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Constrained multi-objective optimization problems (CMOPs) exist widely in the real world, which simultaneously contain multiple constraints to be satisfied and multiple conflicting objectives to be optimized. Therefore, the challage in addressing CMOPs is how to better balance constraints and objectives. To remedy this issue, this paper proposes a novel dual-population based constrained multi-objective evolutionary algorithm to solve CMOPs, in which two populations with different functions are employed. Specifically, the main population considers both objectives and constraints for solving the original CMOPs, while the auxiliary population is used only for optimization of objectives without considering constraints. In addition, a dynamic population size reducing mechanism is proposed, which is used to adjust the size of the auxiliary population, so as to reduce the consumption of computing resoruces in the later stage. Moreover, an independent external archive is set to store feasible solutions found by the auxiliary population, so as to provide high-quality feasible solutions for the main population. The experimental results on 55 benchmark functions show that the proposed algorithm exhibits superior or at least competitive performance compared to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a population-based evolutionary computation model for solving continuous constrained nonlinear optimization problems. The primary goal is achieving better solutions in a specific problem type, regardless of metaphors and similarities. The proposed algorithm assumes that candidate solutions interact with each other to have better fitness values. The interaction between candidate solutions is limited with the closest neighbors by considering the Euclidean distance. Furthermore, Tabu Search Algorithm and Elitism selection approach inspire the memory usage of the proposed algorithm. Besides, this algorithm is structured on the principle of the multiplicative penalty approach that considers satisfaction rates, the total deviations of constraints, and the objective function value to handle continuous constrained problems very well. The performance of the algorithm is evaluated with real-world engineering design optimization benchmark problems that belong to the most used cases by evolutionary optimization researchers. Experimental results show that the proposed algorithm produces satisfactory results compared to the other algorithms published in the literature. The primary purpose of this study is to provide an algorithm that reaches the best-known solution values rather than duplicating existing algorithms through a new metaphor. We constructed the proposed algorithm with the best combination of features to achieve better solutions. Different from similar algorithms, constrained engineering problems are handled in this study. Thus, it aims to prove that the proposed algorithm gives better results than similar algorithms and other algorithms developed in the literature. [ABSTRACT FROM AUTHOR]

The objective of a fixed-cost transportation problem (FCTP) is to minimize the total transportation cost (or to maximize the total profit). FCTPs become more complex due to the presence of uncertain factors like suppliers' capacity, customers' demand, fixed cost and amount of goods transported. In order to deal with its complexity, three mathematical models, namely the expected value, chance-constrained and measure chance models, are basically available in the literature. This paper aims to solve the crisp equivalent models of an uncertain two-echelon FCTP by employing a robust hybrid evolutionary algorithm. The framework of the designed algorithm comprises of parameter-free Jaya algorithm and the efficient quadratic approximation operator. Initially, its efficiency is validated on the 24 constrained benchmark functions from CEC 2006 (Liang et al. in Problem definitions and evaluation criteria for the CEC 2006 special session on constrained real-parameter optimization, 2006) and 30 unconstrained benchmark functions (Rao in Int J Ind Eng Comput 7:19–34, 2016). Friedman rank test and Iman–Davenport test are conducted for both constrained and unconstrained test instances. Additionally, five real-life engineering design problems are solved to further analyze the flexibility of the proposed algorithm. The computational results and convergence graphs justify the robustness of the designed algorithm over state-of-the-art algorithms. Finally, its application is demonstrated on chance-constrained and measure chance two-echelon FCTP models. Near-optimal results are contradicted by genetic algorithm and particle swarm optimization in solving both these models. Further, a few large-scale instances of FCTP are solved by the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a multi-objective bird swarm algorithm (MOBSA) is proposed to cope with multi-objective optimization problems. The algorithm is explored based on BSA which is an evolutionary algorithm suitable for single objective optimization. In this paper, non-dominated sorting approach is used to distinguish optimal solutions and parallel coordinates is applied to evaluate the distribution density of non-dominated solution and further update the external archive when it is full to overflowing, which ensure faster convergence and more widespread of Pareto front. Then, the MOBSA is adopted to optimize benchmark problems. The results demonstrate that MOBSA gets better performance compared with NSGA-II and MOPSO. Since a vehicle power train problem could be treated as a typical multi-objective optimization problem with constraints, with integration of constrained non-dominated solution, MOBSA is adopted to acquire optimal gear ratios and optimize vehicle power train. The results compared with other popular algorithm prove the proposed algorithm is more suitable for constrained multi-objective optimization problem in engineering field. [ABSTRACT FROM AUTHOR]

A novel hybrid many-objective evolutionary algorithm called Reference Vector Guided Evolutionary Algorithm based on hypervolume indicator (H-RVEA) is proposed in this paper. The reference vectors are used in a number of sub-problems to decompose the optimization problem. An adaptation strategy is used in the proposed algorithm to adjust the reference vector distribution. The proposed algorithm is compared over well-known benchmark test functions with five state-of-the-art evolutionary algorithms. The results show H-RVEA's superior performance in terms of the inverted generational distance and hypervolume performance measures than the competitor algorithms. The suggested algorithm's computational complexity is also analysed. The statistical tests are carried out to demonstrate the statistical significance of the proposed algorithm. In order to demonstrate its efficiency, H-RVEA is also applied to solve two real-life constrained many-objective optimization problems. The experimental results indicate that the proposed algorithm can solve the many-objective real-life problems. Note that the source codes of the proposed technique are available at http://dhimangaurav.com/. [ABSTRACT FROM AUTHOR]

Constrained optimization plays an important role in many decision-making problems and various real-world applications. In the last two decades, various evolutionary algorithms (EAs) were developed and still are developing under the umbrella of evolutionary computation. In general, EAs are mainly categorized into nature-inspired and swarm-intelligence- (SI-) based paradigms. All these developed algorithms have some merits and also demerits. Particle swarm optimization (PSO), firefly algorithm, ant colony optimization (ACO), and bat algorithm (BA) have gained much popularity and they have successfully tackled various test suites of benchmark functions and real-world problems. These SI-based algorithms follow the social and interactive principles to perform their search process while approximating solution for the given problems. In this paper, a multiswarm-intelligence-based algorithm (MSIA) is developed to cope with bound constrained functions. The suggested algorithm integrates the SI-based algorithms to evolve population and handle exploration versus exploitation issues. Thirty bound constrained benchmark functions are used to evaluate the performance of the proposed algorithm. The test suite of benchmark function is recently designed for the special session of EAs competition in IEEE Congress on Evolutionary Computation (IEEE-CEC′13). The suggested algorithm has approximated promising solutions with good convergence and diversity maintenance for most of the used bound constrained single optimization problems. [ABSTRACT FROM AUTHOR]

For solving constrained multi-objective optimization problems (CMOPs), an effective constraint-handling technique (CHT) is of great importance. Recently, many CHTs have been proposed for solving CMOPs. However, no single CHT can outperform all kinds of CMOPs. This paper proposes an algorithm, namely, ACHT-M2M, which adaptively allocates the existing CHTs in an M2M framework for solving CMOPs. To be more specific, a CMOP is first decomposed into several constrained multi-objective optimization subproblems by ACHT-M2M. Each subproblem has a subpopulation in a subregion. CHT for each subregion is adaptively allocated according to a proposed composite performance measure. Population for the next generation is selected from subregions by selection operators with different CHTs and the obtained nondominated feasible solutions in each generation are used to update a predefined archive. ACHT-M2M assembles the advantages of different CHTs and makes them cooperate with each other. The proposed ACHT-M2M is finally compared with the other 12 representative algorithms on benchmark CMOPs and the experimental results further confirm the effectiveness of ACHT-M2M for solving CMOPs. [ABSTRACT FROM AUTHOR]

The number of research works on dynamic constrained optimization problems has been increasing rapidly over the past two decades. In this domain, many real-life decision problems need to be solved repeatedly with changing data and parameters. However, no research on dynamic problems with changes in the coefficients of the constraint functions has been reported. In this paper, to deal with such problems, a new evolutionary framework with multiple novel mechanisms is proposed. The new mechanisms are for (1) dealing with both linear and non-linear components in the constraint functions, (2) identifying the rate of change in the coefficients of the variables and (3) updating the population efficiently after every change occurs in the problem. To evaluate the performance of the proposed algorithm, we designed a new set of 13 dynamic benchmark problems, each of which consists of 20 dynamic changes and 3 different scenarios. The results demonstrate that the proposed algorithm significantly contributes in achieving good quality solutions, high feasibility rates and fast convergence in rapidly changing environments. In addition, the framework shows its capability of using different meta-heuristics to solve dynamic problems. [ABSTRACT FROM AUTHOR]

The optimum design of a cam mechanism is a very interesting problem in the contact mechanics today, due to the alternative industrial applications as a mechanism of precision. In this paper, a new evolutionary algorithm called modified adaptive differential evolution (MADE) is introduced for multi-objective optimization of a cam mechanism with offset translating roller follower. The optimization procedure is investigated for three objectives among them minimum congestion, maximum efficiency, and maximum strength resistance of the cam. To enhance the design quality of the mechanism in the optimization process, more geometric parameters and more design constraints are included in the problem formulation. In order to validate the developed algorithm, three engineering design problems are solved. The simulation results for the tested problems indicate the effectiveness and the robustness of the proposed algorithm compared to the various existed optimization methods. Finally, the optimal results obtained for the case study example provide very useful decisions for a cam mechanism synthesis. [ABSTRACT FROM AUTHOR]

This paper proposes the correction of faulty sensors using a synthesis of the greedy sparse constrained optimization GSCO) technique. The failure of sensors can damage the radiation power pattern in terms of sidelobes and nulls. The synthesis problem can recover the wanted power pattern with reduced number of sensors into the background of greedy algorithm and solved with orthogonal matching pursuit (OMP) technique. Numerical simulation examples of linear arrays are offered to demonstrate the effectiveness of getting the wanted power pattern with a reduced number of antenna sensors which is compared with the available techniques in terms of sidelobes level and number of nulls. [ABSTRACT FROM AUTHOR]

This paper aims to improve premature convergence and low precision of the differential evolution algorithm. An adaptive constrained differential evolution algorithm based on a probability interval update mechanism (PIMDE) is proposed. First, the interval probability update mechanism is developed to adaptively generate the mutation probability value, and the hybrid mutation strategy is formulated by effectively selecting complementary mutation strategies based on the mutation probability. Subsequently, the adaptive mechanism is employed to dynamically adjust the mutation strategy and parameters; as a result, the global detection and local search ability of the algorithm can be balanced, and the search efficiency of the algorithm can be enhanced. To verify determine the performance of the proposed algorithm, ten standard constrained optimization problems and five engineering optimization problems are analyzed. As revealed from experimental results, PIMDE can effectively solve COPs with high precision and robustness. [ABSTRACT FROM AUTHOR]

Constrained optimization problems are very important and frequently appear in the real world. The a constrained method is a new transformation method for constrained optimization. In this method, a satisfaction level for the constraints is introduced, which indicates how well a search point satisfies the constraints. The α level comparison, which compares search points based on their level of satisfaction of the constraints, is also introduced. The a constrained method can convert an algorithm for unconstrained problems into an algorithm for constrained problems by replacing ordinary comparisons with the α level comparisons. In this paper, we introduce some improvements including mutations to the nonlinear simplex method to search around the boundary of the feasible region and to control the convergence speed of the method, we apply the a constrained method and we propose the improved α constrained simplex method for constrained optimization problems. The effectiveness of the a constrained simplex method is shown by comparing its performance with that of the stochastic ranking method on various constrained problems. [ABSTRACT FROM AUTHOR]

This paper presents an improved particle swarm optimizer (PSO) for solving mechanical design optimization problems involving problem-specific constraints and mixed variables such as integer, discrete and continuous variables. A constraint handling method called the 'fly-back mechanism' is introduced to maintain a feasible population. The standard PSO algorithm is also extended to handle mixed variables using a simple scheme. Five benchmark problems commonly used in the literature of engineering optimization and nonlinear programming are successfully solved by the proposed algorithm. The proposed algorithm is easy to implement, and the results and the convergence performance of the proposed algorithm are better than other techniques. [ABSTRACT FROM AUTHOR]

In order to solve the dynamic emission economic dispatch (DEED) problem with high-dimensional and large scale constraints, this paper proposed an adaptive multiobjective differential evolutionary algorithm (ADEA). The ADEA designed an adaptive differential crossover module, developed an improving current to best 1 crossover scheme to enhance the diversity of population. These strategies improved the exploration and exploitation ability of the ADEA, it also applied a repair strategy to deal with the equality and inequality constrains in DEED problem. Numerical experiments tested the effectiveness of ADEA on 10-unit system, and compared with several peer algorithms. Simulation results indicate that ADEA has good convergence. The uniformity and ductility of the Pareto front obtained by ADEA is better than that of the compared algorithms. It provides a more efficient scheduling decision-making method for power system dispatcher by a fuzzy decision method. [ABSTRACT FROM AUTHOR]

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 notably effective multi-objective constrained black-box optimization algorithm is proposed. • A constraints-domain-search strategy is designed to rapidly decrease the CV while avoiding a local minimum in the CV space. • A performance-improvement exploration strategy is presented through the integration of constraint penalties. • Comprehensive conclusions that can serve as valuable insights for future algorithm development are included. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm incorporating broadening selection strategy in competitive constraint handling paradigm for finding the optimum solution in constrained problems has been proposed, referred as Broadening Selection Competitive Constraint Handling (BSCCH). Although, competitive constraint handling approaches have proved to be very efficient, but they lack faster convergence due to offspring generation from random individuals. By incorporating selection strategy such as broadening selection in the competitive approach, better results are obtained and convergence rate is improved significantly. Incorporating said strategy, the BSCCH algorithm has been proposed which is generic in nature and can be coupled with various evolutionary algorithms. In this study, the BSCCH algorithm has been coupled with Differential Evolution algorithm as a proof of concept because it is found to be an efficient algorithm in the literature for constrained optimization problems. The proposed algorithm has been evaluated using 24 benchmark functions. The mean closure performance of the BSCCH algorithm is compared against seven selected state-of-the-art algorithms, namely Differential Evolution with Adaptive Trial Vector Generation Strategy and Cluster-replacement-based Feasibility Rule (CACDE), Improved Teaching Learning Based Optimization (ITLBO), Modified Global Best Artificial Bee Colony (MGABC), Stochastic Ranking Differential Evolution (SRDE), Novel Differential Evolution (NDE), Partical Swarm Optimization for solving engineering problems - a new constraint handling mechanism (CVI-PSO) and Ensemble of Constraint Handling Techniques (ECHT). The median convergence traces have been compared with two different algorithms based on differential evolution, i:e: Ensemble of Constraint Handling Techniques (ECHT) and Stochastic Ranking Differential Evolution (SRDE). ECHT is considered to be a flagship ensemble technique till date for constrained optimization problems, whereas SRDE employs a parent selection mechanism for constrained optimization. The proposed algorithm is found to provide better solutions and achieve significantly faster convergence in most of the problems. [ABSTRACT FROM AUTHOR]

The performance of evolutionary algorithms can be seriously weakened when constraints limit the feasible region of the search space. In this paper we present a constrained multi-objective optimization algorithm based on adaptive ε-truncation (ε-T-CMOA) to further improve distribution and convergence of the obtained solutions. First of all, as a novel constraint handling technique ε-truncation technique keeps an effective balance between feasible solutions and infeasible solutions by permitting some excellent infeasible solutions with good objective value and low constraint violation to take part in the evolution, so diversity is improved, and convergence is also coordinated. Next, an exponential variation is introduced after differential mutation and crossover to boost the local exploitation ability. At last, the improved crowding density method only selects some Pareto solutions and near solutions to join in calculation, thus it can evaluate the distribution more accurately. The comparative results with other state-of-the-art algorithms show that ε-T-CMOA is more diverse than the other algorithms and it gains better in terms of convergence in some extent. [ABSTRACT FROM AUTHOR]

In predator–prey algorithm, a relatively small number of predators (“lions”) and a much larger number of prey (“antelopes”) are randomly placed on a two dimensional lattice with connected ends representing an unfolded surface of a torus. The predators are partially or completely biased towards one or more objectives, based on which each predator kills the weakest prey in its neighborhood. A stronger prey created through evolution replaces this prey. In case of constrained problems, the sum of constraint violations serves as an additional objective. Modifications of the basic predator–prey algorithm have been implemented in this paper regarding the selection procedure, apparent movement of the predators, and mutation strategy. Further modifications have been made making the algorithm capable of handling multiple equality and inequality constraints. The final modified algorithm was tested on standard linear/nonlinear and constrained/unconstrained single-objective optimization problems. [ABSTRACT FROM AUTHOR]

• The paper shows the challenge of a direct coding scheme in portfolio optimization. • The paper proposes a compressed coding scheme to deal with mixed variables. • The paper introduces some search operators for portfolio optimization. • The scheme and operators are integrated into three main kinds of algorithms. • Statistical results on 20 instances have shown the superiority. Mixed-Integer Non-Linear Programming (MINLP) is not rare in real-world applications such as portfolio investment. It has brought great challenges to optimization methods due to the complicated search space that has both continuous and discrete variables. This paper considers the multi-objective constrained portfolio optimization problems that can be formulated as MINLP problems. Since each continuous variable is dependent to a discrete variable, we propose a Compressed Coding Scheme (CCS), which encodes the dependent variables into a continuous one. In this manner, we can reuse some existing search operators and the dependence among variables will be utilized while the algorithm is optimizing the compressed variables. CCS actually bridges the gap between the portfolio optimization problems and the existing optimizers, such as Multi-Objective Evolutionary Algorithms (MOEAs). The new approach is applied to two benchmark suites, involving the number of assets from 31 to 2235. The experimental results indicate that CCS is not only efficient but also robust for dealing with the multi-objective constrained portfolio optimization problems. [ABSTRACT FROM AUTHOR]

The objective space of the constrained multiobjective optimization problem (CMOP) is constantly torn by the applied constraints. This makes evolutionary algorithms, which are driven by objectives, face greater difficulties in feasibility, convergence, and diversity. Most evolutionary algorithms will be trapped in local optimums such as a fake Pareto-optimal front or a mutilated Pareto-optimal front. To address this issue, this paper proposes an archive-assisted evolutionary framework with a novel archive structure and cooperative mechanism. A reference line guided archive (RA) is established to record the evolution of the unconstrained solutions. The updated criteria of RA are based on the distance from the individual to the reference line and the unconstrained dominance relation. A specially designed adaptive mating selection operator will select mating parents from RA and the main population according to the convergence of the main population and RA, respectively. The participation of RA in offspring reproduction is conducive to skipping infeasible regions for extensive searches. The performance of the archive-assisted nondominated sorting genetic algorithm (AA-NSGA), which embeds the proposed archive strategy into the nondominated sorting genetic algorithm II, is compared with four state-of-the-art constrained multiobjective evolutionary algorithms (MOEAs). The experimental results on 38 benchmark CMOPs and a reactor network design problem show that the proposed AA-NSGA has a high performance among the existing MOEAs in terms of feasibility, convergence, and diversity. • An archive-assisted coevolutionary framework is proposed. • A reference line guided archive records the evolution of unconstrained solutions. • An adaptive mating selection operator with a new cooperation mechanism is designed. • The competitiveness of the algorithm is verified by comparing with several MOEAs. [ABSTRACT FROM AUTHOR]

Many multi-objective evolutionary algorithms are proposed to handle constrained multi-objective optimization problems. Nevertheless, they often fail to appropriately balance feasibility, convergence and diversity of the population. This paper proposed a dynamic dual-population co-evolution multi-objective evolutionary algorithm (DDCMEA) to solve this issue. In DDCMEA, a dynamic dual-population co-evolution strategy is employed to balance the convergence and the feasibility by dynamically adjusting the offspring number of the two populations. In the early stage of evolution, the algorithm mainly focuses on the convergence and more offspring of the first population are generated. In the late stage of evolution, the algorithm mainly focuses on the feasibility and more offspring of the second population are generated. Finally, feasible solutions with good convergence could be obtained. To further enhance the diversity of the offspring and obtain feasible solutions with a wide spread of distribution, the evolution operators of the genetic algorithm and the differential evolution are chosen as the search engines for the first population and the second population, respectively. The performance of DDCMEA is further tested through thirty-one bench-mark test problems and two real-world problems in comparison with other five state-of-the-art algorithms. The results show the proposed algorithm DDCMEA achieves competitive performance when handling constrained multi-objective optimization problems. • A dual-population MOEA for CMOPs. • Dynamic dual-population co-evolution. • Two evolutionary search engines are applied. • Offspring number dynamically adjusted. [ABSTRACT FROM AUTHOR]

The balance between multiple objectives and various constraints is the key to solving constrained multi-objective optimization problems (CMOPs). When dealing with CMOPs with complex feasible regions, some evolutionary algorithms suffer from great challenges in converging to the constrained Pareto front (CPF) with well-distributed feasible solutions. To address this issue, this paper proposes a dual-population based bidirectional coevolution algorithm, called DBC-CMOEA, which aims to converge to the CPF using promising solutions explored from both feasible and infeasible regions. To do so, DBC-CMOEA maintains two populations and an archive, where the dual-population is complementary in the search process and the archive is used to retain promising feasible and infeasible solutions, thus facilitating information exchange between these two populations. For updating the archive, a nondominated sorting procedure and an angle-based selected scheme are conducted to store infeasible and feasible solutions, as they can help to maintain the diversity of the search and find more feasible regions. To evolve the CPF from the bidirectional side of the feasible region, a novel mating selection strategy is used to choose appropriate mating parents. In comparison with some related constraint multi-objective optimization algorithms on a number of benchmark problems, experimental results show that the proposed algorithm performs better than the state-of-the-art constrained multi-objective evolutionary optimizers. • A cooperative dual-population algorithm is proposed for solving CMOPs. • Constraints are handled in three different ways to explore the search space. • A brand-new selection strategy is applied to select the mating parents. • A novel archive update strategy is designed to retain the promising information. [ABSTRACT FROM AUTHOR]

When solving complex constrained problems, how to efficiently utilize promising infeasible solutions is an essential issue because these promising infeasible solutions can significantly improve the diversity of algorithms. However, most existing constrained multi-objective evolutionary algorithms (CMOEAs) do not fully exploit these promising infeasible solutions. In order to solve this problem, a constrained multi-objective optimization evolutionary algorithm based on the dynamic constraint boundary method is proposed (CDCBM). The proposed algorithm continuously searches for promising infeasible solutions between UPF (the unconstrained Pareto front) and CPF (the constrained Pareto front) during the evolution process by the dynamically changing auxiliary population of the constraint boundary, which continuously provides supplementary evolutionary directions to the main population and improves the convergence and diversity of the main population. Extensive experiments on three well-known test suites and three real-world constrained multi-objective optimization problems demonstrate that CDCBM is more competitive than seven state-of-the-art CMOEAs. [ABSTRACT FROM AUTHOR]

Constrained Multiobjective Problem (CMOP) is widely used in engineering applications, but the current constrained Multiobjective Optimization algorithms (CMOEA) often fails to effectively balance convergence and diversity. For this purpose, a two-stage co-evolution constrained multi-objective optimization evolutionary algorithm (TSC-CMOEA) is presented to solve constrained multi-objective optimization problems. This method divides the search process into two phases: in the first stage, the synchronous co-evolution is used, and the population corresponding to the help problem and the population corresponding to the raw problem cooperate with each other and share the offspring to produce better solutions, so as to quickly cross the infeasible region and approach the Pareto front; The second stage discards the help problem when it fails and maintains only the evolution of the main population to save computing resources and enhance convergence. The combination of synchronous co-evolution and staged strategy allows the population to traverse infeasible regions more efficiently and converge quickly to feasible and non-dominant regions. The test results on benchmark CMOPs show that the convergence and population distribution of TSC-CMOEA is significantly better than those of NSGA-II, NSGA-III, C-MOEA/D, PPS, ToP and CCMO. [ABSTRACT FROM AUTHOR]

Despite the successful application of an extension of the Multi-Objective Evolution Algorithm based on Decomposition (MOEA/D-M2M) to solve unbalanced multi-objective optimization problems (UMOPs), its use in constrained unbalanced multi-objective optimization problems has not been fully explored. In an earlier paper, a definition of UMOPs was suggested that had two necessary conditions: 1) finding a favored subset of the Pareto set is easier than finding an unfavored subset, and 2) the favored subset of the Pareto set dominates a large part of the feasible space. The second condition strongly reduces the fraction of MOPs that are considered UMOPs. In this paper, we eliminate that second condition and consider a broader class of UMOPs. We design an unbalanced constrained multi-objective test suite with three different types of biased constraints, yielding three different types of constrained test problems in which the degree of imbalance is scalable via a set of parameters introduced for each problem. We analyse the characteristics of three types of constraints and the difficulties they present for potential solution algorithms–i.e., NSGA-II, MOEA/D and MOEA/D-M2M, with four constraint-handling techniques. MOEA/D-M2M is shown to significantly outperform the other algorithms on these problems due to its decomposition strategy. [ABSTRACT FROM AUTHOR]

Recently, angle-based approaches have shown promising for unconstrained many-objective optimization problems (MaOPs), but few of them are extended to solve constrained MaOPs (CMaOPs). Moreover, due to the difficulty in searching for feasible solutions in high-dimensional objective space, the use of infeasible solutions comes to be more important in solving CMaOPs. In this paper, an angle based evolutionary algorithm with infeasibility information is proposed for constrained many-objective optimization, where different kinds of infeasible solutions are utilized in environmental selection and mating selection. To be specific, an angle-based constrained dominance relation is proposed for non-dominated sorting, which gives infeasible solutions with good diversity the same priority to feasible solutions for escaping from the locally feasible regions. As for diversity maintenance, an angle-based density estimation is developed to give the infeasible solutions with good convergence a chance to survive for next generation, which is helpful to get across the large infeasible barrier. In addition, in order to utilize the potential of infeasible solutions in creating high-quality offspring, a modified mating selection is designed by considering the convergence, diversity and feasibility of solutions simultaneously. Experimental results on two constrained many-objective optimization test suites demonstrate the competitiveness of the proposed algorithm in comparison with five existing constrained many-objective evolutionary algorithms for CMaOPs. Moreover, the effectiveness of the proposed algorithm on a real-world problem is showcased. • This paper proposes an angle based evolutionary algorithm with infeasibility information for CMaOPs. • A novel constrained dominance relation is proposed for non-dominated sorting. • An angle-based density estimation is developed for diversity maintenance. • A modified mating selection is designed for creating high-quality offspring. [ABSTRACT FROM AUTHOR]

Distributed evolutionary computation has been efficiently used, in last decades, to solve complex optimization problems. Island model (IM) is considered as a distributed population paradigm employed by evolutionary algorithms to preserve the diversification and, thus, to improve the local search. In this article, we study different island model techniques integrated in to particle swarm optimization (PSO) algorithm in order to overcome its drawbacks: premature convergence and lack of diversity. The first IMPSO approach consists in using the migration process in a static way to enhance the police migration strategy. On the other hand, the second approach, called dynamic-IMPSO, consists in integrating a learning strategy in case of migration. The last version called constrained-IMPSO utilizes a stochastic technique to ensure good communication between the sub-swarms. To evaluate and verify the effectiveness of the proposed algorithms, several standard constrained and unconstrained benchmark functions are used. The obtained results confirm that these algorithms are more efficient in solving low-dimensional problems (CEC'05), large-scale optimization problems (CEC'13) and constrained problems (CEC'06), compared to other well-known evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Self-adaptive variants of evolutionary algorithms (EAs) tune their parameters on the go by learning from the search history. Adaptive differential evolution with optional external archive (JADE) and self-adaptive differential evolution (SaDE) are two well-known self-adaptive versions of differential evolution (DE). They are both unconstrained search and optimization algorithms. However, if some constraint handling techniques (CHTs) are incorporated in their frameworks, then they can be used to solve constrained optimization problems (COPs). In an early work, an ensemble of constraint handling techniques (ECHT) is probabilistically hybridized with the basic version of DE. The ECHT consists of four different CHTs: superiority of feasible solutions, self-adaptive penalty, ε -constraint handling technique and stochastic ranking. This paper employs ECHT in the selection schemes, where offspring competes with their parents for survival to the next generation, of JADE and SaDE. As a result, JADE-ECHT and SaDE-ECHT are developed, which are the constrained variants of JADE and SaDE. Both algorithms are tested on 24 COPs and the experimental results are collected and compared according to algorithms' evaluation criteria of CEC'06. Their comparison, in terms of feasibility rate (FR) and success rate (SR), shows that SaDE-ECHT surpasses JADE-ECHT in terms of FR, while JADE-ECHT outperforms SaDE-ECHT in terms of SR. [ABSTRACT FROM AUTHOR]

Many studies have shown that the effective mixture of feasible and infeasible solutions is beneficial to solve the constrained multi-objective optimization problems (CMOPs). So, it is of great importance to fully identify the valuable infeasible solutions that are helpful to find the constrained Pareto fronts. To achieve this, this paper evaluates the potential value of each infeasible solution from different aspects, and forms a criterion to fully identify the valuable infeasible solutions. Comparative analysis shows that compared with the existing constraint handling techniques, the proposed criterion is more effective in identifying valuable infeasible solutions. Accordingly, we embed the proposed criterion in the evolutionary algorithm and design a criterion-based constrained multi-objective algorithm for CMOPs. In the proposed algorithm, an archive preferring feasible solutions is used to record the best solutions, which are then mixed with the valuable infeasible solutions identified by the criterion to deal with the CMOPs. A series of simulation experiments on several benchmarks and engineering problem show the competitive performance of the proposed algorithm. Compared with the other state-of-the-art constrained evolutionary multi-objective optimization algorithms, the proposed algorithm performs better in dealing with different types of CMOPs, and it is the only one that can successfully deal with the problems that the initial population arises in the complex infeasible regions below the Pareto fronts. [ABSTRACT FROM AUTHOR]

A variety of important engineering and scientific tasks may be formulated as non-linear, constrained optimization problems. Their solution often demands high computational power. It may be reached by means of appropriate hardware, software or algorithm improvements. The Evolutionary Algorithms (EA) approach to solution of such problems is considered here. The EA are rather slow methods; however, the main advantage of their application is observed in the case of non-convex problems. Particularly high efficiency is demanded in the case of solving large optimization problems. Examples of such problems in engineering include analysis of residual stresses in railroad rails and vehicle wheels, as well as the Physically Based Approximation (PBA) approach to smoothing experimental and/or numerical data. Having in mind such analysis in the future, we focus our current research on the significant EA efficiency increase. Acceleration of the EA is understood here, first of all, as decreasing the total computational time required to solve an optimization problem. Such acceleration may be obtained in various ways. There are at least two gains from the EA acceleration, namely i) saving computational time, and ii) opening a possibility of solving larger optimization problems, than it would be possible with the standard EA. In our recent research we have preliminarily proposed several new speed-up techniques based on simple concepts. In this paper we mainly develop acceleration techniques based on simultaneous solutions averaging well supported by a non-standard application of parallel calculations, and a posteriori solution error analysis. The knowledge about the solution error is used to EA acceleration by means of appropriately modified standard evolutionary operators like selection, crossover, and mutation. Efficiency of the proposed techniques is evaluated using several benchmark tests. These tests indicate significant speed-up of the involved optimization process. Further concepts and improvements are also currently being developed and tested. [ABSTRACT FROM AUTHOR]

When evolutionary algorithms are employed to solve constrained optimization problems (COPs), how to efficiently make use of the information of some promising infeasible solutions is very important in the process of searching for the optimal feasible solution. In this paper, for selecting and making full use of some better infeasible solutions, a rankings-based fitness function method is designed. Specifically, the final fitness function of each individual is obtained by weighting two rankings, which are got after sorting the population based on the ɛ constraint technique and only based on the objective function, respectively. Furthermore, the weight is dynamically adjusted by considering the proportion of feasible solutions and generation information. By doing this, the tradeoff in constraints and objective can be addressed. Moreover, the promising offspring are generated by three differential evolution strategies with distinct characters to balance diversity and convergence. In addition, 116 benchmark problems from three test suites are used to evaluate the performance of the proposed method. Nine commonly used practical problems are selected to test the potential of the algorithm to solve real-world problems. Experimental results indicate that the proposed method shows superior or competitive to other state-of-the-art methods tailored for COPs. Moreover, the effectiveness of each introduced component in the proposed algorithm is investigated by the ablation study. • A new constrained evolutionary optimization algorithm is developed, denoted as DERFF. • Two rankings are got for each individual by two ways, one prefers constraint and the other prefers objective function. • The weight for summing these two rankings is dynamically adjusted by the proportion of feasible solutions and generation information. • Three evolutionary strategies are combined to balance the convergence and diversity of the population. • Experimental results indicate the effectiveness of the DERFF against other advanced algorithms. [ABSTRACT FROM AUTHOR]

Abstract: A modified Artificial Bee Colony algorithm to solve constrained numerical optimization problems is presented in this paper. Four modifications related with the selection mechanism, the scout bee operator, and the equality and boundary constraints are made to the algorithm with the aim to modify its behavior in a constrained search space. Six performance measures found in the specialized literature are employed to analyze different capabilities in the proposed algorithm such as the ability and cost to generate feasible solutions, the capacity and cost to locate the feasible global optimum solution and the competency to improve feasible solutions. Three experiments, including a comparison with state-of-the-art algorithms, are considered in the test design where twenty four well-known benchmark problems with different features are utilized. The overall results show that the proposed algorithm differs in its behavior with respect to the original Artificial Bee Colony algorithm but its performance is improved, mostly in problems with small feasible regions due to the presence of equality constraints. [Copyright &y& Elsevier]

• The CMOP is divided into a series of sub-problems by objective space partition. • A hybrid constrained handling technique including the unconstrained search mode. • A selection operator based on partition feasible solutions. • Mating-pool selection based on population and external archive. Solving constrained multi-objective optimization problems (CMOPs) is full of challenges due to the difficulties in balancing between feasibility, convergence and distribution. To remedy this issue, this paper proposes a multi-objective differential evolutionary algorithm based on partition selection (MODE-PS). Firstly, MODE-PS divides a CMOP into a series of optimization sub-problems by objective space partition to maintain the distribution. Then, to keep the feasibility of the subspaces, one feasible solution of each subspace is saved to a partition feasible solution set. Next, once there are feasible solutions in one subspace, the individual selection strategy of this subspace is changed from constraint search to non-constraint search. By this way, the convergence is accelerated. Finally, all the feasible solutions are archived and evolved together with the population by a mating-pool selection to balance the feasibility, convergence and distribution. Twenty-two benchmark test problems are used to test the performance of MODE-PS in comparison with five state-of-the-art constrained multi-objective evolution algorithms. Moreover, a real-world problem, i.e., bi-source compressed-air pipeline optimization, is used to test the performance of algorithms. The experimental results have demonstrated the high competitiveness of MODE-PS for solving CMOPs. [ABSTRACT FROM AUTHOR]

Recently, decomposition has gained a wide interest in solving multi-objective optimization problems involving more than three objectives also known as Many-objective Optimization Problems (MaOPs). In the last few years, there have been many proposals to use decomposition to solve unconstrained problems. However, fewer is the amount of works that has been devoted to propose new decomposition-based algorithms to solve constrained many-objective problems. In this paper, we propose the ISC-Pareto dominance (Isolated Solution-based Constrained Pareto dominance) relation that has the ability to: (1) handle constrained many-objective problems characterized by different types of difficulties and (2) favor the selection of not only infeasible solutions associated to isolated sub-regions but also infeasible solutions with smaller CV (Constraint Violation) values. Our constraint handling strategy has been integrated into the framework of the Constrained Non-Dominated Sorting Genetic Algorithm-III (C-NSGA-III) to produce a new algorithm called Isolated Solution-based Constrained NSGA-III (ISC-NSGA-III). The empirical results have demonstrated that our constraint handling strategy is able to provide better and competitive results when compared against three recently proposed constrained decomposition-based many-objective evolutionary algorithms in addition to a penalty-based version of NSGA-III on the CDTLZ benchmark problems involving up to fifteen objectives. Moreover, the efficacy of ISC-NSGA-III on a real world water management problem is showcased. [ABSTRACT FROM AUTHOR]

This article proposes an evolutionary algorithm using multiple penalties and multiple local surrogates (MPMLS) for expensive constrained optimization. In each generation, MPMLS defines and optimizes a number of subproblems. Each subproblem penalizes the constraints in the original problem using a different penalty coefficient and has its own search subregion. A local surrogate is built for optimizing each subproblem. Two major advantages of MPMLS are: 1) it can maintain good population diversity so that the search can approach the optimal solution of the original problem from different directions and 2) it only needs to build local surrogates so that the computational overhead of the model building can be reduced. Numerical experiments demonstrate that our proposed algorithm performs much better than some other state-of-the-art evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization. [ABSTRACT FROM AUTHOR]

When solving constrained multiobjective optimization problems by evolutionary algorithm, the key challenge is how to achieve the balance among convergence, diversity, and feasibility. To deal with this challenge, a purpose-directed two-phase multiobjective differential evolution (PDTP-MDE) algorithm is developed in this paper. The main idea of PDTP-MDE is that the whole evolution process is divided into two sequential phases according to the expected purpose of each stage. To be specific, the first phase aims at keeping the balance between convergence and diversity, while the feasibility is taken as an auxiliary indicator. In this way, the population is capable of exploring different potential areas and avoiding to be trapped into local ones, thus providing more information about convergence and diversity for the later evolution process. Afterwards, the second phase mainly tends to maintain feasibility and diversity by selecting and using some promising infeasible solutions according to the population evolution status. In addition, an archive is maintained after each phase to preserve the superior feasible Pareto solutions found so far. By the above processes, the feasible Pareto front with well convergence and well diversity is obtained. The comprehensive experiments on 42 benchmark problems from three test suites demonstrate the superiority and competitiveness of the proposed PDTP-MDE, in comparison with other state-of-the-art constrained multiobjective evolutionary algorithms. [ABSTRACT FROM AUTHOR]

• Four novel variants of the Cuckoo-Search (CS) algorithm are developed. • Parameter adaptation, dynamic penalties and hybridization with BSA are used. • Enhanced CS variants outperform original CS and other metaheuristic algorithms. • CS variants become more efficient as problem size and/or complexity are increased. • Using dynamic penalties yields better final designs and faster convergence. Although Cuckoo Search (CS) is a quite new nature-inspired metaheuristic optimization algorithm, it has been extensively used in engineering applications, since it has been proven very efficient in solving complex nonlinear problems. In this paper, efficient modifications have been made to the original CS algorithm to enhance its efficiency and robustness. More specifically, constant parameters of the algorithm, such as the probability of the alien egg being discovered by the host bird and the step size of Levy flights have been dynamically tuned. In addition, static and dynamic penalty functions are introduced within the optimization formulation. Finally, a hybrid optimization approach is developed to combine the advantages of CS with those of Bird Swarm Algorithm (BSA). Benchmark problems, widely used in relevant studies, have been solved and the obtained solutions are compared with those previously reported using the standard CS algorithm and other popular evolutionary optimization techniques (i.e., Genetic Algorithms, Particle Swarm Optimization, etc.). [ABSTRACT FROM AUTHOR]

The study proposes a new hybrid multi-objective evolutionary optimisation algorithm based on decomposition and local dominance for meter placement in distribution system state estimation. The evenly distributed qualitative and diverse solutions on the Pareto front are required for a decision-maker for selecting a final optimal solution. Such a Pareto front can be achieved by obtaining the balance between convergence and diversity of multi-objective optimisation algorithm. Therefore, the proposed method combined dominance and decomposition techniques, modelled meter placement as a constrained combinatorial multi-objective optimisation. The meter placement is designed as a trade-off between three objectives that are minimising the cost of the meters, average relative percentage error (ARPE) of voltage magnitude and ARPE of voltage angle. As the meter placement problem is a combinatorial optimisation, the binomial distribution-based Monte Carlo method is utilised to initialise the population, which aims to improve the diversity, as a consequence it improves the convergence, which is a by-product of this method. The results of the proposed method are compared with multi-objective evolutionary algorithm based on decomposition, non-dominated sorting genetic algorithm-II and with multi-objective hybrid particle swarm optimisation-krill herd algorithm, multi-objective hybrid estimation of distribution algorithm-interior point method and demonstrated on PG&E 69-bus distribution system and Practical Indian 85-bus distribution system. [ABSTRACT FROM AUTHOR]