Optimization method and apparatus for a distributed optimization system based on a predefined time

Abstract

The application discloses an optimization method and device of a distributed optimization system based on a predefined time, and relates to the technical field of mathematics and automatic control. The method comprises the following steps: constructing a generalized time benchmark based on the convergence of the predefined time; constructing corresponding distributed predefined time multi-agent systems based on the goal of the distributed optimization problem of resource allocation and consensus; weakening the optimization conditions through the two kinds of multi-agent systems so that the balance point is converged in the case of a non-convex, strictly convex, strongly convex and convex cost function, and the balance point is converged near the equilibrium point in the case of generalized smoothness; associating the balance point with the optimal solution by using the KKT condition, and making the two kinds of multi-agent systems converge to an optimal solution or near the optimal solution under the corresponding conditions by using the related convexity and smoothness; compared with the existing method, the condition of the strongly convex cost function is no longer necessary, and the practical applicability and flexibility are improved.

Description

Technical Field

[0001] This invention relates to the fields of mathematics and automation control technology, and specifically to an optimization method and apparatus for a distributed optimization system based on a predefined time. Background Technology

[0002] In recent years, the rapid development of large-scale systems and networks has sparked great interest in distributed optimization problems. Distributed optimization methods are effective optimization techniques, characterized by reduced communication overhead, strong fault tolerance, operational security, and strong privacy. Therefore, distributed optimization has found applications in various practical problems, such as distributed resource allocation problems, multi-objective distributed optimization problems, and economic scheduling problems.

[0003] Resource allocation-based distributed optimization and consensus-based distributed optimization are two common problems in the field of optimization. Resource allocation-based distributed optimization is a key optimization problem in various applications, such as economic dispatch, power systems, and communication networks. To solve this type of problem, numerous methods have been introduced, such as genetic algorithms, improved particle swarm optimization, and Pareto front methods. With the scaling of networks, fully distributed allocation algorithms have become a very advantageous alternative, providing a resilient, scalable, and privacy-preserving approach to addressing these pressing problems. Consensus-based distributed optimization has also received widespread attention in recent years. Historically, consensus-based distributed optimization has been applied to various environments, such as networked vehicle / UAV coordination / control, distributed estimation and learning, and information processing in sensor networks. Some examples include multi-objective distributed optimization, distributed averaging, and power system control. For instance, considering the inherent intermittency and uncertainty of renewable resources in power systems, a fast economic dispatch algorithm must be designed to ensure efficient and reliable operation.

[0004] In recent years, predefined time-distributed optimization methods have been widely introduced. The settling time in these methods depends on a predefined time convergence method and can be set with arbitrary precision. Strong convexity is an indispensable property in predefined time-distributed methods, but in practical applications, many cost functions often appear in non-convex or even non-convex forms. However, current proposed predefined time-distributed methods only consider cost functions that satisfy the strong convexity property, while many other cost functions tend to appear in non-convex or even non-convex forms. This reduces the system's computational speed and lacks flexibility, failing to meet the needs of current practical applications. Summary of the Invention

[0005] To address the shortcomings of existing technologies that only consider cost functions satisfying strong convexity, which reduces system computation speed and lacks flexibility, thus failing to meet the needs of current practical applications, this invention proposes an optimization method and apparatus for a distributed optimization system based on predefined time. By using two multi-agent systems under a generalized time base generator, distributed optimization problems based on resource allocation and consensus can be solved within a predefined time, thereby resolving the problems existing in the prior art.

[0006] An optimization method for a distributed optimization system based on predefined time includes the following steps:

[0007] Construct a generalized time reference based on predefined time convergence;

[0008] The resource allocation and consensus problems in a distributed optimization system are optimized using a generalized time reference; the optimization process specifically includes the following steps:

[0009] Based on the distributed optimization problem of resource allocation and consensus, corresponding distributed predefined time multi-agent systems are constructed according to the generalized time reference. The optimization conditions are weakened by two types of multi-agent systems so that they converge to the equilibrium point when the cost function is non-convex, strictly convex, convex and strongly convex, and converge to the equilibrium point under the condition of generalized smoothness.

[0010] The equilibrium point is associated with the optimal solution using KKT conditions. By utilizing the correlation convexity and smoothness, the two multi-agent systems are predefined to converge to an optimal solution under the corresponding conditions.

[0011] Furthermore, the construction of the generalized time reference based on predefined time convergence specifically includes the following steps:

[0012] Assume a positive function μ α,D (t,τ) satisfies the following condition:

[0013] satisfy And μ α,D (t,τ) is increasing with respect to t, and decreasing with respect to α and D;

[0014] Satisfying α is large enough Less than a sufficiently small positive number ε;

[0015] Satisfy μ α,D (t,τ)=Dμ α (t,τ), μ(t,τ)μ(τ,s)=μ(t,s), Where μ(t,τ)=μ 1,1 (t,τ);

[0016] The predefined time convergence is defined as: system The correlation evolution operator Φ(t) satisfies the inequality:

[0017] ||Φ(t)||≤μ α,D (t,τ)

[0018] Where A(t) is a generalized time base generator, t represents time, α and D represent constants, and μ(t,τ) and μ(τ,s) are positive functions of the variables within parentheses. α (t,τ) represents the positive function raised to the power of α, and V(t) represents the Lyapunov function with respect to time t. The derivative of the Lyapunov function is represented.

[0019] Furthermore, the aforementioned construction of corresponding distributed predefined time-based multi-agent systems based on resource allocation and consensus as optimization objectives specifically includes:

[0020] The optimization based on resource allocation is expressed as:

[0021] minf(x)=f i (x i )

[0022]

[0023] The consensus-based optimization is represented as:

[0024] minf(x)=f i (x i )

[0025] stx i =x j

[0026] Where x i , q, q i ∈R n f i (x i ):R n →R, x = col(x1,x2,...,x) N ), x N It is the Nth variable, n, N, i, q i q are both positive numbers;

[0027] The multi-agent system constructed based on the distributed optimization objective of resource allocation is then represented as:

[0028]

[0029] in, It is the derivative of the variable, Q is a positive constant, and aij It is the connection weight between two multi-agent systems, z j It is an auxiliary variable;

[0030] A multi-agent system based on a consensus-driven distributed optimization objective is represented as follows:

[0031]

[0032] Among them, y i z i u i and w i All are internal auxiliary variables. Let κ represent the derivative of the corresponding function. For any κ, κA(t) is a generalized time base generator. It is an auxiliary variable.

[0033] Furthermore, the method of weakening the optimization conditions through two multi-agent systems, so that convergence to the equilibrium point is achieved regardless of whether the cost function is non-convex, strictly convex, convex, or strongly convex, specifically includes the following steps:

[0034] The strongly convex cost function is expressed as: A strictly convex cost function is expressed as f(θz1+(1-θ)z2)<θf(z1)+(1-θ)f(z2); a convex cost function is expressed as f(θz1+(1-θ)z2)≤θf(z1)+(1-θ)f(z2); a non-convex cost function does not satisfy any of the above conditions.

[0035] The multi-agent system built on a distributed optimization objective based on resource allocation is simplified as follows:

[0036]

[0037] Where y is an auxiliary variable. These represent matrix L multiplied by auxiliary variables y, z, u, and x, respectively.

[0038] Subtracting the equilibrium point system from this system yields a new system:

[0039]

[0040] in, Both are derivatives of the new variable, x * It is the equilibrium point;

[0041] The new system is transformed into two subsystems using orthogonal transformation:

[0042]

[0043] in, ξ1 and η1 are the derivatives of the new variables after the transformation, T is the transpose, and h is the subtraction of the two functions. It is the transpose of the orthogonal transformation matrix;

[0044] Constructing a new Lyapunov function:

[0045]

[0046] Where ρ, β, and γ are positive constants;

[0047] Differentiating the Lyapunov function:

[0048]

[0049] The variables are then decomposed using Young's inequalities, resulting in the following expression: Format:

[0050]

[0051] Where Λ1 is a constant composed of ρ, β, γ and σ;

[0052] Choosing a suitable generalized time base generator and corresponding coefficients makes it possible to It satisfies the predefined conditions for time convergence;

[0053] Through the new Lyapunov theorem, we obtain... If the system can converge to 0 in a predefined time, then the optimization conditions for the multi-agent system are weakened so that it converges to the equilibrium point regardless of whether the cost function is non-convex, strictly convex, convex, or strongly convex.

[0054] Furthermore, the predefined convergence to the equilibrium point under the condition of generalized smoothness specifically includes the following steps:

[0055] Define generalized smoothness: in, The numbers are positive constants; z1 and z2 are variables.

[0056] Choosing a suitable generalized time base generator and corresponding coefficients allows the derivative of the Lyapunov function to exhibit... Format:

[0057]

[0058] Where Λ2 and Λ3 are constants composed of positive constants ρ, β, γ and σ, and corresponding generalized time base generators and coefficients are selected to satisfy the predefined time convergence condition; V is a Lyapunov function;

[0059] The new Lyapunov theorem yields a predefined convergence to the equilibrium point under the condition of generalized smoothness.

[0060] Furthermore, the step of associating the equilibrium point with the optimal solution using KKT conditions, and leveraging the relevant convexity and smoothness to ensure that the two multi-agent systems converge to an optimal solution under predefined conditions, specifically includes the following steps:

[0061] The KKT conditions are used to associate the equilibrium point with the optimal solution. It is a KKT point if and only if:

[0062]

[0063] In a multi-agent system, x i It converges to the equilibrium point and also converges to the KKT point;

[0064] When the cost function is strictly convex or strongly convex, the KKT point is the unique optimal solution; when the cost function is convex or non-convex, the KKT point is a local optimal solution.

[0065] The present invention also includes an optimization device for a distributed optimization system based on a predefined time, comprising:

[0066] The generalized time reference construction module is used to build a generalized time reference based on predefined time convergence.

[0067] The optimization module is used to optimize resource allocation and consensus issues in a distributed optimization system using a generalized time base; the optimization module specifically includes:

[0068] The convergence unit is used to construct corresponding distributed predefined temporal multi-agent systems based on the generalized time reference, with the goal of a distributed optimization problem involving resource allocation and consensus. By weakening the optimization conditions through two types of multi-agent systems, the system converges to the equilibrium point under the conditions of non-convex, strictly convex, convex, and strongly convex cost functions, and converges to the equilibrium point under the condition of generalized smoothness.

[0069] The optimal solution solving unit is used to associate the equilibrium point with the optimal solution using KKT conditions, and utilizes the correlation convexity and smoothness to make the two multi-agent systems converge to an optimal solution under the corresponding conditions.

[0070] This invention provides an optimization method and apparatus for a distributed optimization system based on a predefined time, which has the following beneficial effects:

[0071] This invention proposes a generalized time base generator, which provides a predefined time closer to the optimal solution than a normal time base generator. New predefined-time multi-agent systems are designed for resource-allocation-based and consensus-based distributed optimization problems, respectively. The proposed predefined-time multi-agent systems converge to the optimal solution within the predefined time. Compared with existing methods, the condition of strong convexity of the cost function is no longer necessary. This invention can generalize the conclusion to strictly convex, convex, and even non-convex cases, where the Lipshitz condition of the derivative of the cost function is no longer necessary. Furthermore, a generalized Lipshitz condition is proposed, which not only retains the properties of the optimization algorithm under the original Lipshitz condition but also reduces computational complexity and improves computational speed in specific cases. Compared with centralized optimization methods, the proposed multi-agent system maintains the advantages of distributed optimization algorithms, effectively alleviates certain necessary conditions, and improves its practical applicability and flexibility. Attached Figure Description

[0072] Figure 1 This is a flowchart of the optimization method for a distributed optimization system based on a generalized time base generator, as described in this embodiment of the invention.

[0073] Figure 2 This is a schematic diagram showing the comparison results between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention;

[0074] Figure 3 This is a schematic diagram showing the comparison between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention;

[0075] Figure 4 This is a schematic diagram showing the comparison between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention;

[0076] Figure 5 This is a schematic diagram showing the comparison results between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention;

[0077] Figure 6 This is a schematic diagram illustrating the running results of the algorithm of the present invention in an embodiment of the present invention;

[0078] Figure 7 This is a schematic diagram showing the comparison results between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention;

[0079] Figure 8 This is a schematic diagram showing the comparison between the existing algorithm (a) and the algorithm (b) of this invention in an embodiment of the present invention. Detailed Implementation

[0080] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments.

[0081] This invention proposes an optimization method for distributed optimization systems based on predefined time. It generates two multi-agent systems under a generalized time base generator, which can solve distributed optimization problems based on resource allocation and consensus within a predefined time. The multi-agent approach improves security and privacy. New definitions of predefined time approximate convergence and predefined time optimization based on the generalized time base generator and generalized smoothness are given. A new Lyapunov theory for predefined time approximate convergence is proposed, and robustness and boundedness theorems for Lyapunov functions based on the generalized time base generator are given. Based on robustness and boundedness theorems, the proposed mathematical model is optimized in predefined time. Specifically, the method includes the following steps:

[0082] S1. Define a new, more widely applicable time base generator using predefined time convergence. The predefined time convergence is defined as:

[0083] system The correlation evolution operator Φ(t) satisfies the inequality: ||Φ(t)‖≤μ α,D (t,τ), where A(t) is the generalized time base generator.

[0084] Assume a positive function μ α,D (t,τ) satisfies:

[0085] (1) And μ α,D (t,τ) is increasing with respect to t, and decreasing with respect to α and D.

[0086] (2) α is large enough that It is less than a sufficiently small positive number ε.

[0087] (3) Satisfies some simple operations: μ α,D (t,τ)=Dμ α (t,τ), μ(T,τ)μ(τ,s)=μ(t,s),

[0088] Where μ(t,τ)=η 1,1 (t,τ).

[0089] Utilizing the properties of the generalized time base generator described above, it is found that as time approaches infinity, V(t) approaches 0, and as long as α is sufficiently large, V(t) can converge to near 0 within a set time. Therefore, this invention can develop the properties of the generalized time base generator: robustness and boundedness of solutions to related nonlinear systems.

[0090] S2. Develop the properties of the generalized time base generator and prove the robustness of the predefined time convergence:

[0091] If it is the system If the system satisfies the predefined time convergence, then the system Under certain conditions, it also satisfies the predefined time convergence property. The boundedness of the solution to the nonlinear system is also proven: if the system... If the system satisfies the predefined time convergence, then the system The solutions are bounded under certain conditions. Therefore, this invention defines a new Lyapunov convergence theorem.

[0092] Robustness of generalized time base generator: If it is the original system If a linear perturbation system satisfies the predefined time convergence property that ||B(t)||≤δ and δ≤α / D, then... It also satisfies the predefined time convergence. A relation is constructed using the form of the solution to the differential equation, satisfying the following: the correlation evolution operator of the linear perturbation system is expressed using the equation of the original system's relational evolution operator. Then, using the predefined time convergence and necessary conditions, this invention shows that the correlation evolution operator of the linear perturbation system also satisfies the predefined time convergence, meaning that A(t) + B(t) becomes a new generalized time base generator.

[0093] Boundedness of solutions to related nonlinear systems: If the original system It satisfies the predefined time convergence property that ||F(t,V)||≤F Max That is, if the nonlinear disturbance is bounded, then the nonlinear system... The solution is bounded. A relation is constructed using the form of the solution to the differential equation, satisfying the condition that the solution of the nonlinear system can be expressed using the equation of the relational evolution operator of the original system. Then, by utilizing predefined time convergence and necessary conditions, this invention can deduce that the solution of the nonlinear system is bounded.

[0094] A new Lyapunov convergence theorem is proposed:

[0095] (1) Original system If a linear perturbation system satisfies the predefined time convergence property that ||B(t)||≤δ and δ≤α / D, then... The solution V(t) will approach 0 as time approaches infinity, and by adjusting the coefficients, V(t) will converge to near 0 within a predefined time.

[0096] (2) Original system It satisfies the predefined time convergence property that ||F(t,V)||≤F Max That is, if the nonlinear disturbance is bounded, then the nonlinear system... The solution V(t) will approach 0 as time approaches infinity, and by adjusting the coefficients, V(t) will converge to near 0 within a predefined time.

[0097] Therefore, by utilizing the above two properties, multi-agent systems can be constructed to solve related optimization problems.

[0098] S3. Addressing resource allocation optimization issues separately:

[0099] minf(x)=f i (x i )

[0100]

[0101] And consensus-based optimization problems:

[0102] minf(x)=f i (x i )

[0103] stx i =x j

[0104] Design a novel predefined time-based multi-agent system.

[0105] To address the distributed optimization problem based on resource allocation, this invention proposes a novel multi-agent system:

[0106]

[0107] To address consensus-based distributed optimization problems, this invention proposes a novel multi-agent system:

[0108]

[0109] Where y i z i u i and w i These are all internal auxiliary variables. Let κ represent the derivative of the corresponding function. For any κ, κA(t) is a generalized time base generator. Next, we will use the generalized time base generator to construct a Lyapunov function and related properties to prove that x converges to the equilibrium point x. *.

[0110] S4. Prove that these two types of multi-agent systems effectively weaken the conditions of the optimization problem, making the cost function:

[0111] Strong convexity:

[0112] Strictly convex: f(θz1+(1-θ)z2)<θf(z1)+(1-θ)f(z2)

[0113] Convex: f(θz1+(1-θ)z2)≤θf(z1)+(1-θ)f(z2)

[0114] Both converge to the equilibrium point in the case of non-convexity (where none of the above conditions are met).

[0115] Take a predefined time-based multi-agent system for solving resource allocation optimization problems as an example:

[0116] Step 1: Simplify the predefined time-based multi-agent system that solves the resource allocation optimization problem into the following equivalent system:

[0117]

[0118] And by subtracting the equilibrium point system from this system, a new system is obtained:

[0119]

[0120] Therefore, all that is needed is to prove We can predefine the convergence time to 0. Then, we need to use orthogonal transformation to convert the new system into two subsystems:

[0121]

[0122] and

[0123]

[0124] Therefore, the discussion of x also turns into a discussion of ξ.

[0125] Step 2: Construct a new Lyapunov function:

[0126]

[0127] Where ρ, β, and γ are positive constants. Next, the derivative of the Lyapunov function is taken:

[0128]

[0129] The variables are then decomposed using Young's inequality, ultimately yielding the following result: Format:

[0130]

[0131] Where Λ1 is a constant composed of ρ, β, γ and σ.

[0132] Step 3: Select a suitable generalized time base generator and corresponding coefficients to make If the predefined time convergence condition is satisfied, then by the new Lyapunov theorem, we can obtain... It is possible to converge to 0 in a predefined time.

[0133] S5. Define generalized smoothness:

[0134] In certain special cases, it can effectively reduce the computational cost of multi-agent systems; the two types of multi-agent systems proposed can also converge to near the equilibrium point under the condition of generalized smoothness.

[0135] Choosing a suitable generalized time base generator and corresponding coefficients allows the derivative of the Lyapunov function to exhibit... Format:

[0136]

[0137] Where Λ2 and Λ3 are constants composed of ρ, β, γ, and σ, and appropriate generalized time base generators and coefficients are selected to satisfy the predefined time convergence condition, then by the new Lyapunov theorem, it can be concluded that the predefined time converges to near 0. Finally, it is only necessary to connect the KKT conditions with the equilibrium point.

[0138] S6. By using the KKT conditions to associate the equilibrium point with the optimal solution, and then by utilizing the relevant convexity and smoothness, the two types of multi-agent systems proposed are predefined to converge to an optimal solution or its vicinity under the corresponding conditions.

[0139] Using KKT conditions: if It is a KKT point if and only if:

[0140]

[0141] Therefore, x in a multi-agent system i Convergence to the equilibrium point also means convergence to the KKT point. In the case of a strictly convex and strongly convex cost function, the KKT point is the unique optimal solution. However, in the case of convex or non-convex functions, the KKT point is a local optimum. Therefore, the proposed multi-agent system will converge to an optimal solution or a vicinity of an optimal solution within a predefined time.

[0142] Finally, the same steps can be followed to obtain the final result for consensus-based distributed optimization problems.

[0143] This invention proposes an embodiment, such as... Figure 1 As shown, this embodiment applies this method to distributed optimization problems based on resource allocation and distributed optimization problems based on consensus. By designing a predefined time-based multi-agent system, the cost function is analyzed to obtain an optimal solution. This method mainly includes the following:

[0144] Step 1: An example of the cost function applied to strictly convex, convex, non-convex, and generalized smoothness is shown below, giving the graph connectivity information a. ij This represents the connection weight between two agents i and j.

[0145]

[0146] Step 2: The initial value of x is given by the instance and has no restrictions. The initial values ​​of the other internal auxiliary variables can be directly set to 0.

[0147] Step 3: Design a predefined time multi-agent system (as shown in the following formula, a predefined time multi-agent system designed based on the distributed optimization problem of resource allocation optimization and the distributed optimization problem based on consensus), and adjust the generalized time base generator and the correlation coefficient.

[0148]

[0149] The optimization problem is solved using the multi-agent approach described above, and the results are shown below:

[0150] This invention addresses the distributed, predefined time optimization problem based on resource allocation using a multi-agent system. By applying strictly convex, convex, and non-convex cost functions, the proposed multi-agent system achieves the following results (e.g., Figure 2 b, Figure 3 b, Figure 4 b) can converge to an optimal solution or a KKT point, and there are already results for multi-agent systems (such as...) Figure 2 a, Figure 3 a, Figure 4 a) cannot achieve this effect. Furthermore, by applying the generalized smooth function, it can be found that the runtime of the multi-agent system proposed in this invention (6.134 s) is... Figure 5 b) The existing multi-agent system runtime is 43.812s. Figure 5 a) The time is shorter, and the convergent solution is closer to the optimal solution within the predefined time. Furthermore, this invention also achieves similar results for consensus-based distributed predefined-time optimization problems, as shown in the following figures. Figures 6 to 8 As shown.

[0151] Based on the same inventive concept, this invention also proposes an optimization device for a distributed optimization system based on a predefined time, comprising:

[0152] The Generalized Time Base Builder module is used to build generalized time bases based on predefined time convergence properties.

[0153] The optimization module is used to optimize resource allocation and consensus issues in a distributed optimization system using a generalized time base; the optimization module specifically includes:

[0154] The convergence unit is used to construct corresponding distributed predefined temporal multi-agent systems based on a generalized time reference, with the goal of a distributed optimization problem involving resource allocation and consensus. By weakening the optimization conditions through two types of multi-agent systems, the system converges to the equilibrium point under the conditions of non-convex, strictly convex, convex, and strongly convex cost functions, and converges to the vicinity of the equilibrium point under the condition of generalized smoothness.

[0155] The optimal solution solving unit is used to associate the equilibrium point with the optimal solution using KKT conditions. It utilizes the correlation convexity and smoothness to make the two multi-agent systems converge to an optimal solution or near the optimal solution under the corresponding conditions.

[0156] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. An optimization method for a distributed optimization system based on predefined time, characterized in that, Includes the following steps: Construct a generalized time reference based on predefined time convergence; Optimize resource allocation and consensus issues in distributed optimization systems using a generalized time reference. The optimization process specifically includes the following steps: Based on the distributed optimization problem of resource allocation and consensus, corresponding distributed predefined time multi-agent systems are constructed according to the generalized time reference. The optimization conditions are weakened by two types of multi-agent systems so that they converge to the equilibrium point when the cost function is non-convex, strictly convex, convex and strongly convex, and converge to the equilibrium point under the condition of generalized smoothness. The equilibrium point is associated with the optimal solution using KKT conditions. By utilizing the correlation convexity and smoothness, the two multi-agent systems are predefined to converge to an optimal solution under the corresponding conditions.

2. The optimization method for a distributed optimization system based on a predefined time, as described in claim 1, is characterized in that... The construction of the generalized time reference based on predefined time convergence specifically includes the following steps: Assume a positive function μ α，D (t, τ) satisfy the following condition: satisfy And μ α，D (t, τ) is increasing with respect to t, and decreasing with respect to α and D; Satisfying α is large enough Less than a sufficiently small positive number ε; meet α，D (t,τ)=Dμ α (t,τ)、μ(t,τ)μ(τ,s)=μ(t,s)、 Where μ(t,τ)=μ 1，1 (t, τ); The predefined time convergence is defined as: system The correlation evolution operator Φ(t) satisfies the inequality: ||Φ(t)||≤μ α，D (t, τ) Where A(t) is a generalized time base generator, t represents time, α and D represent constants, and μ(t, τ) and μ(τ, s) are positive functions of the variables within parentheses. α (t, τ) represents the positive function raised to the power of α, and V(t) represents the Lyapunov function with respect to time t. The derivative of the Lyapunov function is represented.

3. The optimization method for a distributed optimization system based on a predefined time, as described in claim 2, is characterized in that... The aforementioned distributed predefined time-based multi-agent systems, constructed with resource allocation and consensus as optimization objectives, specifically include: The optimization based on resource allocation is expressed as: The consensus-based optimization is represented as: minf(x)=f i (x i ) s.t.x i =x j Where x i , q, q i ∈R n fi(x) i ):R n →R,x=col(x1,x2,...,x N ), x N It is the Nth variable, n, N, i, q i q are both positive numbers; A multi-agent system based on a distributed optimization objective of resource allocation is represented as: in, It is the derivative of the variable, Q is a positive constant, and a ij It is the connection weight between two multi-agent systems, z j It is an auxiliary variable; A multi-agent system based on a consensus-driven distributed optimization objective is represented as follows: Among them, y i z i u i and w i All are internal auxiliary variables. Let κ represent the derivative of the corresponding function. For any κ, κA(t) is a generalized time base generator. It is an auxiliary variable.

4. The optimization method for a distributed optimization system based on a predefined time, as described in claim 3, is characterized in that... The method of weakening the optimization conditions through two multi-agent systems, enabling convergence to the equilibrium point regardless of whether the cost function is non-convex, strictly convex, convex, or strongly convex, specifically includes the following steps: The strongly convex cost function is expressed as: A strictly convex cost function is expressed as f(θz1+(1-θ)z2)<θf(z1)+(1-θ)f(z2); a convex cost function is expressed as f(θz1+(1-θ)z2)≤θf(z1)+(1-θ)f(z2); a non-convex cost function does not satisfy any of the above conditions. The multi-agent system built on a distributed optimization objective based on resource allocation is simplified as follows: Where y is an auxiliary variable. These represent matrix L multiplied by auxiliary variables y, Z, u, and x, respectively. Subtracting the equilibrium point system from this system yields a new system: in, Both are derivatives of the new variable, x * It is the equilibrium point; The new system is transformed into two subsystems using orthogonal transformation: in, ξ1 and η1 are the derivatives of the new variables after the transformation, T is the transpose, and h is the subtraction of the two functions. It is the transpose of the orthogonal transformation matrix; Constructing a new Lyapunov function: Where ρ, β, and γ are positive constants; Differentiating the Lyapunov function: The variables are then decomposed using Young's inequalities, resulting in the following expression: Format: Where Λ1 is a constant composed of ρ, β, γ and σ; Choosing a suitable generalized time base generator and corresponding coefficients makes it possible to It satisfies the predefined conditions for time convergence; Through the new Lyapunov theorem, we obtain... If the system can converge to 0 in a predefined time, then the optimization conditions for the multi-agent system are weakened so that it converges to the equilibrium point regardless of whether the cost function is non-convex, strictly convex, convex, or strongly convex.

5. The optimization method for a distributed optimization system based on a predefined time, as described in claim 4, is characterized in that... The predefined convergence to equilibrium point under the condition of generalized smoothness specifically includes the following steps: Define generalized smoothness: in, The numbers are positive constants; z1 and z2 are variables. Choosing a suitable generalized time base generator and corresponding coefficients allows the derivative of the Lyapunov function to exhibit... Format: Where Λ2 and Λ3 are constants composed of positive constants ρ, β, γ and σ, and corresponding generalized time base generators and coefficients are selected to satisfy the predefined time convergence condition; V is a Lyapunov function; The new Lyapunov theorem yields a predefined convergence to the equilibrium point under the condition of generalized smoothness.

6. The optimization method for a distributed optimization system based on a predefined time, as described in claim 5, is characterized in that... The process of associating the equilibrium point with the optimal solution using KKT conditions, and leveraging the relevant convexity and smoothness, ensures that the two multi-agent systems converge to an optimal solution under predefined conditions. This specifically includes the following steps: The KKT conditions are used to associate the equilibrium point with the optimal solution. It is a KKT point if and only if: In a multi-agent system, x i It converges to the equilibrium point and also converges to the KKT point; When the cost function is strictly convex or strongly convex, the KKT point is the unique optimal solution; when the cost function is convex or non-convex, the KKT point is a local optimal solution.

7. An optimization device for a distributed optimization system based on a predefined time, characterized in that, include: The generalized time reference construction module is used to build a generalized time reference based on predefined time convergence. The optimization module is used to optimize resource allocation and consensus issues in a distributed optimization system using a generalized time base station. Its optimization modules specifically include: The convergence unit is used to construct corresponding distributed predefined temporal multi-agent systems based on the generalized time reference, with the goal of a distributed optimization problem involving resource allocation and consensus. By weakening the optimization conditions through two types of multi-agent systems, the system converges to the equilibrium point under the conditions of non-convex, strictly convex, convex, and strongly convex cost functions, and converges to the equilibrium point under the condition of generalized smoothness. The optimal solution solving unit is used to associate the equilibrium point with the optimal solution using KKT conditions, and utilizes the correlation convexity and smoothness to make the two multi-agent systems converge to an optimal solution under the corresponding conditions.

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