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SPEA2: Improving the Strength Pareto Evolutionary Algorithm for Multiobjective Optimization
Abstract
The Strength Pareto Evolutionary Algorithm (SPEA) is a relatively recent technique for finding or approximating the Pareto-optimal set for multiobjective optimization problems. In different studies ,2 SPEA has shown very good performance in comparison to other multiobjective evolutionary algorithms, and therefore it has been a point of reference in various recent investigations? Furthermore, it has been used in different applications. 4 In this paper, an improved version, namely SPEA2, is...
... Archiving becomes even more relevant in evolutionary multi-objective optimisation (EMO), in which populationbased evolutionary search is performed. The population maintenance (i.e., environmental selection) in EMO can be seen as an archiving process [3,4], where the population is regarded as an archive updated at each generation; that is, new offspring solutions are compared with the ones already in the population, either chunk by chunk, e.g., in the generational evolution of NSGA-II [5] and SPEA2 [6]) or one-by-one, e.g., in the steady-state evolution of SMS-EMOA [7] and MOEA/D [8]. ...
... Environmental selection is usually designed on the basis of two principles: (1) dominated solutions should be removed earlier than nondominated ones and (2) solutions in crowded regions should be removed earlier than ones in sparse regions when all of them are mutually nondominated. Different ways of implementing these two principles resulted in many successful MOEAs during the period of 1999-2002, such as SPEA [21], PAES [32], PESA-II [33], NSGA-II [5] and SPEA2 [6]. However, such a "Pareto dominance + density" criterion does not guarantee a convergent MOEA. ...
... At one iteration t, the generator may generate one or multiple solutions, i.e., ∀t, |S (t) | ≥ 1, possibly using the contents of the old archive A (t−1) , where A (t−1) denotes the archive after updating it with S (t−1) , and A (0) is the empty set. Solutions may be fed to the archive one-at-a-time as in -MOEA [90] and SMS-EMOA [7] or many-at-a-time as in NSGA-II [5] and SPEA2 [6]. There is no requirement that the elements in the sequence are unique. ...
Most multi-objective optimisation algorithms maintain an archive explicitly or implicitly during their search. Such an archive can be solely used to store high-quality solutions presented to the decision maker, but in many cases may participate in the search process (e.g., as the population in evolutionary computation). Over the last two decades, archiving, the process of comparing new solutions with previous ones and deciding how to update the archive/population, stands as an important issue in evolutionary multi-objective optimisation (EMO). This is evidenced by constant efforts from the community on developing various effective archiving methods, ranging from conventional Pareto-based methods to more recent indicator-based and decomposition-based ones. However, the focus of these efforts is on empirical performance comparison in terms of specific quality indicators; there is lack of systematic study of archiving methods from a general theoretical perspective. In this paper, we attempt to conduct a systematic overview of multi-objective archiving, in the hope of paving the way to understand archiving algorithms from a holistic perspective of theory and practice, and more importantly providing a guidance on how to design theoretically desirable and practically useful archiving algorithms. In doing so, we also present that archiving algorithms based on weakly Pareto compliant indicators (e.g., epsilon-indicator), as long as designed properly, can achieve the same theoretical desirables as archivers based on Pareto compliant indicators (e.g., hypervolume indicator). Such desirables include the property limit-optimal, the limit form of the possible optimal property that a bounded archiving algorithm can have with respect to the most general form of superiority between solution sets.
... In [33], the AbYSS algorithm has been compared with two multiobjective optimizers, SPEA2 [35] and NSGA-II [36], using 33 unconstrained and constrained test problems. The comparative analysis showed that AbYSS outperforms the other two algorithms according to the diversity of the solutions, and it obtains very competitive results as regards to the hypervolume metric and the convergence towards the true Pareto fronts. ...
... The MOCell algorithm [37] is an evolutionary algorithm for solving multiobjective optimization problems. In [37], the MOCell algorithm has been compared with two state-of-the-art multiobjective optimizers, SPEA2 [35] and NSGA-II [36]; using 21 unconstrained and constrained test problems, and according three quality indicators. ...
Permanent magnet synchronous machines (PMSM) are competitive motors for in-wheel traction systems of electric vehicles. A new tangentially magnetized permanent magnets machine with outer rotor and unequal stator teeth for in-wheel motor application is proposed in this paper. The analytical calculations of the proposed topology are presented by determining the magnetic flux densities and the iron losses in all parts of the machine. The machine design is optimized using three state-of-the-art multiobjective algorithms which are AbYSS, MOCell and NMOPSO algorithms. Moreover, the optimization procedure is carried out according to three objectives: the maximization of the machine efficiency and the minimization of the mass and ripple torque. The optimization results showed that all the algorithms can find a set of optimal solutions and that the NMPSO algorithm outperforms the other two techniques. The finite element method (FEM) is used to investigate the optimization results. It is observed some magnetic saturation in the rotor yoke and the extremes of magnets. The comparison between the FEM and optimization results proved the rationality of the proposed optimization procedure.
... In [21], the AbYSS algorithm has been compared with two multiobjective optimizers, SPEA2 [22] and NSGA-II [23], using 33 unconstrained and constrained test problems. The comparative analysis showed that AbYSS outperforms the other two algorithms according to the diversity of the solutions, and it obtains very competitive results as regards to the hypervolume metric and the convergence towards the true Pareto fronts. ...
... In [24], the MOCell algorithm has been compared with two state-of-the-art multiobjective optimizers, SPEA2 [22] and NSGA-II [23]; using 21 unconstrained and constrained test problems, and according three quality indicators. The comparative analysis showed that MOCell is very competitive with the other two algorithms considering the hypervolume and convergence measures, and it plainly outperforms as regards to the diversity measure. ...
Permanent magnet synchronous machines (PMSM) are competitive motors for in-wheel traction systems of electric vehicles. A new tangentially magnetized permanent magnets machine with outer rotor and unequal stator teeth for in-wheel motor application is proposed in this paper. The analytical calculations of the proposed topology are presented by determining the magnetic flux densities and the iron losses in all parts of the machine. The machine design is optimized using three state-of-the-art multiobjective algorithms which are AbYSS, MOCell and NMOPSO algorithms.Moreover, the optimization procedure is carried out according to three objectives: the maximization of the machine efficiency and the minimization of the mass and ripple torque. The optimization results showed that all the algorithms can find a set of optimal solutions and that the NMPSO algorithm outperforms the other two techniques. The finite element method (FEM) is used to investigate the optimization results. It is observed some magnetic saturation in the rotor yoke and the magnet’s extremes. The value of the induction in these machine regions is about 1.9 T. The comparison between the FEM and optimization results proved the rationality of the proposed optimization procedure.
... Afterwards, a large number of offspring solutions are generated by genetic operators and filtered out by the surrogate model (Line 10), where a few of them is further evaluated on the real objectives (Line 11) and saved to the archive (Line 12). Lastly, the population for the next generation is obtained by selecting at most N solutions from the archive via the environmental selection strategy of SPEA2 [58] (Line 13), and the number of real-evaluated solutions FE is increased by |Q| (Line 14). ...
... In order to train a pairwise comparison based surrogate model, two issues should be carefully addressed, i.e., how to compare the quality of two solutions and how to pair solutions. For the first issue, the proposed PC-SAEA uses a fitness function to assess the quality of each solution x in terms of both convergence and diversity, which is similar to SPEA2 [58] with shift based den- ...
Multi-objective optimization problems in many real-world applications are characterized by computationally or economically expensive objectives, which cannot provide sufficient function evaluations for evolutionary algorithms to converge. Thus, a variety of surrogate models have been employed to provide much more virtual evaluations. Most existing surrogate models are essentially regressors or classifiers, which may suffer from low reliability in the approximation of complex objectives. In this paper, we propose a novel surrogate-assisted evolutionary algorithm, which employs a surrogate model to conduct pairwise comparisons between candidate solutions, rather than directly predicting solutions' fitness values. In comparison to regression and classification models, the proposed pairwise comparison based model can better balance between positive and negative samples, and may be directly used, reversely used, or ignored according to its reliability in model management. As demonstrated by the experimental results on abundant benchmark and real-world problems, the proposed surrogate model is more accurate than popular surrogate models, leading to performance superiority over state-of-the-art surrogate models.
... In this section, to demonstrate the capabilities of the improved NSGA-II, we perform experiments on the test functions with the original NSGA-II and other methods, such as MOEA/D [19], IBEA [20] and SPEA2 [21], as comparison algorithms. ...
... To prove the capability of the improved NSGA-II, we conduct experiments on the test function with the original NSGA-II and other methods, such as MOEA/D [19], IBEA [20] and SPEA2 [21], as comparison algorithms. ...
Non-dominated sorting genetic algorithm II is a classical multi-objective optimization algorithm but it suffers from poor diversity and the tendency to fall into a local optimum. In this paper, we propose an improved non-dominated sorting genetic algorithm, which aims to address the issues of poor global optimization ability and poor convergence ability. The improved NSGA-II algorithm not only uses Levy distribution for global search, which enables the algorithm to search a wider range, but also improves the local search capability by using the relatively concentrated search property of random walk. Moreover, an adaptive balance parameter is designed to adjust the respective contributions of the exploration and exploitation abilities, which lead to a faster search of the algorithm. It helps to expand the search area, which increases the diversity of the population and avoids getting trapped in a local optimum. The superiority of the improved NSGA-II algorithm is demonstrated through benchmark test functions and a practical application. It is shown that the improved strategy provides an effective improvement in the convergence and diversity of the traditional algorithm.
... Then, MSCEA counted the number of feasible solutions in P1∪ P2. If the number of feasible solutions in P1∪ P2 is not smaller than N, the environmental selection method of SPEA2 [49] is used to select N feasible solutions with the smallest fitness values as the final output. If the number of feasible solutions in P1 ∪ P2 is smaller than N, MSCEA selects N solutions with the smallest constraint violation values as the final output. ...
In recent years, solving constrained multiobjective optimization problems (CMOPs) by introducing simple helper problems has become a popular concept. To date, no systematic study has investigated the conditions under which this concept operates. In this study, we presented a holistic overview of existing constrained multiobjective evolutionary algorithms (CMOEAs) to address three research questions: (1) Why do we introduce helper problems? (2) Which problems should be selected as helper problems? and (3) How do helper problems help? Based on these discussions, we developed a novel helper-problem-assisted CMOEA, where the original CMOP was solved by addressing a series of constraint-centric problems derived from the original problem, with their constraint boundaries shrinking gradually. At each stage, we also had an objective-centric problem that was used to help solve the constraint-centric problem. In the experiments, we investigated the performance of the proposed algorithm on 66 benchmark problems and 15 real-world applications. The experimental results showed that the proposed algorithm is highly competitive compared with eight state-of-the-art CMOEAs.
... In this section, an implementation of MGSAEA, called K-MGSAEA, is proposed for solving CMOPs with both expensive objectives and constraints, where the Kriging model is adopted as the basic surrogate, and the SPEA2 [37] is used as the underlying optimizer. Before elaborating the procedure of K-MGSAEA, a brief description of the Kriging model is first presented. ...
Multi-objective optimization problems (MOPs) with expensive constraints pose stiff challenges to existing surrogate-assisted evolutionary algorithms (SAEAs) in a very limited computational cost, due to the fact that the number of expensive constraints for an MOP is often large. For existing SAEAs, they always approximate constraint functions in a single granularity, namely, approximating the constraint violation (coarse-grained) or each constraint (fine-grained). However, the landscape of constraint violation (CV) is often too complex to be accurately approximated by a surrogate model. Although the modelling of each constraint function may be simpler than that of CV, approximating all the constraint functions independently may result in tremendous cumulative errors and high computational cost. To address this issue, in this paper we develop a multi-granularity surrogate modelling framework for evolutionary algorithms, where the approximation granularity of constraint surrogates is adaptively determined by the position of population in the fitness landscape. Moreover, a dedicated model management strategy is also developed to reduce the impact resulting from the errors introduced by constraint surrogates and prevent the population from trapping into local optima. To evaluate the performance of the proposed framework, an implementation called K-MGSAEA is proposed, and the experimental results on a large number of test problems show that the proposed framework is superior over seven state-of-the-art competitors.
... These algorithms were selected based on their significance and impact on cloud computing and their ability to provide meaningful comparisons and insights into the performance of our proposed scheduling solution. The algorithms used in the evaluation process include MOEA/D [40], NSGA-II [41], OMOPSO [42], and SPEA2 [43], which are integrated and used from the MOEA framework [44], which is a comprehensive opensource software framework for multi-objective evolutionary algorithm research. The MOEA framework's default settings for the evaluated algorithms were utilized. ...
Task scheduling (TS) in cloud computing is a complex problem that involves balancing workload distribution, resource allocation, and power consumption. Existing methods often fail to optimize these objectives simultaneously and efficiently. This paper introduces a novel technique for scheduling independent tasks in cloud computing using multi-objective optimization and deep reinforcement learning (DRL). The proposed technique, DMOTS-DRL, combines Dueling deep Q-networks and dynamic prioritized experience replay to optimize two critical objectives: scheduling completion time (makespan) and power consumption. The performance of DMOTS-DRL is evaluated using CloudSim and compared with several state-of-the-art TS algorithms. The experimental results show that DMOTS-DRL outperforms the other algorithms in reducing makespan, power consumption, and other metrics, demonstrating its effectiveness and reliability for cloud computing services. Specifically, DMOTS-DRL achieves percentage improvements ranging from − 44.04 to − 0.19% in makespan, from − 0.26 to − 27.90% in power consumption, as well as better performance on other metrics such as energy consumption, degree of imbalance, resource utilization, and average waiting time.
... Solution sl a is also said to strictly dominate solution sl b [1]. Many of MOEA algorithms are genetic algorithms including NSGA-II [9], SPEA2 [35] and MOMBI2 [14] that differ from each other mainly in the way that solutions are ranked at every iteration [34], or in their decomposition technique (e.g., in MOEA/DD) [17]. ...
Value-alignment in normative multi-agent systems is used to promote a certain value and to ensure the consistent behavior of agents in autonomous intelligent systems with human values. However, the current literature is limited to incorporation of effective norms for single value alignment with no consideration of agents' heterogeneity and the requirement of simultaneous promotion and alignment of multiple values. This research proposes a multi-value promotion model that uses multi-objective evolutionary algorithms to produce the optimum parametric set of norms that is aligned with multiple simultaneous values of heterogeneous agents and the system. To understand various aspects of this complex problem, several evolutionary algorithms were used to find a set of optimised norm parameters considering two toy tax scenarios with two and five values are considered. The results are analysed from different perspectives to show the impact of a selected evolutionary algorithm on the solution, and the importance of understanding the relation between values when prioritising them.
... Specially, while P winner generates its offspring population Q winner by conventional genetic operators, P loser1 and P loser2 generate their offspring solutions by the newly proposed sparsity-guided genetic operator with the expectation of getting their sparsity to the present best sparsity from two complementary directions. Finally, the proposed SGECF performs the environmental selection strategy in SPEA2 [22] to select the population for the next generation from the union of parent and offspring solutions. At the first step, the proposed SGECF first generates a D × D random matrix dec and a D × D identity matrix mask, where D is the number of decision variables. ...
Sparse large-scale multi-objective optimization problems (LSMOPs) widely exist in real-world applications. Due to the characteristics of high-dimensional search space and sparse Pareto optimal solutions, most existing multi-objective evolutionary algorithms encounter difficulties in solving this kind of optimization problems. In this paper, a sparsity-guided elitism co-evolutionary framework, namely SGECF, is proposed. At each generation, SGECF first performs k-means clustering on non-dominated solutions to calculate the present best sparsity. Then, SGECF divides the population into a winner subpopulation and two loser subpopulations based on the Pareto dominance relationship and the present best sparsity. In the process of offspring reproduction, the winner subpopulation generates offspring solutions using conventional genetic operators, and the two loser subpopulations reproduce offspring solutions under the guidance of the present best sparsity and decision variable importance. In the experiments, the performance of SGECF is investigated on eight benchmark problems and a real-world application. Experimental results show that SGECF is superior over four state-of-the-art algorithms.
... To assess the effectiveness of QDGA, two widely recognized multi-objective evolutionary algorithms were selected for comparison: non-dominated sorting genetic algorithm-II (NSGA-II [39]) and strength Pareto evolutionary algorithm2 (SPEA2 [40]). These algorithms were chosen based on their reputation and their demonstrated ability to effectively solve complex multi-objective optimization problems. ...
In real-world production processes, the same enterprise often has multiple factories or one factory has multiple production lines, and multiple objectives need to be considered in the production process. A dual-population genetic algorithm with Q-learning is proposed to minimize the maximum completion time and the number of tardy jobs for distributed hybrid flow shop scheduling problems, which have some symmetries in machines. Multiple crossover and mutation operators are proposed, and only one search strategy combination, including one crossover operator and one mutation operator, is selected in each iteration. A population assessment method is provided to evaluate the evolutionary state of the population at the initial state and after each iteration. Two populations adopt different search strategies, in which the best search strategy is selected for the first population and the search strategy of the second population is selected under the guidance of Q-learning. Experimental results show that the dual-population genetic algorithm with Q-learning is competitive for solving multi-objective distributed hybrid flow shop scheduling problems.
... Vergilio et al. 22 , based on the idea of taboo search, used forbidden lists to save the generated CITO to avoid repeated search. Assção et al. 23 compared three common multi-objective optimization algorithms based on the Pareto model, namely NSGA-II 24 , SPEA 25 , and PAES 26 , and used them to solve the CITO problem. Zhang et al. 27 proposed a new heuristic algorithm which can reduce the cost of constructed test stubs through an in-depth study of evolutionary algorithms. ...
Class integration testing is an essential issue in software integration testing, and different class integration test order significantly impact the cost of testing. The Class Integration Test Order (CITO) problem is to find an optimal order of class integration test order to reduce the cost of software testing. The existing approaches tend to fall into local optimality when applied to complex systems and fail to achieve a better test order. This paper proposes a CITO generation approach based on deep reinforcement learning Categorical Double DQN (CDDQN) to address this limitation. The process uses the continuous interaction of the agent with the environment generated by inter-class dependencies to learn valuable experience and eventually obtain the optimal class integration test order. Experiments are conducted in eight systems to compare with graph-based, search-based, and reinforcement learning-based approaches. The experimental results show that the approach proposed in this paper can find CITO with lower stubbing complexity for most systems.
... Dominance-based MOEAs utilize the Pareto dominance relationship between solutions to select the promising solutions in each iteration. The popular algorithms include nondominated sorting genetic algorithm (NSGA)-II [8], strength Pareto evolutionary algorithm 2 (SPEA2) [9] and grid-based evolutionary algorithm [10,11]. Recent studies show that the classical dominance-based MOEAs become inefficient minimize ( ) = f 1 ( ), f 2 ( ), … f m ( ) T (1) subject to ∈ intersection method. ...
The multiobjective evolutionary algorithm based on decomposition (MOEA/D) proposed in 2007 has shown to be effective in solving multiobjective optimization problems. However, the fixed neighborhood size prevents MOEA/D series algorithm from effective exploring and exploiting. Meanwhile, since the newly generated offspring can replace the parent solutions in the neighborhood without any constraints, the current solutions of multiple subproblems may converge to a certain region or even a certain point. Obviously, the population diversity is significantly deteriorate. To alleviate these problems, this paper proposes an enhanced MOEA/D with adaptive neighborhood operator and extended distance-based environmental selection (MOEA/D-ANED) to solve many-objective optimization problems (MaOPs). The basic idea is that the adaptive neighborhood operator based on utility assignment is helpful for the algorithm to select a more promising solution, thereby enhancing the search performance in many-objective optimization. In addition, the extended distance-based environmental selection strategy is developed to considerably improve the accuracy and effectiveness of updating the parent solutions of different subproblems. Experimental results on two widely used test suits demonstrate the competitiveness of the proposed algorithm in terms of solution accuracy compared with eight state-of-the-art multiobjective optimization algorithms.
... Two other algorithms of NSGA3 and SPEA2 will also be employed as comparisons. NSGA3 is currently widely recognized as the best available algorithm for optimization problems with more than three objectives [45]. SPEA2 is a technique for finding or approximating the Pareto-optimal set for multi-objective optimization problems and good to solve the nonlinear and discontinuous combinatorial problem [46]. ...
The anti-aircraft system plays an irreplaceable role in modern combat. An anti-aircraft system consists of various types of functional entities interacting to destroy the hostile aircraft moving in high speed. The connecting structure of combat entities in it is of great importance for supporting the normal process of the system. In this paper, we explore the optimizing strategy of the structure of the anti-aircraft network by establishing extra communication channels between the combat entities. Firstly, the thought of combat network model (CNM) is borrowed to model the anti-aircraft system as a heterogeneous network. Secondly, the optimization objectives are determined as the survivability and the accuracy of the system. To specify these objectives, the information chain and accuracy chain are constructed based on CNM. The causal strength (CAST) logic and influence network (IN) are introduced to illustrate the establishment of the accuracy chain. Thirdly, the optimization constraints are discussed and set in three aspects: time, connection feasibility and budget. The time constraint network (TCN) is introduced to construct the timing chain and help to detect the timing consistency. Then, the process of the multi-objective optimization of the structure of the anti-aircraft system is designed. Finally, a simulation is conducted to prove the effectiveness and feasibility of the proposed method. Non-dominated sorting based genetic algorithm-II (NSGA2) is used to solve the multi-objective optimization problem and two other algorithms including non-dominated sorting based genetic algorithm-III (NSGA3) and strength Pareto evolutionary algorithm-II (SPEA2) are employed as comparisons. The deciders and system builders can make the anti-aircraft system improved in the survivability and accuracy in the combat reality.
... We compared the proposed DOS-CEA with nine MaOEAs to investigate the performance of the proposed algorithm. Among them, four MaOEAs are objective set decomposition based methods that use the NSGA-II [8] with different objective set decomposition strategies, i.e., random [30] , fixed [30] , shift [30] , and conflict degree [31] , called NSGA-IIrandom, NSGA-II-fixed, NSGA-II-shift, and NSGA-IIconflict in this paper. We also consider other five popular MaOEAs, i.e., NSGA-III [32] , MOEA/D [12] , RVEA [33] , PREA [16] , and MaOEA-IGD [17] . ...
Evolutionary algorithm is an effective strategy for solving many-objective optimization problems. At present, most evolutionary many-objective algorithms are designed for solving many-objective optimization problems where the objectives conflict with each other. In some cases, however, the objectives are not always in conflict. It consists of multiple independent objective subsets and the relationship between objectives is unknown in advance. The classical evolutionary many-objective algorithms may not be able to effectively solve such problems. Accordingly, we propose an objective set decomposition strategy based on the partial set covering model. It decomposes the objectives into a collection of objective subsets to preserve the nondominance relationship as much as possible. An optimization subproblem is defined on each objective subset. A coevolutionary algorithm is presented to optimize all subproblems simultaneously, in which a nondominance ranking is presented to interact information among these sub-populations. The proposed algorithm is compared with five popular many-objective evolutionary algorithms and four objective set decomposition based evolutionary algorithms on a series of test problems. Numerical experiments demonstrate that the proposed algorithm can achieve promising results for the many-objective optimization problems with independent and harmonious objectives.
... EMO algorithms [4,5] have been applied to problems with multiple objectives for the task of finding a well-representative set of Paretooptimal solutions. These methods [6,7] have been successful in solving a wide variety of problems with two or three objectives. However, these methodologies are criticized for their excessive computational expense, and they often tend to suffer while solving problems with objectives higher than three [8,9]. ...
Most of the practical applications that require optimization often involve multiple objectives. These objectives, when conflicting in nature, pose both optimization as well as decision-making challenges. An optimization procedure for such a multi-objective problem requires computing (computer-based search) and decision making to identify the most preferred solution. Researchers and practitioners working in various domains have integrated computing and decision-making tasks in several ways, giving rise to a variety of algorithms to handle multi-objective optimization problems. For instance, an a priori approach requires formulating (or eliciting) a decision maker’s value function and then performing a one-shot optimization of the value function, whereas an a posteriori decision-making approach requires a large number of diverse Pareto-optimal solutions to be available before a final decision is made. Alternatively, an interactive approach involves interactions with the decision maker to guide the search towards better solutions (or the most preferred solution). In our tutorial and survey paper, we first review the fundamental concepts of multi-objective optimization. Second, we discuss the classic interactive approaches from the field of Multi-Criteria Decision Making (MCDM), followed by the underlying idea and methods in the field of Evolutionary Multi-Objective Optimization (EMO). Third, we consider several promising MCDM and EMO hybrid approaches that aim to capitalize on the strengths of the two domains. We conclude with discussions on important behavioral considerations related to the use of such approaches and future work.
... • selectSPEA2 [129]: selects K individuals from a fitness assignment strategy which incorporates density information. ...
The growing usage of machine learning solutions (movie recommendation, speech recognition, fraud detection and so on) pushes the demand for having more efficient tools to build them. Indeed, building a machine learning model is a tedious task. The practitioner requires to preprocess the data, builds the features, selects the machine learning algorithms and tunes its hyper-parameters. Historically, these steps are handmade, but more recent tools called AutoML for Automatic Machine Learning have blossomed and, propose to perform these tasks automatically. Thus, AutoML eases the research of models and permits a gain of time for the experts but, also aims to help the non-experts to build a model without having to understand all the underlying mechanisms. In this work, we analyze the best known optimization methods used by the AutoML tools, and notice that among these methods, the evolutionary algorithms are very promising when it comes to improve the research of models. Indeed, the evolutionary algorithms ease the tuning of the exploration versus exploitation trade-offs, are inherently capable of handling any sort of candidates (fix and variable sizes), can tackle multiple objectives and can be easily parallelized. However, they have been barely studied on the AutoMLs, especially when it concerns the choice of the components such as the mutations or the algorithms. In this work, we first define a modular AutoML and a range of new components designed to study their impacts when used to automatically solve the classification problems. Then, we come up with a method to accelerate all the optimization processes based on evolutionary algorithms for large datasets. Finally, we propose a solution to automatically tackle the time series classification problems which, to the best of our knowledge, have never been studied before.
... Recently, an increasing number of researchers have used meta-heuristic algorithms to solve multi-objective optimization problems (MOPs) [11]. The most commonly used meta-heuristics are NSGA-II [12] and the strength Pareto evolutionary algorithm 2 (SPEA2) [13]. Recently, many researchers have focused on the artificial bee colony (ABC) algorithm because of its specific global search ability and suitable local searchability. ...
With the continuous development of science and technology, electronic devices have begun to enter all aspects of human life, becoming increasingly closely related to human life. Users have higher quality requirements for electronic devices. Electronic device testing has gradually become an irreplaceable engineering process in modern manufacturing enterprises to guarantee the quality of products while preventing inferior products from entering the market. Considering the large output of electronic devices, improving the testing efficiency while reducing the testing cost has become an urgent problem to be solved. This study investigates the electronic device testing machine allocation problem (EDTMAP), aiming to improve the production of electronic devices and reduce the scheduling distance among testing machines through reasonable machine allocation. First, a mathematical model was formulated for the EDTMAP to maximize both production and the scheduling distance among testing machines. Second, we developed a discrete multi-objective artificial bee colony (DMOABC) algorithm to solve EDTMAP. A crossover operator and local search operator were designed to improve the exploration and exploitation of the algorithm, respectively. Numerical experiments were conducted to evaluate the performance of the proposed algorithm. The experimental results demonstrate the superiority of the proposed algorithm compared with the non-dominated sorting genetic algorithm II (NSGA-II) and strength Pareto evolutionary algorithm 2 (SPEA2). Finally, the mathematical model and DMOABC algorithm were applied to a real-world factory that tests radio-frequency modules. The results verify that our method can significantly improve production and reduce the scheduling distance among testing machines.
... Most algorithms first consider convergence and then diversity when dealing with CMOPs, which means that solutions with higher convergence are more likely to survive to the next generation. To illustrate, NSGA-II [31] and SPEA2 [32] first emphasize non-dominated solutions and then consider less crowded non-dominated solutions to keep the population diverse; decomposition-based methods, such as MOEA/D, also consider non-dominated solutions first and then diverse ones. It has been observed that despite the great success of these EMO algorithms in optimizing many real-world MOPs, they have encountered difficulties in optimizing certain problems and require greater emphasis on diversity preservation [33]. ...
In recent years, many effective constrained multi-objective evolutionary algorithms (CMOEAs) have been proposed and successfully applied to address constrained multi-objective optimization problems (CMOPs). Nevertheless, few CMOEAs have fully explored CMOPs with imbalanced objectives or constraints. In this study, we propose a hybrid algorithm called M2M-IEpsilon to handle CMOPs with such characteristics, which combines an improved epsilon constraint-handling method (IEpsilon) with a multi-objective to multi-objective (M2M) decomposition strategy. The M2M decomposition mechanism divides a population into a set of sub-populations, which strengthens the diversity of the population. The IEpsilon constraint-handling method enables individuals with small constraint violation values to survive to the next generation, thus leading to a search for promising regions. In addition, a series of imbalanced CMOPs, named IM-CMOPs, is designed to verify the performance of the proposed M2M-IEpsilon algorithm. The comprehensive experimental results indicate that the proposed method can solve such imbalanced CMOPs well and perform significantly better than the other eight new or classical CMOEAs. Finally, we used the proposed M2M-IEpsilon to optimize the wellbore trajectory problem, and it achieved good results compared with previously developed algorithms.
... Moreover, the elitism mechanism is employed in NSGA-II to improve the probability of finding better solutions. Zitzler et al. [18] proposed an improved strength Pareto evolutionary algorithm (SPEA2), which employs a nearest-neighbor density estimation technique to guide the search process more precisely. Hashemi et al. [19] proposed an effective approach based on Pareto dominance to solve the multilabel feature selection problem. ...
Neighborhood selection is an important part of a multiobjective evolutionary algorithm based on decomposition (MOEA/D) because the impetus for population evolution mainly comes from its neighborhood. However, the fixed neighborhood size used in MOEA/D may deteriorate the performance of the algorithm due to an unreasonable allocation of computational resources. To further improve the performance of MOEA/D, this paper proposes a multiobjective decomposition evolutionary algorithm with optimal history-based neighborhood adaptation and a dual-indicator selection strategy. The optimal history-based neighborhood adaptation strategy is applied to alleviate the imbalance between exploration and exploitation in the search process, while the dual-indicator selection strategy is developed to enhance the population diversity. The performance of the proposed algorithm is evaluated on the DTLZ and WFG series test problems. Experimental results show that the proposed algorithm performs competitively in comparison with several MOEA/D variants.
... To achieve the tradeoff between objective functions and constraints, and choose "good" infeasible solutions for the next generation, a natural way is to use the information of objective functions. Fortunately, we can use the method in SPEA2 [42] to calculate the fitness of infeasible solutions without considering any constraints, and the ranking of infeasible solution x i can be obtained (S F i ) through Lines 1 of Algorithm 2, a smaller value of S F i represents the better performance of x i . Afterward, the other ranking of infeasible solution x i can be obtained (S NSCV i ), in which the number of satisfied constraints (NS) and CV are used as two objectives (minimize CV, and maximize NS) for sorting based on Pareto dominance, as shown in Line 3-10 of Algorithm 2. Subsequently, to balance the importance of objectives and constraints, we use a new fitness function to evaluate infeasible solutions. ...
When solving constrained multi-objective optimization problems (CMOPs), multiple conflicting objectives and multiple constraints need to be considered simultaneously, which are challenging to handle. Although some recent constrained multi-objective evolutionary algorithms (CMOEAs) have been developed to solve CMOPs and have worked well on most CMOPs. However, for CMOPs with small feasible regions and complex constraints, the performance of most algorithms needs to be further improved, especially when the feasible region is composed of multiple disjoint parts or the search space is narrow. To address this issue, an efficient global diversity CMOEA (EGDCMO) is proposed in this paper to solve CMOPs, where a certain number of infeasible solutions with well-distributed feature are maintained in the evolutionary process. To this end, a set of weight vectors are used to specify several subregions in the objective space, and infeasible solutions are selected from each subregion. Furthermore, a new fitness function is used in this proposed algorithm to evaluate infeasible solutions, which can balance the importance of constraints and objectives. In addition, the infeasible solutions are ranked higher than the feasible solutions to focus on the search in the undeveloped areas for better diversity. After the comparison tests on three benchmark cases and an actual engineering application, EGDCMO has more impressive performance compared with other constrained evolutionary multi-objective algorithms.
... However, an initial selection of weights does not guarantee the desired solution, requiring creators to adjust these during the design process. An additional disadvantage of using a weighted sum method is that it cannot find certain Pareto optimal solutions in the case of a non-convex objective space [41], a problem that other methods such as evolutionary algorithms could overcome [71]. ...
Adaptive user interfaces can improve experiences in Cross Reality (XR) applications by changing interface elements according to the user's context. Although extensive work explores diferent adaptation policies, XR creators often struggle with their implementation, which involves laborious manual scripting. The few available tools are underdeveloped for realistic XR settings where is often necessary to consider conflicting aspects that affect an adaptation. We fill this gap by presenting AUIT, a toolkit that facilitates the design of optimization-based adaptation policies. AUIT allows creators to flexibly combine policies that address common objectives in XR applications, such as element reachability, visibility, and consistency. In contrast to using rules or scripts, specifying adaptation policies via adaptation objectives simplifies the design process and enables creative exploration of adaptations. After creators decide which objectives to use, a multi-objective solver finds appropriate adaptations in real-time. A study showed that AUIT allowed creators of XR applications to quickly and easily create high-quality adaptations.
... Since the birth of evolutionary algorithm, due to its heuristic search strategy, it has been used in multi-objective problem optimization and has achieved excellent performance. Current multi-objective evolutionary algorithms can be divided into four categories: a. Multi-objective evolutionary algorithms based on Pareto dominance, represented by SPEA, PESA algorithms and their improved algorithms (Zitzler et al., 2001;Lalitesh & Prawendra, 2020); b. Multiobjective evolution based on decomposition Algorithm. ...
In the multi-objective optimization algorithm, the parameter strategy has a huge impact on the performance of the algorithm, and it is difficult to set a set of parameters with excellent distribution and convergence performance in the actual optimization process. Based on the MOEA/D algorithm framework, this paper construct an improved dual-population co-evolution MOEA/D algorithm by adopt the idea of dual-population co-evolution. The simulation test of the benchmark functions shows that the proposed dual-population co-evolution MOEA/D algorithm have significant improvements in IGD and HV indicators compare with three other comparison algorithms. Finally, the application of the LTE base station power allocation model also verifies the effectiveness of the proposed algorithm.
... MOEAs and a hot research direction of scholars. Non-dominated Sorting Genetic Algorithm II, NSGA-II (Deb et al., 2002); Strength Pareto Evolutionary Algorithm II, SPEA-II (Zitzler et al., 2001); Pareto Envelope-Based Selection Algorithm II, PESA-II (Corne et al., 2001); etc. However, with the increase of the objective dimension of the optimization problem, the dominance relationship between individuals will become less and less obvious, and the selection pressure will drop sharply, resulting in the performance of the algorithm in high-dimensional optimization problems is far less excellent than that in low-dimensional optimization problems. ...
In recent years, multi-objective optimization algorithms, especially many-objective optimization algorithms, have developed rapidly and effectively.Among them, the algorithm based on particle swarm optimization has the characteristics of simple principle, few parameters and easy implementation. However, these algorithms still have some shortcomings, but also face the problems of falling into the local optimal solution, slow convergence speed and so on. In order to solve these problems, this paper proposes an algorithm called MUD-GMOPSO, A Many-Objective Practical Swarm Optimization based on Mixture Uniform Design and Game mechanism. In this paper, the two improved methods are combined, and the convergence speed, accuracy and robustness of the algorithm are greatly improved. In addition, the experimental results show that the algorithm has better performance than the four latest multi-objective or high-dimensional multi-objective optimization algorithms on three widely used benchmarks: DTLZ, WFG and MAF.
In this work, we present a new combined measure for improving the quality of digital image segmentation. The measure has two components. The first is information redundancy, and the second is variation of information. The new measure makes it possible to obtain an image partition that provides a compromise between the conflicting objectives of minimizing the number of information-important segments and minimizing the information dissimilarity between the original and segmented images. We found the conditions for the minimum of the combined measure. The conducted computational experiment showed an improvement in segmentation results when using the proposed combined measure in comparison with the previously used information redundancy measure.KeywordsImage SegmentationSegmentation QualityInformation Redundancy MeasureVariation of InformationCombined Measure
In recent years, multi-objective optimization techniques have gained popularity as a powerful approach to the solution of complex engineering problems that involve multiple competing objectives. This popularity can be attributed to the fact that multi-objective optimization techniques can solve complex engineering problems. Engineers are able to think about many goals at the same time using this method, which enables them to discover the best solution that fits all of their requirements. Yet, there are obstacles connected with setting objectives, dealing with high computational costs, comprehending complicated Pareto fronts, coping with uncertainty, and having limited accessibility to data from the real world. The purpose of this paper is to present a literature review of multi-objective optimization techniques for the solution of complex engineering problems. The review covers a variety of techniques that are available for multi-objective optimization as well as their applications in a variety of engineering fields. It also draws attention to the difficulties involved with their application and the necessity of giving serious consideration to these difficulties while attempting to solve actual engineering problems that occur in the real world. The research comes to the conclusion that multi-objective optimization is an effective method that can assist engineers in solving complicated engineering issues by concurrently taking into consideration numerous competing objectives.
In this paper, we present the application of our fuzzy rule-based modeling technique with genetically optimized accuracy-interpretability trade-off to time-series-dynamics discovery, modeling, and forecasting. The so-called Box-Jenkins’ benchmark, i.e., measurement-based time series describing the behavior of an industrial gas furnace is considered. We employ – as a multi-objective evolutionary optimization algorithm – our generalization – characterized by a higher spread and a better balanced distribution of solutions – of the well-known SPEA2 method. Our approach has been compared with several alternative techniques applied to the same time series data.
In recent years, many improved multi-objective particle swarm optimization algorithms have been proposed to solve multi-objective optimization problems, but there are still some shortcomings. Therefore, in order to enhance the convergence and diversity of the algorithm, the multi-objective particle swarm optimization based on archive control strategy (ACMOPSO) is proposed. Firstly, by introducing the objective extreme optimal solution of particles in the population as the learning sample, the search ability of the particles and the diversity of the population can be enhanced. Secondly, the regional division control strategy is used in the algorithm to control the external archive, which effectively improves the quality of the solutions in the external archive and the search ability of the population. Finally, the numerical experiments of the proposed ACMOPSO and MOEAIGDNS, SPEA2, NSGA-II, MOPSO, and MPSOD are carried out on five test functions. The numerical experiments show that the overall performance of the proposed ACMOPSO is better than that of the other five algorithms.
An Optimal power flow (OPF) is non‐linear and constrained multi‐objective problem. OPF problems are expensive and evolutionary algorithms (EAs) are computationally complex to obtain uniformly distributed and global Pareto front (PF). Therefore, here, hybrid two‐phase algorithm integrated with parameter less constraint technique is applied to solve OPF problem. Proposed technique combines single and multi‐objective EAs to find better convergence and evenly distributed PF. For the validation and effectiveness of proposed algorithm, various conflicting objective functions are formulated and implemented on IEEE 30 and 300‐bus network. Each case is independently run twenty times. Hyper volume indicator technique is employed to find the best PF, and the best‐compromised solution is obtained by using fuzzy decision‐making technique. Recently, maximum integration of wind and solar power is highly encouraged. Complexity of OPF is increased with the integration of uncertain renewable energy resources. Hence, 30‐bus test system is modified by replacing some conventional generators with the wind and solar generation. Uncertainties in wind, solar and load demand are modelled by appropriate probability distribution functions. Simulation results show that the proposed method can find the near global PF of highly complex problems subject to satisfying all the operational constraints.
This article is about investigating the potential of multi-harmonic design of permanent magnet synchronous machines. Thereby, multi-harmonic refers to the temporal periodic characteristics of linked fluxes and currents of the stator phases in order to ensure high-efficient operation with very low torque ripple. At first, a suitable model allowing for multi-harmonic currents evaluation is derived. Consequently, an interior-rotor IPM machine is studied. In order to have a fair comparison with conventional PMSMs featuring sinusoidal currents, an optimization scenario is defined where the same outer dimensions as well as magnet and copper volume for both design variants are considered. The use of multiple current harmonics requires a dedicated strategy for evaluating machine designs, as the conventional
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-grid analysis cannot be utilized. The introduced approach is applied and results of the optimization are illustrated. Prototypes were manufactured and machine designs were subsequently evaluated through experiments. A valuable lesson learned during practical implementation regarding rotor angle determination is highlighted. Measurement results reveal that adding more degrees of freedom in terms of multiple harmonics can give significant improvements for performance measures efficiency and torque ripple. This particularly holds if the harmonics are already considered throughout the machine design and optimization process, rather than only at post-processing stage.
In this paper, a graph-based optimization problem related to epidemics control is studied. This multiobjective optimization problem requires determining which graph nodes to protect in order to limit the number of nodes affected by a threat spreading in the graph. In the paper, a classifier-based evolutionary algorithm is proposed in which population initialization, crossover, mutation and local search can use a previously trained classifier to decide which graph nodes to protect. The classifier is trained on a dataset collected by running an optimization algorithm on small problem instances and it is subsequently reused for solving multiple, larger instances of the same optimization problem. Selecting graph nodes to protect using a previously trained classifier allows the algorithm to find better solutions to the optimization problem compared to an algorithm not using a classifier. In the experiments, several machine learning models were evaluated and a multilayer perceptron (MLP) neural network was selected as the best performing classifier. The proposed optimization method was tested and found to work effectively when the MLP classifier was trained on data collected when solving problem instances with 1000 graph nodes. In tests on problem instances with up to 20000 nodes the proposed method outperformed an evolutionary optimizer not using any classifier-based components thereby showing that knowledge can be transferred from smaller problem instances to larger ones.
Evolving intelligent systems (EISs), particularly, the zero-order ones have demonstrated strong performance on many real-world problems concerning data stream classification, while offering high model transparency and interpretability thanks to their prototype-based nature. Zero-order EISs typically learn prototypes by clustering streaming data online in a “one pass” manner for greater computation efficiency. However, such identified prototypes often lack optimality, resulting in less precise classification boundaries, thereby hindering the potential classification performance of the systems. To address this issue, a commonly adopted strategy is to minimise the training error of the models on historical training data or alternatively, to iteratively minimise the intra-cluster variance of the clusters obtained via online data partitioning. This recognises the fact that the ultimate classification performance of zero-order EISs is driven by the positions of prototypes in the data space. Yet, simply minimising the training error may potentially lead to overfitting, whilst minimising the intra-cluster variance does not necessarily ensure the optimised prototype-based models to attain improved classification outcomes. To achieve better classification performance whilst avoiding overfitting for zero-order EISs, this paper presents a novel multi-objective optimisation approach, enabling EISs to obtain optimal prototypes via involving these two disparate but complementary strategies simultaneously. Five decision-making schemes are introduced for selecting a suitable solution to deploy from the final non-dominated set of the resulting optimised models. Systematic experimental studies are carried out to demonstrate the effectiveness of the proposed optimisation approach in improving the classification performance of zero-order EISs.
In this paper, we propose a multi-operator differentia evolution variant that incorporates three diverse mutation strategies in MOEA/D. Instead of exploiting the local region, the proposed approach continues to search for optimal solutions in the entire objective space. It explicitly maintains diversity of the population by relying on the benefit of clustering. To promowe convergence, the solutions close to the ideal position, in the objective space are given preference in the evolutionary process. The core idea is to ensure diversity of the population by applying multiple mutation schemes and a faster convergence rate, giving preference to solutions based on their proximity to the ideal position in the MOEA/D paradigm. The performance of the proposed algorithm is evaluated by two popular test suites. The experimental results demonstrate that the proposed approach outperforms other MOEA/D algorithms.
With growing environmental concerns and tough carbon-neutral objectives, logistics providers have to consider not only economic benefits but also environmental impact in the delivery process. This study proposes a novel multi-objective optimization model for the vehicle routing problem with drone delivery. The proposed model involves improving delivery efficiency and reducing environmental impact by extending the conventional ground vehicle (i.e. truck) delivery model with the implementation of drone delivery as well as the optimization of the total energy consumption of trucks. Drones need to collaborate with trucks to serve customers because of their limited flight endurance. Moreover, the fact that flight endurance is dynamic and influenced by the loading rate of drones is also considered to satisfy practical application scenarios. An extended non-dominated sorting genetic algorithm is presented to solve the proposed model. A new encoding and decoding method is incorporated to represent multiple feasible routes of drones and trucks, several crossover and mutation operators are integrated to accelerate the algorithmic convergence, and a multi-dimensional local search strategy is employed to enhance the diversity of population. Finally, the experimental results demonstrate that the presented algorithm is effective in obtaining high-quality non-dominated solutions by comparing it with three other baseline multi-objective algorithms.
The role of carefully constructed benchmark suites in algorithm design and testing is critical. Within the continuous multi-objective optimisation domain, existing suites include the general purpose ZDT, DTLZ and WFG suites, and more recent ones specifically designed to explore the impacts of a particular problem characteristic. However, the relationship between existing suites is not clear, and the field would benefit from a "stock-take" assessment. This paper investigates the coverage of current continuous multi-objective suites using the Instance Space Analysis methodology. Exploratory landscape analysis is used to measure critical features of each problem suite. Thereafter, we generate a 2D visualisation of the existing problem instances by locating them in the instance space, assessing their diversity, and identifying whether there are sparse areas of value to fill with new problem instances. Our findings show that the current suites are restricted in diversity when representing the entire problem instance space. We propose and evaluate three problem construction methods: problem tuning, toolkit hybridisation, and new function injection. Problem tuning is shown to generate problems surrounding existing instances, while hybridisation creates problems falling between existing suites. Furthermore, utilising the insights afforded by Instance Space Analysis, we show how problem features can be identified to inform the creation of new functions which fill gaps towards the boundaries of the instance space.
Multi-objective Genetic Programming (MGP) plays a prominent role in generating Pareto optimal classifier sets and making trade-offs among multiple classes adaptively. However, the existing MGP algorithms show poor performance and are difficult to implement when dealing with imbalanced classification problems. This work proposes a new MGP-based algorithm designed for imbalanced classification. Firstly, an efficient evolutionary strategy with nondominated sorting, environmental selection, and an archiving mechanism is designed to optimize the false positive rate, the false negative rate and reduce the size of the resulting tree. Then, a weighted ensemble decision is made according to each classifier’s performance in the majority and minority classes to obtain final classification results. Experimental results on 21 binary-class datasets and 17 multi-class datasets show that the proposed method outperforms existing ones in several commonly used imbalanced classification metrics.
The framework of decomposition-based multi-objective evolutionary algorithms(MOEA/D) has evolved for more than ten years, and it has become irreplaceable tool for solving multi-objective optimization problems. In recent years, many scholars have investigated improved strategies from different directions. This paper gives a systematic comparison of six different components for decomposition-based algorithms, including framework analysis, weight vector generation scheme, aggregation evaluation function construction, reproduction operator, individual selection and update strategy, and the characteristics and application scope of various algorithms are also analyzed in detail in the survey. Different from previous survey on decomposition-based multi-objective evolutionary algorithms, a more detailed classification and experimental comparison are elaborated in the proposed paper.
An important factor in constrained multi-objective evolutionary algorithms (CMOEAs) is how to make optimal use of the information of feasible and infeasible solutions. To fully utilize these promising solutions, this paper proposes an archive-based two-stage evolutionary algorithm, called AT-CMOEA, for solving constrained multi-objective optimization problems (CMOPs). AT-CMOEA divides the evolutionary process into two stages – exploration and exploitation. The purpose of the exploration stage is to encourage a broader exploration of the search space to discover some promising regions, which is achieved by two populations with different priorities of constraints and objectives. In the exploitation stage, the goal is to obtain a set of well-distributed Pareto-optimal solutions by utilizing the useful archived information found during the exploration stage, allowing the search resources to be focused on the region of interest. Then, the two populations cooperatively converge to the constrained Pareto front (CPF). Comprehensive experiments on a series of benchmark test problems and three real-world CMOPs demonstrate the competitiveness of our method when compared with other representative algorithms in terms of effectiveness and reliability in finding a set of well-distributed optimal solutions.
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