator that achieves local estimation consistency, and it is defined as follows:

[0072]

[0073] Then the set Defined as It can be observed that when Sometimes, .

[0074] definition As The dual variables, where .make as well as .

[0075] Two selection matrices were introduced. Used for filtering In Defined as:

[0076]

[0077] in as well as .

[0078] Used for filtering In Defined as:

[0079]

[0080] Therefore, we have

[0081]

[0082] make , , as well as You can get Meanwhile, the pseudo-gradient mapping under partial information should be:

[0083]

[0084] Therefore, the KKT conditions are as follows:

[0085]

[0086] The centralized form of the algorithm is:

[0087]

[0088] , , And three heterogeneous step size matrices:

[0089]

[0090] Decomposing the centralized algorithm, we can obtain the distributed form of the algorithm as follows (EDPDPG-PI):

[0091]

[0092] The beneficial effects of this invention are:

[0093] This invention first models the joint path and destination planning problem, transforming it into a non-cooperative game model. This model includes the objective function of each electric vehicle, global coupling constraints, and local constraints. Second, it uses pseudo-gradients to transform the game model into a variational inequality (VI) problem, introducing edge-based consistency constraints and a heterogeneous step size mechanism. Based on fixed-point iteration and proximal gradient operator theory, a distributed solution algorithm under complete information is proposed. Then, by introducing a global estimate of the plans of other users, a distributed solution algorithm under partial information is proposed. This invention avoids the construction of double random matrices through edge-based consistency constraints, and can maintain solution accuracy while meeting low computational requirements when more users participate in the game model. Attached Figure Description

[0094] Figure 1 This is a traffic network diagram assumed in the embodiments of the present invention;

[0095] Figure 2 The relevant errors were shown. The trajectory convergence process diagram;

[0096] Figure 3 This demonstrates the impact of different weighting factors of the travel time function on user actions under full information settings;

[0097] Figure 4 This shows the impact of different weighting factors of the driving time function on user actions under partial information settings;

[0098] Figure 5 This shows the load on roads and charging stations in the region when the problem reaches a Nash equilibrium (GNE). Detailed Implementation

[0099] The present invention adopts the following technical solution:

[0100] A solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm, including... There are N electric vehicles = {1, ..., n}, where "electric vehicle" also refers to the user. road as well as One charging station A charging station refers to a parking lot equipped with a charging station, characterized by including the following steps:

[0101] Model the planning problem of joint paths and destinations, and establish the objective function for each user. Based on the maximum traffic load of the road and the maximum power supply load of the charging station, the global coupling constraints of the planning problem of the joint path and destination are obtained. At the same time, according to probability, each user also needs to satisfy its own local constraints.

[0102] The above game theory model is transformed into a variational inequality problem. The solution to this variational inequality problem (called the variational GNE, v-GNE) has an economic interpretation without price discrimination, and simultaneously, when all local Lagrange multipliers... When consensus is reached in steady state, the solution to the variational inequality problem is also the solution to the original game model;

[0103] against To address consistency issues, an edge-based consistency constraint is introduced, and a distributed solution algorithm under complete information is designed based on fixed-point iteration and proximal gradient operator theory. Furthermore, a global estimate of the plans of other users is introduced. A distributed solution algorithm based on partial information is proposed.

[0104] As a preferred embodiment of the present invention, the problem of planning the combined path and destination specifically includes:

[0105] Model building

[0106] Each user Decide on your own plans: ,in, Indicates user The probability of choosing each path Indicates user The probability of choosing each charging station.

[0107] Each user The objective function is:

[0108]

[0109] in Indicates deviation from user The associated costs of habitual choices User Estimated travel time User Estimated service costs (such as charging fees, parking fees), It is a weighting factor representing the time term. Each user The three parts of the objective function are defined as follows:

[0110] (4) Deviating from the user The associated costs of habitual choices

[0111]

[0112] in and Indicates user Based on past experience, the probability of choosing the preferred destination and route. and These represent the weighting factors for these two preferences, respectively.

[0113] (5) Travel time

[0114]

[0115] in It indicates a road The strictly monotonically increasing function of the boarding traffic flow is defined as:

[0116]

[0117] in Indicates road Travel time under uncongested conditions With roads The length is related to the speed limit. Indicates road Traffic capacity, . Indicates road The projected traffic volume is defined as:

[0118]

[0119] in This indicates the traffic flow of non-charging vehicles.

[0120] (6) Service Costs

[0121]

[0122] in Indicates user energy demand, This indicates the parking fee. The price function of energy is defined as:

[0123]

[0124] in It is a price coefficient. It is a charging station The charging capacity, It is a charging station Total energy demand is projected. User It is a charging station Energy demand.

[0125] To ensure that the game model has GNE, when the user From the starting point Depart and reach a feasible destination. When, its probability is ,user The following constraints must be met:

[0126]

[0127] The above constraints can be understood as user Leave the starting point The probability of reaching the 1st is equal to 1. The probability of a charging station is equal to The probability of entering an intersection and leaving an intersection should be the same. Meanwhile, the user... You can only go to a specific set of destinations, that is Because the chargers there are compatible with their electric vehicles. Therefore, users... Need to meet In addition, users The sum of the probabilities of going to all charging stations needs to satisfy... .

[0128] Considering the maximum capacity constraints of roads and charging stations, the global coupling constraints of the joint path and destination planning problem can be modeled as follows:

[0129]

[0130] in Indicate destination The largest energy supply, Indicates road Maximum traffic flow.

[0131] As a preferred technical solution of the present invention, the game theory model is transformed into a variational inequality problem and the relationship between the solution of the variational inequality problem and the solution of the game theory model is explained, specifically including:

[0132] The game theory model described above can be summarized into another general game theory model as follows:

[0133]

[0134] in The constraints of the general game model are as follows: ,in The feasible set is defined as follows:

[0135]

[0136] in Indicates user Local constraints that need to be satisfied Define the global coupling constraints that all users must satisfy; , as well as The general game theory model can be written as:

[0137]

[0138] Furthermore, the Karush-Kuhn-Tucker conditions for the general game theory model need to satisfy:

[0139]

[0140] The pseudo-gradient mapping of the general game theory model is represented as:

[0141]

[0142] The variational inequality problem can then be expressed as:

[0143]

[0144] in The Karush-Kuhn-Tucker conditions for variational inequality problems must satisfy:

[0145]

[0146] By observing the Karush-Kuhn-Tucker conditions of general game theory models and variational inequality problems, it is inferred that when , In this case, any solution to the variational inequality problem is a solution to the game theory model.

[0147] As a preferred embodiment of the present invention, the distributed solution algorithm under complete information specifically includes:

[0148] Distributed solution algorithms under complete information

[0149] definition and for each edge Introducing operators , making ,in It is an operator that achieves consistency in local Lagrange multipliers, defined as:

[0150]

[0151] set Defined as Therefore, we can obtain when Sometimes, , .

[0152] definition As The dual variables, where .make , , as well as Therefore, the Karush-Kuhn-Tucker (KKT) conditions are as follows:

[0153]

[0154] Based on the KKT conditions, using fixed-point iteration and applying the proximal gradient operator, the intensive form of the algorithm is:

[0155]

[0156] in , , The three heterogeneous step size matrices in the above formula are represented as follows:

[0157]

[0158] Decomposing the centralized algorithm, we can obtain the distributed form of the distributed solution algorithm under complete information as follows (EDPDPG-FI):

[0159]

[0160] As a preferred embodiment of the present invention, the distributed solution algorithm based on the partial information specifically includes:

[0161] Distributed solution algorithm with partial information

[0162] Considering a more practical application, where some users cannot know the plans of all other users, this invention introduces an estimate of the plans of other users for each user: Accordingly, define Therefore, users The local objective function can be written as All users' local estimates need to reach a consensus in steady state, i.e. .

[0163] Similar to EDPDPG-FI, for each edge Operators were introduced Make

[0164]

[0165] in It is an operator that achieves local estimation consistency, and it is defined as follows:

[0166]

[0167] Then the set Defined as It can be observed that when Sometimes, .

[0168] definition As The dual variables, where .make as well as .

[0169] Two selection matrices were introduced. Used for filtering In Defined as:

[0170]

[0171] in as well as .

[0172] Used for filtering In Defined as:

[0173]

[0174] Therefore, we have

[0175]

[0176] make , , as well as You can get Meanwhile, the pseudo-gradient mapping under partial information should be:

[0177]

[0178] Therefore, the KKT conditions are as follows:

[0179]

[0180] The centralized form of the algorithm is:

[0181]

[0182] , , And three heterogeneous step size matrices:

[0183]

[0184] Decomposing the centralized algorithm, we can obtain the distributed form of the algorithm as follows (EDPDPG-PI):

[0185] .

[0186] Example 1

[0187] consider Figure 1 The traffic network shown has 8 intersections, 4 charging stations, and 27 roads. It is assumed that there are an average of 16 users distributed throughout the area.

[0188] The experimental parameters are as follows. We... Set parameters for all users and Let the adjustment parameter in the travel time function be... In the absence of congestion, the time to travel through each road is... ,in It is the length of each road, and the maximum speed in this area is... Assume the capacity of all roads is... Its maximum capacity The non-electric vehicle traffic flow on each road is randomly selected from (10, 30) (unit: Weighting factors Uniformly distributed between (20, 70) (unit: User energy demand The values ​​are evenly distributed between (30, 70) (unit: kWh). The parameter settings for the charging station are shown in Table 1.

[0189]

[0190] Table 1. Parameter Description

[0191] Figure 3 and Figure 4 The effects of different weighting factors of the travel time function on user actions are shown under both full and partial information settings. Users 7 and 15 both start near intersection 4 and cannot reach charging station 4. User 7's parameters are... User 15's parameters are Among them, in Figure 3 and Figure 4 In the above, except for D1-user7, D2-user7, D3-user7, D2-user15, and D3-user15, the remaining D1-user15, D4-user7, and D4-user15 almost coincide with the horizontal axis.

[0192] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm, including... There are N electric vehicles = {1, ..., n}, where "electric vehicle" also refers to the user. road as well as One charging station A charging station refers to a parking lot equipped with charging stations, or it can also refer to a user's destination. Its characteristics are... Includes the following steps: Model the planning problem of joint paths and destinations, and establish the objective function for each user. Based on the maximum traffic load of the road and the maximum power supply load of the charging station, the global coupling constraints of the planning problem of the joint path and destination are obtained. At the same time, according to probability, each user also needs to satisfy its own local constraints. The above game theory model is transformed into a variational inequality problem. The solution to this variational inequality problem has an economic interpretation without price discrimination, and simultaneously, when all local Lagrange multipliers... When consensus is reached in steady state, the solution to the variational inequality problem is also the solution to the original game model; against To address consistency issues, an edge-based consistency constraint is introduced, and a distributed solution algorithm under complete information is designed based on fixed-point iteration and proximal gradient operator theory. Furthermore, a global estimate of the plans of other users is introduced. This paper proposes a distributed solution algorithm under partial information, transforming the game model into a variational inequality problem and explaining the relationship between the solution of the variational inequality problem and the solution of the game model. Specifically, it includes: The game theory model described above can be summarized into another general game theory model as follows: in The constraints of the general game model are as follows: ,in The feasible set is defined as follows: in Indicates user Local constraints that need to be satisfied Define the global coupling constraints that all users must satisfy; , as well as The general game theory model can be written as: Furthermore, the Karush-Kuhn-Tucker conditions for the general game theory model need to satisfy: The pseudo-gradient mapping of the general game theory model is represented as: The variational inequality problem can then be expressed as: in The Karush-Kuhn-Tucker conditions for variational inequality problems must satisfy: By observing the Karush-Kuhn-Tucker conditions of general game theory models and variational inequality problems, it is inferred that when , In this case, any solution to the variational inequality problem is a solution to the game theory model.

2. The solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm as described in claim 1, characterized in that, The aforementioned problem of planning joint routes and destinations specifically includes: Each user Decide on your own plans: ,in, , indicating user The probability of choosing each path , indicating user The probability of choosing each charging station; Each user The objective function is: in Indicates deviation from user The associated costs of habitual choices User Estimated travel time User Expected service costs It is a weighting factor representing a time term; each user The three parts of the objective function are defined as follows: (1) Deviating from the user The associated costs of habitual choices in and Representing users respectively Based on its past experience, the probability of choosing the preferred destination and route. and These represent the weighting factors for these two preferences, respectively. (2) Travel time in It indicates a road The strictly monotonically increasing function of the boarding traffic flow is defined as: in Indicates road Travel time under uncongested conditions With roads The length is related to the speed limit. Indicates road Traffic capacity, , representing the travel time function Adjustment parameters, Indicates road The projected traffic volume is defined as: in This indicates the traffic flow of non-charging vehicles; (3) Service Costs in Indicates user energy demand, This indicates the parking fee. The price function of energy is defined as: in It is a price coefficient. It is a charging station The charging capacity, It is a charging station Total energy demand is projected. User It is a charging station Energy demand; To ensure that the game model has GNE, when the user From the starting point Depart and reach a feasible destination. When, its probability is ,user The following constraints must be met: The above constraints can be understood as user Leave the starting point The probability of reaching the 1st is equal to 1. The probability of a charging station is equal to The probability of entering an intersection and leaving an intersection should be the same; at the same time, the user You can only go to a specific set of destinations, that is Because the chargers there are compatible with their electric vehicles; therefore, users Need to meet In addition, users The sum of the probabilities of going to all charging stations needs to satisfy... ; Considering the maximum capacity constraints of roads and charging stations, the global coupling constraints of the joint path and destination planning problem can be modeled as follows: in Indicate destination The largest energy supply, Indicates road Maximum traffic flow.

3. The solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm as described in claim 1, characterized in that, The distributed solution algorithm under complete information specifically includes: definition and for each edge Introducing operators , making ,in It is an operator that achieves consistency in local Lagrange multipliers, defined as: set Defined as Therefore, we can obtain the current Sometimes, , ; definition As The dual variables, where ,make , , as well as Therefore, the Karush-Kuhn-Tucker conditions are as follows: Based on the Karush-Kuhn-Tucker conditions, using fixed-point iteration and applying the proximal gradient operator, the intensive form of the algorithm is: in , , The three heterogeneous step size matrices in the above formula are represented as follows: Decomposing the centralized algorithm, we can obtain the distributed form of the distributed solution algorithm under complete information as follows: 。 4. The solution to the joint path and destination planning problem based on a distributed generalized Nash equilibrium algorithm as described in claim 1, characterized in that, The distributed solution algorithm based on the aforementioned partial information specifically includes: Considering a more practical application, where some users cannot know the plans of all other users, an estimate of the plans of other users is introduced for each user: Accordingly, define Therefore, users The local objective function can be written as All users' local estimates need to reach a consensus in steady state, that is ; For each edge Introducing operators Make in It is an operator that achieves local estimation consistency, and it is defined as follows: Then the set Defined as It can be observed that when Sometimes, ; definition As The dual variables, where ;make as well as ; Introducing two selection matrices and , Used for filtering In Defined as: in as well as ; Used for filtering In Defined as: Therefore, there is make , , as well as You can get Meanwhile, the pseudo-gradient mapping under partial information is: Therefore, the Karush-Kuhn-Tucker conditions are as follows: The centralized form of the algorithm is: , , The three heterogeneous step size matrices in the above formula are represented as follows: Decomposing the centralized algorithm, we can obtain the distributed form of the distributed solution algorithm with partial information as follows: 。

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