Constraint Handling Technique Research Articles

An adaptive uniform search framework for constrained multi-objective optimization

This paper proposes an adaptive uniform search framework designed for constrained multi-objective optimization. The framework comprises three key components: a global uniform exploration strategy, a local greedy exploitation strategy, and a search switch mechanism. These components work together to facilitate comprehensive exploration of promising areas while maintaining a balance between global exploration and local exploitation. Specifically, the global uniform exploration strategy ensures even distribution within promising areas, preventing any oversights during exploration. The local greedy exploitation strategy divides these areas into sub-areas and employs a feasibility-led constraint handling technique to enhance efficiency in identifying optimal solutions. Additionally, the search switch dynamically adjusts the search strategy between global exploration and local exploitation. Numerical simulations on various benchmark suites and real-world problem demonstrate the strong performance of the framework in addressing constrained multi-objective optimization problems. The comparison results show that compared with eight recently proposed algorithms, the proposed framework is more robust in solving diverse constrained multi-objective optimization problems.

Multi-objective optimal input design for grey-box identification modelling of ship manoeuvring motion

ABSTRACT Grey-box modelling has been widely used to predict the ship manoeuvring motion. In order to design input signals for the grey-box model of ship manoeuvring motion, a multi-objective optimal input design (MOID) method is proposed in this paper. In the proposed method, the MOID is regarded as a multi-objective optimization problem (MOP), where both of D-optimality criterion and condition number are taken as two-objective functions to improve the accuracy and robustness of identifying the grey-box model. The correlation factor is also presented as a constraint for reducing the parameter drift. The MOP is solved by the non-dominated sorting genetic algorithm-II (NSGA-II) with a constraint handling technique. The technique for order preference by similarity to an ideal solution (TOPSIS) is also adopted to screen out an optimal solution from the Pareto solutions. The effectiveness of MOID signals is validated by Monte Carlo analysis and grey-box identification of the MMG model.

Boosting constrained teaching–learning based optimization algorithm through group discussion

Teaching–learning based optimization algorithm (TLBO) is an efficient population-based nature-inspired meta-heuristic designed initially to solve unconstrained optimization problems. TLBO being a swam intelligence based algorithm is renowned for fast convergence. This characteristic, however, occasionally causes it to converge too soon to a non-globally optimal solution. Group discussion techniques are used in educational institutions all over the world to help participants to exchange ideas and information and to improve their skills. This work imitates the outlined strategy, therefore innovating TLBO. Here, a group of virtual students is selected corresponding to each student in the learner phase of TLBO to produce diversity in the population and to balance the exploration and exploitation capabilities of it, hence reducing the likelihood of early convergence. GTLBO stands for the new algorithm that incorporates group discussion. It is essential to incorporate some constraint handling techniques (CHTs) into the TLBO framework in order to handle constrained optimization problems (COPs). The most recent, effective, and enhanced version of the widely used CHT stochastic ranking (SR) is called hybrid stochastic ranking (HSR). HSR is added to the GTLBO structure as a selection operator for managing COPs; the resulting setup is referred to as HSR-GTLBO. Additionally, the suggested algorithm’s group size for discussion is sensitive, resulting in a number of constrained variants. The efficacy of these variations is assessed with the suit CEC 2017’s constrained problems. In conformity with the parameters settlement, comparison, and ranking criteria of CEC 2017, the simulation results of the proposed versions are compared and ranked using the most advanced state-of-the-art algorithms. As a result, in the competition, the newly developed HSR-GTLBO variants placed first, second, and third.

Efficient implicit constraint handling approaches for constrained optimization problems

Many real-world optimization problems, particularly engineering ones, involve constraints that make finding a feasible solution challenging. Numerous researchers have investigated this challenge for constrained single- and multi-objective optimization problems. In particular, this work extends the boundary update (BU) method proposed by Gandomi and Deb (Comput. Methods Appl. Mech. Eng. 363:112917, 2020) for the constrained optimization problem. BU is an implicit constraint handling technique that aims to cut the infeasible search space over iterations to find the feasible region faster. In doing so, the search space is twisted, which can make the optimization problem more challenging. In response, two switching mechanisms are implemented that transform the landscape along with the variables to the original problem when the feasible region is found. To achieve this objective, two thresholds, representing distinct switching methods, are taken into account. In the first approach, the optimization process transitions to a state without utilizing the BU approach when constraint violations reach zero. In the second method, the optimization process shifts to a BU method-free optimization phase when there is no further change observed in the objective space. To validate, benchmarks and engineering problems are considered to be solved with well-known evolutionary single- and multi-objective optimization algorithms. Herein, the proposed method is benchmarked using with and without BU approaches over the whole search process. The results show that the proposed method can significantly boost the solutions in both convergence speed and finding better solutions for constrained optimization problems.

Covariance matrix adaptation evolution strategy based on ensemble of mutations for parking navigation and maneuver of autonomous vehicles

Previous works have demonstrated the use of human-inspired frameworks to address autonomous vehicle parking. In such frameworks, autonomous vehicle parking is formulated as a two-stage optimization problem where the first stage involves the generation of initial waypoints which are consequently used to reduce the autonomous parking task into a local motion planning task. In the second stage, the local motion planning is solved as a classical optimization problem. However, the overall algorithmic performance in the second stage depends on the planning horizon used to formulate the local planning task. In this work, we show first that the state-of-the-art algorithms often employed in the second stage fail with an increase in the planning horizon. Motivated by the need for algorithms that are scalable to different planning horizons, this work proposes Covariance Matrix Adaptation Evolution Strategy (CMA-ES) with an ensemble of mutations for the second stage. Specifically, the proposed algorithm features an ensemble of Gaussian- and Cauchy-based mutations to facilitate an efficient blend of both exploitation and exploration that is crucial for both short and long horizons. Furthermore, to handle the associated constraints, Superiority of Feasible (SF) constraint handling technique is incorporated into the proposed algorithm. Performance analysis based on 24 parking missions conducted with planning horizons of 2 and 4 shows that the proposed algorithm is scalable to different planning horizons compared with three commonly employed state-of-the-art algorithms.

Toward unique electrical ladder network model synthesis of a transformer winding high-frequency modeling using K-means and metaheuristic-based method

PurposeAlthough, numerous optimization algorithms have been devoted to construct an electrical ladder network model (ELNM), they suffer from some frail points such as insufficient accuracy as well as the majority of them are unconstrained, which result in optimal solutions that violate certain security operational constraints. For this purpose, this paper aims to propose a flexible-constraint coyote optimization algorithm; the novelty lies in these points: penalty function is introduced in the objective function to discard any unfeasible solution, an advanced constraint handling technique and empirical relationship between the physical estimated parameters and their natural frequencies.Design/methodology/approachFrequency response analysis (FRA) is very significant for transformer winding diagnosis. Interpreting results of a transformer winding FRA is quite challenging. This paper proposes a new methodology to synthesize a nearly unique ELNM physically and electrically coupled for power transformer winding, basing on K-means and metaheuristic algorithm. To this end, the K-means method is used to cluster the setting of control variables, including the self-mutual inductances/capacitances, and the resistances parameters. Afterward, metaheuristic algorithm is applied to determine the cluster centers with high precision and efficiency.FindingsFRA is performed on a power transformer winding model. Basing on the proposed methodology, the prior knowledge in selecting the initial guess and search space is avoided and the global solution is ensured. The performance of the abovementioned methodology is compared using evaluation expressions to verify its feasibility and accuracy.Originality/valueThe proposed method could be generalized for diagnosis of faults in power transformer winding.

Adaptive constraint handling technique selection for constrained multi-objective optimization

Constrained multi-objective optimization problems involve the optimization of multiple conflicting objectives simultaneously subject to a number of constraints, which pose a great challenge for the existing algorithms. When utilizing evolutionary algorithms to solve them, the constraint handling technique (CHT) plays a pivotal role in the environmental selection. Several CHTs, such as penalty functions, superiority of feasible solutions, and ϵ-constraint methods, have been developed. However, there are still some issues with the existing methods. On the one hand, different CHTs are typically better suited to specific problem and selecting the most appropriate CHT for a given problem is crucial. On the other hand, the suitability of CHTs may vary throughout different stages of the optimization process. Regrettably, limited attention has been given to the adaptive selection of CHTs. In order to address this research gap, we develop an adaptive CHT selection method based on deep reinforcement learning, allowing for the selection of CHTs that are tailored to different evolutionary states. In the proposed method, we adopt the deep Q-learning network to evaluate the impact of various CHTs and operators on the population state during evolution. Through a dynamic evaluation, the network adaptively outputs the most appropriate CHT and operator portfolio based on the current state of the population. Specifically, we propose novel state representation and reward calculation methods to accurately capture the performance of diverse actions across varying evolutionary states. Furthermore, to enhance network training, we introduce a two-stage training method that facilitates the collection of diverse samples. Moreover, this adaptive selection method can be easily integrated into the existing methods. The proposed algorithm is tested on 37 test problems, the optimal results can be achieved on 19 instances in terms of the inverted generational distance metric. Experimental results verify the proposed method generalizes well to different types of problems.

Two-Stage Multiobjective Evolution Strategy for Constrained Multiobjective Optimization

For the past many years, several constrained multiobjective evolutionary algorithms (CMOEAs) have been designed for solving constrained multi-objective optimization problems (CMOPs). In these CMOEAs, some constraint-handling techniques (CHTs) were proposed to balance the convergence and constrained satisfaction, however, they still face some serious challenges, such as premature convergence to the local optimal region and labor-intensive tuning of parameters for a specific CMOP. Furthermore, most of the existing CHTs are derived by solving constrained single-objective optimization. The information hidden from the feasible non-dominated set (FNDS) has not been fully utilized. This study proposed a novel parameter-less constraint handling technique, which divides the entire population into three mutually exclusive subsets dynamically: FNDS, the subset dominated by FNDS, and the subset not dominated by FNDS. According to the proposed division of labor, it is not necessary to balance the convergence and constrained satisfaction in each subset. To avoid being entrapped in local optima, the proposed algorithm adopts a two-stage strategy to solve CMOPs. In the first stage, the proposed algorithm focuses solely on converging toward the unconstrained Pareto front without considering the constrained satisfaction. In the second stage, the FNDS constraint handling technique is adopted to guide the population converging toward constrained Pareto front effectively. The performance of the proposed algorithm was compared to that of nine state-of-the-art CMOEAs, and the comparison results show that the proposed algorithm performs significantly better on the CF, MW, and LIRCMOP test suites.

Wind Power Bidding Based on an Ensemble Differential Evolution Algorithm with a Problem-Specific Constraint-Handling Technique

The intermittent nature of wind power generation induces great challenges for power bidding in the electricity market. The deployment of battery energy storage can improve flexibility for power bidding. This paper investigates an optimal power bidding strategy for a wind–storage hybrid power plant in the day-ahead electricity market. To handle the challenges of the uncertainties of wind power generation and electricity prices, the optimal bidding problem is formulated as a risk-aware scenario-based stochastic programming, in which a number of scenarios are generated using a copula-based approach to represent the uncertainties. These scenarios consider the temporal correlation of wind power generation and electricity prices between consecutive time intervals. In the stochastic programming, a more practical but nonlinear battery operation cost function is considered, which leads to a nonlinear constrained optimization problem. To solve the nonlinear constrained optimization problem, an ensemble differential evolution (EDE) algorithm is proposed, which makes use of the merits of an ensemble of mutant operators to generate mutant vectors. Moreover, a problem-specific constraint-handling technique is developed. To validate the effectiveness of the proposed EDE algorithm, it is compared with state-of-the-art DE-based algorithms for constrained optimization problems, including a constrained composite DE (C2oDE) algorithm and a novel DE (NDE) algorithm. The experimental results demonstrate that the EDE algorithm is much more reliable and much faster in finding a better bidding strategy against benchmarking algorithms. More precisely, the average values of the success rate are 0.893, 0.667, and 0.96 for C2oDE, NDE, and EDE, respectively. Compared to C2oDE and NDE, the average value of the mean number of function evaluations to succeed with EDE is reduced by 76% and 59%, respectively.

Drag Minimization of Low Subsonic Airfoil with Constrained Genetic Algorithm

Drag minimization of low subsonic airfoil was conducted with constrained genetic algorithm (CGA). To cope with the constraints, each of these two different types of constraint handling techniques (CHTs), namely Superiority of Feasible Individual (SoF) and Generalized Multiple Constraint Ranking (G-MCR) were employed to the CGA and compared. From three independent runs for each CHT, it was obtained that G-MCR performed significantly better than SoF, indicating that G-MCR, a novel type of CHT, provides better exploration of the design space to obtain better solution. The obtained best airfoil designs were compared with a baseline airfoil and analyzed. The best optimum airfoil increases the aerodynamic efficiency by 21.4%. It was observed that the reduction of drag only occurs locally, so that a robust optimization is required in the future.

A Constraint-Handling Technique for Decomposition-Based Constrained Many-Objective Evolutionary Algorithms

A Constraint-Handling Technique for Decomposition-Based Constrained Many-Objective Evolutionary Algorithms

Constrained evolutionary optimization based on dynamic knowledge transfer

While utilizing evolutionary algorithms (EAs) to solve constrained optimization problems (COPs), seeking feasible solutions that satisfy the constraints is the primary concern. In some cases, utilizing the knowledge of the objective may facilitate the exploration of feasible regions and the exploitation of global optima. To strike a balance between objective and constraints, we attempt to address two problems, namely whether transferring the knowledge of objective in the early evolutionary stage and how to dynamically transfer the knowledge of objective to constraints. First, whether transferring in the early stage is determined by the characteristics of the objective, which has been classified to simple and complex based on the variation of population distribution. Second, how to transfer is solved by designing suitable constraint handling techniques (CHTs). To deal with COPs with simple objective, an objective-oriented CHT is proposed, where an indicator called Knowledge Transfer Rate (KTR) representing the relationship between the objective and constraints is mined and used. For COPs with complex objective, a constraint-oriented CHT is proposed, which includes a constraint-driven strategy and a hybrid-driven strategy. The proposed method is executed on three benchmark test suites, namely IEEE CEC2006, CEC2010, and CEC2017, it achieves superior or at least competitive performance in comparison with other state-of-the-art methods. Furthermore, the proposed method is successfully implemented in a redundant robotic manipulator motion planning problem.

On the Advantages of Searching Infeasible Regions in Constrained Evolutionary-Based Multi-Objective Engineering Optimization

Abstract Solving a multiple-criteria optimization problem with severe constraints remains a significant issue in multi-objective evolutionary algorithms. The problem primarily stems from the need for a suitable constraint handling technique. One potential approach is balancing the search in feasible and infeasible regions to find the Pareto front efficiently. The justification for such a strategy is that the infeasible region also provides valuable information, especially in problems with a small percentage of feasibility areas. To that end, this paper investigates the potential of the infeasibility-driven principle based on multiple constraint ranking-based techniques to solve a multi-objective problem with a small feasibility ratio. By analyzing the results from intensive experiments on a set of test problems, including the realistic multi-objective car structure design and actuator design problem, it is shown that there is a significant improvement gained in terms of convergence by utilizing the generalized version of the multiple constraint ranking techniques.

A novel multi-objective evolutionary algorithm with a two-fold constraint-handling mechanism for multiple UAV path planning

Most multiple unmanned aerial vehicle (UAV) path planning problems are often treated as constrained single-objective optimization problems. How to consider them as constrained multi-objective optimization problems (CMOPs) have seldom been explored in this field. To fill this gap, this paper firstly constructs multiple UAV path planning problem as a CMOP with two objectives and five constraints. Then a novel multi-objective evolutionary algorithm with a two-fold constraint-handling mechanism is proposed for multiple UAV path planning. To cope with constraints effectively, a constraint-handling technique based on a progressive weight vector strategy is proposed. Besides, a constraint repair technique that considers the flying environment is designed to further guide the algorithm to find feasible promising regions. Eight multiple UAV path planning test instances with different solving difficulties are constructed. Subsequently, they are used to validate the performance of the proposed algorithm. The experimental results demonstrate that the proposed algorithm is superior over four compared algorithms in terms of obtaining a set of better-distributed Pareto optimal solutions.

A Two-Phase Constraint-Handling Technique for Constrained Optimization

A two-phase constraint-handling technique is integrated into the evolutionary algorithms to solve constrained optimization problems (called TPDE) in this article. In phase one, denoted as the exploration phase, an exterior penalty function method with the dynamic penalty coefficients is developed to compare any two candidate solutions, which aims to push the population into the feasible region. To reduce the computational burden, in phase two, denoted as the exploitation phase, an interior penalty function method with the dynamic penalty coefficients is developed, which enhances the search ability by using the information of constraints in feasible solutions. During the optimization process, differential evolution is adopted as the search algorithm to produce the offspring population. Experiment results on four benchmark test suites, namely, IEEE CEC 2006, IEEE CEC 2010, IEEE CEC 2017, and IEEE CEC 2020, indicate that TPDE is competitive with other popular algorithms.

Multiobjective-Based Constraint-Handling Technique for Evolutionary Constrained Multiobjective Optimization: A New Perspective

Multiobjective-based constraint-handling techniques are popular in evolutionary constrained single-objective optimization. However, most of these techniques run into troubles when dealing with constrained multiobjective optimization problems (CMOPs). That is, they have difficulty optimizing too many objective functions, are ineffective in maintaining population diversity, or are challenged in establishing appropriate additional objective functions. As a remedy to these limitations, we propose a novel technique called NRC for handling CMOPs. The novelty of NRC lies in its three sorting procedures: nondominated sorting, reversed nondominated sorting, and constrained crowding distance sorting, which are performed in sequence to provide driving forces toward the Pareto front (PF) of a transformed unconstrained multiobjective optimization problem (treating the overall constraint violation as an additional objective function), the boundary front, and the constrained PF, respectively. With the combination of these three different forces, NRC can conveniently approach the desired PF from diverse search directions. The effectiveness of NRC is experimentally verified. Also, we incorporate NRC into a two-archive mechanism and develop a novel constrained multiobjective evolutionary algorithm, called NRC2. Comprehensive experiments on 49 benchmark CMOPs and 21 real-world ones demonstrate that NRC2 is significantly superior or comparable to six state-of-the-art constrained evolutionary multiobjective optimizers on most test instances.

An adaptive GLNPSO method for inventory replenishment supply chain problem with multiple-warehouse policy and budget consideration

This paper presents an efficient computation tool for a multiple-warehouse inventory replenishment lot-sizing problem with supplier selection, quantity discounts, and budget constraint. The cost model of the considered inventory replenishment issue is formulated as a mixed-integer nonlinear programming (MINLP) problem. With the proposed MINLP problem being an NP-hard problem, the MINLP cost model was optimized by using a novel global–local neighbor particle swarm optimization algorithm (TPAGLNPSO), inspired by two-phase self-adaptive inertia weight and improved constraint-handling technique. Apart from introducing the velocity index to switch the search scheme from exploration to exploitation, the constraint-handling technique was utilized to guide and guarantee the particle search toward undiscovered regions in the solution space. The stability analysis for the proposed novel TPAGLNPSO was also conducted using the stochastic process theorem. The corresponding parameter selection ranges were obtained based on the convergent conditions. Finally, eight industry cases were provided as examples to evaluate the performance of the proposed TPAGLNPSO method, after which a sensitivity analysis for the MINLP cost model was performed. The results demonstrate the competitive performance of the proposed TPAGLNPSO optimizer in terms of solution quality (with near-optimal results) than the optimal solutions obtained from Lingo software and computational effort (CPU time). Such findings can provide practitioners marginal insights into the effective management of the supply chain inventory replenishment problem.

Neural network-enabled discovery of mapping between variables and constraints for autonomous repair-based constraint handling in multi-objective structural optimization

Repairing infeasible solutions is a seldomly used constraint handling technique (CHT) in optimization because, with this technique, users must define ways to prevent constraint violation, which is difficult or impossible in many situations. To overcome this challenge, in this paper we propose an approach based on artificial neural networks to automatically discover variable-constraint mapping that specifies which variables must be modified if a constraint is violated. We embed this procedure into a recent repair-based CHT that uses members of the current population, either feasible or infeasible, to determine the new value for the repaired variables. Here, the CHT is modified to repair each constraint separately using a different solution that does not violate the same constraint. The entire technique is fully autonomous, meaning that a user does not provide any insights into the problem. The modified CHT is implemented in the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and compared with four other multi-objective optimization algorithms and a few CHTs. Four test problems are considered: a mathematical benchmark problem, two truss problems and structural optimization of a chemical tanker. The tanker case is a real-world optimization problem with 94 variables and 376 nonlinear constraints. A minimum of 30 independent runs are performed with each algorithm, and various statistical results are shown. With the proposed automated mapping, the modified repair-assisted NSGA-II obtains significantly better results than NSGA-II with other CHTs on all test problems while outperforming all other algorithms and CHTs for the tanker.

A leader supply-demand-based optimization for large scale optimal power flow problem considering renewable energy generations

The supply-demand-based optimization (SDO) is among the recent stochastic approaches that have proven its capability in solving challenging engineering tasks. Owing to the non-linearity and complexity of the real-world IEEE optimal power flow (OPF) in modern power system issues and like the existing algorithms, the SDO optimizer necessitates some enhancement to satisfy the required OPF characteristics integrating hybrid wind and solar powers. Thus, a SDO variant namely leader supply-demand-based optimization (LSDO) is proposed in this research. The LSDO is suggested to improve the exploration based on the simultaneous crossover and mutation mechanisms and thereby reduce the probability of trapping in local optima. The LSDO effectiveness has been first tested on 23 benchmark functions and has been assessed through a comparison with well-regarded state-of-the-art competitors. Afterward, Three well-known constrained IEEE 30, 57, and 118-bus test systems incorporating both wind and solar power sources were investigated in order to authenticate the performance of the LSDO considering a constraint handling technique called superiority of feasible solutions (SF). The statistical outcomes reveal that the LSDO offers promising competitive results not only for its first version but also for the other competitors.

Deep reinforcement learning assisted co-evolutionary differential evolution for constrained optimization

Solving constrained optimization problems (COPs) with evolutionary algorithms (EAs) is a popular research direction due to its potential and diverse applications. One of the key issues in solving COPs is the choice of constraint handling techniques (CHTs), as different CHTs can lead to different evolutionary directions. Combining EAs with deep reinforcement learning (DRL) is a promising and emerging approach for solving COPs. Although DRL can help solve the problem of pre-setting operators in EAs, neural networks need to obtain diverse training data within a limited number of evaluations in EAs. Based on the above considerations, this work proposes a DRL assisted co-evolutionary differential evolution, named CEDE-DRL, which can effectively use DRL to help EAs solve COPs. (1) This method incorporates co-evolution into the extraction of training data for the first time, ensuring the diversity of samples and improving the accuracy of the neural network model through information exchange between multiple populations. (2) Multiple CHTs are used for offspring selection to ensure the algorithm’s generality and flexibility. (3) DRL is used to evaluate the population state, taking into account feasibility, convergence, and diversity in the state setting and using the overall degree of improvement as a reward. The neural network selects suitable parent populations and corresponding archives for mutation. Finally, (4) to avoid premature convergence and local optima, an adaptive operator selection and individual archive elimination mechanism is added. Comparisons with state-of-the-art algorithms on benchmark functions CEC2010 and CEC2017 show that the proposed method performs competitively and produced robust solutions. The results of the application test set CEC2020 show that the proposed algorithm is also effective in real-world problems.