Constrained Multi-objective Optimization Research Articles

Evolutionary Algorithm with Dynamic Population Size for Constrained Multiobjective Optimization

The core task of constrained multiobjective optimization is to achieve a tradeoff between exploration and exploitation as well as a tradeoff between constraints and objectives. We present an effective evolutionary algorithm with a dynamic population size (DPSEA) to achieve these two tradeoffs. In order to balance exploration and exploitation, the population size of DPSEA decreases continually as generation increases. In this manner, a bigger population size can encourage exploration in the early stage, while a smaller one can promote exploitation in the later stage. Aiming to balance constraints and objectives, a two-stage environmental selection strategy is proposed. In this strategy, objective information and constraint information is used to select two sets of solutions, respectively. Note that the sizes of these two sets are also adjusted dynamically. In this way, the information of both objectives and constraints can be used. Moreover, a novel mating selection strategy is designed to select promising parents. By assembling the above processes, DPSEA is able to achieve the two tradeoffs which are critical to constrained multiobjective optimization. Experiments on three sets of benchmark test functions owning difficult characteristics validate that DPSEA is competitive against some state-of-the-art constrained multiobjective optimization evolutionary algorithms.

Navigation of a Fuzzy-Controlled Wheeled Robot Through the Combination of Expert Knowledge and Data-Driven Multiobjective Evolutionary Learning.

This article proposes a navigation scheme for a wheeled robot in unknown environments. The navigation scheme consists of obstacle boundary following (OBF), target seeking (TS), and vertex point seeking (VPS) behaviors and a behavior supervisor. The OBF behavior is achieved by a fuzzy controller (FC). This article formulates the FC design problem as a new constrained multiobjective optimization problem and finds a set of nondominated FC solutions through the combination of expert knowledge and data-driven multiobjective ant colony optimization. The TS behavior is achieved by new fuzzy proportional-integral-derivative (PID) and proportional-derivative (PD) controllers that control the orientation and speed of the robot, respectively. The VPS behavior is proposed to shorten the navigation route by controlling the robot to move toward a new subgoal determined from the vertex point of an obstacle. A new behavior supervisor that manages the switching among the OBF, TS, and VPS behaviors in unknown environments is proposed. In the navigation of a real robot, a new robot localization method through the fusion of encoders and an infrared localization sensor using a particle filter is proposed. Finally, this article presents simulations and experiments to verify the feasibility and advantages of the navigation scheme.

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Constrained multiobjective optimization problems commonly arise in real-world applications. In the presence of constraints and multiple conflicting objectives, finding a set of feasible solutions which can best tradeoff different objectives is quite challenging. In this study, an ε-constrained multiobjective differential evolution using linear population size expansion is proposed to solve such a kind of problems. First, the ε-constraint-handling method, which originally solves constrained optimization problems with only one objective, is further improved to handle constraints in a multiobjective optimization way. Second, to achieve a better approximation to the feasible Pareto front, a linear population size expansion strategy is developed. Once enough feasible solutions have been found, the population size will be linearly increased to find more promising solutions. As a result, a simple yet efficient constrained multiobjective differential evolution is proposed. Experiments are conducted to evaluate the performance of the proposed algorithm on 35 benchmark test functions with different numbers of constraints and objectives. Obtained results are compared with seven state-of-the-art algorithms. Empirical results and comparisons demonstrate that our proposed algorithm achieves better or at least comparable performance to the competitors, and is capable of obtaining a set of representative feasible solutions for the selected real-world constrained multiobjective optimization problems, especially for highly constrained problems.

A multi-objective natural aggregation algorithm for optimizing user allocation matrix in visible light communication

Allocating users to LED panels in a visible light communication (VLC) environment is a non-convex multi-objective constrained optimization problem. The computational complexity and accuracy of the optimization are the key factors to deal with in a VLC environment. In this manuscript, initially the user allocation problem is formulated as a single objective optimization problem by using sum-utility function of data rate and blockage probability. Then these two contradictory objective functions are simultaneously optimized by a new multi-objective natural aggregation algorithm (MONAA). This proposed algorithm is inspired by the aggregation behavior of cockroaches which help them to determine the darkest spot in a search space. The performance of this multi-objective algorithm is compared with NSGA-II, MOPSO, MOEA/D and MOSOS on eight unconstrained and six constrained mathematical function optimizations. In single objective and multi-objective cases, superior performance is demonstrated by the natural aggregation algorithm for solving the user allocation matrix optimization.

Variable speed multi-task allocation for mobile crowdsensing based on a multi-objective shuffled frog leaping algorithm

In multi-task studies of mobile crowdsensing, the possibility that a user may adopt another travel mode when completing the current task to the next is ignored. In addition, existing methods tend to allocate more tasks to the users with high reputation, which causes that few tasks will be assigned to new users with low reputation. In order to cover these shortages, a constrained multi-objective optimization model of variable speed multi-task allocation is established, which aims to maximize the user rewards and minimize the task completion time simultaneously. Meanwhile, the maximum number of fully paid tasks positively correlated with reputation is set for each user. To solve the constructed model, a three-stage multi-objective shuffled frog leaping algorithm is proposed, which introduces an objective anchored hybrid initialization operator based on heuristic information, a region mining strategy for the archive individuals, a discrete leaping rule to enhance the interaction of individual information and a constraint handling operator to reduce the loss of individual information. The performance of the proposed algorithm is evaluated by comparing it with five state-of-the-art algorithms on both real-world and synthetic instances. Experimental results show that the proposed algorithm can find a set of Pareto optimal allocation solutions with better convergence and distributions.

Status update frequency optimization for In-vehicle LAN route based on deep Q-learning

The popularization of the automated fleet requires a reliable transmission of In-vehicle Local Area Network (LAN) with high mobility under complex environment. Therefore, In-vehicle communication system needs to collect the network status of the vehicles in real time to ensure the availability of routing decisions. The high update frequency of the network status will increase the availability of the routing decisions. However, it will also incur congestion of the In-vehicle LAN. Therefore, it is of great practical interest to optimize the network status update frequency for the In-vehicle LAN to achieve the balance between route availability and network congestion. We consider this important problem in this paper and make the following contributions. Firstly, we analyze the network routing feature information to establish the multi-feature dynamic queue model and age of multi-feature information model, which describes the relationship between the network status and the routing availability. Secondly, this status update frequency selection problem is formulated as a multi-objective constrained optimization and solved by deep Q-learning with feature selection mechanism to improve the timeliness of In-vehicle LAN. Numerical results demonstrate that comparing with the existing algorithms, the drop rate of In-vehicle LAN decreases about 7.68% and the convergence speed of the proposed method improves about 8.3%.

Constrained multi-objective differential evolution algorithm with ranking mutation operator

There are feasible and infeasible solutions in Constrained Multi-objective Optimization Problems (CMOPs). The feasible solutions with lower rank should be given more chances to generate offspring, while infeasible and worse solutions with higher rank should have fewer chances in these CMOPs. A constrained multi-objective Differential Evolution (DE) algorithm is developed by considering the selection pressure. The population is ranked based on the non-dominated crowd sort and constrained dominated principle. Then, a tournament operator is designed to extend the conventional mutation operator to boost the exploitation. The performances of the proposed algorithm are assessed on nineteen benchmark functions and industrial applications. Five representative algorithms are selected to make comparisons. The experiments have demonstrated that the algorithm can find well-distributed Pareto front, and the result of the performance indicator is superior.

Battery energy storage system for enhancing the electrolyzer capacity factor in small-scale WindtH2 system with a smoothing control strategy: Constrained multi-objective Pareto optimization and case study in Algeria

Stochastic nature of wind energy prevents the electrolyzer in wind-to-hydrogen (WindtH2) system to accomplish high capacity factor without the assistance of the battery energy storage system (BESS). Furthermore, design process focuses on the reliability of the system and its components to achieve low production cost. The goal of this investigation is to develop a decision making tool for solving constrained multi-objective optimization (CMO) problems to minimize the cost and power losses, and maximize the reliability of a small-scale WindtH2 system subjected to various constraints. The proposed algorithm, which combines a CMO model with a smoothing control strategy (SCS), solves the CMO problem to find Pareto-optimal solutions. Dynamic SCS smoothes the power provided to the electrolyzer by the assistance of the BESS. It favors the electrolyzer full-capacity operation for enhancing its capacity factor. Minimum/maximum electrolyzer power input and safe thresholds of battery and H2 tank state-of-charge are the main operational constraints taken into consideration by the CMO model. The four conflicting objectives considered in the optimization model are the Levelized Cost of H2 (LCOH), Total H2 Deficit (THD), Energy Dump Possibility (EDP) and a new indicator namely Electrolyzer Capacity Factor (ECF). Solution vector consists of seven decision variables including battery cycles spent and electrolyzer operation hours. The developed algorithm is applied to generate Pareto solutions of a case for industrial usage. Pareto results show: (1) The proposed algorithm with the associated indicators is useful for the design process; (2) The SCS significantly helps to improve the ECF; (3) Consistent relationships between the four objectives; (4) WindtH2 system consisting of wind turbine farm of 836.5 kW, electrolyzer of 276 kW, battery of 765 kWh and H2 Tank of 10.8 kg gives a LCOH of 27.01 $/kg, ECF of 20.5%, EDP of 8.19% and THD of 41.72%.

Characterization of constrained continuous multiobjective optimization problems: A feature space perspective

Despite the increasing interest in constrained multiobjective optimization in recent years, constrained multiobjective optimization problems (CMOPs) are still insufficiently understood and characterized. For this reason, the selection of appropriate CMOPs for benchmarking is difficult and lacks a formal background. We address this issue by extending landscape analysis to constrained multiobjective optimization. By employing four exploratory landscape analysis techniques, we propose 29 landscape features (of which 19 are novel) to characterize CMOPs. These landscape features are then used to compare eight frequently used artificial test suites against a recently proposed suite consisting of real-world problems based on physical models. The experimental results reveal that the artificial test problems fail to adequately represent some realistic characteristics, such as strong negative correlation between the objectives and the overall constraint violation. Moreover, our findings show that all the studied artificial test suites have advantages and limitations, and that no “perfect” suite exists. Additionally, the effectiveness of the proposed features at predicting algorithm performance is demonstrated for two multiobjective optimization algorithms. Benchmark designers can use the obtained results to select or generate appropriate CMOP instances based on the characteristics they want to explore.

Cooperative Multiobjective Evolutionary Algorithm With Propulsive Population for Constrained Multiobjective Optimization

Convergence, diversity and feasibility are three important issues when solving constrained multiobjective optimization problems (CMOPs). To deal with the balance among convergence, diversity and feasibility well, this article proposes a cooperative multiobjective evolutionary algorithm with propulsive population (CMOEA-PP) for solving CMOPs. CMOEA-PP has two populations, including propulsive population and normal population, and these two populations work cooperatively. Specifically, propulsive population focuses on convergence. Normal population gives priority to feasibility and is obligated to maintain diversity. To cross through the infeasible region and reach the Pareto front (PF), propulsive population does not consider constraints in the early stage and only considers constraints in the later stage. To further accelerate the speed of convergence, propulsive population only searches for corner solutions and center solutions, while normal population searches for the whole PF. As a result, propulsive population can cross through the infeasible region because of the lack of attention to feasibility. In addition, propulsive population also can guide and accelerate the convergence of the evolutionary process. Comprehensive experiment results on several sets of benchmark problems demonstrate that CMOEA-PP is better than existing state-of-the-art competitors.

Constrained multiobjective differential evolution algorithm with infeasible-proportion control mechanism

When solving constrained multiobjective optimization problems, it is crucial to deal with several conflicting objectives and various constraints simultaneously. To address these two issues, a constrained multiobjective differential evolution algorithm with an infeasible-proportion control mechanism is presented in this paper. Specifically, two populations are cooperatively employed in the evolution process. The first population is used to solve the original problem, that is, to explore the constrained Pareto front, while the second population is created to search for high-quality objective function information. Furthermore, differential evolution with two specific mutation strategies is employed to update each population. Cooperative strategies can not only balance diversity and convergence but also realize information exchange between the two populations provide new evolutionary directions. In addition, an infeasible-proportion control mechanism is used to gradually decrease the proportion of infeasible solutions in the second population. This enables the main population to pass through the infeasible region barrier in the early evolution stage and facilitates the search for a constrained Pareto front in the later evolution stage. Systematic experiments on 65 benchmark test functions show that the proposed algorithm is superior to or at least comparable to five well-established constrained multiobjective optimization methods.

Performance optimization of hard rock tunnel boring machine using multi-objective evolutionary algorithm

The hard rock Tunnel Boring Machine (TBM) is a complex engineering equipment coupled with multiple sub-systems for underground tunnel excavation in complex geological environments. Resetting the operational and structural parameters of TBM according to different geological conditions usually requires engineers to spend a lot of time dealing with the interaction between various subsystems, which is a tedious and time-consuming job. To facilitate setting the operational and structural parameters of TBM, we present a constrained multi-objective optimization model and its solving method in this paper. To be specific, three performance indices, i.e. minimizing the system construction period, construction energy consumption and construction cost of TBM, are firstly considered as the three objectives in the proposed model. Secondly, two push and pull search (PPS) based algorithms, including PPS-MOEA/D and PPS-KnEA, are suggested to solve the formulated constrained multi-objective optimization problem. Finally, to verify the performance of the developed method, the presented method is compared with several popular constrained multi-objective evolutionary algorithms by tackling the established optimization model. The experimental results reveal that the presented method has the best performance among the comparison algorithms, and the overall performance of the algorithm with PPS is better than other algorithms without PPS, which indicates the superiority of PPS framework in solving practical optimization problems.

A machine learning approach for propeller design and optimization: Part I

This paper introduces a machine learning approach for optimizing propellers. The method aims to improve the computational cost of optimization by reducing the number of evaluations required to find solutions. This is achieved by directing the search towards design clusters with good performance, i.e. high propulsive efficiency and low cavitation. Three types of clusters are expected. The first cluster constitutes designs with performance of interest, i.e. high efficiency and low cavitation. The second cluster constitutes designs with performance not of interest, i.e. low efficiency and high cavitation. The third cluster constitutes designs whose performance cannot be estimated with the Boundary Element Methods (BEM) that we use in this study. In simple cases with single objective optimization to maximize efficiency, these clusters can be identified a-priori with unsupervised classifiers provided that orthogonally independent parameters are used as demonstrated in this paper. For multi-objective constrained optimization, to maximize efficiency and minimize cavitation, for example, supervised classifiers may be required to learn the clusters. Classical design variables such as chordlength, pitch, skew, rake, thickness distribution and camber of hydrofoils cannot be used to identify these clusters because of multicollinearity. Thus, a new orthogonal parametric model is proposed where the parameters are directly derived from the propeller blade mesh. As the blade surface mesh is used as boundary conditions to solve the governing equations, the orthogonal parameters are expected to have a stronger correlation with performance predictions of BEM or Computational Fluid Dynamics (CFD) than classical design variables. We demonstrate that design clusters with good performance can be identified with few BEM evaluations. Furthermore, the method synergizes explainable supervised and unsupervised learning to advice search algorithms and quickly guide them to lucrative regions in the design space. However, reducing the cost of optimization results in a trade-off with completeness of the search; this is also investigated in this paper. The method is demonstrated on a simple fully wetted flow case of the benchmark Wageningen B-4 70 propeller with P/D=1.0, as the geometry and open-water curves are readily accessible allowing back of the envelope verification and validation of our results.

Coevolutionary Framework-Based Constrained Multi-Objective Optimization for Optimal Carbon-Energy Combined Flow of a Power Grid

A new constrained multi-objective optimization coevolutionary algorithm (CCMO) based on the NSGA-II algorithm is proposed to cope with the efficient optimization of multiple objectives containing constraints in the optimal combined carbon-energy flow (OCECF). The algorithm improves the convergence of the population by evolving a new auxiliary population that shares effective information with the original population for weak cooperation, offering significant performance advantages. Applying the algorithm for reactive power control on two different-sized IEEE benchmark systems (IEEE-57 and IEEE-300 bus systems), respectively, minimizes carbon emissions and voltage deviations in the grid. Simulation results show that the CCMO algorithm has significant advantages in terms of the convergence speed and Pareto front.

Constrained Multi-Objective Optimization with a Limited Budget of Function Evaluations

This paper proposes the Self-Adaptive algorithm for Multi-Objective Constrained Optimization by using Radial Basis Function Approximations, SAMO-COBRA. This algorithm automatically determines the best Radial Basis Function-fit as surrogates for the objectives as well as the constraints, to find new feasible Pareto-optimal solutions. SAMO-COBRA is compared to a wide set of other state-of-the-art algorithms (IC-SA-NSGA-II, SA-NSGA-II, NSGA-II, NSGA-III, CEGO, SMES-RBF) on 18 constrained multi-objective problems. In the first experiment, SAMO-COBRA outperforms the other algorithms in terms of achieved Hypervolume (HV) after being given a fixed small evaluation budget on the majority of test functions. In the second experiment, SAMO-COBRA outperforms the majority of competitors in terms of required function evaluations to achieve 95% of the maximum achievable Hypervolume. In addition to academic test functions, SAMO-COBRA has been applied on a real-world ship design optimization problem with three objectives, two complex constraints, and five decision variables.

An Evolutionary Multitasking Optimization Framework for Constrained Multiobjective Optimization Problems

When addressing constrained multiobjective optimization problems (CMOPs) via evolutionary algorithms, various constraints and multiple objectives need to be satisfied and optimized simultaneously, which causes difficulties for the solver. In this article, an evolutionary multitasking (EMT)-based constrained multiobjective optimization (EMCMO) framework is developed to solve CMOPs. In EMCMO, the optimization of a CMOP is transformed into two related tasks: one task is for the original CMOP, and the other task is only for the objectives by ignoring all constraints. The main purpose of the second task is to continuously provide useful knowledge of objectives to the first task, thus facilitating solving the CMOP. Specially, the genes carried by parent individuals or offspring individuals are dynamically regarded as useful knowledge due to the different complementarities of the two tasks. Moreover, the useful knowledge is found by the designed tentative method and transferred to improve the performance of the two tasks. To the best of our knowledge, this is the first attempt to use EMT to solve CMOPs. To verify the performance of EMCMO, an instance of EMCMO is obtained by employing a genetic algorithm as the optimizer. Comprehensive experiments are conducted on four benchmark test suites to verify the effectiveness of knowledge transfer. Furthermore, compared with other state-of-the-art constrained multiobjective optimization algorithms, EMCMO can produce better or at least comparable performance.

Constrained Multi-Objective Optimization for UAV-Enabled Mobile Edge Computing: Offloading Optimization and Path Planning

An unmanned aerial vehicle (UAV) is employed to sequentially visit the specific waypoints and provide offloading services for nearby devices. Most of the current works optimized the UAV-enabled offloading according to a single criterion while neglecting necessary optimizations and constraints for flight safety of the UAV. This motivates us to study the optimization problem of the UAV from a multi-objective viewpoint by considering the UAV’s flight safety. A constrained multi-objective optimization problem (CMOP) involving two objective functions about the energy-efficient offloading and safe path planning is formulated for the UAV. To solve the formulated CMOP, we present a constrained decomposition-based multi-objective evolution algorithm. To further improve the algorithm, we particularly utilize the infeasible individuals with great objective values, which provide useful information for improving the optimized objective values during the evolution process. Finally, experimental results demonstrate that compared with the existing works, our scheme is beneficial to simultaneously reduce energy consumption and ensure safe flight for the UAV.

A tri-population based co-evolutionary framework for constrained multi-objective optimization problems

Balancing between the optimization of objective functions and constraint satisfaction is essential to handle constrained multi-objective optimization problems (CMOPs). Recently, various methods have been presented to enhance the performance for the constrained multi-objective optimization evolutionary algorithms (CMOEAs). However, most of them encounter difficulties when dealing with the CMOPs with complex feasible regions. To overcome this drawback, this paper proposes a tri-population based co-evolutionary framework (TriP): i) the first and second populations are evolved through a weak co-evolutionary relation for the original and unconstrained problems respectively to handle CMOPs with relatively simple constraints; and ii) the third population is evolved solely for the constraint relaxed problem with constraint relaxation technique. The cooperation of three populations preserve the advantages of weak co-evolution and constraint relaxation. Experiments on six benchmark CMOPs with 65 instances and diverse features are performed. Compared to 9 state-of-the-art CMOEAs, the proposed framework yields highly competitive performance and the best versatility. In addition, the effectiveness of the proposed framework on handling real-world CMOPs is also verified.

Indicator-Based Evolutionary Algorithm for Solving Constrained Multiobjective Optimization Problems

To prevent the population from getting stuck in local areas and then missing the constrained Pareto front fragments in dealing with constrained multiobjective optimization problems (CMOPs), it is important to guide the population to evenly explore the promising areas that are not dominated by all examined feasible solutions. To this end, we first introduce a cost value-based distance into the objective space, and then use this distance and the constraints to define an indicator to evaluate the contribution of each individual to exploring the promising areas. Theoretical studies show that the proposed indicator can effectively guide population to focus on exploring the promising areas without crowding in local areas. Accordingly, we propose a new constraint handling technique (CHT) based on this indicator. To further improve the diversity of population in the promising areas, the proposed indicator-based CHT divides the promising areas into multiple subregions, and then gives priority to removing the individuals with the worst fitness values in the densest subregions. We embed the indicator-based CHT in evolutionary algorithm and propose an indicator-based constrained multiobjective algorithm for solving CMOPs. Numerical experiments on several benchmark suites show the effectiveness of the proposed algorithm. Compared with six state-of-the-art constrained evolutionary multiobjective optimization algorithms, the proposed algorithm performs better in dealing with different types of CMOPs, especially in those problems that the individuals are easy to appear in the local infeasible areas that dominate the constrained Pareto front fragments.

A fuzzy constraint handling technique for decomposition-based constrained multi- and many-objective optimization

The challenge in solving constrained multi-objective optimization problems (CMOPs) is how to balance minimizing objectives and satisfying constraints, especially when the infeasible region is very large. To address this issue, this work proposes a fuzzy constraint handling technique, which uses the fuzzy set theory to accurately characterize the difference between solutions on objective function values and constraint violation degrees. On this basis, a new concept, called “fuzzy advantage”, is introduced to comprehensively quantify the degree to which one solution is better than others, allowing the infeasible solutions with promising fitness to survive. The proposed method is integrated with a decomposition-based multi-objective evolutionary algorithm to verify its effectiveness. Compared with nine state-of-the-art MOEAs on a number of test problems and a real-world optimization problem, the proposed algorithm shows high competitiveness in solving a variety of CMOPs.