A large-scale particle swarm optimization method balancing convergence and diversity with dual search spaces

Abstract

The present application belongs to the technical field of computing intelligence and optimization algorithm, and particularly relates to a particle swarm optimization algorithm suitable for large-scale optimization problems and a system implementation thereof, which is used for solving problems such as insufficient diversity, easy premature convergence and unbalanced exploration-development in large-scale optimization problems. The present application constructs a new velocity update structure with four types of explicit learning operators, and introduces a joint entropy measure of the target space and the decision space, so that the algorithm can maintain effective distribution characteristics and search ability in a high-dimensional space.

Description

Technical Field

[0001] This invention belongs to the field of computational intelligence and optimization algorithm technology, and specifically relates to a particle swarm optimization method and system for solving large-scale optimization problems. Background Technology

[0002] Swarm optimization algorithms (such as Particle Swarm Optimization (PSO) and Differential Evolution (DE)) are widely used in engineering optimization, machine learning hyperparameter tuning, and large-scale system regulation due to their advantages of few parameters, ease of implementation, and fast convergence. However, when the problem dimension reaches hundreds or even thousands, swarm algorithms generally face performance degradation caused by "Large-Scale Optimization Problems (LSOPs)". The main challenges of LSOPs are: (1) The search space expands exponentially (curse of dimensionality) (2) Populations struggle to balance convergence and diversity. (3) The algorithm struggles to maintain effective global search capabilities with limited computing resources. Therefore, improving the performance of swarm intelligence algorithms in LSOP is an important research direction in the field of intelligent optimization.

[0003] Particle Swarm Optimization (PSO) is widely used in LSOP due to its simple structure and fast convergence speed. However, basic PSO is difficult to maintain effective search in high-dimensional problems. The reasons are as follows: firstly, the particles converge to each other at an increased rate, leading to premature convergence; secondly, diversity decreases rapidly, causing the algorithm to get stuck in local optima; and thirdly, it is difficult to manage the population distribution in both the decision space and the objective space at the same time.

[0004] To address the above problems, existing research has proposed two main categories of improvement directions: (1) Cooperative Coevolution (CC) reduces dimensionality. Representative algorithms include CCPSO-SK, CCPSO2, and Differential Grouping (DG, DG2, RDG, MDG, etc.). The main idea is to decompose high-dimensional variables into several sub-components and optimize them separately. However, the above algorithms have limitations. For example, variable grouping is highly dependent: if the interaction relationship between variables is not found correctly, it is difficult to improve performance; in addition, the correctness of grouping cannot be guaranteed in black-box optimization; even after grouping, the dimensionality of the sub-problems may still be large.

[0005] (2) Modify the velocity update structure to enhance convergence and diversity. Existing strategies include: stochastic exemplar selection mechanisms (such as CSO, SLPSO, RCI-PSO, Level-based PSO); hybrid algorithms (PSO + DE, PSO + CuckooSearch, etc.); and explicit explore-explore decoupling (APSO-DEE, PSO-DBCD). The core idea of ​​these algorithms is to add a diversity enhancement term to the velocity update equation to avoid population aggregation. However, a core problem exists: it is still impossible to simultaneously and explicitly perform diversity preservation operations in both the decision space and the target space.

[0006] Deficiencies of existing technology Defect 1: Inability to simultaneously perform diversity preservation in both the decision space and the target space. Most existing PSO improvements manage diversity in only one of the following dimensions: target space (fitness space): such as LSD and ELDM, both based on distance calculations; decision space (variable space): such as differential mutation improvements. The drawbacks are also quite obvious. For example, spatial information is incomplete, and single-space diversity preservation cannot truly reflect the actual distribution of the population; it easily leads to the problem of "high apparent diversity, but actual search duplication"; furthermore, it cannot guarantee effective exploration in ultra-high dimensions (1000D+).

[0007] Defect 2: The Exemplar randomization mechanism is prone to selecting crowded regions, leading to the failure of diversity maintenance. For algorithms such as CSO, RCI-PSO, and Level-based PSO, due to the random selection of the exemplar, if the population itself is unevenly distributed, it is easy to repeatedly select particles from "crowded regions", thus resulting in the failure to truly improve diversity.

[0008] Defect 3: Distance-based diversity measures (such as LSD and ELDM) cannot reflect true disorder. LSD / ELDM only focuses on the "uniformity" of particles in the target space, but truly good diversity should manifest as a chaotic distribution, covering all sub-intervals. Therefore, a high LSD / ELDM does not equate to rich population information, which can easily lead to pseudo-diversity and increase the probability of getting trapped in local optima.

[0009] Defect 4: Hybrid algorithms introduce a large number of parameters, leading to high algorithm complexity and poor robustness. For example, algorithms such as PSO-DE and PSO+Cuckoo Search are prone to introducing external operators, resulting in: the need to adjust new parameters, a significant increase in complexity, and a decrease in algorithm robustness.

[0010] Defect 5: Existing exploration-exploitation decoupling methods still cannot clearly define "diversity learning". For example, APSO-DEE uses the diversity of the target space to replace the diversity of the decision space, and cannot clearly explain "why this diversity prompts the decision space to be updated". Summary of the Invention

[0011] The shortcomings of existing technologies directly affect the global search capability, search efficiency, and computational performance of swarm intelligence algorithms in large-scale optimization (LSOP). This invention aims to address these shortcomings by proposing a new algorithmic mechanism; therefore, the technical problems to be solved include, but are not limited to, the following five aspects: Technical Problem 1: It is impossible to perform effective diversity preservation in both the target space and the decision space simultaneously. Most existing PSO improvements only perform diversity preservation in one dimension of the goal space or decision space, thus failing to fully reflect the true distribution characteristics of the population in different spaces. Since goal space diversity mainly reflects fitness differences, while decision space diversity reflects the spatial distribution of solution vectors, maintaining diversity in only a single dimension leads to the following technical problems: (1) The search particles may appear to be "dispersed" in the target space, but are actually highly concentrated in the decision space; (2) It is difficult to effectively identify potential global optimal regions, resulting in premature convergence; (3) There is an inconsistency between search behavior and diversity indicators, which affects the interpretability of the algorithm.

[0012] Therefore, the technical problems to be solved by the present invention include: How to construct a search mechanism that can simultaneously maintain diversity in both the decision space and the target space, so that diversity management is consistent, real-time and interpretable, and ensures a clear causal relationship between search behavior and diversity information.

[0013] The objective of this invention is to: construct a new speed update structure; enable the algorithm to perform diversity management in two spaces simultaneously; and clarify the causal relationship between the corresponding operation items and the search behavior.

[0014] Technical Issue 2: The random exemplar selection mechanism lacks directionality and cannot truly improve diversity. Existing algorithms (such as CSO, RCI-PSO, and Level-based PSO) often employ a random exemplar selection mechanism. However, when the population is already crowded, random exemplar selection can easily lead to the selection of crowded regions again. This will result in: (1) The exemplar selection lacks information guidance; (2) Populations are highly likely to repeatedly explore local areas; (3) The diversity maintenance mechanism is merely a formality, resulting in low efficiency in actual exploration; Therefore, the technical problems to be solved by the present invention include: This invention aims to construct an information-guided mechanism based on population distribution, ensuring that exemplar selection is not based on randomness but on quantifiable diversity characteristics. It also seeks to utilize Shannon entropy or similar disorder indicators to identify sparse and crowded regions, thereby implementing a distributed exemplar selection strategy of "sparse regions guiding crowded regions" to significantly improve exploration quality. The invention proposes to: introduce a "sparse region → guiding crowded region" strategy based on Shannon entropy; and ensure that exemplar selection is based on diversity information, not randomness.

[0015] Technical Issue 3: Distance-based diversity measures cannot accurately reflect the true amount of information. Traditional diversity assessment methods (such as LSD and ELDM) rely on spatial distance between samples. However, distance information typically only characterizes geometric homogeneity and cannot reflect the overall disorder or information content of the population. Especially in high-dimensional cases, distance metrics exhibit the following shortcomings: (1) "Uniformity" does not equal "rich information"; (2) A high diversity value may simply mean that the population is evenly distributed within a narrow area; (3) It is difficult to identify hidden clustering trends or recurring patterns.

[0016] Therefore, the technical problems to be solved by the present invention include: This invention aims to assess population diversity by introducing a diversity measure that can more accurately represent the "information content" of a population, rather than just its "geometric uniformity." It also seeks to quantify the degree of population disorder using methods such as frequency domain disorder and probability distribution entropy, and further utilize this indicator to guide the migration direction and search intensity of the particle swarm, thereby improving search robustness in large-scale scenarios.

[0017] Technical Issue 4: Existing hybrid algorithms have complex structures, numerous parameters, and poor robustness. Numerous PSO studies have attempted to improve performance by introducing external operators (such as DE, Cuckoo Search, GA, etc.), but these hybrid structures lead to: (1) The overall complexity of the algorithm has increased significantly; (2) External operators introduce multiple additional parameters, requiring extensive experimental parameter tuning; (3) The lack of theoretical explanation for cross-operator behavior reduces the reproducibility of the algorithm; (4) The robustness is unstable in different benchmarks or application scenarios.

[0018] Therefore, the technical problems to be solved by the present invention include: How to improve population diversity, convergence, and exploration capabilities solely through the internal structure of PSO (such as difference vectors, entropy-based guidance mechanisms, etc.) without introducing external operators; how to reduce the structural complexity and parameter tuning cost of the algorithm while maintaining stable performance in high-dimensional optimization tasks.

[0019] Technical Issue 5: The existing PSO "exploration-development-diversity" mechanism lacks interpretability and a clear structure. The speed update structure of traditional PSO contains multiple implicit behaviors, and the relationships between its different learned terms are unclear; furthermore, various improvement strategies, by superimposing multiple operators or random terms, further reduce the interpretability of the structure. This leads to: (1) The source of exploration and development cannot be clearly distinguished; (2) There is no clear mathematical relationship between diversity enhancement terms and velocity update behavior; (3) It is difficult to quantify the contribution of different components to performance.

[0020] Therefore, the technical problems to be solved by the present invention include: How to construct a velocity update framework with a clear structure that includes the following four learning operators with clear mathematical meaning and behavioral interpretation: Inertia term: responsible for trend maintenance and acceleration adjustment; Convergence learning term: used for focused search of the global optimum or ideal exemplar; Decision space diversity learning term: guides particles to jump out of the clustered region of the decision space; Target space diversity learning term: ensures sufficient diversity in the fitness space.

[0021] By constructing an explicit and interpretable speed update mechanism, a quantifiable and verifiable behavioral chain can be formed among exploration, development, and diversity, thereby improving the overall controllability, interpretability, and performance of the algorithm.

[0022] The technical solution adopted in this invention is a large-scale particle swarm optimization method that balances convergence and diversity in two search spaces, comprising the following steps: Step 1: Initialize the particle swarm and set the algorithm parameters; Step 2: Determine if the termination condition is met. If it is, output the optimal solution; otherwise, proceed to Step 3. Step 3: Select the particle to be updated and its corresponding example particle from both the target space and the decision space. Step 4: Based on the degree of dispersion of the population in the decision space, adaptively calculate the adaptive factor for perturbation of diversity in the decision space, and construct an adaptive difference vector based on the factor. Step 5: Update the velocity of the selected particles to be updated according to the decoupled velocity update model. The velocity update model integrates an inertia term, a convergence guidance term, a decision space diversity term, and a target space diversity term. Step 6: Update the particle positions based on the updated velocity; Step 7: Evaluate the new population, update individual and global optimum information, and return to Step 2.

[0023] The process of selecting particles to be updated and their demonstration particles from the target space in step 3 includes: dividing the target space into multiple intervals based on the fitness values ​​of particles in the current population, and counting the number of particles in each interval; calculating the Shannon entropy of the population distribution based on the number of particles in each interval; sorting according to the crowding of the interval where each particle is located, selecting particles from the interval with high crowding as particles to be updated, and randomly selecting diversity demonstration particles from the interval with low crowding.

[0024] The number of intervals in the target space is equal to the population size, and the formula for calculating the Shannon entropy is: , in represents the proportion of particles in the i-th interval to the total population.

[0025] The process of selecting particles to be updated and their demonstration particles from the decision space in step 3 includes: dividing the current population into multiple subgroups according to a preset group size set; in each subgroup, selecting non-optimal particles whose fitness ranking in the entire population meets preset conditions as particles to be updated, and using the optimal particle of its subgroup as its convergence demonstration particle; the element values ​​in the group size set are dynamically adjusted as the optimization process progresses, with a smaller group size used in the early stages of optimization to enhance convergence, and a larger group size used in the later stages of optimization to maintain diversity.

[0026] The formula for calculating the adaptive factor in step 4 is as follows:

[0027] Where |Divc| is the absolute value matrix of the difference vectors of all particles to be updated; the adaptive difference vector is the product of the original difference vector and the adaptive factor.

[0028] The decoupled speed update model in step 5 is specifically as follows: , , in, It is the inertial term; For convergent terms; For decision space diversity; For the diversity term of the target space, It is an adaptive difference vector; To convergence demonstration particles; As a demonstration particle for diversity, All in The numbers are randomly generated within the range; r1, r2, r3, and r4 are random numbers, and φ is a preset target space diversity control parameter.

[0029] The target spatial diversity control parameter φ takes values ​​in the range of (0, 5).

[0030] The algorithm parameters in step 1 include: population size, maximum number of function evaluations, target space diversity control parameters, and dynamic grouping size set.

[0031] The large-scale particle swarm optimization method and system based on dual search space co-evolution (LSSO-EDDE) provided by this invention has the following significant advantages compared with the prior art: 1. It fundamentally improves the overall performance of the algorithm in large-scale optimization problems, achieving an efficient and explicit balance between convergence and diversity.

[0032] Existing technologies either focus only on the diversity of a single space (such as the decision space) or employ implicit, coupled methods to manage the balance between two spaces, leading to premature convergence or low search efficiency in complex high-dimensional spaces. The core improvement of this invention lies in proposing a novel, decoupled speed update structure.

[0033] (1) Explicit and simultaneous control: This structure achieves explicit, independent and simultaneous control of inertia, convergence, decision space diversity and target space diversity for the first time through four independent operators. This means that the algorithm can independently adjust the convergence pressure and the diversity maintenance strength of different dimensions according to their respective state information in each iteration, avoiding the dilemma of "paying attention to one thing but losing attention to another" in traditional methods.

[0034] (2) Precise Balancing: Because the components are independent of each other, the algorithm can more finely adjust the balance between exploration (dominated by D_d and D_o) and development (dominated by C). For example, in the early stages of optimization, the weights of D_d and D_o can be increased to fully explore; in the later stages, their influence can be weakened, allowing C to dominate the fine search. This dynamic and precise control mechanism is not available in existing technologies, directly leading to a significant improvement in the final solution accuracy and robustness.

[0035] (3) Performance Verification: This mechanism directly leads to a significant improvement in the final solution accuracy and robustness. This is mainly reflected in the following: On 30 functions across two large-scale benchmark sets, CEC2013 and 2022, as shown in Tables 1, 2, and 3, the mean error (Mean) and standard deviation (Std) of LSSO-EDDE and peer algorithms after running a large number of independent experiments are presented. The gray best result markers and statistical test symbols (+ / = / –) intuitively show that LSSO-EDDE has higher accuracy (smaller Mean) and more stable results (smaller Std, i.e., better robustness) on most problems. The Friedman test in Table 3 ranks the algorithms as a whole, with LSSO-EDDE ranking second and first, respectively, significantly outperforming the other seven mainstream advanced algorithms, demonstrating its excellent overall accuracy and stability (robustness).

[0036] Furthermore, the algorithm exhibits strong convergence in the later stages, from Figure 2 , Figure 3 The convergence curves clearly show that LSSO-EDDE exhibits stronger and more stable convergence capabilities in the later stages of optimization (after the number of function evaluations exceeds 1E+06), and the curves are smooth and stable. This intuitively demonstrates its: higher solution accuracy (lower final position of the curve); better robustness (smooth curve, indicating consistent performance in each run, unaffected by drastic randomness); and effective balancing ability (a gentle curve in the early stages indicates exploration, while a steep drop in the later stages indicates effective convergence, reflecting good balance). This proves that its precise balancing mechanism can continue to be effective in long-term searches, avoiding premature convergence and thus obtaining solutions with higher accuracy.

[0037] 2. The algorithm's ability to escape local optima and discover global optima is significantly enhanced, thanks to the diversity protection mechanism based on Shannon entropy introduced in the target space.

[0038] Existing distance-based diversity metrics (such as LSD and ELDM) aim to achieve a "uniform distribution" of particles in the target space, but this is not the optimal diversity state and may lead to insufficient search motivation. The improvement of this invention lies in proposing the Shannon entropy population diversity metric (SDM-SE) and the corresponding velocity update strategy (VUS-OS).

[0039] (1) Better diversity orientation: SDM-SE guides the population towards “chaotic distribution” (i.e., particles in each interval) by calculating the distribution entropy of particles in different fitness intervals. This contains richer search information than “uniform distribution” and can more effectively reveal unknown high-potential areas.

[0040] (2) Efficient information utilization: The VUS-OS strategy actively guides particles in dense regions to learn from particles in sparse regions, promoting the vertical flow of information between different performance levels of the population. This not only prevents all particles from rushing to the currently seemingly optimal region, but also effectively utilizes "scout" particles that are temporarily poor in performance but may be located on the path to the global optimum, thereby greatly enhancing the algorithm's global exploration capability and ability to escape local extrema.

[0041] 3. It effectively maintains a broad exploration of the solution structure in the decision space and avoids search blind spots caused by the curse of dimensionality, thanks to the adaptive difference mutation strategy.

[0042] In high-dimensional decision spaces, random perturbations can easily become too large (breaking convergence) or too small (ineffective exploration). The improvement of this invention lies in the design of a differential mutation strategy (VUS-DS) with an adaptive factor β.

[0043] (1) Intelligent disturbance regulation: adaptive factor It can intelligently sense the dispersion of the population in the decision space. When the population is dispersed (exploration phase), the β value is larger, increasing the mutation intensity to accelerate exploration; when the population converges (development phase), the β value automatically decreases to reduce perturbation and avoid interfering with convergence. This adaptability ensures that the algorithm can apply "just the right" perturbation at any stage.

[0044] (2) Advantages of parameter-free operation: This mechanism dynamically adjusts according to the population state, eliminating the need for users to preset fixed mutation probabilities or step sizes. This reduces the requirements for user experience and improves the ease of use and robustness of the algorithm. At the same time, by introducing differential information between particles, it injects new search directions into the population, effectively expanding the search range in the decision space.

[0045] 4. Dynamic and controllable optimization of the convergence process was achieved, and convergence pressure was cleverly managed through grouping and elite retention mechanisms.

[0046] Fixed learning strategies are difficult to adapt to the needs of different stages of the optimization process. The improvement of this invention lies in the adoption of a convergence strategy of dynamic grouping and random elite retention (VUS-C).

[0047] (1) Dynamic convergence pressure: The size s of the subgroup is dynamically adjusted by using a predefined set S. Small subgroups are used in the early stages to enhance competition and convergence speed, while large subgroups are used in the later stages to maintain population diversity. This allows the convergence pressure to change intelligently over time, taking into account both rapid localization in the early stages and fine-grained search in the later stages.

[0048] (2) High-quality convergence guarantee: The random elite retention mechanism (only updating non-elite and lower-ranked particles) ensures that high-quality particles have a higher probability of not being destroyed, protecting the discovered excellent solutions, while guiding poorer particles to learn towards local optima. This mechanism ensures that the algorithm can steadily advance towards high-quality solution regions while converging quickly, improving the reliability of the optimization process.

[0049] 5. The overall algorithm maintains high performance while also possessing good scalability and practicality.

[0050] (1) Excellent scalability: As shown in the scalability test in the paper, when the problem dimension increases from 1000 to 3000, LSSO-EDDE still maintains its leading advantage over the comparison algorithm. This proves that the series of mechanisms proposed in this invention are very effective in solving the "curse of dimensionality" problem, and the algorithm performance does not decrease sharply with the increase of dimension.

[0051] (2) Parameter robustness and ease of implementation: Parameter sensitivity analysis shows that the algorithm is not sensitive to the key parameters φ and N_pop and can maintain good performance over a wide range, which reduces the difficulty and cost of parameter tuning. At the same time, the algorithm has a clear structure, and each component can be implemented and verified independently, which is convenient for those skilled in the art to program and integrate into various optimization systems (such as the cloud task scheduling scenario successfully applied in the paper).

[0052] This invention systematically solves the core challenge of balancing convergence and diversity in large-scale particle swarm optimization through a series of innovative improvements that work in synergy. These improvements are not simply a collection of improvements, but rather form an organic whole, working together from multiple levels such as structural design, metric criteria, adaptive control, and dynamic management. Ultimately, the algorithm of this invention exhibits significantly superior comprehensive performance compared to existing technologies in terms of solution accuracy, robustness, global search capability, and practicality. Attached Figure Description

[0053] Figure 1 This is a flowchart of the method of the present invention.

[0054] Figure 2 The graph shows the convergence curves of the method of this invention (LSSO-EDDE) and the comparison algorithm on the CEC 2013 test function set.

[0055] Figure 3 This is a convergence curve of the method of this invention (LSSO-EDDE) and the comparison algorithm on the CEC 2022 test function set.

[0056] Figure 4 This diagram illustrates the calculation of the number of particles located in the k-th interval.

[0057] Figure 5This is a schematic diagram of the motion of particles in the target space to maintain diversity.

[0058] Figure 6 This is the trajectory diagram of the adaptive factor β. Detailed Implementation

[0059] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0060] This invention proposes a large-scale group optimizer called LSSO-EDDE, the core of which lies in a novel speed update structure that models convergence and diversity protection as independent operators:

[0061] in, Let be the velocity of the i-th particle at time t; : Inertia term, inherited from standard PSO, used to maintain the original motion tendency of particles; : Convergence guiding term, used to guide particles to move towards the region of high-quality solutions, thereby enhancing the convergence capability of the algorithm; : Decision space diversity protection term, used to maintain the dispersion of particles in the solution parameter space (decision space) and avoid getting trapped in local optima; : Target space diversity protection term, used to maintain a wide distribution of particles in the fitness value space (target space) to ensure a broad search range.

[0062] Target Space Diversity Protection Strategy (VUS-OS) Existing distance-based diversity metrics (such as LSD and ELDM) aim for a uniform distribution of particles in the target space, but this is not optimal diversity. Ideal diversity should be a "chaotic distribution," where particles are distributed as uniformly as possible across all possible fitness ranges. Therefore, this invention proposes a diversity measure based on Shannon entropy.

[0063] 1. Diversity measure based on Shannon entropy (SDM-SE): (1) Interval partitioning: SDM-SE is based on Shannon entropy. In generation t, the target space is uniformly partitioned into two segments based on the maximum and minimum fitness values ​​(max and min) of all particles in the current population. Each interval Population size.

[0064] (2) Particle counting: For each particle, based on its fitness value Determine the interval i to which it belongs. Count the number of particles contained in each interval i, denoted as num(i), and calculate the proportion of particles in each interval i to the total population. :

[0065] (3) Entropy Calculation: Calculate the Shannon entropy H(P(t)) of the current population P(t) to measure diversity. Calculate the distribution entropy of particles in each interval.

[0066]

[0067] in, For the first The number of particles in each interval.

[0068] 2. Entropy-guided speed update (VUS-OS) (1) Identify the particles to be updated and the example particles: First, calculate the crowding density of the interval where each particle is located. (i.e., the number of particles in that interval), plus a random perturbation. ∈(0,1) to facilitate information exchange. Secondly, according to Sort the particles. After selecting the rank, use rand. The particles to be updated are stored in the set. .

[0069] For each particle to be updated, randomly select one from the top-ranked (i.e., sparser) particles as its diversity demonstration particle. Store in a collection .

[0070] (2) Speed ​​update strategy: Select particles located in the dense region and guide them to move towards particles in the sparse region:

[0071] in, As a demonstration particle for diversity from sparse regions; Define parameters for users to control the intensity of diversity guidance; A random number within a certain range. ϕ is a target space diversity control parameter, set by the user, used to adjust the intensity of diversity guidance. According to theoretical analysis, its optimal range is ϕ∈(0,5), and experiments show that performance is best when ϕ=0.3∼0.5.

[0072] Decision Space Diversity Protection Strategy (VUS-DS): This strategy injects new perturbation information into the decision space by introducing an adaptive differential mutation vector into the velocity update.

[0073] (1) Adaptive difference mutation vector: For the i-th particle, randomly select two other different particles from the population. Construct difference vectors :

[0074] Where D represents the dimension of the optimization problem.

[0075] (2) Introducing adaptive factors This factor is used to adaptively adjust the influence of the difference vector based on the population state. First, the mean, mean(∣Divc∣), and maximum value (max(∣Divc∣)) of the absolute value matrix of the difference vectors of all particles to be updated are calculated to obtain the adaptive factor. :

[0076]

[0077] The range of β is restricted to (0, 1). When the population is dispersed in the decision space (exploration phase), β is larger, which enhances the perturbation; when the population converges, β is smaller, which weakens the perturbation and avoids interfering with convergence.

[0078] (3) Speed ​​update:

[0079] in, Random numbers within a certain range. This strategy enhances exploration in the early stages and adaptively weakens it in the later stages to avoid interference with convergence.

[0080] Convergence Guidance Strategy (VUS-C): This strategy applies controllable convergence pressure through dynamic grouping and elite retention mechanisms.

[0081] (1) Dynamic grouping: Preset a group size set The entire optimization process is divided into |S| stages based on the maximum number of function evaluations maxFEs.

[0082] At each stage, a value is selected from S to serve as the current subgroup size s. Smaller s is used in the early stages of optimization (more groups, stronger competition, faster convergence), while larger s is used in the later stages (fewer groups, better diversity).

[0083] (2) Grouping and Elite Retention: Divide the population P(t) into / s subgroups, in which the particle with the best fitness is selected as the convergence demonstration particle of that subgroup.

[0084] For a particle, fitness is determined only if it is not the best particle in its subgroup and its fitness ranking in the entire population. Only when the time is right will it be selected as a particle to be updated and stored in the set. And the optimal particle of its subgroup is used as the convergence demonstration particle. Store in a collection For the selected particles to be updated, the update rate is calculated using the following formula:

[0085] in It is the optimal particle in the subgroup. For the first The position of each particle at time t; Let be the velocity of the i-th particle at time t; Let be the position of the i-th particle in the d-th dimension.

[0086] (3) Dynamic group size: The group size s is dynamically adjusted as the optimization process progresses. In the early stage, small groups enhance convergence, while in the later stage, large groups maintain diversity.

[0087] Complete Algorithm Flow (LSSO-EDDE) (1) Speed ​​and position updates: (2)

[0088] (3)

[0089] (4) Among them, All in Randomly generated within the range; It is an adaptive difference vector; To convergence demonstration particles; Demonstrating particles for diversity.

[0090] Basic steps and principles of the method The core principle of the method of this invention is that, through a decoupled and modular velocity update structure, the evolutionary direction of the particle swarm is clearly divided into four independent and cooperative components: inertia maintenance, convergence guidance, decision space diversity protection, and target space diversity protection, thereby achieving an explicit, precise, and adaptive balance between convergence and diversity.

[0091] Step S101: Algorithm Initialization Set algorithm parameters: population size Maximum number of function evaluations (maxFEs), target space diversity control parameter ϕ, and dynamic grouping size set. .

[0092] Within the decision space of the problem, the particle swarm is randomly initialized. and initialize the velocities of each particle. (0)=0.

[0093] Calculate the fitness value of each particle in the initial population. And determine the global historical best position. .

[0094] Step S102: Determine whether the termination condition is met. Typically, the termination condition is that the current number of evaluations of the cumulative function, FEs, reaches the preset maxFEs. If this condition is met, the process jumps to step S110 and outputs the result; otherwise, it continues to step S103.

[0095] Step S103: Select update particle and demonstration particle (dual path selection) This step executes two selection mechanisms in parallel to prepare data for subsequent speed updates: S103a: Target Space Diverse Paths (Executing VUS-OS Selection) (1) Calculate Shannon Entropy (SDM-SE): Based on the fitness value of the current population, the target space is uniformly divided into [a number of segments]. For each particle, there are intervals based on its fitness value. Determine the interval i to which it belongs. Count the number of particles num(i) contained in each interval i. The diagram for calculating the number of particles in each interval is shown below. Figure 4 As shown. Calculate the proportion of particles in each interval i to the total population. Calculate the distribution entropy of particles in each interval. A schematic diagram of the motion of particles in the target space to maintain diversity is shown below. Figure 5 As shown.

[0096] (2) Identify particles to be updated and diversity demonstration particles: First, calculate the crowding of the interval where each particle is located. And add a random perturbation. ∈(0,1) to facilitate information exchange. Secondly, according to... Sort the particles. After selecting the rank, use rand. The particles to be updated are stored in the set. For each particle to be updated, randomly select one from the top-ranked (i.e., sparser) particles as its diversity demonstration particle. Store in a collection .

[0097] S103b: Convergence Bootstrap Path (Execute VUS-C Selection) This step corresponds to the "convergence guidance mechanism for dynamic grouping and elite retention" in the claim.

[0098] (1) Dynamic grouping: Based on the current optimization stage, a set of group sizes is preset. The population P(t) is divided into / s subgroups (2) Selecting particles to be updated and convergent demonstration particles: For a particle, only if it is not the best particle in its subgroup and its fitness ranking in the entire population is it considered to be updated. Only when the time is right will it be selected as a particle to be updated and stored in the set. And the optimal particle of its subgroup is used as the convergence demonstration particle. Store in a collection .

[0099] Step S104: Determine the final update set and calculate the difference vector. Determine the final set of particles that need to perform velocity updates in this generation. , = This ensures that the updated particles are guided by both convergence and diversity. Each particle in the process randomly selects two other distinct particles from the population. Construct difference vectors .

[0100] Step S105: Calculate the adaptive factor β First, calculate the mean, mean(∣Divc∣), and maximum value (max(∣Divc∣)) of the absolute value matrix of the difference vectors of all particles to be updated, and obtain the adaptive factor. , , The range of β is restricted to (0, 1). When the population is dispersed in the decision space (exploration phase), β is larger, increasing perturbation; when the population converges, β is smaller, reducing perturbation and avoiding interference with convergence. The trajectory of β is shown in the figure. Figure 6 As shown.

[0101] Step S106: Construct adaptive difference vector for For each particle in the array, calculate its adaptive difference vector.

[0102] Step S107: Perform a comprehensive speed update for Each particle in the process is updated according to its d-th dimension velocity. This update integrates all four components:

[0103] in, It is the inertial term; For convergent terms; For decision space diversity; For the diversity of the target space.

[0104] Step S108: Update particle positions Update the particle's position based on the updated velocity: The updated location is handled to ensure it remains within the feasible domain of the problem.

[0105] Step S109: Assessment and Update Calculate the fitness value of the new population P(t+1) and update the function evaluation counters FEs. Update the individual best position of each particle in its historical data. and the global historical best position Once completed, return to step S102 for the next iteration.

[0106] Step S110: Output optimization results When the termination condition is met, output the global historical best position. This is the optimal solution found by the method of the present invention.

[0107] System / Device Implementation The present invention can also be implemented in the form of an optimized computing system or device. The device includes at least one processor and a memory communicatively connected to said at least one processor.

[0108] Memory: Stores computer program instructions that can be executed by a processor. This memory can be non-volatile, such as a solid-state drive or read-only memory.

[0109] Processor: When executing the computer program instructions, it is configured to perform all operations of the above method steps S101 to S110. Specifically, the processor is further configured to: As the population management unit, it is responsible for performing the initialization of step S101.

[0110] As the selection and control unit, it is responsible for executing steps S102, S103, S104, and S105, completing the termination condition judgment, the dual-path selection of the particle and the demonstration particle, and the calculation of the adaptive factor.

[0111] As a speed update calculation unit, it is responsible for executing steps S106 and S107. Its core task is to implement the four-component decoupled speed update formula.

[0112] As the location update and evaluation unit, it is responsible for executing steps S108 and S109.

[0113] As the result output unit, it is responsible for executing step S110.

[0114] Each software unit of the system corresponds one-to-one with the aforementioned method steps, working together to achieve an efficient and balanced search function in large-scale optimization problems.

[0115] Table 1. Comparison of LSSO-EDDE and other algorithms on a large-scale test function set in 2013 (dimension set to 1000).

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[0118] Table 2 Comparison results of LSSO-EDDE and other algorithms on a large-scale test function set in 2022 (dimension set to 1000).

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[0121] Table 3 Friedman Test Rankings

[0122] 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. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A large-scale particle swarm optimization method balancing convergence and diversity in two search spaces, characterized in that, Includes the following steps: Step 1: Initialize the particle swarm and set the algorithm parameters; Step 2: Determine if the termination condition is met. If it is, output the optimal solution; otherwise, proceed to Step 3. Step 3: Select the particle to be updated and its corresponding example particle from both the target space and the decision space. Step 4: Based on the degree of dispersion of the population in the decision space, adaptively calculate the adaptive factor for perturbation of diversity in the decision space, and construct an adaptive difference vector based on the factor. Step 5: Update the velocity of the selected particles to be updated according to the decoupled velocity update model. The velocity update model integrates an inertia term, a convergence guidance term, a decision space diversity term, and a target space diversity term. Step 6: Update the particle positions based on the updated velocity; Step 7: Evaluate the new population, update individual and global optimum information, and return to Step 2.

2. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 1, characterized in that, The process of selecting particles to be updated and their demonstration particles from the target space in step 3 includes: dividing the target space into multiple intervals based on the fitness values ​​of particles in the current population, and counting the number of particles in each interval; calculating the Shannon entropy of the population distribution based on the number of particles in each interval; sorting according to the crowding of the interval where each particle is located, selecting particles from the interval with high crowding as particles to be updated, and randomly selecting diversity demonstration particles from the interval with low crowding.

3. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 2, characterized in that, The number of intervals in the target space is equal to the population size, and the formula for calculating the Shannon entropy is: , in represents the proportion of particles in the i-th interval to the total population.

4. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 1, characterized in that, The process of selecting particles to be updated and their demonstration particles from the decision space in step 3 includes: dividing the current population into multiple subgroups according to a preset group size set; in each subgroup, selecting non-optimal particles whose fitness ranking in the entire population meets preset conditions as particles to be updated, and using the optimal particle of its subgroup as its convergence demonstration particle; the element values ​​in the group size set are dynamically adjusted as the optimization process progresses, with a smaller group size used in the early stages of optimization to enhance convergence, and a larger group size used in the later stages of optimization to maintain diversity.

5. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 1, characterized in that, The formula for calculating the adaptive factor in step 4 is as follows: , Where |Divc| is the absolute value matrix of the difference vectors of all particles to be updated; the adaptive difference vector is the product of the original difference vector and the adaptive factor.

6. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 1, characterized in that, The decoupled speed update model in step 5 is specifically as follows: ， ， in, It is the inertial term; For convergent terms; For decision space diversity; For the diversity term of the target space, It is an adaptive difference vector; To convergence demonstration particles; As a demonstration particle for diversity, All in The numbers are randomly generated within the range; r1, r2, r3, and r4 are random numbers, and φ is a preset target space diversity control parameter.

7. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 6, characterized in that, The target spatial diversity control parameter φ takes values ​​in the range of (0, 5).

8. The large-scale particle swarm optimization method for balancing convergence and diversity in a dual search space as described in claim 1, characterized in that, The algorithm parameters in step 1 include: population size, maximum number of function evaluations, target space diversity control parameters, and dynamic grouping size set.

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