Constrained Multi-objective Optimization Research Articles

Constrained multiobjective robust optimization of a bistable mechanism for inertial switch

Constrained multiobjective robust optimization of a bistable mechanism for inertial switch

A novel competitive constrained dual-archive dual-stage evolutionary algorithm for constrained multiobjective optimization

In constrained multi-objective optimization problems (CMOPs), the key problem is how to balance convergence and diversity simultaneously and find as many feasible solutions as possible in the reduced search space. To address this issue, a novel competitive constrained dual-archive dual-stage evolutionary algorithm, named c-DADSEA, is proposed with a convergence-driven archive (CA) and a diversity-driven archive (DA). To improve the algorithm’s convergence, instead of simply excluding infeasible solutions, an adaptive penalty function is novelly proposed to maintain competitively infeasible solutions in CA. To overcome the difficulty in finding the true Pareto front for the dual-population framework in some problems, a dual-stage framework is introduced to make DA search in different directions at different stages (i.e., Forward and Backward), enhancing the diversity of the algorithm. DA evolves without considering the constraints to explore all regions as much as possible in the Forward stage. While in the Backward stage, DA degenerates with increasing constraints to search the regions around the true Pareto front by using ϵ-constrained method. Moreover, to balance convergence and diversity, different benchmark rankings are designed for the individuals in different archives. The experiments on three benchmark CMOPs test suites demonstrate that c-DADSEA is better than seven other state-of-the-art constrained evolutionary algorithms, especially for the CMOPs with a constrained Pareto front away from the unconstrained Pareto front.

A multi-preference-based constrained multi-objective optimization algorithm

When tackling constrained multi-objective optimization problem, evolutionary algorithms grapple with the simultaneous need to optimize the conflict objectives and satisfy constraints. The preference for the algorithm in processing objectives and constraints directly affects the feasibility, diversity and convergence of population. At the same time, the decision regarding algorithmic preference should be based on the characteristics of the current evolution. Therefore, preference decision of algorithm is a intractable challenge throughout the optimization. To address this issue, a multi-preference-based constrained multi-objective optimization algorithm is proposed in this paper, operating under the aegis of three evolutionary models. The preferences are determined by the analysis of the evolution states in concert with the actual characteristic of the population, and are implemented through the reasonable scheduling evolutionary models. When the preference changes, the algorithm identifies the useful knowledges carried by the previous populations and transfers them to the current populations. In addition, a shift-based update strategy and a new co-evolution strategy are designed for different models in the proposed algorithm, respectively. Compared to five state-of-the-art multi-objective algorithms on two benchmark suits and two real-world applications, the proposed algorithm performs better than its peers, especially in diversity difficulty and convergence difficulty multi-objective optimization problems.

Coordinated Multi-UAV Reconnaissance Scheme for Multiple Targets

This study addresses dynamic task allocation challenges in coordinated surveillance involving multiple unmanned aerial vehicles (UAVs). A significant concern is the increased UAV flight distance resulting from the assignment of new missions, leading to decreased reconnaissance efficiency. To tackle this issue, we introduce a collaborative multi-target and multi-UAV reconnaissance scheme. Initially, the multitasking constrained multi-objective optimization framework (MTCOM) is employed to optimize task allocation and reconnaissance time in static scenarios. Subsequently, in case of emergency, we iteratively refine the outcomes of static task allocation through an enhanced auction-based distributed algorithm, effectively reducing UAV flight costs in response to new missions, UAV withdrawal, or damage. Simulation results demonstrate the efficacy of our proposed multi-UAV and multi-target cooperative reconnaissance scheme in resolving dynamic task allocation issues. Additionally, our approach achieves a 5.4% reduction in UAV flight distance compared to traditional allocation methods. The main contribution of this paper is to consider a dynamic scenario model involving UAV damage and the emergence of new reconnaissance areas. Then we propose an innovative collaborative multi-target and multi-UAV reconnaissance scheme to address this issue and, finally, conduct experimental simulations to verify the effectiveness of the algorithm.

A Multistage Algorithm for Solving Multiobjective Optimization Problems With Multiconstraints

There are usually multiple constraints in constrained multi-objective optimization. Those constraints reduce the feasible area of the constrained multi-objective optimization problems (CMOPs) and make it difficult for current multi-objective optimization algorithms (CMOEAs) to obtain satisfactory feasible solutions. In order to solve this problem, this paper studies the relationship between constraints, then obtains the priority between constraints according to the relationship between the Pareto Front (PF) of the single constraint and their common PF. Meanwhile, this paper proposes a multi-stage CMOEA and applies this priority, which can save computing resources while helping the algorithm converge. The proposed algorithm completely abandons the feasibility in the early stage to better explore the objective space, and obtains the priority of constraints according to the relationship; Then the algorithm evaluates a single constraint in the medium stage to further explore the objective space according to this priority, and abandons the evaluation of some less-important constraints according to the relationship to save the evaluation times; At the end stage of the algorithm, the feasibility will be fully considered to improve the quality of the solutions obtained in the first two stages, and finally get the solutions with good convergence, feasibility, and diversity. The results on five CMOP suites and three real-world CMOPs show that the algorithm proposed in this paper can have strong competitiveness in existing constrained multi-objective optimization.

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.

A Competitive and Cooperative Swarm Optimizer for Constrained Multiobjective Optimization Problems

Solving multi-objective optimization problems through meta-heuristic methods gets considerable attention. Based on the classical variation operators, several enhanced operators, as well as multi-objective optimization evolutionary algorithms, have been developed. Among these operators, the competitive swarm optimizer exhibits promising performance. However, it encounters difficulties when tackling constrained multi-objective optimization problems with large objective spaces or complex infeasible regions. In this paper, a competitive and cooperative swarm optimizer is proposed, which contains two particle update strategies: i) the competitive swarm optimizer provides faster convergence speed to accelerate the approximation of the Pareto front; and ii) the cooperative swarm optimizer suggests a mutual-learning strategy to enhance the ability to jump out of local feasible regions or local optima. We also present a new algorithm for constrained multi-objective optimization problems. The results on four benchmark suites with 47 instances demonstrate the superiority of our approach compared with other state-of-the-art methods. Additionally, its effectiveness on large scale constrained multi-objective optimization problems has also been verified.

Evolutionary Multitasking with Global and Local Auxiliary Tasks for Constrained Multi-Objective Optimization

Evolutionary Multitasking with Global and Local Auxiliary Tasks for Constrained Multi-Objective Optimization

An Instance Space Analysis of Constrained Multiobjective Optimization Problems

Constrained multi-objective optimization problems (CMOPs) are generally more challenging than unconstrained problems. This in part can be attributed to the infeasible region generated by the constraint functions, the interaction between constraints and objectives, or both. In this paper, we explore the relationship between the performance of constrained multi-objective evolutionary algorithms (CMOEAs) and the instance characteristics of CMOP using Instance Space Analysis (ISA). To do this, we extend recent work on Landscape Analysis features for characterising CMOPs. Specifically, we introduce new features to describe the multi-objective-violation landscape, formed by the interaction between constraint violation and multi-objective fitness. Detailed evaluation of the algorithm footprints, spanning eight CMOP benchmark suites and fifteen CMOEAs, demonstrates that ISA effectively captures the strength and weakness of the CMOEAs. We conclude that two characteristics, the isolation of non-dominate set and the correlation between constraints and objectives evolvability, have the greatest impact on algorithm performance. However, the current benchmarks problems lack of diversity to represent the real-world problems and to fully reveal the efficacy of CMOEAs evaluated.

Multi-dimensional management framework on fresh groundwater lens of Kish Island in the Persian Gulf, Iran

Groundwater in arid and semi-arid coastal aquifers is vulnerable to seawater intrusion and quality deterioration despite being one of the most reliable sources of water supply due to the increasing number of development plans and competition between water consumers. A multi-dimensional groundwater management framework is developed to trade-off between groundwater abstraction, allocation equity, groundwater quality, and energy considerations in the reverse osmosis (RO) filtration process in the fresh groundwater lens of Kish Island, Iran. An arid island confined in the Persian Gulf is modeled using 3D simulation and three well-occupied multi-objective evolutionary optimization algorithms. Four objectives include: maximizing the groundwater abstraction, minimizing the Gini coefficient (allocation inequity), minimizing the total energy required to pass saline water through the RO membrane to reach the standard total dissolved solids (TDS), and minimizing the average TDS concentration of water abstraction positions from 11 management zones have been considered over a 50-year management horizon. Solutions obtained in the simulation-based constrained multi-objective optimization framework allow managers to choose from 587 Pareto optimal solutions. They provide an abstraction scheme with a range of 1.44 to 4.53 MCM/yr, a Gini coefficient of 0 to 0.98, filtration energy of 988,562 to 1,935,760 kWh/yr, and an average TDS of 19,663 to 21,351 mg/L. The Pareto optimal solutions can help decision-makers decide on the multi-dimensional problems of sustainable coastal groundwater management and show patterns among different objectives.

A multi-population evolutionary algorithm for multi-objective constrained portfolio optimization problem

A multi-population evolutionary algorithm for multi-objective constrained portfolio optimization problem

A dual-population evolutionary algorithm based on dynamic constraint processing and resources allocation for constrained multi-objective optimization problems

Constrained multi-objective optimization problems (CMOPs) contain the satisfaction of various constraints and optimization of multiple objectives simultaneously, thus they are extremely challenging. Although many constrained multi-objective evolutionary algorithms (CMOEAs) have been proposed, they ignore the information of each constraint, which might help utilize more various infeasible solutions to improve the search ability of the population. Therefore, this paper proposes a new dual-population CMOEA to solve CMOPs, in which a dynamic constraint processing mechanism and a dynamic resource allocating scheme are designed. To be specific, the proposed algorithm evolves two populations, which adopt different mechanisms to handle constraints respectively. The main population directly optimizes all constraints to find the feasible Pareto optimal solutions, which can improve the feasibility. The auxiliary population adopts a dynamic constraint processing mechanism, which gradually increases the number of constraints being processed, so as to fully utilize various infeasible solutions to help find feasible regions. Moreover, a new dynamic resource allocating scheme is proposed to reasonably allocate the limited computational resources to the two populations according to their performance feedback. Experimental results on three test suites and ten practical problems show that the proposed algorithm has a better or competitive performance compared with several state-of-the-art CMOEAs.

Interactive niching-based two-stage evolutionary algorithm for constrained multiobjective optimization

When solving constrained multiobjective optimization problems (CMOPs), how to maintain diversity without losing convergence is a major challenge, because some small discrete feasible regions make the population hard to find a complete feasible Pareto Front. To this end, an interactive niching-based two-stage evolutionary algorithm for constrained multiobjective optimization, named INCMO, is proposed in this study. At each stage of INCMO, both two populations are employed to focus on different evolution purposes. Specifically, the main population optimizes the original CMOP, while the auxiliary population only considers the objectives by ignoring all constraints. INCMO uses two different niching mechanisms in two stages. In the early stage, each population independently uses the niching technique to generate offspring, to maintain diversity. In addition, two offspring populations are shared to realize migration between two populations. In the later stage, two populations cooperate to generate offspring based on an interactive niching technique. To be specific, two populations are merged and then divided into different niches. In this case, individuals from two populations will be assigned to the same niche. After that, individuals in the niche produce offspring through crossover and mutation. The process of generating offspring also promotes more effective migration between the two populations on the basis of increasing the diversity of population. Compared with other well-established methods, the proposed method in this study shows better performance on four test suites including 47 benchmark problems.

Multi-objective constrained black-box optimization algorithm based on feasible region localization and performance-improvement exploration

Over the past decade, surrogate-assisted evolutionary algorithms have demonstrated their effectiveness across various computationally expensive real-world domains. Nevertheless, the focus of surrogate-assisted multi-objective evolutionary algorithms has primarily centered on non-constrained optimization problems. There has been relatively limited exploration into addressing expensive constrained multi-objective optimization problems, which inherently require a delicate equilibrium between convergence, diversity, and feasibility. To bridge this gap, this paper concentrates on constrained multi-objective optimization problems where both objectives and constraints involve substantial computational costs. In response, a novel data-driven constrained multi-objective evolutionary algorithm is introduced, leveraging feasible region localization and performance-improvement exploration. For feasible region localization, a constraints-domain-search strategy is presented to locate the feasible region quickly. To enhance performance-improvement exploration, a progressive enhancement of convergence and diversity is achieved through the incorporation of constraint penalties. With the help of exploration and exploitation, the proposed algorithm can balance convergence, diversity, and feasibility while working within a limited number of function evaluations. By comparing the proposed algorithm with state-of-the-art algorithms on 66 mathematical problems and a resource-intensive black-box problem, its outstanding performance for solving multi-objective constrained black-box problems is validated.

A constrained multi-objective evolutionary algorithm with Pareto estimation via neural network

The main challenge in addressing constrained multi-objective optimization problems (CMOPs) lies in achieving a balance among convergence, diversity, and feasibility. To address this issue, this paper proposes a constrained multi-objective evolutionary algorithm with Pareto estimation via neural network named CMOEA-PeNN. In order to exploit and explore the decision space, the proposed algorithm employs a dual-population mechanism, which is trained with a self-organizing map (SOM). Firstly, the population distribution structure in decision space is mapped to objective space while preserving neighborhood information, and then the neuron weight is utilized to estimate the Pareto front (PF). Secondly, a novel approach is devised to preserve the feasibility of the population and enhance the estimation of the Pareto front by SOM. The achievement scalarizing function (ASF) is employed to choose promising solutions. This strategy could guide the population toward the optimal solution while exploring the small feasible regions. Finally, the performance of CMOEA-PeNN is compared with five state-of-the-art constrained multi-objective evolutionary algorithms (CMOEAs) on three widely used benchmark problems and a real-world problem. The experimental results show that CMOEA-PeNN could archive competitive performance in solving CMOPs.

A coevolutionary constrained multi-objective algorithm with a learning constraint boundary

When solving constrained multi-objective optimization problems, the balance of convergence, diversity, and feasibility plays a pivotal role. To address this issue, this paper proposes a coevolutionary constrained multi-objective algorithm with learning constraint boundary (CCMOLCB). Firstly, the constrained multi-objective problems are transformed by adding an additional objective using the constraint violation degree. Then, the transformed problem is solved by an improved coevolutionary framework which employs two populations. The main population explores the objective space and repairs infeasible solutions to maintain the feasibility of population. Meanwhile, the feasibility and diversity of solutions are balanced by using a dynamic weight coefficient during the evolution, it changes as the number of iterations increases. The subordinate population selects solutions by taking into consideration the learning constraint boundary (LCB). This boundary guarantees convergence of solutions by constraining the search range of the main population, thereby enhancing the environmental selection pressure. The performance of CCMOLCB is compared with seven state-of-the-art constrained multi-objective evolutionary algorithms on five test suites. The experimental results illustrate that CCMOLCB exhibits competitive performance when dealing with this family of problems.

Dual population approximate constrained Pareto front for constrained multiobjective optimization

For constrained multiobjective optimization problems (CMOPs), the ultimate goal is to obtain a set of well-converged and well-distributed feasible solutions to approximate the constrained Pareto front (CPF). Various constraints may change the position and/or shape of the CPF. This poses great challenges to the approximation of the CPF. This is especially true when the CPF mainly lies on constraint boundaries (i.e., CPF and unconstrained PF have little or even no intersection). To tackle this issue, we propose a novel dual population algorithm for approximating the CPF from both sides of the constraint boundaries. Specifically, Population1 uses the constrained-domination principle to approximate the CPF from the sides of feasible regions only; Population2 adopts an improved ϵ-constrained method to approximate the CPF from both the feasible as well as infeasible regions. Offspring generated by both populations are merged and combined with Population1 and Population2. In addition, some selected members of Population1 and Population2 are permitted to migrate to the combined populations to facilitate knowledge sharing. Systematic experiments carried out on three benchmark test suites and 10 real-world CMOPs show the proposed algorithm achieved superior or competitive performance, especially for CMOPs where the CPF is mainly located at constraint boundaries. Therefore, on the basis of dual population, approximating CPFs from both sides of feasible and infeasible regions contributes an alternative approach to solving CMOPs.

Operation Optimization of Wind/Battery Storage/Alkaline Electrolyzer System Considering Dynamic Hydrogen Production Efficiency

Hydrogen energy is regarded as a key path to combat climate change and promote sustainable economic and social development. The fluctuation of renewable energy leads to frequent start/stop cycles in hydrogen electrolysis equipment. However, electrochemical energy storage, with its fast response characteristics, helps regulate the power of hydrogen electrolysis, enabling smooth operation. In this study, a multi-objective constrained operation optimization model for a wind/battery storage/alkaline electrolyzer system is constructed. Both profit maximization and power abandonment rate minimization are considered. In addition, some constraints, such as minimum start/stop times, upper and lower power limits, and input fluctuation limits, are also taken into account. Then, the non-dominated sorting genetic algorithm II (NSGA-II) algorithm and the entropy method are used to optimize the operation strategy of the hybrid energy system by considering dynamic hydrogen production efficiency, and through optimization to obtain the best hydrogen production power of the system under the two objectives. The change in dynamic hydrogen production efficiency is mainly related to the change in electrolyzer power, and the system can be better adjusted according to the actual supply of renewable energy to avoid the waste of renewable energy. Our results show that the distribution of Pareto solutions is uniform, which indicates the suitability of the NSGA-II algorithm. In addition, the optimal solution indicates that the battery storage and alkaline electrolyzer can complement each other in operation and achieve the absorption of wind power. The dynamic hydrogen production efficiency can make the electrolyzer operate more efficiently, which paves the way for system optimization. A sensitivity analysis reveals that the profit is sensitive to the price of hydrogen energy.

Adaptive design of the magnetron injection gun using constrained multiobjective optimization

The design of the magnetron injection gun (MIG) is an inverse problem with implicit objective functions and complex constraints. The computer-aided design using an optimization algorithm is a promising method to improve the performance of the MIG and reduce design complexity and time consumption. However, the dependence on the initial structure makes it difficult to realize the real sense of using algorithms to design. To address this issue, this paper proposes an adaptive design method for the MIG. The design problem is equivalent to a constrained multiobjective optimization problem. A pseudo-random structure generation algorithm is developed based on the geometric constraints and the first-order MIG design trade-off equations. The fully parameterized MIG structure is randomly generated by the decision variables composed of the scale coefficients ranging from (0,1) and the other structural parameters. This transformation can increase the effectiveness of the randomly generated MIG structures and make it easier to set the value range of decision variables. The proposed adaptive design method is further verified by successfully designing the MIG for an 800 GHz gyrotron via the co-simulation of MATLAB and CST.

An adaptive tradeoff evolutionary algorithm with composite differential evolution for constrained multi-objective optimization

Convergence, diversity, and feasibility are crucial factors in solving constrained multi-objective optimization problems (CMOPs). Their imbalance can result in the algorithm failing to converge well to the Pareto front, especially when dealing with complex CMOPs. To address this issue, we propose an adaptive tradeoff evolutionary algorithm (ATEA), which can adjust the environment selection strategy based on the characteristics of problem, aiming to achieve a balance between convergence and diversity while ensuring feasibility of the population. The ATEA divides the search process into three phases: In the extended exploration phase, a global search is conducted using a guided constraint relaxation technique to enable the population to quickly traverse the infeasible region and approach the feasible region. In the tradeoff exploration phase, constraints are further detected and estimated to retain more feasible individuals and competing infeasible individuals, allowing the population to accurately identify all possible feasible regions and gradually expand towards the feasible boundary. The exploitation phase explores under-explored regions in the earlier phases with the aim of accelerating the convergence of the population and escaping from the local optima. Extensive experiments conducted on four benchmark test suites demonstrate that ATEA exhibits superior performance in three benchmark test suites compared with six other state-of-the-art algorithms.