Multi-objective transportation network optimization method based on adaptive constraint relaxation strategy

Abstract

The present invention discloses a multi-objective transportation network optimization method based on an adaptive constraint relaxation strategy, comprising the following steps: inputting multiple objectives for transportation network optimization; initializing a convergence archive and a guidance archive; initializing a population; performing iterations: selecting a parent from the convergence archive using a tournament selection mechanism; mutating the selected parent to generate offspring by simulating binary crossover and polynomial mutation; updating the archive using an adaptive constraint relaxation method, while simultaneously updating the convergence archive using a local convergence index and a diversity comprehensive selection method; and outputting an optimized transportation network optimization solution after the iterations are completed. This application enhances the adaptability of constraint processing through the ACR method, achieving a dynamic balance between exploration and utilization; and introduces a diversity comprehensive selection mechanism to maintain the diversity of solutions, ensuring thorough exploration and robust convergence.

Description

Technical Field

[0001] The present invention belongs to the technical field of traffic network optimization, and in particular relates to a multi-objective traffic network optimization method based on an adaptive constraint relaxation strategy. Background Art

[0002] Constrained multi-objective optimization problems (CMOPs) exist in the field of transportation network optimization, requiring the simultaneous consideration of multiple conflicting objectives and constraints. Key conflicting objectives include minimizing travel time versus reducing operating costs, maximizing traffic efficiency versus ensuring fairness, infrastructure investment versus environmental protection, and responding to dynamic demand versus system stability. Core constraints include road capacity and flow limits, budget and resource constraints, and user demand volatility. Traffic demand varies significantly between peak and off-peak hours, requiring optimization models to accommodate dynamic adjustments, such as through robust optimization to address demand uncertainty.

[0003] These problems are inherently complex due to the presence of multiple objective functions and a large number of constraints. Solving them efficiently is crucial to finding solutions that are not only optimal but also feasible in real-world scenarios.

[0004] In recent years, significant progress has been made in developing algorithms for solving CMOPs. A popular approach within the evolutionary computation (EC) community utilizes a co-evolutionary framework that incorporates auxiliary archives to help quickly identify solutions on the unconstrained Pareto front, thereby steering the evolutionary process away from local optima. However, these auxiliary archives traditionally do not consider constraints, instead focusing on the unconstrained Pareto front (UPF) to guide the evolutionary process. This strategy has been shown to be effective on several benchmark problems. However, because the auxiliary archives do not consider constraints, these methods can lead to unnecessary waste of computational resources.

[0005] Furthermore, existing research has a significant gap in addressing the trade-off between solution diversity and convergence. While these archives help quickly identify potential solutions, they often rarely consider the diversity of solutions in the decision space, leading to a poor solution distribution, especially for problems characterized by a discontinuous Pareto front (PF). Furthermore, reliance on these auxiliary archives can inadvertently overlook solutions residing on the constrained Pareto front (CPF), disseminating incorrect information and wasting precious computational resources. Summary of the Invention

[0006] The need to balance these two conflicting objectives (convergence and diversity) is key to this application. This application uses a nuanced approach that accounts for the stochastic nature of constraints and promotes adaptive local convergence to overcome shortcomings found in current state-of-the-art algorithms. Most previous constrained multi-objective (CMOEA) algorithms struggle to obtain the full PF when multiple separate feasible regions exist due to a lack of population diversity in the decision space. Furthermore, many algorithms tend to introduce leading profiles in the final stages, such as Population 2, which deviates from the true CPF. In this case, Population 2 wastes computational resources and may even provide erroneous information.

[0007] To address these drawbacks and promote a harmonious balance between solution diversity and convergence, thereby improving the effectiveness and efficiency of the optimization process, this application fully considers the nuances associated with CMOP and integrates innovative strategies to more cleverly guide the evolution process towards the optimal solution while ensuring a robust representation of the solution space.

[0008] To achieve the above objectives, the multi-objective transportation network optimization method based on the adaptive constraint relaxation strategy disclosed in this application includes the following steps:

[0009] Input multiple objectives of transportation network optimization;

[0010] Initialize convergence file and boot files ; Convergence File and boot files Serves as a repository of solutions, with the former focusing on convergence and the latter on diversity; calculates the fitness of solutions in the convergence archive to guide the selection process, ensuring that solutions with higher convergence potential are prioritized;

[0011] Population initialization;

[0012] Iterations are performed: parents are selected from the convergence archive using a tournament selection mechanism; the selected parents are mutated to generate offspring by simulating binary crossover and polynomial mutation; the archive is updated using an adaptive constraint relaxation method, i.e., in early generations, various solutions are explored without considering constraints, and a multi-objective evolutionary algorithm is used to update the bootstrap archive for rapid search of the Pareto frontier; in later generations, an adaptive constraint relaxation method is used to converge, aiming to improve the utilization of function evaluation; simultaneously, the convergence archive is updated using a local convergence indicator and a diversity comprehensive selection method;

[0013] After the iteration is completed, the optimized traffic network optimization solution is output.

[0014] Furthermore, the method of using the adaptive constraint relaxation method to converge includes: determining whether to consider the constraint ; The constraints The calculation expression is:

[0015] ;

[0016] ;

[0017] ;

[0018] in and Representative Convergence Profile the average distances between the median solutions and the distances between the bootstrap archive and the convergence archive; Indicates that the distance between two files is less than the preset threshold. are the i-th and j-th solutions in the convergence archive, respectively, yes The Euclidean distance of yes The plan in .

[0019] Furthermore, the multi-objective evolutionary algorithm includes at least one of a feasibility-driven multi-objective evolutionary algorithm, a target and feasibility balanced multi-objective evolutionary algorithm, an evolutionary operator modified multi-objective evolutionary algorithm, and a hybrid strategy-based multi-objective evolutionary algorithm; the early generation is a stage in which the number of iterations is less than or equal to half of the maximum number of iterations.

[0020] Furthermore, the local convergence index is expressed as follows:

[0021] ;

[0022] ;

[0023] ;

[0024] ;

[0025] ;

[0026] express Neighborhood solution set of ; and are local dominance relations and constraint dominance relations respectively, is the local dominant intensity, are the constraint violation degrees of the i-th and j-th solutions, respectively.

[0027] Furthermore, the diversity comprehensive selection process includes:

[0028] After obtaining the local optimal solution through the local convergence index, the population size is truncated to ; Calculated using a method based on crowding distance:

[0029] ;

[0030] in Represents the solution and The Euclidean distance of

[0031] When the number of remaining solutions is greater than When will be discarded; this process is iterated until the population size is equal to .

[0032] The beneficial effects of this application include:

[0033] Develop ACR methods to enhance the adaptability of constraint handling and achieve a dynamic balance between exploration and exploitation;

[0034] A diversity comprehensive selection mechanism is introduced to maintain the diversity of solutions and ensure thorough exploration and robust convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0035] Figure 1 Pseudo code of this application method.

[0036] Figure 2 Flowchart of the method of this application. DETAILED DESCRIPTION

[0037] The present invention will be further described below with reference to the accompanying drawings, but the present invention is not limited in any way. Any changes or substitutions made based on the teachings of the present invention fall within the scope of protection of the present invention.

[0038] Constrained multi-objective evolutionary algorithms (CMOEAs) play a key role in solving complex optimization problems that require balancing multiple objectives and constraints. These algorithms are essential in fields such as engineering, economics, and environmental science, where solutions must be both optimal and feasible. However, existing CMOEAs often struggle to maintain a balance between convergence and diversity, especially when faced with complex constraints and incoherent feasible regions. This imbalance can lead to suboptimal solutions and inefficient resource utilization.

[0039] To address these challenges, this application introduces an adaptive constraint relaxation (ACR) method that dynamically adjusts constraint handling based on the evolutionary state. This adaptability enables this application to efficiently navigate the search space, balancing the needs of early exploration and late exploitation. By doing so, this application effectively avoids common pitfalls such as premature convergence and local optima. In addition, this application employs a diversity-integrated selection mechanism that uses local convergence metrics to maintain a diverse set of solutions. This mechanism ensures that the algorithm thoroughly explores the solution space, preventing it from being trapped in a narrow area and promoting a well-distributed Pareto front.

[0040] Before introducing the embodiments of the present application, some terms involved in the present application are explained.

[0041] 1. Minimizing CMOP can be expressed as follows:

[0042] ;

[0043] in It is the decision space The decision vector in yes The target vector. and are the lower and upper bounds respectively. and separately It is inequality and equality constraints, the number of which is and . Constraint violation It is usually calculated as follows:

[0044] ;

[0045] Overall constraint violation Defined as

[0046] ;

[0047] Constrained multi-objective optimization problems include unconstrained Pareto front (UPF) and constrained Pareto front (CPF). There are two disconnected CPFs, and solution A is non-dominated with respect to both CPF and UPF.

[0048] Given a decision variable vector (i.e., a solution), it is feasible only if the overall constraint violation is zero. For two feasible solutions, and , if for and for, It is called dominant A solution is called Pareto optimal when no other feasible solution dominates it. All Pareto optimal solutions in the decision space constitute a Pareto optimal set.

[0049] The graph of all Pareto optimal solutions in the objective space is the Pareto front (PF). To distinguish them, the terms constrained PF (CPF) and unconstrained PF (UPF) are often used to represent the PF for constrained and unconstrained multi-objective optimization problems. When solving a CMOP, the goal is to approximate the PF with a set of well-converged and well-distributed feasible solutions.

[0050] The algorithm pseudo code provided in this application is referenced Figure 1 . refer to Figure 2 The structure of this framework is the same as that of a typical EA, including population initialization, offspring generation, and environment selection. The main differences are two aspects: this application uses two archives for co-evolution, and is divided into two stages. The framework starts with initializing two key archives: the convergence archive ( ) and the boot file ( These archives serve as repositories of solutions, with the former focusing on convergence and the latter on diversity. The fitness of the solutions in the convergence archive is calculated to guide the selection process, ensuring that solutions with higher convergence potential are prioritized.

[0051] The algorithm proceeds in an iterative manner, using a tournament selection mechanism to select parents from the converged archive. The selected parents are mutated to generate offspring through simulated binary crossover (SBX) and polynomial mutation (PM). The framework is divided into two different stages: in the early generation ( ), the normal MOEA is used to update the bootstrap archive. This phase emphasizes exploring various solutions without considering constraints, in order to quickly search the Pareto front. In later generations, the focus shifts to using adaptive constraint relaxation methods to achieve convergence, aiming to improve the utilization of function evaluations.

[0052] Homologically, the convergence profile is updated using a local convergence metric and a diversity-based environment selection method. This ensures that the final solution is not only convergent but also diverse, thus addressing the dual goals of the optimization process. By carefully balancing exploration and exploitation in these two phases, this application effectively navigates the complex landscape of CMOP, providing a robust and adaptive method that improves the quality and diversity of the obtained solutions.

[0053] In one embodiment, guiding the solution update in the archive - adaptive constraint relaxation method includes:

[0054] Using auxiliary archives that don't consider constraints can effectively help cross infeasible regions, thereby improving the CMOEA's search capabilities. However, in the later stages of evolution, once the UPF differs from the CPF, the information from these auxiliary archives not only misleads the algorithm but also wastes computational resources. To address this issue, this application proposes an adaptive constraint relaxation (ACR) method. The main idea of ​​ACR is to control whether the leading archive considers constraints by evaluating the current evolutionary state.

[0055] Specifically, in the first stage, the leading archive helps promote the algorithm to converge to the true Pareto front without considering the constraints. When the evolution enters the second stage, the algorithm will be based on The value of determines whether the constraint is considered. Specifically, The calculation expression is:

[0056] ;

[0057] ;

[0058] ;

[0059] in and Represents the average distance between solutions in the convergence archive and the distance between the bootstrap profile and the convergence profile. Indicates that the distance between the two archives is less than a given threshold. In this case, more exploration is needed, so the bootstrap archive will not be considered for constraints. Conversely, the bootstrap archive is far from the converged archive and the constraints should be considered to avoid wasting computational resources.

[0060] ACR dynamically adjusts constraint handling based on the state of evolution. Initially, it relaxes constraints to enhance exploration, allowing the algorithm to traverse infeasible regions that may contain promising solutions. By dynamically shifting focus, ACR facilitates efficient exploration of the search space and converges toward feasible regions as the process progresses. This ensures that the search remains adaptive and responsive to the problem environment. Furthermore, ACR optimizes the use of computing resources by adaptively directing effort at different stages of the search, thereby improving overall algorithmic efficiency.

[0061] Here, we discuss the key role played by the Convergence Archive in ACREA. This archive is a repository of solutions that have demonstrated good convergence properties. This application analyzes the solution update mechanism in this archive, focusing on the strategies implemented to maintain a harmonious balance between convergence and diversity.

[0062] In the context of convergence profiles, the local convergence metric is an important tool for guiding the evolution process. In this section, we delve deeper into the functionality of this metric, showing how it can help guide the solution search towards the optimal frontier while avoiding local optima.

[0063] In one embodiment, the selection method based on the local convergence index in the convergence profile includes:

[0064] The local convergence index proposed for multi-modal multi-objective optimization problems can effectively enhance the population diversity in the decision space. However, since it is proposed for unconstrained problems, there are serious problems in directly applying the local convergence index to constrained optimization problems. , the solution is the current non-dominated solution. In this case, performing environment selection using the local convergence metric will save the solution However, for constrained optimization problems, the solution , because it is not feasible.

[0065] To avoid this problem, both global constraint violations and local dominance information should be considered simultaneously. Specifically, constraint violations should be compared globally, while convergence information should be considered locally, as in the conventional definition of a local convergence metric. Based on this motivation, in one embodiment, this application proposes a new CMOP local convergence metric, which can be expressed as follows:

[0066] ;

[0067] ;

[0068] ;

[0069] ;

[0070] ;

[0071] Specifically, express Neighborhood solution set of ; and They are local dominance relationship and constraint dominance relationship respectively. It can be regarded as the local dominant intensity.

[0072] By adopting the improved local convergence index, this application can select the potential feasible optimal solution. , and ,only and Can survive because is not feasible. , and ,only It can be retained according to the local convergence index. It can be seen that the solution No neighbors. Then, if the original local convergence metric is used, the solution is the local optimal solution, so it can survive. However, due to For unconstrained problems, it is infeasible and non-dominated, which will release wrong information to mislead the evolution. By adopting a novel local convergence indicator, will not survive because it is dominated by the other solution constraints.

[0073] The diversity-integrated mating selection process is a key step in this application, which helps to generate offspring with good diversity properties, thereby ensuring rich diversity in the solution space, promoting more extensive exploration and more robust representation of potential solutions.

[0074] After obtaining the local optimal solution through the local convergence index, the population size needs to be truncated to . Using the crowding distance based method:

[0075] ;

[0076] in Represents the solution and The Euclidean distance of .

[0077] Specifically, when the number of remaining solutions is greater than When the maximum The solution will be discarded. This process will be iterated until the population size is equal to .

[0078] In the experiment of this application, the population size of the algorithm is Set to 100, the maximum number of function evaluations Set as All experiments are run on PlatEMO with an AMD R9-5900X @ 3.70 GHz and 64G RAM.

[0079] We choose the inverse generation distance (IGD) to evaluate the performance of the algorithm. Specifically, given A set of points uniformly sampled along the PF and a set of feasible target vectors obtained by CMOEA, IGD values The calculation is as follows:

[0080] ;

[0081] in, yes and Note that for CMOEA, a smaller IGD value is required. It should be noted that only feasible solutions can be used to calculate the IGD value.

[0082] Hypervolume (HV) evaluates the convergence and diversity of an algorithm by computing the volume formed between the discovered solution and a reference point. This metric is particularly valuable for evaluating the performance of an algorithm on unknown problems because it does not rely on prior knowledge of the true Pareto front.

[0083] In one embodiment, the multi-objective evolutionary algorithm includes at least one of a feasibility-driven multi-objective evolutionary algorithm, a target and feasibility balance multi-objective evolutionary algorithm, an evolutionary operator modified multi-objective evolutionary algorithm, and a hybrid strategy-based multi-objective evolutionary algorithm.

[0084] Feasibility-driven CMOEAs typically prioritize feasible solutions over infeasible solutions, primarily leveraging feasibility information during the evolutionary search process. Overall, while feasibility-driven CMOEAs are effective in ensuring solution feasibility, they can have limited exploration capabilities and may perform poorly on complex or highly constrained problems.

[0085] Balancing Objectives and Feasibility: CMOEA advocates for maintaining a dynamic balance between objectives and constraints, initially favoring optimization of the objective and later shifting to optimizing the constraints. This allows for dynamic adjustment of the focus between objectives and constraints, leading to a more balanced search process. However, their complexity and resource requirements can be limiting factors, making it difficult to find the optimal balance between feasibility and objective improvement, especially in highly complex problems.

[0086] Evolutionary Operator Modification (CMOEA) focuses on the design of evolutionary operators, customizing them to effectively navigate complex environments, thereby enhancing exploration and convergence. This approach often leads to higher-quality solutions. However, the increased complexity and need for problem-specific customization can increase computational costs and limit general applicability.

[0087] CMOEA based on hybrid strategies uses multiple constraint handling techniques (CHTs). Hybrid strategies can be embodied at different stages of the algorithm, across different populations, or in response to different tasks.

[0088] The above-mentioned multi-objective evolutionary algorithm is the existing technology in this field and will not be described in detail in this application.

[0089] In summary, the above embodiment is one implementation method of the present invention, but the implementation method of the present invention is not limited to the described embodiment. Any other changes, modifications, substitutions, combinations, and simplifications that deviate from the spirit and principles of the present invention should be equivalent replacement methods and are included in the scope of protection of the present invention.

Claims

1. A multi-objective transportation network optimization method based on an adaptive constraint relaxation strategy, characterized by: The following steps are involved: Input multiple objectives of transportation network optimization; Initialize convergence file and boot files ; Convergence File and boot files Serving as a repository of solutions, the former focuses on convergence and the latter on diversity; Calculating the fitness of solutions in the convergence archive to guide the selection process, ensuring that solutions with higher convergence potential are prioritized; Population initialization; Iterations are performed: parents are selected from the converged archive using a tournament selection mechanism; the selected parents are mutated to generate offspring by simulating binary crossover and polynomial mutation; An adaptive constraint relaxation method is used to update the archive. That is, in the early generations, various solutions are explored without considering the constraints, and a multi-objective evolutionary algorithm is used to update the bootstrap archive to quickly search the Pareto frontier. In the later generations, an adaptive constraint relaxation method is used to converge, aiming to improve the utilization rate of function evaluation. At the same time, the convergence archive is updated using a local convergence indicator and a diversity comprehensive selection method. After the iteration is completed, the optimized traffic network optimization solution is output.

2. The multi-objective transportation network optimization method based on the adaptive constraint relaxation strategy according to claim 1 is characterized in that: The method of using the adaptive constraint relaxation method to converge includes: determining whether to consider the constraint ; The constraints The calculation expression is: ； ； ； in and Representative Convergence Profile the average distances between the median solutions and the distances between the bootstrap archive and the convergence archive; Indicates that the distance between two files is less than the preset threshold. are the i-th and j-th solutions in the convergence archive, respectively, yes The Euclidean distance of Is the boot file The plan in .

3. The multi-objective transportation network optimization method based on the adaptive constraint relaxation strategy according to claim 1 is characterized in that: The multi-objective evolutionary algorithm includes at least one of a feasibility-driven multi-objective evolutionary algorithm, a target and feasibility balanced multi-objective evolutionary algorithm, an evolutionary operator modified multi-objective evolutionary algorithm, and a hybrid strategy-based multi-objective evolutionary algorithm; the early generation is a stage in which the number of iterations is less than or equal to half of the maximum number of iterations.

4. The multi-objective transportation network optimization method based on the adaptive constraint relaxation strategy according to claim 2 is characterized in that: The local convergence index It is expressed as follows: ； ； ； ； ； express Neighborhood solution set of ; and are local dominance relations and constraint dominance relations respectively, is the local dominant intensity, are the constraint violation degrees of the i-th and j-th solutions, respectively.

5. The multi-objective transportation network optimization method based on the adaptive constraint relaxation strategy according to claim 4 is characterized in that: The diversity comprehensive selection process includes: After obtaining the local optimal solution through the local convergence index, the population size is truncated to ; Calculated using a method based on crowding distance: ； in Represents the solution and The Euclidean distance of When the number of remaining solutions is greater than When will be discarded; this process is iterated until the population size is equal to .

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