Constraint Optimization Problem Research Articles

Proactive Agent Behaviour in Dynamic Distributed Constraint Optimisation Problems

In multi-agent systems, the Dynamic Distributed Constraint Optimisation Problem (D-DCOP) framework is pivotal, allowing for the decomposition of global objectives into agent constraints. Proactive agent behaviour is crucial in such systems, enabling agents to anticipate future changes and adapt accordingly. Existing approaches, like Proactive Dynamic DCOP (PD-DCOP) algorithms, often necessitate a predefined environment model. We address the problem of enabling proactive agent behaviour in D-DCOPs where the dynamics model of the environment is unknown. Specifically, we propose an approach where agents learn local autoregressive models from observations, predicting future states to inform decision-making. To achieve this, we present a temporal experience-sharing message-passing algorithm that leverages dynamic agent connections and a distance metric to collate training data. Our approach outperformed baseline methods in a search-and-extinguish task using the RoboCup Rescue Simulator, achieving better total building damage. The experimental results align with prior work on the significance of decision-switching costs and demonstrate improved performance when the switching cost is combined with a learned model.

Minimization of fuel cost for generator systems with valve point effect using modified PSO

The goal of this research is to lay the groundwork for understanding the particle swarm optimization technique. The applicability of this technique is proven by solving a variety of Constraint and unconstraint optimization problems. During the operation of the power system, economic load dispatch is a vital optimization activity for allocating generation between the committed units to divide the power while meeting various system constraints economically. In this work, the modified PSO method's application to the electrical power system's economic load dispatch problem on 3-generator, 5-generator and 13-generator systems are also investigated. In this research work Exterior penalty method is applied to convert constraint optimization problem into unconstraint optimization problem

Toward fast belief propagation for distributed constraint optimization problems via heuristic search

Toward fast belief propagation for distributed constraint optimization problems via heuristic search

Enhancing constraint programming via supervised learning for job shop scheduling

Constraint programming (CP) is a powerful technique for solving constraint satisfaction and optimization problems. In CP solvers, the variable ordering strategy used to select which variable to explore first in the solving process has a significant impact on solver effectiveness. To address this issue, we propose a novel variable ordering strategy based on supervised learning, which we evaluate in the context of job shop scheduling problems. Our learning-based methods predict the optimal solution of a problem instance and use the predicted solution to order variables for CP solvers. Unlike traditional variable ordering methods, our methods can learn from the characteristics of each problem instance and customize the variable ordering strategy accordingly, leading to improved solver performance. Our experiments demonstrate that training machine learning models is highly efficient and can achieve high accuracy. Furthermore, our learned variable ordering methods perform competitively compared to four existing methods. Finally, we showcase the benefits of integrating machine learning-based variable ordering methods with conventional domain-based approaches through tie-breaking.

An Estimation of Distribution Based Algorithm for Continuous Distributed Constraint Optimization Problems

Continuous Distributed Constraint Optimization Problem(C-DCOP) is a constraint processing framework for continuous variables problems in multi-agent systems. There is a constraint cost function between two mutually restrictive agents in C-DCOP. The goal of the C-DCOP solving algorithm is to keep the sum of constraint cost functions in an extreme state. In a C-DCOP, each function is defined by a set of continuous variables. At present, some C-DCOP solving algorithms have been proposed, but there are some common problems such as the limitation of constraints cost function form, easy to fall into local optimum, and lack of anytime attribute. Aiming at these thorny problems, we propose a parallel optimization algorithm named Estimation of Distribution Based Algorithm for Continuous Distributed Constraint Optimization Problems (EDA-CD). In EDA-CD, each solution is regarded as an individual, and the distribution of agent value is jointly described by all outstanding individuals. Firstly, all agents cooperate to hold a distributed population. Secondly, each agent calculates the mean and variance of its variables to build probability models in parallel. Finally, the agent evaluates the fitness of samples and updates the probability model through cooperative communication on Breadth First Search (BFS) pseudo-tree. We theoretically prove that EDA-CD is an anytime algorithm. The extensive experimental results on four types of benchmark problems show that the proposed EDA-CD outperforms the state-of-the-art C-DCOP algorithms and has about 20% improvement in solution quality.

A decentralized multi-agent framework for urban flood management

The management of water resources in urban environments is challenging due to the existence of numerous stakeholders and the large number of affected people. In this paper, agents in an urban environment are categorized as active, passive, and monitoring agents, and a new mechanism is proposed to determine the flood control motivation functions (FCMFs) of active agents. An FCMF shows the motivation of an agent for investing and participating in a flood control activity. Also, for the first time, the effectiveness of a conflict resolution model based on asynchronous distributed constraint optimization (ADOPT) is evaluated for urban flood management. To evaluate the effectiveness of the proposed framework, it is applied to the eastern drainage catchment of Tehran, the capital city of Iran. The FCMFs of active stakeholders are modeled using some mathematical functions. To determine the FCMFs of active agents, an economic optimum point is suggested. This point is determined based on the construction costs of flood control infrastructures and flood damage costs. The utility functions of passive agents are modeled using a regression analysis. Flood management scenarios include a combination of water transfer canals, pipes, and ponds with real-time operating methods. ADOPT is used for solving Distributed Constraint Optimization Problems (DCOPs). The goal in DCOPs is to assign values to these variables in an approach that maximizes a global objective function while satisfying the constraints of each agent. The results show that by using the ADOPT-based conflict resolution model, the average depth of flooding is reduced from 24 cm to about 10 cm, and this is associated with the satisfaction of more than 90 percent of stakeholders. A decline in Average Flood Depth (AFD) has been observed with increased usage of technological methods such as constructing ponds or transferring water outside the basin. However, stakeholder desirability is critical and both technical and stakeholder interests should be taken into account to ensure project viability.

A Population-Based Search Approach to Solve Continuous Distributed Constraint Optimization Problems

Distributed Constraint Optimization Problems (DCOPs) are an efficient framework widely used in multi-agent collaborative modeling. The traditional DCOP framework assumes that variables are discrete and constraint utilities are represented in tabular forms. However, the variables are continuous and constraint utilities are in functional forms in many practical applications. To overcome this limitation, researchers have proposed Continuous DCOPs (C-DCOPs), which can model DCOPs with continuous variables. However, most of the existing C-DCOP algorithms rely on gradient information for optimization, which means that they are unable to solve the situation where the utility function is a non-differentiable function. Although the Particle Swarm-Based C-DCOP (PCD) and Particle Swarm with Local Decision-Based C-DCOP (PCD-LD) algorithms can solve the situation with non-differentiable utility functions, they need to implement Breadth First Search (BFS) pseudo-trees for message passing. Unfortunately, employing the BFS pseudo-tree results in expensive computational overheads and agent privacy leakage, as messages are aggregated to the root node of the BFS pseudo-tree. Therefore, this paper aims to propose a fully distributed C-DCOP algorithm to solve the utility function form problem and avoid the disadvantages caused by the BFS pseudo-tree. Inspired by the population-based algorithms, we propose a fully decentralized local search algorithm, named Population-based Local Search Algorithm (PLSA), for solving C-DCOPs with three-fold advantages: (i) PLSA adopts a heuristic method to guide the local search to achieve a fast search for high-quality solutions; (ii) in contrast to the conventional C-DCOP algorithm, PLSA can solve utility functions of any form; and (iii) compared to PCD and PCD-LD, PLSA avoids complex message passing to achieve efficient computation and agent privacy protection. In addition, we implement an extended version of PLSA, named Population-based Global Search Algorithm (PGSA), and empirically show that our algorithms outperform the state-of-the-art C-DCOP algorithms on three types of benchmark problems.

An effective theoretical and experimental analysis method for the improved slime mould algorithm

Metaheuristic intelligent optimization algorithms are effective methods for solving high-dimensional nonlinear complex optimization problems. The slime mould algorithm is a novel intelligent optimization algorithm proposed in 2020. However, the basic slime mould algorithm still has some shortcomings, such as slow convergence rate, easy to fall into local extremum, and unbalanced exploration and development capability. To further improve and expand the optimization ability and application scope of the slime mould algorithm, and enhance its performance in solving large-scale complex optimization problems, this paper proposes a slime mould algorithm (PPMSMA) based on Gaussian perturbation and phased position update, positive variation, and multi-strategy greedy selection. Firstly, Gaussian perturbation and phased position update mechanism are introduced to avoid the difficulty of the algorithm to jump out of the local extrema and also to speed up the convergence of the algorithm. Then, a positive variation strategy based on the sine cosine mechanism is introduced to move the variation of the population towards a better direction. Finally, a multi-strategy greedy selection mechanism is introduced, which effectively improves the search ability of the algorithm. The analysis and research on the optimization ability and performance of metaheuristic algorithms mainly include two aspects: theoretical analysis and experimental testing. Theoretical analysis has always been a relatively weak link in the research of metaheuristic algorithms, and there is currently no clear and effective method formed. For experimental testing, although there are more methods, they often lack systematization and adequacy. In this paper, a more complete, fine-grained and systematic approach to theoretical and experimental analysis is proposed. In the theoretical analysis part, the time complexity and spatial complexity of the PPMSMA algorithm are analytically proved to be the same as the basic slime mould algorithm, and the probability measure method is used to prove that PPMSMA algorithm can converge to the global optimal solution. In the simulation experiment section, the PPMSMA algorithm is compared with multiple sets of 10 representative comparison algorithms on the CEC2017 complex test function set suite for optimization accuracy analysis, Friedman comprehensive ranking analysis, average optimization rate analysis of PPMSMA relative to other algorithms, convergence curve analysis, and Wilcoxon rank-sum test analysis. To further examine the scalability of the improved algorithm in solving large-scale optimization problems, PPMSMA is compared with the above 10 comparative algorithms under 1000 dimensional conditions in the large-scale global optimization test set CEC2010, and the solution stability of each algorithm is analyzed through violin plots. The results show that the PPMSMA algorithm has significantly improved convergence performance, optimization accuracy, and solution stability in both high-dimensional and large-scale complex problems, and has significant advantages compared to multiple sets of 10 representative comparative algorithms. Finally, PPMSMA and 10 other comparative algorithms are used to solve engineering design optimization problems with different complexities. The experimental results validate the universality, reliability, and superiority of PPMSMA in handling engineering design constraint optimization problems.

Self-Adaptive Forensic-Based Investigation Algorithm with Dynamic Population for Solving Constraint Optimization Problems

The Forensic-Based Investigation (FBI) algorithm is a novel metaheuristic algorithm. Many researches have shown that FBI is a promising algorithm due to two specific population types. However, there is no sufficient information exchange between these two population types in the original FBI algorithm. Therefore, FBI suffers from many problems. This paper incorporates a novel self-adaptive population control strategy into FBI algorithm to adjust parameters based on the fitness transformation from the previous iteration, named SaFBI. In addition to the self-adaptive mechanism, our proposed SaFBI refers to a novel updating operator to further improve the robustness and effectiveness of the algorithm. To prove the availability of the proposed algorithm, we select 51 CEC benchmark functions and two well-known engineering problems to verify the performance of SaFBI. Experimental and statistical results manifest that the proposed SaFBI algorithm performs superiorly compared to some state-of-the-art algorithms.

An improved composite particle swarm optimization algorithm for solving constrained optimization problems and its engineering applications

<abstract><p>In the last few decades, the particle swarm optimization (PSO) algorithm has been demonstrated to be an effective approach for solving real-world optimization problems. To improve the effectiveness of the PSO algorithm in finding the global best solution for constrained optimization problems, we proposed an improved composite particle swarm optimization algorithm (ICPSO). Based on the optimization principles of the PSO algorithm, in the ICPSO algorithm, we constructed an evolutionary update mechanism for the personal best position population. This mechanism incorporated composite concepts, specifically the integration of the $ \varepsilon $-constraint, differential evolution (DE) strategy, and feasibility rule. This approach could effectively balance the objective function and constraints, and could improve the ability of local exploitation and global exploration. Experiments on the CEC2006 and CEC2017 benchmark functions and real-world constraint optimization problems from the CEC2020 dataset showed that the ICPSO algorithm could effectively solve complex constrained optimization problems.</p></abstract>

Dynamic ε-multilevel hierarchy constraint optimization with adaptive boundary constraint handling technology

Real-world optimization problems are often difficult to solve because of the complexity of the objective function and the large number of constraints that accompany it. To solve such problems, we propose Adaptive Dynamic ε-Multilevel Hierarchy Constraint Optimization (εMHCO). Firstly, we propose the dynamic constraint tolerance factor ε which can change dynamically with the feasible ratio and the number of iterations in the current population. This ensures a reasonable proportion of virtual feasible solutions in the population. Secondly, we propose adaptive boundary constraint handling technology (ABCHT). It can reshape the current individual position adaptively according to the size of constraint violation and increase the diversity of the population. Finally, we propose multi-level hierarchy optimization, whose multiple population structure is beneficial to solve real-world constraint optimization problems (COPs). To validate and analyze the performance of εMHCO, numerical experiments are conducted on the latest real-world test suite CEC’2020, which contains a set of 57 real-world COPs, and compared with four state-of-the-art algorithms. The results show that εMHCO is significantly superior to, or at least comparable to the state-of-the-art algorithms in solving real-world COPs. Meanwhile, the effectiveness and feasibility of εMHCO are verified on the real-world problem of the pipeline inner detector speed control.

Bald Eagle Search Optimization Algorithm Combined with Spherical Random Shrinkage Mechanism and Its Application

Over the last two decades, stochastic optimization algorithms have proved to be a very promising approach to solving a variety of complex optimization problems. Bald eagle search optimization (BES) as a new stochastic optimization algorithm with fast convergence speed has the ability of prominent optimization and the defect of collapsing in the local best. To avoid BES collapse at local optima, inspired by the fact that the volume of the sphere is the largest when the surface area is certain, an improved bald eagle search optimization algorithm (INMBES) integrating the random shrinkage mechanism of the sphere is proposed. Firstly, the INMBES embeds spherical coordinates to design a more accurate parameter update method to modify the coverage and dispersion of the population. Secondly, the population splits into elite and non-elite groups and the Bernoulli chaos is applied to elite group to tap around potential solutions of the INMBES. The non-elite group is redistributed again and the Nelder-Mead simplex strategy is applied to each group to accelerate the evolution of the worst individual and the convergence process of the INMBES. The results of Friedman and Wilcoxon rank sum tests of CEC2017 in 10, 30, 50, and 100 dimensions numerical optimization confirm that the INMBES has superior performance in convergence accuracy and avoiding falling into local optimization compared with other potential improved algorithms but inferior to the champion algorithm and ranking third. The three engineering constraint optimization problems and 26 real world problems and the problem of extracting the best feature subset by encapsulated feature selection method verify that the INMBES’s performance ranks first and has achieved satisfactory accuracy in solving practical problems.

Fast DOA Estimation Algorithms via Positive Incremental Modified Cholesky Decomposition for Augmented Coprime Array Sensors.

This paper proposes a fast direction of arrival (DOA) estimation method based on positive incremental modified Cholesky decomposition atomic norm minimization (PI-CANM) for augmented coprime array sensors. The approach incorporates coprime sampling on the augmented array to generate a non-uniform, discontinuous virtual array. It then utilizes interpolation to convert this into a uniform, continuous virtual array. Based on this, the problem of DOA estimation is equivalently formulated as a gridless optimization problem, which is solved via atomic norm minimization to reconstruct a Hermitian Toeplitz covariance matrix. Furthermore, by positive incremental modified Cholesky decomposition, the covariance matrix is transformed from positive semi-definite to positive definite, which simplifies the constraint of optimization problem and reduces the complexity of the solution. Finally, the Multiple Signal Classification method is utilized to carry out statistical signal processing on the reconstructed covariance matrix, yielding initial DOA angle estimates. Experimental outcomes highlight that the PI-CANM algorithm surpasses other algorithms in estimation accuracy, demonstrating stability in difficult circumstances such as low signal-to-noise ratios and limited snapshots. Additionally, it boasts an impressive computational speed. This method enhances both the accuracy and computational efficiency of DOA estimation, showing potential for broad applicability.

Exploiting Functional Constraints in Automatic Dominance Breaking for Constraint Optimization

Dominance breaking is a powerful technique in improving the solving efficiency of Constraint Optimization Problems (COPs) by removing provably suboptimal solutions with additional constraints. While dominance breaking is effective in a range of practical problems, it is usually problem specific and requires human insights into problem structures to come up with correct dominance breaking constraints. Recently, a framework is proposed to generate nogood constraints automatically for dominance breaking, which formulates nogood generation as solving auxiliary Constraint Satisfaction Problems (CSPs). However, the framework uses a pattern matching approach to synthesize the auxiliary generation CSPs from the specific forms of objectives and constraints in target COPs, and is only applicable to a limited class of COPs. This paper proposes a novel rewriting system to derive constraints for the auxiliary generation CSPs automatically from COPs with nested function calls, significantly generalizing the original framework. In particular, the rewriting system exploits functional constraints flattened from nested functions in a high-level modeling language. To generate more effective dominance breaking nogoods and derive more relaxed constraints in generation CSPs, we further characterize how to extend the system with rewriting rules exploiting function properties, such as monotonicity, commutativity, and associativity, for specific functional constraints. Experimentation shows significant runtime speedup using the dominance breaking nogoods generated by our proposed method. Studying patterns of generated nogoods also demonstrates that our proposal can reveal dominance relations in the literature and discover new dominance relations on problems with ineffective or no known dominance breaking constraints.

Scheduling operations in a large hospital by multiple agents

The scheduling of operations in a large hospital is performed jointly by several groups of people, each with its own objective and constraints. It is a two-phase process, starting with the allocation of operating rooms to wards, and followed by the scheduling of operations in each operating room of the hospital on each day. The final schedule must satisfy all inter-ward hard constraints, such as the allocation of anesthetists, nurses, and equipment to operations that are taking place in parallel, and ideally, it should also address soft constraints such as taking the urgency and complexity of operations into consideration.This study contributes to the ongoing effort of adapting multi-agent optimization models and algorithms to real-world applications by modeling the problems in both phases as distributed constraint optimization problems (DCOPs), with different properties. The first phase includes partially cooperative ward-representing agents, allocating operating rooms for daily usage among themselves. In the second phase, ward-representing agents interact with agents representing constraining elements, in order to generate daily operation schedules for each operating room, thus forming a unique bipartite constraint graph. On one side are the ward representatives, while on the other are the agents representing the constraining resources. Each agent has a non-trivial local problem to solve, and its solution serves as the proposed assignment in the distributed algorithm.The study begins by discussing the properties required of the algorithms needed to solve the two phases. It then proposes adjustments to existing distributed partially cooperative algorithms and local search algorithms to solve these problems, and compares the results of different variants of these algorithms. The results obtained for both phases emphasize that successful collaboration is predicated on two requirements: that agents hold consistent information regarding their peers’ states and that the degree of exploration undertaken by the algorithm is restricted in order to produce high-quality solutions.

Proposed residual stress model for hot‐rolled wide flange steel cross sections

AbstractResidual stresses in hot‐rolled wide flange steel cross sections may lead to premature yielding, accelerated corrosion and brittle fracture of steel members. The above phenomena lead to a loss of a steel member's stiffness and resistance under mechanical loading. Available residual stress models are mostly based on residual stress measurements that date back to 1950s. Some of the drawbacks of these models relate to the lack of consistency in the considered parameters that affect the residual stress development within a steel cross section. Motivated by this, this paper proposes a new residual stress model for hot‐rolled wide flange steel cross sections. The proposed residual stress model relies on a dataset of 80 experiments that are complemented by additional measurements done as part of the present study. A parabolic residual stress distribution, which is deduced from a constraint optimization problem, is fitted to the assembled data. The proposed residual stress distributions are generalized with the aid of rigorous statistical analyses. The results suggest that the residual stress model is highly dependent on the cross‐sectional area and the depth‐to‐width ratio of a hot‐rolled wide flange steel cross section. The proposed residual stress model reduces the error, on average, by 60‐70% compared to available residual stress models (e.g., European Convention for Constructional Steelwork model), regardless of the cross‐sectional geometry of the hot‐rolled cross section.

Learning to select SAT encodings for pseudo-Boolean and linear integer constraints

Many constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT). However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.

Study on reservoir optimal operation based on coupled adaptive ε constraint and multi strategy improved Pelican algorithm

The optimal operation of reservoir groups is a strongly constrained, multi-stage, and high-dimensional optimization problem. In response to this issue, this article couples the standard Pelican optimization algorithm with adaptive ε constraint methods, and further improves the optimization performance of the algorithm by initializing the population with a good point set, reverse differential evolution, and optimal individual t-distribution perturbation strategy. Based on this, an improved Pelican algorithm coupled with adaptive ε constraint method is proposed (ε-IPOA). The performance of the algorithm was tested through 24 constraint testing functions to find the optimal ability and solve constraint optimization problems. The results showed that the algorithm has strong optimization ability and stable performance. In this paper, we select Sanmenxia and Xiaolangdi reservoirs as the research objects, establish the maximum peak-cutting model of terrace reservoirs, apply the ε-IPOA algorithm to solve the model, and compare it with the ε-POA (Pelican algorithm coupled with adaptive ε constraint method) and ε-DE (Differential Evolution Algorithm) algorithms, the results indicate that ε. The peak flow rate of the Huayuankou control point solved by the IPOA algorithm is 12,319 m3/s, which is much lower than the safe overflow flow rate of 22,000 m3/s at the Huayuankou control point, with a peak shaving rate of 44%, and other algorithms do not find effective solutions meeting the constraint conditions. This paper provides a new idea for solving the problem of flood control optimal operation of cascade reservoirs.

Energy-Efficient Power Allocation for Full-Duplex Device-to-Device Underlaying Cellular Networks with NOMA

Full-duplex (FD), Device-to-Device (D2D) and non-orthogonal multiple access (NOMA) are promising wireless communication techniques to improve the utilization of spectrum resources. Meanwhile, introducing FD, D2D and NOMA in cellular networks is very challenging due to the complex interference problem. To deal with the complex interference of FD D2D underlaying NOMA cellular networks, power allocation (PA) is extensively studied as an efficient interference management technique. However, most of the previous research works on PA to optimize energy efficiency only consider the system framework of partially joint combining techniques of FD, D2D and NOMA, and the constraints of optimization problem are very different. In this paper, in order to further improve the energy efficiency of a system, a dual-layer iteration power allocation algorithm is proposed to eliminate the complex interference. The outer-layer iteration is to solve the non-linear fractional objective function based on Dinkelbach, and the inner-layer iteration is to solve the non-convex optimization problem based on D.C. programming. Then, the non-convex and non-linear fractional objective function is transformed into a convex function to solve the optimal power allocation. In this approach, FD D2D users reuse the spectrum with downlink NOMA cellular users. Imperfect self-interference (SI) cancellation at the FD D2D users and the successive interference cancellation (SIC) at the strong NOMA user are considered in the system framework. The optimization problem is constructed to maximize the system’s energy efficiency with the constraints of successful SIC, QoS requirements, the maximum transmit power of BS and FD D2D users. Numerical results demonstrate that the proposed algorithm outperforms the traditional orthogonal multiple access (OMA) in terms of energy efficiency with a higher system sum rate.

Airfoil shape optimization using genetic algorithm coupled deep neural networks

To alleviate the computational burden associated with the computational fluid dynamics (CFD) simulation stage and improve aerodynamic optimization efficiency, this work develops an innovative procedure for airfoil shape optimization, which is implemented through coupling the genetic algorithm (GA) optimizer with the aerodynamic coefficients prediction network (ACPN) model. The ACPN is established using a fully connected neural network with the airfoil geometry as the input and aerodynamic coefficients as the output. The results show that the ACPN's mean prediction accuracy for the lift and drag coefficient is high up to about 99.02%. Moreover, the prediction time of each aerodynamic coefficient is within 5 ms, four orders of magnitude faster compared to the CFD solver (3 min). Taking advantage of the fast and accurate prediction, the proposed ACPN model replaces the expensive CFD simulations and couples with GA to force the airfoil shape change to maximize the lift–drag ratio under multiple constraints. In terms of time efficiency, optimized airfoils can be fast obtained within 25 s. Even considering an extra 50 h spent on data preparing and 20 s for model training, the overall calculation cost is reduced by a remarkable 62.1% compared to the GA-CFD optimization method (5.5 days). Furthermore, the GA-ACPN model improves the lift–drag ratio with and without constraint by 51.4% and 55.4% for NACA0012 airfoil, respectively, while 50.3% and 60.0% improvement achieved by the GA-CFD optimization method. These results indicate that the GA-ACPN optimization approach significantly enhances the optimization efficiency and has great potential to address varying constraint optimization problems.