A Distributed Online Optimization Method with Complex Inequality Constraints Based on Scale-Difference Uniform Quantization

Abstract

This invention discloses a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization. Within the primal-dual optimization framework, each node calculates its local intermediate update and constructs a difference vector between itself and the previous reference state. After normalization using a scale factor, uniform quantization is performed to obtain a low-bit integer message. The receiving end decodes and reconstructs the message based on the reference state, synchronously updating the result to the current reference state, followed by consistency fusion, projection, and dual variable updates. Compared to directly quantizing state variables, this invention reduces the dynamic range of the transmitted signal through "difference quantization + state synchronization," effectively suppressing error propagation under finite bit conditions, reducing communication overhead, and maintaining optimization performance under complex inequality constraints. It is applicable to scenarios such as smart grid scheduling and wireless sensor network positioning.

Description

Technical Field

[0001] This invention belongs to the fields of distributed optimization, networked collaborative control, edge intelligence and low-bit communication technology, and in particular, it is a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization. Background Technology

[0002] In networked systems such as smart grid dispatching, wireless sensor network target localization, multi-agent collaborative decision-making, and edge resource optimization, multiple nodes typically need to collaborate to complete global optimization tasks based on local observation information. Compared with centralized solutions, distributed optimization methods can reduce the computational and communication burden on central nodes and have better scalability and robustness, thus gaining widespread application.

[0003] However, in actual deployment, communication links between nodes are typically limited by bandwidth, power consumption, latency, and hardware processing capabilities, making it difficult to transmit high-precision continuous real-valued vectors in each iteration. If full-precision communication is still used, the communication overhead will be excessive, severely restricting the online running efficiency of distributed optimization algorithms.

[0004] To reduce communication overhead, existing technologies typically employ direct quantization to compress and transmit state variables, gradient information, or intermediate updates. However, in distributed primordial-dual optimization frameworks with complex inequality constraints, simply quantizing the current variable directly can lead to quantization errors that propagate and accumulate over multiple rounds of neighbor interactions. This can affect the consistency maintenance between nodes, the stability of primordial variable updates, and the ability of dual variables to adjust for constraint violations, resulting in increased system oscillations, decreased convergence accuracy, and deteriorated long-term constraint control performance.

[0005] Especially in online optimization scenarios, the objective function and constraint functions change continuously over time, and nodes need to continuously adjust decision variables based on new local observations. In this case, if the quantization scheme cannot simultaneously achieve low-bit communication and error suppression, it will be difficult to achieve a smooth, stable, and deployable distributed collaborative optimization process under complex inequality constraints.

[0006] Therefore, there is an urgent need for a new low-bit distributed online optimization technology to reduce the dynamic range of the signal to be transmitted, suppress the propagation of quantization errors, and achieve stable collaborative updates of the original and dual variables under complex inequality constraints, by improving the quantization object, reconstruction mechanism, and state synchronization method under bandwidth constraints. Summary of the Invention

[0007] The purpose of this invention is to address the deficiencies or shortcomings of the existing technology by providing a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization.

[0008] The technical solution to achieve the objective of this invention is: a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization. This method is applied to a communication network containing multiple nodes, where each node maintains an original variable, a dual variable, and a reference state. The reference state is used to share a unified evolutionary trajectory at both the encoding and decoding ends, serving as a reference benchmark for difference quantization in the next iteration. The method includes the following steps:

[0009] Step 1: At the current iteration time, each node calculates the local intermediate update amount based on the local loss function gradient and constraint function information;

[0010] Step 2: Each node constructs a difference vector based on the difference between the local intermediate update and the reference state at the previous time step, and processes the difference vector using a scaling factor to obtain a normalized difference vector.

[0011] Step 3: Each node quantizes the normalized difference vector to obtain a low-bit integer quantized message and sends it to the adjacent node.

[0012] Step 4: Each node decodes and reconstructs the received low-bit integer quantization message and the reference state from the previous moment to obtain the reconstructed state;

[0013] Step 5: Each node updates the reconstructed state to the reference state at the current moment;

[0014] Step 6: Each node performs a consistent fusion of the reconstruction states of its own node and its neighboring nodes based on a preset weight matrix to obtain a fusion vector, and updates the original variables for the next time step based on the fusion vector.

[0015] Step 7: Each node updates its dual variable based on the constraint violation amount corresponding to the current original variable.

[0016] Furthermore, in step 1, the local intermediate update is obtained by updating the current original variable along the gradient direction of the local loss function and the constrained subgradient direction weighted by the dual variable.

[0017] Furthermore, the normalized difference vector in step 2 is constructed as follows:

[0018]

[0019] In the formula, This represents the normalized difference vector at the current time, i.e., time t. This represents the reference state of node i at the previous time step, i.e., time step t-1. This represents the local intermediate update amount at time t. This represents the scale factor corresponding to the current iteration time.

[0020] Furthermore, in step 3, uniform quantization is used, and its symmetric integer set is: , where K is the upper limit of the quantization level.

[0021] Furthermore, in the uniform quantization, the single transmission bit width b satisfies .

[0022] Furthermore, in step 4, after receiving the low-bit integer quantization message, the node decodes and reconstructs the state based on the reference state and scale factor from the previous time step to obtain the reconstructed state:

[0023]

[0024] In the formula, Represents a node The reconstructed state at time t, The reference state is the state of the previous time step. Indicates a low-bit integer quantization message. is the scale factor.

[0025] Furthermore, in step 6, updating the original variables at the next moment includes: projecting the fused vector onto a preset convex constraint set to obtain the original variables at the next moment.

[0026] Further, in step 7, the dual variable is updated based on the positive part violation of the corresponding inequality constraint function; wherein, the update method for the s-th dual variable component is as follows:

[0027]

[0028] In the formula, Indicates the dual step size, Let represent the s-th inequality constraint function. This represents the positive part operator, i.e. , It has no practical meaning and is only used as an indicator variable. Indicates the original variable at the next moment. Denotes the s-th dual variable component. It represents the total number of components of the dual variable.

[0029] Furthermore, the method also includes:

[0030] Step 8: By making the normalized difference vector satisfy the preset dynamic range constraint, the single-round quantization error is controlled within the preset upper bound.

[0031] Furthermore, step 8 specifically includes:

[0032] For the normalized difference vector Apply unsaturated control conditions to satisfy:

[0033] When the above conditions are met, the quantization error It has a deterministic upper bound:

[0034] In the formula, K is the upper limit of the quantization level. Represents a node The reconstructed state at time t, This represents the local intermediate update amount at time t. is the scale factor.

[0035] Compared with the prior art, the significant advantages of this invention are:

[0036] (1) This invention does not directly quantize the current original variable or gradient, but quantizes the difference between the current local intermediate update and the reference state at the previous time, which can effectively reduce the dynamic range of the signal to be transmitted.

[0037] (2) The present invention introduces a scale normalization mechanism before differential quantization, which improves the ability of low-bit messages to express subtle changes in state under finite bit communication width.

[0038] (3) The present invention uses the reference state and low-bit integer message to perform decoding and reconstruction at the receiving end, and updates the reconstruction result to the new reference state synchronously, which helps to reduce the propagation and amplification of quantization error in multiple iterations.

[0039] (4) This invention, through the collaborative design of consistency fusion, projection update and dual variable update, takes into account both system objective optimization and constraint control in scenarios with complex inequality constraints.

[0040] (5) The present invention uses low-bit integer messages to replace high-precision continuous real-value messages, which can significantly reduce network communication load and is suitable for distributed collaborative scenarios with limited bandwidth, limited power consumption or time sensitivity.

[0041] (6) This invention improves the smoothness, robustness and engineering deployability of the iteration process by setting quantizer non-saturation control and time-varying scale factor.

[0042] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0043] Figure 1 This is an overall flowchart of the method of the present invention in one embodiment.

[0044] Figure 2 This is a block diagram showing the modular composition of a single node in one embodiment.

[0045] Figure 3 This is a schematic diagram illustrating the interaction of encoding, transmission, decoding, state synchronization, and consistency fusion in one embodiment.

[0046] Figure 4 This is a schematic diagram of a system in one embodiment of the present invention applied to a dynamic economic dispatch scenario of a smart grid. Detailed Implementation

[0047] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0048] It should be noted that if the embodiments of the present invention involve directional indicators (such as up, down, left, right, front, back, etc.), the directional indicators are only used to explain the relative positional relationship and movement of the components in a certain specific posture (as shown in the figure). If the specific posture changes, the directional indicators will also change accordingly.

[0049] Furthermore, if the embodiments of this invention involve descriptions such as "first" or "second," these descriptions are for descriptive purposes only and should not be construed as indicating or implying their relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first" or "second" may explicitly or implicitly include at least one of those features. Additionally, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. If the combination of technical solutions is contradictory or impossible to implement, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by this invention.

[0050] In one embodiment, a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization is provided. This method is applied to a communication network containing multiple nodes, where each node maintains an original variable, a dual variable, and a reference state. The reference state is used to share a unified evolutionary trajectory at the encoding and decoding ends, serving as a reference benchmark for difference quantization in the next iteration. Figure 1 and Figure 3 The method includes the following steps:

[0051] Step 1: At the current iteration time, each node calculates the local intermediate update amount based on the local loss function gradient and constraint function information;

[0052] Step 2: Each node constructs a difference vector based on the difference between the local intermediate update and the reference state at the previous time step, and processes the difference vector using a scaling factor to obtain a normalized difference vector.

[0053] Step 3: Each node quantizes the normalized difference vector to obtain a low-bit integer quantized message and sends it to the adjacent node.

[0054] Step 4: Each node decodes and reconstructs the received low-bit integer quantization message and the reference state from the previous moment to obtain the reconstructed state;

[0055] Step 5: Each node updates the reconstructed state to the reference state at the current moment;

[0056] Step 6: Each node performs a consistent fusion of the reconstruction states of its own node and its neighboring nodes based on a preset weight matrix to obtain a fusion vector, and updates the original variables for the next time step based on the fusion vector.

[0057] Step 7: Each node updates its dual variable based on the constraint violation amount corresponding to the current original variable.

[0058] Furthermore, in one embodiment, in step 1, the local intermediate update is obtained by updating the current original variable along the gradient direction of the local loss function and the constrained subgradient direction weighted by the dual variable.

[0059] Furthermore, in one embodiment, the normalized difference vector in step 2 is constructed as follows:

[0060]

[0061] In the formula, This represents the normalized difference vector at the current time, i.e., time t. This represents the reference state of node i at the previous time step, i.e., time step t-1. This represents the local intermediate update amount at time t. This represents the scale factor corresponding to the current iteration time.

[0062] Furthermore, in one embodiment, the quantization in step 3 employs uniform quantization, whose symmetric integer set is: , where K is the upper limit of the quantization level.

[0063] Preferably, in some embodiments, during the uniform quantization, the single transmission bit width b satisfies .

[0064] Furthermore, in one embodiment, in step 4, after receiving the low-bit integer quantization message, the node decodes and reconstructs the state based on the reference state and scale factor from the previous time step to obtain the reconstructed state:

[0065]

[0066] In the formula, Represents a node The reconstructed state at time t, The reference state is the state of the previous time step. Indicates a low-bit integer quantization message. is the scale factor.

[0067] Furthermore, in one embodiment, step 6, updating the original variables at the next time step includes: projecting the fused vector onto a preset convex constraint set to obtain the original variables at the next time step.

[0068] Furthermore, in one embodiment, in step 7, the dual variable is updated based on the positive part violation of the corresponding inequality constraint function; wherein, the update method for the s-th dual variable component is as follows:

[0069]

[0070] In the formula, Indicates the dual step size, Let represent the s-th inequality constraint function. This represents the positive part operator, i.e. , It has no practical meaning and is only used as an indicator variable. Indicates the original variable at the next moment. Denotes the s-th dual variable component. It represents the total number of components of the dual variable.

[0071] Furthermore, in one embodiment, the method further includes:

[0072] Step 8: By making the normalized difference vector satisfy the preset dynamic range constraint, the single-round quantization error is controlled within the preset upper bound.

[0073] Preferably, in some embodiments, step 8 specifically includes:

[0074] For the normalized difference vector Apply unsaturated control conditions to satisfy:

[0075] When the above conditions are met, the quantization error It has a deterministic upper bound:

[0076] In the formula, K is the upper limit of the quantization level. Represents a node The reconstructed state at time t, This represents the local intermediate update amount at time t. is the scale factor.

[0077] Preferably, in some embodiments, the above-mentioned scale factor adopts a time-varying decay sequence.

[0078] Preferably, in some embodiments, the preset weight matrix in step 6 is a double random matrix adapted to the network topology.

[0079] Preferably, in some embodiments, combined with Figure 4 The method of this invention is applied to the dynamic economic dispatch of smart grids, wherein the original variables correspond to the output variables of each generator unit, the local loss function corresponds to the generator unit's power generation cost, and the constraints include upper and lower limits of output constraints and system supply and demand balance coupling constraints.

[0080] In one embodiment, combined Figure 2 This paper provides a distributed online optimization system, which includes multiple nodes, each node comprising:

[0081] The first module (local gradient update module) is used to calculate the local intermediate update amount based on the local loss function gradient and constraint function information;

[0082] The second module (difference construction module) is used to construct a difference vector based on the local intermediate update amount and the reference state at the previous time step.

[0083] The third module (scale normalization module) is used to normalize the difference vector using a scale factor;

[0084] The fourth module (uniform quantization encoding module) is used to perform uniform quantization on the normalized difference vector to generate low-bit integer quantized messages;

[0085] The fifth module (communication module) is used to send and receive the low-bit integer quantization message;

[0086] The sixth module (decoding and reconstruction module) is used to restore the reconstruction state based on the received low-bit integer quantization message and the reference state of the previous time step.

[0087] The seventh module (state synchronization module) is used to synchronize and update the reconstructed state to the reference state at the current moment;

[0088] The eighth module (consistency fusion module) is used to perform consistency fusion on the reconstruction state of the current node and its neighboring nodes based on a preset weight matrix.

[0089] The ninth module (projection update module) is used to obtain the original variables for the next time step based on the fusion result;

[0090] The tenth module (dual update module) is used to update the dual variable based on the constraint violation.

[0091] Specific limitations regarding the distributed online optimization system can be found in the above section on the limitations of distributed online optimization methods with complex inequality constraints based on scale-difference uniform quantization, and will not be repeated here. Each module in the aforementioned distributed online optimization system can be implemented entirely or partially through software, hardware, or a combination thereof. These modules can be embedded in or independent of the processor in a computer device, or stored in the memory of a computer device in software form, so that the processor can call and execute the corresponding operations of each module.

[0092] In one embodiment, an electronic device is provided, including a processor, a memory, and a computer program stored in the memory and executable by the processor, wherein the computer program, when executed, implements the steps of the method:

[0093] Step 1: At the current iteration time, each node calculates the local intermediate update amount based on the local loss function gradient and constraint function information;

[0094] Step 2: Each node constructs a difference vector based on the difference between the local intermediate update and the reference state at the previous time step, and processes the difference vector using a scaling factor to obtain a normalized difference vector.

[0095] Step 3: Each node quantizes the normalized difference vector to obtain a low-bit integer quantized message and sends it to the adjacent node.

[0096] Step 4: Each node decodes and reconstructs the received low-bit integer quantization message and the reference state from the previous moment to obtain the reconstructed state;

[0097] Step 5: Each node updates the reconstructed state to the reference state at the current moment;

[0098] Step 6: Each node performs a consistent fusion of the reconstruction states of its own node and its neighboring nodes based on a preset weight matrix to obtain a fusion vector, and updates the original variables for the next time step based on the fusion vector.

[0099] Step 7: Each node updates its dual variable based on the constraint violation amount corresponding to the current original variable;

[0100] Step 8: By making the normalized difference vector satisfy the preset dynamic range constraint, the single-round quantization error is controlled within the preset upper bound.

[0101] For specific constraints on each step, please refer to the constraints on the distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization mentioned above, which will not be repeated here.

[0102] In one embodiment, a computer-readable storage medium is provided having a computer program stored thereon, the computer program being implemented when executed by a processor:

[0103] Step 1: At the current iteration time, each node calculates the local intermediate update amount based on the local loss function gradient and constraint function information;

[0104] Step 2: Each node constructs a difference vector based on the difference between the local intermediate update and the reference state at the previous time step, and processes the difference vector using a scaling factor to obtain a normalized difference vector.

[0105] Step 3: Each node quantizes the normalized difference vector to obtain a low-bit integer quantized message and sends it to the adjacent node.

[0106] Step 4: Each node decodes and reconstructs the received low-bit integer quantization message and the reference state from the previous moment to obtain the reconstructed state;

[0107] Step 5: Each node updates the reconstructed state to the reference state at the current moment;

[0108] Step 6: Each node performs a consistent fusion of the reconstruction states of its own node and its neighboring nodes based on a preset weight matrix to obtain a fusion vector, and updates the original variables for the next time step based on the fusion vector.

[0109] Step 7: Each node updates its dual variable based on the constraint violation amount corresponding to the current original variable;

[0110] Step 8: By making the normalized difference vector satisfy the preset dynamic range constraint, the single-round quantization error is controlled within the preset upper bound.

[0111] For specific constraints on each step, please refer to the constraints on the distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization mentioned above, which will not be repeated here.

[0112] As a specific example, the invention will be described in detail in some embodiments.

[0113] Example 1

[0114] This embodiment provides a distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization, applied to a time-varying communication network consisting of multiple nodes. Nodes in the network exchange information via communication links. Each node only obtains local loss function information and local constraint information, and maintains its own original variables. Dual variables and reference status ,in t represents the node number and t represents the iteration time.

[0115] In this embodiment, each node is first initialized. Preferably, the initial primitive variables of each node are... Set as the zero vector or a preset initial value, dual variable Set as zero vector, reference state Set to the zero vector or initial reconstruction state. After initialization, each node iteratively updates according to the following steps.

[0116] 1. Calculation of local intermediate update volume

[0117] At the current iteration time t, node Calculate the local intermediate update based on the current original variables, the gradient of the local loss function, and the subgradient information corresponding to the constraint function. Preferably, the local intermediate update amount satisfies:

[0118]

[0119] in, Indicates the original step size. Represents a node The gradient of the local loss function. This represents the subgradient term corresponding to the s-th inequality constraint. Represents a node The s-th dual variable component, where p represents the number of inequality constraints.

[0120] By using the above method, the descent direction of the local loss function and the constraint penalty direction are coupled to the same intermediate update quantity, thus providing a foundation for subsequent quantization communication and distributed consensus updates.

[0121] 2. Construction of Difference Vector and Scale Normalization

[0122] Unlike directly quantizing the current original variable or gradient, in this embodiment, each node uses the difference between the current local intermediate update and the reference state at the previous time step as the quantization object. Specifically, the node... Construct the difference vector:

[0123]

[0124] in, This represents the reference state of node i at the previous time step.

[0125] To reduce the dynamic range of the signal to be quantized, a scaling factor is further utilized. The difference vector is normalized to obtain a normalized difference vector:

[0126]

[0127] in, This represents the scaling factor corresponding to the current iteration time. Preferably, the scaling factor is a positive number and can be set as a time-varying decay sequence to retain greater dynamic response capability in the early stages of the algorithm and improve quantization resolution in the later stages.

[0128] By setting the quantization object to the difference relative to the reference state at the previous moment, the amplitude range of the signal to be transmitted can be significantly reduced, thereby improving the accuracy of expressing state changes with a finite number of bits.

[0129] 3. Uniform quantization of normalized difference vector

[0130] After obtaining the normalized difference vector After that, node Uniform quantization is performed on it to generate a low-bit integer quantized message. Preferably, the symmetric quantization set is set as follows:

[0131]

[0132] Where K represents the upper limit of quantization levels. Node For the normalized difference vector Each component is quantized independently using a component-wise uniform quantization process to obtain an integer quantized message:

[0133]

[0134] in, This represents the component-wise uniform quantization operator.

[0135] Preferably, the component-wise uniform quantization method is as follows: when the input component falls into the interval near zero, it is quantized to 0; when the input component falls into the corresponding integer interval, it is quantized to the corresponding integer; when the input component exceeds the maximum quantization range, it is quantized to +K or -K.

[0136] Furthermore, if the total number of quantization levels is 2K+1, then the bit width of a single transmission satisfies: .

[0137] Therefore, each node only needs to send low-bit integer quantization messages in each iteration. This significantly reduces the communication burden by eliminating the need to send high-precision continuous real-valued vectors.

[0138] 4. Decoding and reconstruction are synchronized with the reference state update.

[0139] node After receiving the integer quantization message, based on the reference state of the previous time step... and scale factor Decode and reconstruct the state to obtain the reconstructed state:

[0140]

[0141] in, Represents a node The reconstructed state at time t.

[0142] After the reconstruction is completed, to ensure that the latest reconstruction result is still used as the reference benchmark for difference quantization in the next iteration, each node will synchronously update its current reconstruction state to the reference state at the current moment, that is:

[0143]

[0144] That is to say .

[0145] With the above settings, the encoder and decoder can share a unified reference state evolution trajectory, so that subsequent iterations still construct differences around the latest reconstruction results, thereby suppressing the continuous accumulation of quantization errors.

[0146] 5. Consistency Fusion and Projection Update

[0147] After obtaining the reconstruction status of the current node and its neighbors, the node... According to the preset weight matrix Perform consistency fusion to obtain the fusion vector:

[0148]

[0149] in, For a weight matrix adapted to the current network topology, preferably, It is a double random matrix, and N represents the total number of nodes in the network.

[0150] Obtain the fusion vector Then, node i projects the fused vector onto a preset set of convex constraints. To obtain the original variables for the next time step:

[0151]

[0152] in, Represents a set The projection operator, the set It is a bounded convex set that contains the feasible region of the original problem.

[0153] Through consistency fusion, each node can collaboratively update using the reconstructed state of its neighbors; through projection update, it can be ensured that the original variables are always within the pre-defined bounded convex set, thereby enhancing the feasibility and stability of the algorithm.

[0154] 6. Dual variable update

[0155] Obtain the original variables at the next moment. After that, node Update the dual variable based on the constraint violation amount corresponding to the current original variable. Preferably, the update method for the s-th dual variable component is as follows:

[0156]

[0157] in, Indicates the dual step size, Let represent the s-th inequality constraint function. This represents the positive part operator, i.e. .

[0158] By mapping the inequality constraint violation amount to the adjustment amount of the dual variable, when the degree of a constraint violation is large, the corresponding dual variable will increase, thereby imposing a stronger penalty on the constraint in subsequent iterations, so as to balance the optimization objective and long-term constraint control.

[0159] 7. Quantizer non-saturation control

[0160] To avoid the quantizer operating in the saturation region for an extended period, preferably, the normalized difference vector... Apply unsaturated control conditions to satisfy:

[0161]

[0162] When the above conditions are met, the quantization error It has a deterministic upper bound:

[0163]

[0164] This allows for control over the maximum amplitude of single-round quantization error, reducing state distortion caused by quantization saturation and improving system smoothness and robustness.

[0165] 8. Iterative execution

[0166] Each node repeats the above steps until a preset termination condition is met. The termination condition can be any one or a combination of the following: reaching a preset number of iterations, reaching a preset time length, the change in the objective function being lower than a threshold, or the constraint violation being satisfied.

[0167] Through the above steps, this embodiment realizes a closed-loop iterative process of "local intermediate update calculation—difference construction—scale normalization—integer quantization—decoding and reconstruction—reference state synchronization—consistency fusion—projection update—dual update". Compared with the method of directly quantizing the current state quantity, this embodiment can effectively reduce the dynamic range of the signal to be transmitted, reduce communication bit overhead, and suppress the propagation of quantization error in multiple iterations.

[0168] Example 2

[0169] This embodiment applies the above method to a smart grid dynamic economic dispatch scenario. Each node corresponds to a different generator set, and the local decision variable for each node is the active power output of the generator set. The local loss function is the generation cost function of the corresponding generator set. Local constraints include upper and lower limits for generator set output, and global coupling constraints include supply and demand balance constraints to ensure that the total system output meets load requirements.

[0170] In this scenario, each generator node first calculates its local intermediate update based on the current output status of the unit, the gradient of the cost function, and the constraint function information. Then, using the difference between the current intermediate update and the reference state at the previous moment as the quantization object, it performs scale normalization and uniform quantization to form a low-bit integer quantized message, which is then sent to neighboring nodes. The receiving node decodes and reconstructs the integer quantized message based on the reference state, and then obtains the unit output decision for the next moment through consistency fusion and projection update. Finally, it updates the dual variable based on the violation of the supply and demand balance constraint.

[0171] In this way, each generator node does not need to transmit high-precision continuous status data. It can achieve collaborative optimization by using only low-bit integer quantization messages. This reduces the total power generation cost of the entire network and reduces communication bandwidth usage while meeting the constraints of generator capacity and system supply and demand balance.

[0172] Example 3

[0173] This embodiment applies the above method to a target localization scenario in a wireless sensor network. Each sensor node constructs a localization loss function based on local observation data, using the target position estimate as the original variable. In each iteration, each node calculates an intermediate update based on local observations and constraint information, then constructs the difference between this intermediate update and the reference state at the previous time step. This difference is then scaled and uniformly quantized to form a low-bit integer quantized message, which is sent to neighboring nodes.

[0174] The receiving node decodes and reconstructs the integer quantized message based on the reference state, and synchronously updates the decoding result to the current reference state. Subsequently, it uses the reconstructed states of neighboring nodes to perform consistent fusion and projection updates, ultimately obtaining a new target location estimate. Since the transmitted information is low-bit residual information rather than high-precision continuous state quantities, it can significantly reduce the wireless link communication burden and extend the network lifespan, while maintaining the stability of the multi-node cooperative positioning process.

[0175] In summary, compared to directly quantizing the current state variable, this invention reduces the dynamic range of the signal to be transmitted by using "differential quantization + scale normalization + reference state synchronization". It can suppress the propagation of quantization error under finite bit communication conditions, reduce communication overhead, and take into account the performance of complex inequality constraint control and distributed online optimization. It is suitable for scenarios such as smart grid scheduling, wireless sensor network positioning, and edge collaborative optimization.

[0176] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization, characterized in that, This method is applied to a communication network containing multiple nodes, where each node maintains an original variable, a dual variable, and a reference state. The reference state is used to share a unified evolutionary trajectory at the encoding and decoding ends, serving as a reference benchmark for difference quantization in the next iteration. The method includes the following steps: Step 1: At the current iteration time, each node calculates the local intermediate update amount based on the local loss function gradient and constraint function information; Step 2: Each node constructs a difference vector based on the difference between the local intermediate update and the reference state at the previous time step, and processes the difference vector using a scaling factor to obtain a normalized difference vector. Step 3: Each node quantizes the normalized difference vector to obtain a low-bit integer quantized message and sends it to the adjacent node. Step 4: Each node decodes and reconstructs the received low-bit integer quantization message and the reference state from the previous moment to obtain the reconstructed state; Step 5: Each node updates the reconstructed state to the reference state at the current moment; Step 6: Each node performs a consistent fusion of the reconstruction states of its own node and its neighboring nodes based on a preset weight matrix to obtain a fusion vector, and updates the original variables for the next time step based on the fusion vector. Step 7: Each node updates its dual variable based on the constraint violation amount corresponding to the current original variable.

2. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, characterized in that, In step 1, the local intermediate update is obtained by updating the current original variable along the gradient direction of the local loss function and the constrained subgradient direction weighted by the dual variable.

3. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, characterized in that, The normalized difference vector in step 2 is constructed as follows: In the formula, This represents the normalized difference vector at the current time, i.e., time t. This represents the reference state of node i at the previous time step, i.e., time step t-1. This represents the local intermediate update amount at time t. This is the scale factor corresponding to the current iteration time.

4. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, characterized in that, In step 3, uniform quantization is used, and its symmetric integer set is: , where K is the upper limit of the quantization level.

5. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 4, characterized in that, In the uniform quantization, the single transmission bit width b satisfies .

6. In the distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, in step 4, after receiving the low-bit integer quantization message, the node decodes and reconstructs the state according to the reference state and scale factor of the previous time step to obtain the reconstructed state: In the formula, Represents a node The reconstructed state at time t, The reference state is the state of the previous time step. Indicates a low-bit integer quantization message. is the scale factor.

7. In the distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, step 6, updating the original variables at the next time step, includes: The fused vector is projected onto a preset set of convex constraints to obtain the original variables at the next time step.

8. In the distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, in step 7, the dual variable is updated according to the positive part violation of the corresponding inequality constraint function; wherein, The update method for the s-th dual variable component is as follows: In the formula, Indicates the dual step size, Let represent the s-th inequality constraint function. This represents the positive part operator, i.e. , It has no practical meaning and is only used as an indicator variable. Indicates the original variable at the next moment. Denotes the s-th dual variable component. It represents the total number of components of the dual variable.

9. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 1, characterized in that, The method further includes: Step 8: By making the normalized difference vector satisfy the preset dynamic range constraint, the single-round quantization error is controlled within the preset upper bound.

10. The distributed online optimization method with complex inequality constraints based on scale-difference uniform quantization according to claim 9, step 8 specifically includes: For the normalized difference vector Apply unsaturated control conditions to satisfy: When the above conditions are met, the quantization error It has a deterministic upper bound: In the formula, K is the upper limit of the quantization level. Represents a node The reconstructed state at time t, This represents the local intermediate update amount at time t. is the scale factor.

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