Optimization problem solving method and system

Abstract

The invention discloses a method and a system for solving an optimization problem, and relates to the field of computer algorithms. The system comprises a problem analysis modeling module, a hybrid intelligent optimization algorithm engine, a constraint adaptive mechanism, a parallel computing architecture and a result evaluation and parameter learning module. Aiming at the problems of media combinatorial optimization, KOC matrix construction and the like in marketing, a solution strategy is dynamically selected by intelligently analyzing a target function and complex constraints (mutual exclusion / binding / budget) of a non-independent object (with audience overlap) and combining an improved genetic algorithm and an adaptive particle swarm algorithm: minute-level solution is realized for medium and small-scale problems by adopting exhaustion / Monte Carlo; large scale problems are accelerated by distributed parallel computing. And an interpretable result is output by utilizing a visualization technology (a combined contribution histogram and an audience overlapping thermodynamic diagram), so that the global optimal solution precision and the solution efficiency are remarkably improved, and high-reliability support is provided for marketing decision making.

Description

Technical Field

[0001] This invention relates to the field of computer algorithm technology, and more specifically, to a method and system for solving optimization problems. Background Technology

[0002] In today's context of rapid technological and economic development, various industries have an urgent need for efficient solutions to optimization problems. Among these, finding the optimal local combination of multiple non-independent objects is a common optimization problem. Examples include selecting 15 media outlets from 100 to reach the widest possible audience for an advertisement, or choosing 10 product formulations from 50 to attract the largest user base for further development. These problems often involve constraints such as combinatorial width (the number of objects in the combinatorial group), combinatorial depth (the minimum number of objects needed to cover each individual object), order (e.g., A must precede B), mandatory entry (e.g., the combinatorial group must contain A, D, and J), mutual exclusion (e.g., A and K cannot appear simultaneously, or K cannot appear simultaneously with S and W), and binding (e.g., if C appears, F must also appear). However, existing optimization problem-solving techniques have significant shortcomings. Traditional exhaustive search methods, when faced with more than 30 candidates and thousands of data points, can become computationally unsustainable for most personal computers, requiring hours or even days of waiting. Furthermore, they frequently crash due to memory overflows or excessively large cache files, leading to the adoption of simple but unscientific methods such as substituting individual optimalities for overall optimalities. This severely hinders the widespread application of such optimization problems and increases the risk of erroneous decisions. For large-scale problems, such as finding the optimal KOC (Key Opinion Consumer) combination or building an efficient media matrix in a large data space, traditional gradient-based optimization methods, such as gradient descent, are prone to getting trapped in local optima when dealing with complex nonlinear and nonconvex functions, making it difficult to find the global optimum and resulting in a significant deviation between the final result and the actual optimal value. Meanwhile, some common heuristic algorithms, such as genetic algorithms and particle swarm optimization, while capable of exploring a wider solution space to some extent, suffer from low computational efficiency, slow convergence speed, and sensitivity to parameter settings. Furthermore, when dealing with optimization problems with multiple constraints and high-dimensional data, existing methods often struggle to balance constraint satisfaction and objective optimization, failing to effectively balance the relationship between the two. This makes the solutions unsatisfactory for the stringent requirements of practical applications. Moreover, existing systems and methods typically lack universality and adaptability to different types of optimization problems, requiring extensive manual adjustments and customization for specific issues. This not only increases the complexity and difficulty of applications but also consumes significant time and human resources.

[0003] Therefore, it is necessary to invent a method and system for solving the above-mentioned optimal combination problem. Summary of the Invention

[0004] To overcome the aforementioned deficiencies of the prior art, the present invention provides a method and system for solving optimization problems, addressing the problem that existing systems and methods mentioned in the background art generally lack universality and adaptability to different types of optimization problems, and require a large amount of manual adjustment and customization for specific problems. This not only increases the complexity and difficulty of the application, but also consumes a lot of time and manpower.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method and system for solving optimization problems, comprising a problem analysis and modeling module, a hybrid intelligent optimization algorithm engine, a constraint handling and adaptive adjustment mechanism, a parallel computing and distributed architecture module, a result evaluation and verification module, and a parameter adaptive learning and optimization module. The problem analysis and modeling module is used to receive the description of the optimization problem, analyze the objective function, constraints, and decision variables, and transform them into a standard mathematical model. The hybrid intelligent optimization algorithm engine integrates an improved genetic algorithm, an adaptive particle swarm optimization algorithm, and an algorithm combining simulated annealing and tabu search, and can intelligently select and combine algorithms according to the characteristics and scale of the problem. As a further description of the above technical solution, the problem parsing and modeling module can recognize various forms of problem input, including mathematical expressions, text descriptions and specific data formats, and accurately classify and label problem elements; As a further description of the above technical solution, the improved genetic algorithm in the hybrid intelligent optimization algorithm engine has a dynamic crossover and mutation probability adjustment mechanism, and the adaptive particle swarm optimization algorithm can dynamically adjust the inertia weight and learning factor. As a further description of the above technical solution, the constraint processing and adaptive adjustment mechanism adopts the improved Lagrange multiplier method to process equality constraints, the dynamic penalty function method to process inequality constraints, and can adaptively adjust the strategy and parameters according to the solution progress and the quality of the solution. As a further description of the above technical solution, the parallel computing and distributed architecture module utilizes parallel computing technology to decompose computing tasks, and adopts distributed architecture and data management technology to process high-dimensional data and massive samples, thereby improving solution efficiency. As a further description of the above technical solution, the result evaluation and verification module uses multiple evaluation indicators such as the optimality of the objective function value, the degree of constraint satisfaction, and the stability of the solution to verify the results, and displays the results through visualization technology. As a further description of the above technical solution, the evaluation indicators of the result evaluation and verification module specifically include the relative error or absolute error with the known optimal solution, the number and degree of constraint violation, and the change of the solution after a small perturbation. As a further description of the above technical solution, the parameter adaptive learning and optimization module automatically adjusts the parameters of the optimization algorithm according to the algorithm running status and problem characteristics, such as the crossover probability, mutation probability and selection pressure of the genetic algorithm, and the inertia weight and learning factor of the particle swarm optimization algorithm. As a further description of the above technical solution, a method for solving an optimization problem system includes the following steps: Step 1: Problem Input and Analysis. Users input an optimization problem description through the system's interface. The input can be a mathematical expression, text description, or a specific data format. The problem analysis and modeling module analyzes the input, identifies the objective function, constraints, and decision variables, determines the problem type, and transforms it into a standard mathematical model. Step Two: Parameter Setting and Algorithm Selection. Users set relevant parameters in the system based on the nature and requirements of the problem, such as algorithm execution parameters and constraint handling parameters. The hybrid intelligent optimization algorithm engine intelligently selects appropriate algorithms or combinations of algorithms for solving the problem based on its type, scale, and user-defined parameters. Step 3: Constraint handling and solution. The constraint handling and adaptive adjustment mechanism processes the constraints in the problem, transforming equality and inequality constraints into forms suitable for solution. The hybrid intelligent optimization algorithm engine begins the solution process according to the selected algorithm. During the solution process, the algorithm parameters are adaptively adjusted based on the characteristics of the problem and the operation of the algorithm, such as the crossover probability and mutation probability of the genetic algorithm, and the inertia weight and learning factor of the particle swarm optimization algorithm. Step 4: Parallel computing acceleration. For large-scale complex problems, the parallel computing and distributed architecture modules decompose the computational task into multiple subtasks and distribute them across multiple computing nodes or processors for parallel processing. During the computation process, distributed storage and data management technologies are used to ensure efficient data access and processing, thereby improving solution efficiency. Step 5: Result Evaluation and Verification. After the solution is obtained, the result evaluation and verification module uses various evaluation indicators to assess the results, including the optimality of the objective function value, the degree of constraint satisfaction, and the stability of the solution. The reliability and validity of the results are verified through comparative analysis with known optimal solutions, benchmark solutions, or historical data. Step Six: Result Output and Visualization. Present the solution results to the user in an intuitive way, including the optimal solution and objective function value in text format, as well as the results displayed using visualization techniques such as bar charts, cumulative line charts, and scatter plots, to help users better understand and analyze the results. If the user is not satisfied with the results, they can return to the previous steps, adjust the parameters, or select a different algorithm to solve the problem again.

[0006] Compared with the prior art, the beneficial effects of the present invention are: 1. This invention automatically determines the nature of the problem, objective function, constraints, decision variables, and data volume; automatically estimates the computational power requirements and solution time; and intelligently matches the appropriate algorithm. For smaller-scale combinatorial optimization problems, an exhaustive search algorithm is directly used; for medium-scale combinatorial optimization problems, it automatically switches to more computationally efficient genetic algorithms, Monte Carlo algorithms, or particle swarm optimization algorithms. Both of these scenarios can be solved on a typical personal computer in minutes, greatly facilitating use by general marketing and research personnel (such as optimizing advertising campaigns or product concept testing combinations). For large-scale combinatorial optimization problems, such as selecting a large number of social media accounts to push promotional information or building an efficient media matrix, the system automatically recommends running the solution on a server equipped with parallel computing and distributed architecture modules, and intelligently suggests various settings and parameters for user convenience.

[0007] 2. This invention, through the synergistic effect of a hybrid intelligent optimization algorithm engine and constraint processing and adaptive adjustment mechanism, enables the system to explore the solution space more effectively, avoid getting trapped in local optima, and thus find a solution closer to the global optimum. In practical applications, compared with traditional methods, it can significantly reduce the error between the objective function value and the optimal value, improve the accuracy and reliability of the solution results, and provide stronger support for decision-making in various fields. The application of parallel computing and distributed architecture modules allows the system to fully utilize computing resources, process complex computational tasks in parallel, and significantly shorten the solution time. Especially for large-scale complex optimization problems, such as selecting a large number of social media accounts to push promotional information or building an efficient media matrix, results can be obtained in a shorter time, meeting the timeliness requirements of practical applications, improving work efficiency, and providing new possibilities for the direct deployment and use of such algorithms in enterprise data management platforms (DMP), customer middleware (DMP), or real-time bidding systems (RTB) to achieve more efficient intelligent decision-making.

[0008] 3. The system and solution method of this invention can be extended to various types of optimization problems without requiring extensive customization and modification for different problems. Whether it's linear programming, nonlinear programming, or integer programming, all problems can be solved within the same system, reducing application complexity and cost. Simultaneously, the parameter adaptive learning and optimization module enables the system to automatically adjust algorithm parameters according to the characteristics of different problems, further improving the system's adaptability and self-learning ability. The result evaluation and verification module provides users with comprehensive and intuitive analysis of the solution results through various evaluation indicators and visualization techniques. Users can clearly understand the quality of the solution results, constraint satisfaction, and solution stability, which helps in a deeper understanding of the problem and making scientific decisions, increasing user trust in the solution results and their application capabilities. Attached Figure Description

[0009] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this invention. For those skilled in the art, other drawings can be obtained based on these drawings.

[0010] Figure 1 A system diagram illustrating an optimization problem solving system provided by the present invention; Detailed Implementation

[0011] The following specific embodiments illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0012] To enable those skilled in the art to better understand the present application, the present application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0013] See attached document Figure 1 The optimization problem solving system of this embodiment consists of the following key parts: Problem Analysis and Modeling Module: This module receives user-input descriptions of optimization problems, which can be in the form of mathematical expressions, text descriptions, or specific data formats. It accurately analyzes the objective function, constraints, and decision variables within the problem, classifying and labeling them according to the problem's characteristics and type. For example, for a media placement optimization problem, it can accurately identify the weighted average reach rate as the objective function, target audience demographics, available media time slots, and total placement cost as constraints, and the cost per thousand impressions for each media and the placement volume for each media as decision variables. By transforming the problem into a standardized mathematical model, it lays a solid foundation for the subsequent solution process, ensuring the system can accurately understand and process the problem.

[0014] Hybrid Intelligent Optimization Algorithm Engine: This engine integrates multiple advanced intelligent optimization algorithms, including but not limited to improved genetic algorithms, Monte Carlo algorithms, adaptive particle swarm optimization algorithms, and algorithms combining simulated annealing and tabu search. These algorithms are carefully designed and optimized, each possessing unique search strategies and advantages. For example, the improved genetic algorithm introduces a dynamic crossover and mutation probability adjustment mechanism, which can automatically optimize genetic operations based on the evolutionary state of the population, enhancing population diversity and search capabilities; the adaptive particle swarm optimization algorithm can dynamically adjust inertia weights and learning factors based on particle flight trajectories and search performance, enabling particles to search more effectively in the solution space. During the solution process, the engine can intelligently select and combine these algorithms according to the characteristics and scale of the problem, fully leveraging their advantages to achieve an efficient solution process.

[0015] Constraint Handling and Adaptive Adjustment Mechanism: This mechanism employs a series of innovative methods to address constraints in optimization problems. For equality constraints, an improved version of the efficient Lagrange multiplier method is used to quickly and accurately find solutions that satisfy the equality constraints. For inequality constraints, a method based on a dynamic penalty function is used to dynamically adjust the penalty intensity according to the degree of constraint violation, guiding the search process closer to the feasible solution region. Simultaneously, this mechanism also possesses adaptive adjustment capabilities, dynamically adjusting the constraint handling strategy and parameters based on the problem's progress and the quality of the current solution, ensuring that both constraint conditions are satisfied and good search efficiency and accuracy are maintained throughout the entire solution process.

[0016] Parallel Computing and Distributed Architecture Module: To improve solution efficiency, especially for large-scale complex optimization problems, this module utilizes parallel computing techniques and a distributed architecture. It decomposes complex computational tasks into multiple subtasks and distributes them across multiple computing nodes or processors for parallel processing. For example, when evaluating the objective function values ​​and constraints of a large number of candidate solutions, computation can be performed simultaneously on multiple computing units, significantly reducing computation time. By employing distributed storage and data management technologies, it can effectively handle high-dimensional data and massive samples, ensuring the system's stability and efficiency when dealing with large-scale problems.

[0017] The Result Evaluation and Verification Module: After the solution process is completed, this module performs a comprehensive and in-depth evaluation and verification of the results. It employs various advanced evaluation metrics, such as the optimality of the objective function value (e.g., relative or absolute error compared to the known optimal solution), constraint satisfaction metrics (statistically analyzing the number and degree of constraint violations), and solution stability metrics (re-evaluating changes in the results after applying minor perturbations to the solution). Simultaneously, by comparing and analyzing with existing benchmark solutions, historical data, or other reference results, the reliability and effectiveness of the solution are further verified. Furthermore, this module utilizes visualization techniques, such as creating bar charts, cumulative line graphs, and scatter plots, to intuitively display key information and characteristics of the solution results, helping users better understand and analyze the results.

[0018] The parameter adaptive learning and optimization module automatically learns and adjusts the parameters of the optimization algorithm based on its running status and the characteristics of the problem throughout the solution process. For example, it can monitor indicators such as population diversity and convergence speed of the genetic algorithm. When it detects a decrease in population diversity or a slow convergence speed, it automatically adjusts parameters such as crossover probability, mutation probability, and selection pressure to promote population evolution and the generation of new solutions. For the particle swarm optimization algorithm, it dynamically adjusts parameters such as inertia weight and learning factor based on the particle's trajectory and search performance, enabling the particles to search the solution space more effectively. Through this adaptive learning and optimization mechanism, the system can continuously improve its solution performance and better adapt to optimization problems of different types and scales.

[0019] The steps for solving optimization problems are as follows: Problem Input and Analysis: Users input an optimization problem description through the system's interface. The input can be a mathematical expression, a text description, or a specific data format. The Problem Analysis and Modeling module analyzes the input, identifies the objective function, constraints, and decision variables, determines the problem type, and transforms it into a standard mathematical model.

[0020] Parameter settings and algorithm selection: Users set relevant parameters in the system according to the nature and requirements of the problem, such as algorithm running parameters and constraint handling parameters. The hybrid intelligent optimization algorithm engine intelligently selects appropriate algorithms or combinations of algorithms for solving the problem based on its type, scale, and user-set parameters.

[0021] Constraint Handling and Solution: The constraint handling and adaptive adjustment mechanism processes the constraints in the problem, transforming equality and inequality constraints into forms suitable for solution. The hybrid intelligent optimization algorithm engine begins the solution process according to the selected algorithm. During the solution process, it adaptively adjusts algorithm parameters based on the characteristics of the problem and the algorithm's performance, such as the crossover probability and mutation probability of the genetic algorithm, and the inertia weight and learning factor of the particle swarm optimization algorithm.

[0022] Parallel computing acceleration: For large-scale complex problems, parallel computing and distributed architecture modules decompose the computational task into multiple subtasks, which are then distributed across multiple computing nodes or processors for parallel processing. During the computation process, distributed storage and data management technologies are used to ensure efficient data access and processing, thereby improving solution efficiency.

[0023] Results Evaluation and Verification: After the solution is obtained, the results evaluation and verification module uses various evaluation indicators to assess the results, including the optimality of the objective function value, the degree of constraint satisfaction, and the stability of the solution. The reliability and validity of the results are verified through comparative analysis with known optimal solutions, benchmark solutions, or historical data.

[0024] Results Output and Visualization: The solution results are presented to the user in an intuitive way, including the optimal solution and objective function value in text form, as well as the results displayed through visualization techniques such as bar charts, line charts, and scatter plots, to help users better understand and analyze the results. If the user is not satisfied with the results, they can return to the previous steps, adjust the parameters, or select a different algorithm to solve the problem again.

[0025] Example: Users input an optimization problem description through the system's user-friendly interface. For example, in a media advertising optimization problem, a user might input something like selecting 7 media outlets from 30 candidate media (M1, M2, ..., M30) for a combined campaign. The reach (R_i) and cost per campaign (C_i) of each media outlet are known, as well as the audience overlap coefficient (O_ij, representing the proportion of people simultaneously reached by both media outlets) between any two media outlets (i,j). The goal is to find the combination of 7 media outlets that maximizes the total unique reach (i.e., the total reach after deduplication) while keeping the total cost within budget B. Alternatively, users can upload a structured data file containing the aforementioned media attributes (reach, cost) and an overlap coefficient matrix.

[0026] The problem analysis and modeling module parses the input. Identify decision variables: Define a binary variable X_i for each media outlet (1 indicates selection, 0 indicates no selection).

[0027] Identify the objective function: Maximize the total number of unique reachers. This is typically not a simple linear summation (Σ (R_i * X_i)) because of overlap (O_ij). The module needs to model it as a more complex function, such as using the inclusion-exclusion principle or other estimation methods to represent the deduplicated total reach as f(X) = Σ(R_i * X_i) - Σ Σ (O_ij * min(R_i, R_j) * X_i * X_j) + ... (higher-order overlap terms).

[0028] Identify constraints: Combination width constraint: Σ X_i = 7 (select 7 companies).

[0029] Total budget constraint: Σ (C_i * X_i) <= B.

[0030] (Optional) Other constraints: such as mutual exclusion constraints (media M1 and M5 cannot be selected at the same time, X1 + X5 <= 1), mandatory inclusion constraints (must include M2, X2 = 1), etc.

[0031] Problem type: This is an integer programming (especially 0-1 programming) problem with a complex nonlinear objective function and linear constraints.

[0032] Transform into a standard model: Transform the objective function Max f(X) into Min -f(X), and combine all constraints to form a standard optimization model within the system.

[0033] Applications of hybrid intelligent optimization algorithms: The hybrid intelligent optimization algorithm engine intelligently selects algorithms or combinations based on the problem type (integer programming, nonlinear objective) and scale (choose 3 out of 10, small scale but complex objective function). For example, it may initiate an improved genetic algorithm.

[0034] Initialize the population: Randomly generate a certain number of individuals (chromosomes), each representing a possible 7-media combination scheme (e.g., a binary string of length 30 with exactly 7 ones).

[0035] Calculate fitness: Fitness(X) = f(X) (the objective function value, i.e., the estimated total number of independent reach). If a program violates the budget constraint (Σ (C_i * X_i) > B), a large penalty term is imposed (e.g., Fitness(X) = f(X) - Penalty * (Σ (C_i * X_i) - B)), severely reducing its fitness.

[0036] Genetic manipulation: Selection (e.g., tournament selection): Select individuals with high fitness (representing a good media combination) to enter the mating pool.

[0037] Crossover (e.g., single-point crossover): Based on dynamic crossover probability, two parent individuals are randomly selected, and a portion of their chromosomes is exchanged to generate offspring. For example, chromosomes are cut at random locations, and segments are exchanged. Dynamic probability ensures high diversity in the early stages, while later stages favor superior individuals.

[0038] Mutation (e.g., bit flipping mutation): Based on the dynamic mutation probability, certain gene positions on the chromosomes of offspring individuals are randomly changed (from 0 to 1 or 1 to 0). After mutation, it is necessary to check whether the constraint of selecting 7 individuals is still satisfied (if not, it is repaired or discarded). The dynamic probability increases when the population converges to escape local optima.

[0039] Iteration and Termination: Repeated selection, crossover, and mutation generate a new population. The algorithm dynamically adjusts the crossover and mutation probabilities based on the population's evolutionary state (such as changes in average fitness and the number of generations the optimal solution has not been updated). It stops when the maximum number of iterations is reached or the optimal solution becomes stable.

[0040] Constraint handling and adaptive adjustment: Equality Constraint (Example): Suppose the problem adds a constraint: "If media M9 (premium magazine) is selected, it must also be published in the weekend edition (additional cost C_w)." This can be modeled as: X9 - X_w = 0 (where X_w is an auxiliary binary variable representing whether the weekend edition is published). The improved Lagrange multiplier method is used to handle this equality constraint. Construct the Lagrange function L(X, λ) = -f(X) + λ(X9 - X_w). During the search process, the algorithm adaptively adjusts the value of the Lagrange multiplier λ and the update step size according to the degree to which the current solution violates the constraint (X7 - X_w), so that the solution satisfies X9 = X_w.

[0041] Inequality constraint (example): Core constraint Σ (C_i * X_i) <= B (total budget constraint). The system uses a method based on a dynamic penalty function. Define the penalty term P(X) = max(0, (Σ (C_i * X_i) - B))^2. Modify the fitness function to Fitness(X) = f(X) - μ * P(X). The penalty coefficient μ is dynamically adjusted: Initially: μ is set relatively small, allowing the algorithm to explore some slightly over-budget but potentially promising solutions (which may be satisfied by fine-tuning the cost).

[0042] Later stage: If it is found that many individuals in the population are over budget or the optimal solution is continuously over budget, then μ is gradually increased to impose a more severe penalty on over-budget solutions and strongly guide the search to the budget region.

[0043] The system also monitors the prevalence and severity of constraint violations and adaptively adjusts the form of the penalty function (such as linear penalty, quadratic penalty) and the growth strategy.

[0044] Parallel computing and distributed processing: For larger-scale problems, such as selecting 50 people from 1,000 potential KOCs to form a promotion team, the goal is to maximize the team's total influence and the diversity of the audience reached (the objective function is complex), while satisfying constraints such as total cooperation costs, complementary fan profiles of each KOC, and balanced platform distribution.

[0045] Task decomposition: The parallel computing and distributed architecture modules decompose the massive solution space evaluation task. For example, the master node generates a batch of candidate KOC combination schemes (e.g., 100 schemes).

[0046] Parallel evaluation: These schemes are distributed across different computing nodes (CPU cores or servers). Each node is responsible for calculating the objective function values ​​(influence, diversity) and constraint violations (total cost, profile complementarity, platform distribution) for one or more assigned schemes. The computation involves querying the database to obtain detailed data for each KOC and calculating complex overlap / complementarity metrics.

[0047] Data Management: KOC attribute data (number of followers, profile tags, cooperation costs, platform distribution) is stored in a distributed database or file system (such as HDFS). Compute nodes efficiently pull the required data fragments from the distributed storage for computation based on the KOC ID included in the solution.

[0048] Results Summarization: Each node returns its calculation results (target value of the scheme, constraint violation value) to the master node. The master node summarizes the results for genetic operations such as selection and crossover. This method greatly accelerates the evaluation process of large-scale combinations.

[0049] Results evaluation and validation: Objective function optimality: The report measures the total number of unique reach (media case) or the combined influence / diversity score (KOC case) of the optimal solution found. If a known optimal solution exists (e.g., exhaustive results for small-scale problems) or an industry benchmark, the relative error |(Found - Best) / Best| is calculated.

[0050] Constraint Satisfaction: Check if all constraints are satisfied. Report the number of constraints violated (should be 0) and details of key constraint satisfaction (e.g., actual total cost vs. budget B).

[0051] Solution stability: Make a small perturbation to the optimal media mix (e.g., replace one media outlet with similar cost or coverage) and recalculate the reach. Observe whether the difference between the new reach and the original optimal value is within an acceptable range (small change). For KOC combinations, replace a small number of KOCs with similar attributes and observe the change in effect. Visual presentation: Media combination case study: The bar chart comparing the combined performance shows the reach, cost, and net contribution of each media outlet to the total independent reach of the optimal combination (considering overlap) for each outlet. It also displays key metrics comparisons of 1-2 other high-performing alternative combinations.

[0052] Budget usage and reach relationship line graph: Shows the maximum number of independent reach achievable under different budget levels B (fine-tuned near the optimal combination), helping to understand the marginal benefits of the budget.

[0053] Audience overlap heatmap: Displays the audience overlap coefficient O_ij between each pair of media within the optimal combination, intuitively showing the synergy / redundancy within the combination.

[0054] KOC combination case study: Profile Coverage Radar Chart: Shows the breadth of coverage of the optimal KOC combination across different profile dimensions of the target users (such as age, interests, region, and consumption level).

[0055] Platform Distribution Stacked Bar Chart: Shows the distribution of KOCs across major social media platforms (WeChat, Weibo, Douyin, Xiaohongshu, etc.).

[0056] Influence-Cost Scatter Plot: Plot all candidate KOCs as a scatter plot based on influence (vertical axis) and cost (horizontal axis), highlighting the KOCs selected for the optimal combination and observing their distribution characteristics (such as high cost-effectiveness areas).

[0057] Results output and visualization: The system outputs the final optimal solution (selected media / KOC list), optimal objective function value (maximum unique reach / influence score), actual cost, and other key evaluation metrics in clear text format. It also generates various visualization charts (bar charts, line charts, heatmaps, radar charts, scatter plots) mentioned above to help marketers: Understand why this combination is optimal (the contribution of each component, synergistic effect).

[0058] Assess the portfolio’s risk and robustness (stability test results).

[0059] Conduct a sensitivity analysis (the impact of budget changes).

[0060] Compare the characteristics of different excellent solutions.

[0061] If the user is not satisfied with the results (e.g., costs exceed expectations, goals are not achieved), they can go back and adjust parameters (e.g., increase the budget, modify constraint weights), try different algorithm combinations, or fine-tune the problem description and solve it again.

[0062] In conclusion, the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A system for solving optimization problems, characterized in that: It includes a problem analysis and modeling module, a hybrid intelligent optimization algorithm engine, a constraint handling and adaptive adjustment mechanism, a parallel computing and distributed architecture module, a result evaluation and verification module, and a parameter adaptive learning and optimization module. The problem analysis and modeling module is used to receive the optimization problem description, analyze the objective function, constraints and decision variables, and transform them into a standard mathematical model. The hybrid intelligent optimization algorithm engine integrates an improved genetic algorithm, an adaptive particle swarm optimization algorithm, and an algorithm combining simulated annealing and tabu search, and can intelligently select and combine algorithms according to the characteristics and scale of the problem.

2. The optimization problem solving system according to claim 1, characterized in that: The problem parsing and modeling module can recognize various forms of problem input, including mathematical expressions, text descriptions, and specific data formats, and accurately classify and label problem elements.

3. The optimization problem solving system according to claim 2, characterized in that: The improved genetic algorithm in the hybrid intelligent optimization algorithm engine has a dynamic crossover and mutation probability adjustment mechanism, and the adaptive particle swarm optimization algorithm can dynamically adjust the inertia weight and learning factor.

4. The optimization problem solving system according to claim 3, characterized in that: The constraint handling and adaptive adjustment mechanism employs an improved Lagrange multiplier method to handle equality constraints and a dynamic penalty function method to handle inequality constraints. It can also adaptively adjust strategies and parameters based on the solution progress and solution quality.

5. The optimization problem solving system according to claim 4, characterized in that: The parallel computing and distributed architecture module utilizes parallel computing technology to decompose computing tasks and employs distributed architecture and data management technology to process high-dimensional data and massive samples, thereby improving solution efficiency.

6. The optimization problem solving system according to claim 5, characterized in that: The result evaluation and verification module uses multiple evaluation indicators such as the optimality of the objective function value, the degree of constraint satisfaction, and the stability of the solution to verify the results, and displays the results through visualization technology.

7. The optimization problem solving system according to claim 6, characterized in that: The evaluation metrics of the result evaluation and verification module specifically include the relative or absolute error with the known optimal solution, the number and degree of constraint violations, and the changes in the solution after small perturbations.

8. The optimization problem solving system according to claim 7, characterized in that: The parameter adaptive learning and optimization module automatically adjusts the parameters of the optimization algorithm based on the algorithm's running status and problem characteristics.

9. The method for solving the optimization problem system according to any one of claims 1-8, characterized in that, Includes the following steps: Step 1: Problem Input and Analysis. Users input an optimization problem description through the interface provided by the system. The input can be in the form of mathematical expressions, text descriptions, or specific data formats. The problem analysis and modeling module analyzes the input, identifies the objective function, constraints, and decision variables, determines the problem type, and transforms it into a standard mathematical model. Step 2: Parameter setting and algorithm selection. Users set relevant parameters in the system according to the nature and requirements of the problem. The hybrid intelligent optimization algorithm engine intelligently selects a suitable algorithm or combination of algorithms to solve the problem based on the type and scale of the problem and the parameters set by the user. Step 3: Constraint handling and solution. The constraint handling and adaptive adjustment mechanism processes the constraints in the problem, transforming equality and inequality constraints into forms suitable for solution. The hybrid intelligent optimization algorithm engine begins the solution process according to the selected algorithm. During the solution process, the algorithm parameters are adaptively adjusted based on the characteristics of the problem and the operation of the algorithm. Step 4: Parallel computing acceleration. For large-scale complex problems, the parallel computing and distributed architecture modules decompose the computational task into multiple subtasks and distribute them across multiple computing nodes or processors for parallel processing. During the computation process, distributed storage and data management technologies are used to ensure efficient data access and processing, thereby improving solution efficiency. Step 5: Result Evaluation and Verification. After the solution is completed, the result evaluation and verification module uses a variety of evaluation indicators to evaluate the results. By comparing and analyzing with known optimal solutions, benchmark solutions or historical data, the reliability and validity of the results are verified. Step Six: Result Output and Visualization. Present the solution results to the user in an intuitive way to help the user better understand and analyze the results. If the user is not satisfied with the results, they can return to the previous steps, adjust the parameters, or select a different algorithm to solve the problem again.

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