Charging strategy planning method based on distributed proximal gradient algorithm

Abstract

The application provides a charging strategy planning method based on a distributed proximal gradient algorithm, abstracts an electric vehicle charging strategy planning problem as a generalized Nash equilibrium (GNE) problem based on a non-directional connected graph, constructs an optimization problem of the generalized Nash equilibrium problem in combination with an objective function of each intelligent agent (electric vehicle) and constraint conditions of the intelligent agent, converts the optimization problem into a dual problem in combination with a Lagrange multiplier method, obtains KKT conditions of the dual problem, establishes a connection between the KKT conditions and a variation problem, and solves the original optimization problem by solving the variation problem. An edge-based communication model is adopted, and a heterogeneous step length is adopted to avoid conservatism, so that the optimization problem can converge faster and more accurately, and the electric vehicle charging strategy planning problem can be solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of distributed optimization, in particular to a charging strategy planning method based on a distributed proximal gradient algorithm. BACKGROUND

[0002] The research on generalized Nash equilibrium (GNE) can be traced back to the social equilibrium problem. In recent years, more and more research results based on generalized Nash equilibrium (GNE) of large-scale multi-agent network have been proposed. Due to the possible application of generalized Nash equilibrium (GNE) problem in economics, engineering and operations research, etc., this problem has become a research hotspot. It has also been widely studied in the fields of smart grid, mobile ad hoc network, social network, resource allocation, etc.

[0003] In recent years, the electric vehicle field has developed rapidly. The large-scale electric vehicles (agents) accessing the power grid have caused not small load to the power grid, which has affected the safe and stable operation of the power grid. Therefore, how to reasonably plan the charging strategy of electric vehicles to reduce the load on the power grid is a problem that needs to be faced at present.

[0004] The present application constructs the electric vehicle (agent) charging strategy planning problem as a generalized Nash equilibrium problem, and designs a distributed primal-dual proximal gradient algorithm to solve the problem. In the above generalized Nash equilibrium problem, the objective of each agent (charging vehicle) is to minimize its private objective function under the feasible constraints including coupling constraints and local constraints. Therefore, it should be pointed out that the private objective function and constraints of a certain agent are affected by the decision information of other agents. Here, a generalized Nash equilibrium problem model composed of m agents can be briefly described as:

[0005]

[0006] where x i represents the decision information of agent i, and x -i represents the decision information of other agents except agent i. SUMMARY

[0007] In view of the deficiencies in the prior art, the present application provides a charging strategy planning method based on a distributed proximal gradient algorithm to solve the problem of unreasonable electric vehicle charging strategy planning in the related art.

[0008] The symbol definition involved in the present application; all vectors in this patent are column vectors by default. n and R n represent n-dimensional real Euclidean space and n-dimensional non-negative real Euclidean space, respectively.

[0009] For matrix A∈R m×n, A T denotes its transpose. For a positive definite matrix G and two vectors x, y, <x, y> G = <Gx, y> and ||·|| G denote the G-induced inner product and G-norm, respectively.

[0010] denotes the block diagonal matrix with A1,..., A m as the main diagonal elements; x = col(x1,..., x m ) denotes the stacked vector of x1,..., x m .

[0011] For a proper closed convex function f, its subdifferential is denoted as:

[0012]

[0013] The proximal operator is defined as The conjugate function is denoted as:

[0014]

[0015] For a non-empty closed convex set C ∈ R n , proj C denotes the projection operator onto C,

[0016] proj C (x) = arg min y∈C ||x-y|| 2 .

[0017] The indicator function δ C is defined as:

[0018] N Ω (x) = {y ∈ R n | y T (v-x) ≤ 0, for all y ∈ Ω} denotes the cone of x on Ω. θ max (·) denotes the maximum eigenvalue of a matrix, and 0 denotes a zero vector or matrix of arbitrary dimension.

[0019] For an operator T and any constant α > 0, if T satisfies <Tx-Ty, x-y> ≤ α||Tx-Ty||, (where D is a non-empty set), then T is α-cocoercive.

[0020] The application provides a charging strategy planning method based on a distributed proximal gradient algorithm, comprising the following steps:

[0021] S1, abstract the problem of the charging strategy planning method as a generalized Nash equilibrium problem based on an undirected connected graph in a general case, and construct a target function and constraint condition of each agent as an optimization problem of the generalized Nash equilibrium problem;

[0022] S2, convert the optimization problem into a dual problem by combining a Lagrange multiplier method, and obtain a KKT condition of the dual problem;

[0023] S3, establish a connection between the KKT condition and a variation problem, and solve the original optimization problem by solving the variation problem;

[0024] S4, design a distributed original dual proximal gradient algorithm based on the KKT condition under a complete information condition in which each agent can know decision information of all other agents or a partial information condition in which each agent cannot know decision information of other agents, so as to obtain an optimal solution of the optimization problem.

[0025] Optionally, the step of abstracting the problem of the charging strategy planning method as a generalized Nash equilibrium problem based on an undirected connected graph in a general case, and constructing a target function and constraint condition of each agent as an optimization problem of the generalized Nash equilibrium problem comprises the following steps.

[0026] Suppose that m electric vehicles come to a charging station to charge in a time period T, P i，t is a total available power of the EVCS, and then the following needs to be met:

[0027]

[0028] The electric vehicle battery power model can be expressed as:

[0029]

[0030] where E i，t is a battery power of the agent at time t, Δt is a time step, χ c is a charging efficiency, C i is a maximum capacity of the battery. Each electric vehicle arrives at the charging station with a power E i，a at time a and leaves the charging station with a power E i，d at time b, and therefore the required power of the agent is expressed as:

[0031]

[0032] In addition, the following conditions should be met during charging:

[0033] E i，a ≤ E i，t ≤ E i，d

[0034] Each electric vehicle is regarded as an agent in the GNE model, and each electric vehicle decides its charging strategy to minimize its cost function under the above constraints:

[0035]

[0036]

[0037]

[0038]

[0039] E i，a ≤E i，t ≤E i，d

[0040] where β t is the electricity price at time t, and α t is the electricity price influence factor at time t. For a certain vehicle agent i, and are the upper and lower bounds of the charging power of the vehicle.

[0041] Optionally, the optimization problem is converted into a dual problem by combining the Lagrange multiplier method and the penalty function, and KKT conditions of the dual problem are obtained, including:

[0042] The objective function and the constraint condition of each agent are further expressed as:

[0043]

[0044] The set Ω i is a non-empty closed convex set, and there exists y∈R n such that y∈int(D i );

[0045] In addition, for a given x -i , the objective function f i (x i , x -i ) is convex and continuously differentiable at x i ;

[0046] The KKT conditions of the dual problem are obtained as follows:

[0047]

[0048] For each agent i, if and only if there exists a multiplier such that the KKT conditions of the dual problem are established, then for a given is the optimal solution of the constraints of the agent.

[0049] Optionally, the original optimization problem is solved by solving the variational problem by establishing a connection between the KKT conditions and the variational problem, comprising:

[0050] The gradient of the agent i The pseudo-gradient is defined as follows:

[0051]

[0052] Consider the variational inequality:

[0053]

[0054] Similarly, if and only if there exists a global multiplier The KKT conditions of the variational problem are obtained,

[0055]

[0056] By comparing the KKT conditions of the original optimization problem and the variational problem, it can be concluded that the optimal solution of the variational problem is also the optimal solution of the original optimization problem.

[0057] Based on the KKT conditions (7), we design distributed primal-dual proximal gradient algorithms under full information and partial information, respectively.

[0058] In order to solve (7) in a distributed manner, we introduce local variables λ i to estimate the global shared variable λ g . For each edge (i, j), define the following operator:

[0059]

[0060] In this way, the local variables λ i and λ j can be constrained to be;

[0061]

[0062] Further, define the set C (i，j) = {(z1, z2) ∈ R c × R c | z1 + z2 = 0},

[0063] The linear operator N (i，j) : λ→(E ij λ i , E ji λ j ) (i，j)∈ε .

[0064] Thus, condition (7) can be re-expressed as

[0065]

[0066] where λ = col(λ1,..., λ m ), z = col(z (i，j)∈ε ).

[0067] Optionally, a distributed algorithm under full information (Algorithm 1 D-PDPG-FI) is shown as follows:

[0068] Initialization: Given arbitrary a proper positive step size τ i , σ i , ω (i，j) °

[0069] Iteration: Compute as follows:

[0070] Loop k = 0, 1, 2...

[0071] Loop i = 1 to m

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078] Loop end set k→k+1, continue loop until a pre-set condition is reached (e.g., maximum iteration number).

[0079] Loop end

[0080] Final result returned:

[0081] Distributed algorithm under partial information:

[0082] Note that in the computation of the gradient each agent i needs to access the decision information of all other agents. To ensure privacy, this section aims to develop another distributed algorithm for the case of partial information.

[0083] Consider that each agent cannot obtain the true decision information. Assume that each agent i has a noisy observation of the decision information of all other agents. That is, *Local estimation of

[0084]

[0085] Let be the estimation of all other agents' decision information.

[0086] For example: is the estimation of agent i on agent j's decision information, in this case, each agent i only uses its estimation to replace the real decision information (x i , x -i ) to calculate the gradient. Therefore, the consistency x (i) = x (j) needs to be satisfied. The design process is similar to Algorithm 1. Here define the matrix:

[0087]

[0088] Linear mapping:

[0089] Therefore, there are R i x (i) = x i , S i x (i) = x -i ,

[0090]

[0091] Let Define a new pseudo-gradient:

[0092]

[0093] Then, the optimal condition in the case of partial information in compact form can be described as:

[0094]

[0095] Then, a fully distributed algorithm in the case of partial information is described in Algorithm 2 (D-PDPG-PI) below.

[0096] Initialization: Given any a proper positive step size τ i , σ i , ω(i, j), π(i, j).

[0097] Iteration: Calculate as follows:

[0098] Loop k = 0, 1, 2…

[0099] Loop i = 1 to m

[0100]

[0101]

[0102]

[0103]

[0104]

[0105]

[0106]

[0107]

[0108]

[0109] Loop end

[0110] Set k→k+1, continue the loop until the preset condition (for example: maximum iteration number) is reached.

[0111] Loop end, the final result is returned:

[0112] Compared with the prior art, the present application has the following beneficial effects:

[0113] The electric vehicle charging strategy planning problem is abstracted as a generalized Nash equilibrium (GNE) problem based on an undirected connected graph, an optimization problem of the generalized Nash equilibrium problem is constructed in combination with the objective function of each agent (electric vehicle) and the constraint condition thereof, the optimization problem is converted into a dual problem in combination with the Lagrange multiplier method, the KKT condition of the dual problem is obtained, the KKT condition is associated with a variation problem, and the original optimization problem is solved by solving the variation problem. An edge-based communication model is adopted, and a heterogeneous step length is adopted to avoid conservatism, so that the optimization problem can converge faster and more accurately, and the electric vehicle charging strategy planning problem can be solved. BRIEF DESCRIPTION OF DRAWINGS

[0114] The accompanying drawings, which are incorporated into and form a part of the specification, illustrate an embodiment consistent with the present application and, together with the description, serve to explain the principles of the application.

[0115] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or prior art description. Obviously, for those skilled in the art, other drawings can also be obtained based on these drawings without any creative effort.

[0116] Figure 1 A flowchart of the present application;

[0117] Figure 2 A parameter setting table in an embodiment of the present application;

[0118] Figure 3 A connection relationship diagram of the charging station and the electric vehicle in an embodiment of the present application;

[0119] Figure 4 A convergence display diagram of the algorithm in an embodiment of the present application;

[0120] Figure 5 A charging strategy of each vehicle in a charging period in an embodiment of the present application;

[0121] Figure 6 A total charging amount at each time in an embodiment of the present application;

[0122] Figure 7 A charging power iteration process of each vehicle at a specific time in an embodiment of the present application. DETAILED DESCRIPTION

[0123] In order to make the purpose, technical solutions and advantages of the embodiments of the present application more clear, the following will combine the drawings in the embodiments of the present application to clearly and completely describe the technical solutions in the embodiments of the present application. Obviously, the described embodiments are a part of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without any creative effort are within the protection scope of the present application. The function units with the same and similar structures and functions in the embodiments of the present application have the same and similar structures and functions.

[0124] Finding the optimal solution of the related variational inequality (VI) is an effective method. Considering the generalized Nash equilibrium of the variation, a generalized Nash equilibrium problem can be converted into VI, and the solution of VI is regarded as the solution of the original problem. The solution of VI involves two cases. One is the full information case, in which each agent can know the decision information of all other agents. The other is that each agent cannot know the decision information of other agents, but can only make a local estimation of the decision information of all other agents. This case is called partial information. According to the two cases, the optimal solution of the agent charging strategy planning problem is calculated.

[0125] See Figure 1 The present application provides an embodiment of an electric vehicle charging station (EVCS) with m agents arriving at the charging station to charge at time period T, the present application provides a method of determining the charging strategy of each agent to minimize the cost function of each agent under the constraints of the charging station and the agents. i，t The total available power of the charging station EVCS is Then the following constraints need to be satisfied:

[0126]

[0127] The battery power model of the agent can be represented as:

[0128]

[0129] Where E i，t is the battery power of the agent at time t, Δt is the time step, χ c is the charging efficiency, C i is the maximum capacity of the battery. Each agent arrives at the charging station with battery power E i，a at time a and leaves the charging station with battery power E i，d at time b, therefore the required battery power of each agent is:

[0130]

[0131] In addition, the following constraints should be maintained during charging:

[0132] E i，a ≤ E i，t ≤ E i，d

[0133] Each agent is considered as an agent in a generalized Nash equilibrium model, under the above constraints, each agent determines its own charging strategy to minimize its cost function, the charging strategy is as follows:

[0134]

[0135]

[0136]

[0137]

[0138] E i，a ≤ E i，t ≤ E i，d

[0139] Where β t is the electricity price at time t, α tis the electricity price influence factor at time t. For a certain vehicle agent, and are the upper and lower bounds of the charging power of the vehicle.

[0140] The constraint condition in the above formula is processed:

[0141]

[0142] Constraint condition E i，a ≤ E i，t ≤ E i，d is expressed as:

[0143]

[0144] Therefore, the problem model is re-expressed as:

[0145]

[0146]

[0147] p i，t ∈ Ω i，t

[0148]

[0149] -p i Δt ≤ 0

[0150]

[0151] Regarding the constraint -Δtp i ≤ 0, that is, p i，t ≤ 0, in the following constraint condition, this constraint condition is combined into the coupling constraint for consideration.

[0152] The pseudo-gradient of the objective function is:

[0153] F(p) = col(F1(p), …, F m (p))

[0154] F i (p) = α t Δt 2 p i + β t Δt

[0155] After considering the above electric vehicle charging strategy planning as a general Nash equilibrium problem, in order to solve the generalized Nash equilibrium, the invention distributes the original dual proximal gradient algorithm, one is the case of complete information, and the other is the case of partial information.

[0156] Distributed algorithm under complete information:

[0157] To enable distributed solution, a local variable λ is introduced. i To estimate the globally shared variable λ g .

[0158] For each edge (i, j), define the following operator:

[0159]

[0160] Thus, the local variable λ i and λ j It can then be constrained as:

[0161]

[0162] Further, define set C (i，j) ={(z1, z2)∈R c ×R c |z1+z2=0}, linear operator N (i.j) :λ→(E ij λ i E ji λ j ) (i，j)∈ε .

[0163] Therefore, the KKT conditions for the variational problem can be reformulated as:

[0164]

[0165] Where.λ=col(λ1,...,λ m ), z = col(z) (i，j)∈ε ).

[0166] Note 1: As can be seen from equation (6), the KKT conditions for the variational problem can be initially expressed as:

[0167]

[0168] From this, we can deduce that:

[0169]

[0170] in It is based on the edge variable z of agents i and j. (i，j)，i and z (i，j)，j A set that consists of.

[0171] Using the proximal gradient method, the compact recurrence relation for solving (7) is obtained as follows:

[0172]

[0173] where Λ is the relaxation matrix, Γ,∑ and W are three step block diagonal matrices composed of local fixed steps, which are specified as:

[0174]

[0175]

[0176] W = diag{ω (i，j) I} (i，j)∈ε

[0177]

[0178] By Moreau decomposition, we can get the local recursion about and , as follows:

[0179]

[0180] Specifically, the following Algorithm 1 (D-PDPG-FI) shows a distributed algorithm under full information:

[0181] Initialization: Given any proper positive step size τ i , σ i , ω (i，j) .

[0182] Iteration: Compute as follows:

[0183] Loop k = 0, 1, 2...

[0184] Loop i = 1 to m

[0185]

[0186]

[0187]

[0188]

[0189]

[0190]

[0191] Loop end

[0192] Set k→k+1, continue the loop until the preset condition (e.g., the maximum number of iterations) is reached.

[0193] Loop end

[0194] Final result returns:

[0195] Distributed algorithm under partial information:

[0196] Note that in computing the gradient , each agent i needs to access the decision information of all other agents. To guarantee privacy, this section aims to develop another distributed algorithm for the partial information case.

[0197] Consider that each agent cannot obtain the true decision information. Assume that each agent i has a local estimate of x * :

[0198]

[0199] Let be the estimate of the decision information of all other agents.

[0200] For example: is the estimate of agent i on the decision information of agent j, in which case each agent i only uses its estimate instead of the true decision information (x i , x -i ) to compute the gradient. Therefore, consistency x (i) = x (j) needs to be satisfied. The design process is similar to Algorithm 1. Here, define the matrix:

[0201]

[0202] Linear mapping Therefore, there are R i x (i) = x i , S i x (i) = x -i ,

[0203]

[0204] Let define a new pseudo-gradient:

[0205]

[0206] Then, the optimal condition under partial information in compact form can be described as:

[0207]

[0208] Note 2: According to E ij x(i) +E ji x (j) =0 can be deduced

[0209]

[0210] It consists of edge variables i and j. The set is composed of. Then, using the proximal gradient method from (9), the compact recurrence relation for solving (9) is obtained as follows:

[0211]

[0212] in It is based on a fixed step size π (i，j) It is a block diagonal step size matrix with main diagonal elements.

[0213] The first two lines of equation (10) are from It was split off.

[0214] Then, Algorithm 2 (D-PDPG-PI) below describes a fully distributed algorithm for the partial information case.

[0215] Initialization: Given any Appropriate positive step size τ i , σ i ω (i，j) , π (i，j) .

[0216] Iteration: computation as follows:

[0217] Cycle k = 0, 1, 2...

[0218] Loop i = 1 to m

[0219]

[0220]

[0221]

[0222]

[0223]

[0224]

[0225]

[0226]

[0227]

[0228] Loop end

[0229] Set k→k+1, continue the loop until the preset condition (for example: maximum iteration number) is reached.

[0230] Loop end

[0231] The final result is returned:

[0232] According to the foregoing, it can be seen that the application abstracts the electric vehicle charging strategy planning problem into a generalized Nash equilibrium (GNE) problem based on an undirected connected graph, constructs an optimization problem of the distributed generalized Nash equilibrium problem in combination with the objective function of each agent (electric vehicle) and the constraint condition thereof, converts the optimization problem into a dual problem in combination with the Lagrange multiplier method and the penalty function, obtains the KKT condition of the dual problem, establishes a connection through the KKT condition and the variation problem, and solves the original optimization problem by solving the variation problem. Distributed primal-dual proximal gradient algorithms under complete information and partial information are respectively invented, both of which adopt a communication model based on edges, adopt heterogeneous steps to avoid conservatism, and can make the optimization problem converge faster and more accurately, so as to solve the electric vehicle charging strategy planning problem. Meanwhile, the application is not limited to planning of electric vehicle charging strategies, but is applicable to agents that also need to be charged.

[0233] Referring to Figure 3 , further, it is assumed that:

[0234] T=24h, Δt=1h, χ c =0.8, C i =199kw, β t =0.8$ / kw, other parameters of the model are shown in Figure 2 .

[0235] In algorithm 1, τ i =0.08, σ i =0.003, ω (i，j) =0.08,

[0236] In algorithm 2, τ i =0.08, σ i =0.003, ω (i，j) =0.08, π (i，j) =0.05.

[0237] Randomly select in the local feasible set and set the initial value of to 0. Convergence is shown in Figure 4 , and the charging strategy of each vehicle is shown in Figure 5As shown, the total charging amount at each time is as follows Figure 6 As shown, the charging power iteration process for each vehicle at a certain time is as follows Figure 7 As shown.

[0238] It should be noted that, in the present document, relational terms such as "first" and "second", and the like can be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Moreover, the terms "comprises", "comprising", or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but can include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element proceeded by "comprises... a" does not, without more constraints, exclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element. The above specification, examples and data are illustrative and not limiting. Changes in or substitutions of the examples can be made without departing from the scope of the application, which is defined by the appended claims.

Claims

1. A charging strategy planning method based on a distributed proximal gradient algorithm, characterized in that, The method comprises the following steps: S1, abstracting the problem of the charging strategy planning as a generalized Nash equilibrium problem based on an undirected connected graph, constructing a target function and constraint conditions of each agent as an optimization problem of the generalized Nash equilibrium problem, comprising: Set in time period There are A vehicle electric vehicle to come to the charging station charging, P i，t The charging power of the intelligent agent at time t, the total available power of EVCS is Then need to meet: ， The electric vehicle battery power model is: ， where E i，t is the battery level of the agent at time t, Δt is the time step, χ c is the charging efficiency, C i is the maximum capacity of the battery, each electric vehicle arrives at the charging station at time a with battery level and leaves the charging station at time b with battery level Therefore, the demand battery level of the agent is represented as: ， In addition, the following conditions should be met during charging: E i，a ≤E i，t ≤E i，d , Each electric vehicle is regarded as an agent in the GNE model, and each electric vehicle decides its charging strategy under the above constraint conditions to minimize its cost function: ， where β t is the electricity price at time t, α t is the electricity price influence factor at time t, for a certain vehicle agent, and are the upper and lower bounds of the charging power of the vehicle; S2, converting the optimization problem into a dual problem by combining the Lagrange multiplier method, obtaining the KKT conditions of the dual problem, comprising: The target function and constraint conditions of each agent are further expressed as: （1） set is a nonempty closed convex set, there exists such that ; Furthermore, for a given , the objective function is convex and continuously differentiable at ; The KKT conditions of the dual problem are obtained as: For each agent iff there exists a multiplier such that the KKT conditions for the dual problem hold, then for a given , is the optimal solution to the agent's objective function; S3, establishing a connection between the KKT conditions and the variational problem, and solving the original optimization problem by solving the variational problem; S4, based on the KKT conditions, designing a distributed primal-dual proximal gradient algorithm under the condition that each agent can know the decision information of all other agents (complete information) or each agent cannot know the decision information of other agents (partial information), so as to obtain the optimal solution of the optimization problem.

2. The method of claim 1, wherein, The connection between the KKT conditions and the variational problem comprises: For the agent of the gradient , define the following pseudo-gradient: （3） Consider the variational inequality: （4） Also, there exists a global multiplier which gives the KKT conditions of the variational inequality, By comparing the KKT conditions of the original optimization problem and the variational problem, it can be concluded that the optimal solution of the variational problem is also the optimal solution of the original optimization problem.

3. The method of claim 2, wherein, The KKT conditions of the dual problem are obtained as: Introduce local variable λ i to estimate global shared variable ; for each edge , define the following operator: ， In this way, the local variable λ i and λ j can be constrained as follows: （6） Further, a set of definitions , Linear operator ; Therefore, the KKT conditions of the variational problem can be represented as: where λ = col(λ1,..., λ m ), z = col(z (i，j)∈ε ); Initialization: Given arbitrary , a proper positive step size τ i , σ i , ω (i，j) ; Iteration: Calculation As follows: Circulating , Circulating to , ， End of cycle setting the cycle continues until a preset condition is reached; Loop end The final result returns: .

4. The method of claim 2, wherein, The KKT conditions of the dual problem are obtained as: Each agent has a corresponding local estimate: ， Let be the estimate of all other agent decision information; Definition matrix: , Linear mapping , ， ； Let , Define a new pseudo-gradient: (8) Then, the optimal condition in the compact form of the partial information condition can be described as: Initialization: Given any , the appropriate positive step size ; Iteration: Calculation As follows: Circulating , Circulating to , ， Loop end; Setting , the loop continues until a preset condition is reached; The loop ends and the final result is returned: .

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