Penalty Boundary Intersection Research Articles

A Bi-Objective Evolutionary Algorithm for Multimodal Multiobjective Optimization

Multimodal multi-objective optimization problems (MMOPs) possess multiple Pareto optimal sets corresponding to the identical Pareto optimal front (PF). To handle MMOPs, we propose a bi-objective evolutionary algorithm (BOEA), which transforms an MMOP into a bi-objective optimization problem. This problem is constructed by the penalty boundary intersection technique and a diversity indicator to obtain multiple Pareto optimal sets. The first objective reflects the population convergence and factors in the population diversity in the objective space, while the other objective concentrates more on the population diversity in the decision space. Furthermore, an environmental selection strategy is designed to choose the offspring solutions, which consists of non-dominated sorting based on the transformed optimization problem and hierarchical clustering for selecting promising solutions. Experiments on 34 MMOPs demonstrate that BOEA performs better than selected state-of-the-art representatives, including 22 MMOPs from CEC2019 and 12 imbalanced MMOPs. In addition, the effectiveness of BOEA is further validated by six feature selection problemsin real-world applications.

A novel feature selection algorithm using decomposition based multi-objective guided honey badger algorithm (MO-GHBA) and NSGA-III

In solving two or more objective problems, multi-objective evolutionary algorithms (MOEAs) have proven their effective performance. In most of the MOEAs based feature selection algorithms, more optimal solutions are obtained around the Pareto front's center because of the deficiency in selection features. With the complicated Pareto fronts, it is difficult to select features using the penalty boundary intersection decomposition approach that can provide stable selection pressures. With the adaptive penalty boundary intersection (APBI) decomposition approach, a novel wrapper based feature selection technique named MOGHBNS3/D is proposed in this paper based on the hybrid of Multi Objective Guided Honey Badger Algorithm and Non-dominated sorting genetic algorithm III. Classification accuracy improvement and removal of redundant features are considered to be the multi-objective optimization functions of the proposed multi-objective feature selection technique. An external archive is employed as a repository for storing the non-dominated solutions in the MOGHBNS3/D. To enhance the selection pressures of the external archive, the penalty values are adjusted adaptively using the APBI mechanism. The Wavelet kernel Extreme Learning Machine is used to classify the selected features. To analyze the performance of the proposed feature selection technique, eighteen benchmark datasets are utilized and results are compared with the four well known multi-objective techniques in terms of accuracy, precision, recall, F1 score, coverage error, hamming loss, ranking loss and training time. The experimental results show that the proposed method can select optimal features with improved classification accuracy, diversity and convergence while the removal of redundancy.

A Comparative Study of Three NSGA Versions for the Multi-Objective Sustainable Process Plans Optimization in RMS

Reconfigurable manufacturing system (RMS) is a paradigm that considers many of the issues the economy and society face today. This paper addresses an environmental-oriented multi-objective process plan generation problem in a reconfigurable manufacturing context. Four objective functions are minimized: the total production cost, the total production time, the amount of greenhouse gas emitted by machines, and the hazardous liquid wastes. To solve the problem, three adapted versions of the well-known non-dominated sorting genetic algorithm (NSGA) approach, namely the penalty boundary intersection (PBI) reference point (RP) with the NSGA-II (PBI-R-NSGA-II), NSGA-III with a selection and elimination operator (SE-NSGA-III), and NSGA-III with the similarity coefficient (SC-NSGA-III) are evaluated and compared. Moreover, the impacts of perturbation ratio on SC-NSGA-III's performance as the number of iterations increases is examined. Finally, the efficacity of the three approaches is demonstrated by rich experimental results and analyzes using two metrics, respectively cardinality of the mixed Pareto fronts (CMPF) and max Pareto front error (MPFE).

Many-objective cloud manufacturing service selection and scheduling with an evolutionary algorithm based on adaptive environment selection strategy

Cloud manufacturing service selection and scheduling (CMSSS) problem has obtained wide attentions in recent years. However, most existing methods describe this problem as single-, bi-, or tri-objective models. Little work deals with this problem in four or more objectives simultaneously. This paper investigated CMSSS problem in consideration of the interests of users, cloud platform and service providers. An eight-objective CMSSS optimization model is constructed for the problem. Meanwhile, a many-objective evolutionary algorithm with adaptive environment selection (MaOEA-AES) is designed to address the problem. Specifically, diversity-based population partition technology is used to divide the population into multiple subregions to maintain the population diversity, and an adaptive penalty boundary intersection (APBI) distance is designed to select elitist solutions in different stages of evolutionary process. The proposed algorithm is tested on 2 cases with 5 and 8 objectives in CMSSS problems and each of them has sixteen experimental groups with different problem scales. The experiment results show that MaOEA-AES is competitive to resolve the MaO-CMSSS model compared with eight state-of-the-art evolutionary algorithms in convergence and diversity.

A Decomposition-Based Evolutionary Algorithm with Correlative Selection Mechanism for Many-Objective Optimization.

Decomposition-based evolutionary algorithms have been quite successful in dealing with multiobjective optimization problems. Recently, more and more researchers attempt to apply the decomposition approach to solve many-objective optimization problems. A many-objective evolutionary algorithm based on decomposition with correlative selection mechanism (MOEA/D-CSM) is also proposed to solve many-objective optimization problems in this article. Since MOEA/D-SCM is based on a decomposition approach which adopts penalty boundary intersection (PBI), a set of reference points must be generated in advance. Thus, a new concept related to the set of reference points is introduced first, namely, the correlation between an individual and a reference point. Thereafter, a new selection mechanism based on the correlation is designed and called correlative selection mechanism. The correlative selection mechanism finds its correlative individuals for each reference point as soon as possible so that the diversity among population members is maintained. However, when a reference point has two or more correlative individuals, the worse correlative individuals may be removed from a population so that the solutions can be ensured to move toward the Pareto-optimal front. In a comprehensive experimental study, we apply MOEA/D-CSM to a number of many-objective test problems with 3 to 15 objectives and make a comparison with three state-of-the-art many-objective evolutionary algorithms, namely, NSGA-III, MOEA/D, and RVEA. Experimental results show that the proposed MOEA/D-CSM can produce competitive results on most of the problems considered in this study.

Hybrid Evolutionary Multi-Objective Optimization Algorithm for Helping Multi-Criterion Decision Makers

ABSTRACT Obtaining a specific region from the efficient front for multi-objective and practical optimization problems helps decision-makers. Reference point approaches are suggested to reach the region of interest. Many evolutionary algorithms integrated with a reference point idea to obtain apart from the efficient front solutions close to the reference solution. This paper integrated a penalty boundary intersection (PBI) with the non-dominated sorting genetic algorithm (NSGA-II) to reach the efficient front close to the reference point. The proposed approach allows theoretically convergent solutions to be found. Also, using the advantage of PBI, our algorithm is able to reach a specific part for a set multi and many-objective optimization problems effectively. The proposed algorithm compared with other evolutionary algorithms, such as the original reference-based NSGA-II (R-NSGA-II), r-NSGA-II, and g-NSGA-II. The R-metric values show that the proposed algorithm outperforms the compared algorithms. Using the proposed algorithm for solving the multi-objective engineering design problem helps find solutions according to the decision-makers interest.

An adaptive decomposition evolutionary algorithm based on environmental information for many-objective optimization

The performance of traditional penalty boundary intersection (PBI) decomposition-based evolutionary algorithm is totally determined by the penalty factor. The fixed penalty factor causes the imbalance between the convergence and the diversity when solving many-objective problems. So, an adaptive decomposition evolutionary algorithm based on environmental information (MaOEA/ADEI) is proposed to solve the imbalance. The penalty factor of PBI decomposition is determined by the environmental information (include distribution information of weight vectors and population). Furthermore, the parent individual selection strategy is introduced to select promising individuals for variation and the weight vectors adaption strategy is used to handle problems with scaled objectives. Comparisons with 4 algorithms on 24 benchmark instances are used to test the property of MaOEA/ADEI. The experimental results show MaOEA/ADEI performs best on 14 test instances.

Effect of Objective Normalization and Penalty Parameter on Penalty Boundary Intersection Decomposition-Based Evolutionary Many-Objective Optimization Algorithms.

An objective normalization strategy is essential in any evolutionary multiobjective or many-objective optimization (EMO or EMaO) algorithm, due to the distance calculations between objective vectors required to compute diversity and convergence of population members. For the decomposition-based EMO/EMaO algorithms involving the Penalty Boundary Intersection (PBI) metric, normalization is an important matter due to the computation of two distance metrics. In this article, we make a theoretical analysis of the effect of instabilities in the normalization process on the performance of PBI-based MOEA/D and a proposed PBI-based NSGA-III procedure. Although the effect is well recognized in the literature, few theoretical studies have been done so far to understand its true nature and the choice of a suitable penalty parameter value for an arbitrary problem. The developed theoretical results have been corroborated with extensive experimental results on three to 15-objective convex and non-convex instances of DTLZ and WFG problems. The article, makes important theoretical conclusions on PBI-based decomposition algorithms derived from the study.

Investigating the equivalence between PBI and AASF scalarization for multi-objective optimization

Abstract Scalarization refers to a generic class of methods to combine multiple conflicting objectives into one in order to find a Pareto optimal solution to the original problem. Augmented achievement scalarizing function (AASF) is one such method used popularly in the multi-criterion decision-making (MCDM) field. In evolutionary multi-objective optimization (EMO) literature, scalarization methods such as penalty boundary intersection (PBI) are commonly used to compare similar solutions within a population. Both AASF and PBI methods require a reference point and a reference direction for their calculation. In this paper, we aim to analytically derive and understand the commonalities between these two metrics and gain insights into the limitations of their standard parametric forms. We show that it is possible to find an equivalent modified AASF formulation for a given PBI parameter and vice versa for bi-objective problems. Numerical experiments are presented to validate the theory developed. We further discuss the challenges in extending this to higher objectives and show that it is still possible to achieve limited equivalence along symmetric reference vectors. The study connects the two philosophies of solving multi-objective optimization problems, provides a means to gain a deeper understanding of both these measures, and expands their parametric range to provide more flexibility of controlling the search behavior of the EMO algorithms.

Variable-Length Pareto Optimization via Decomposition-Based Evolutionary Multiobjective Algorithm

Optimization problems with variable-length decision space are a class of challenging optimization problems derived from some real-world applications, such as the composite laminate stacking problem and the sensor coverage problem. Unlike other optimization problems, the solutions in these problems might be represented as the vectors with different variable size (i.e., dimensionality). So far, some research efforts have been done on the use of evolutionary algorithms (EAs) for solving single objective variable-length optimization problems. In fact, the variable-length problem difficulty can also exist in multiobjective optimization. However, such challenging problems have not yet gained much attention in the area of evolutionary multiobjective optimization. To facilitate the research on the variable-length Pareto optimization, we first suggest a systematic toolkit for constructing benchmark multiobjective test problems with variable-length feature in this paper. Then, we also propose a variable-length multiobjective EA based on a two-level decomposition strategy, which decomposes a multiobjective optimization problem in terms of the penalty boundary intersection search directions and the dimensionality of variables. The performance of our proposed algorithm and the other three state-of-the-art algorithms on these problems are compared. To further show the effectiveness of our proposed algorithm, some experimental results on a bi-objective laminate stacking optimization problem are also reported and analyzed.

A new dominance-relation metric balancing convergence and diversity in multi- and many-objective optimization

Maintaining a good balance between convergence and diversity in many-objective optimization is a key challenge for most Pareto dominance-based multi-objective evolutionary algorithms. In most existing multi-objective evolutionary algorithms, a certain fixed metric is used in the selection operation, no matter how far the solutions are from the Pareto front. Such a selection scheme directly affects the performance of the algorithm, such as its convergence, diversity or computational complexity. In this paper, we use a more structured metric, termed augmented penalty boundary intersection, which acts differently on each of the non-dominated fronts in the selection operation, to balance convergence and diversity in many-objective optimization problems. In diversity maintenance, we apply a distance-based selection scheme to each non-dominated front. The performance of our proposed algorithm is evaluated on a variety of benchmark problems with 3 to 15 objectives and compared with five state-of-the-art multi-objective evolutionary algorithms. The empirical results demonstrate that our proposed algorithm has highly competitive performance on almost all test instances considered. Furthermore, the combination of a special mate selection scheme and a clustering-based selection scheme considerably reduces the computational complexity compared to most state-of-the-art multi-objective evolutionary algorithms.

A comparative study of multi-objective expected improvement for aerodynamic design

Multi-objective optimization in aerodynamics plays an important role in revealing trade-offs between conflicting objectives in order to discover important knowledge and insight for better future design. Of interest here is the use of Kriging surrogate models incorporated into a sequential Bayesian optimization (BO) strategy. In this paper, we studied four variants of multi-objective BO (MOBO) techniques that are based on expected improvement (EI), that is, Euclidean-based EI (EEI), expected hypervolume improvement (EHVI), ParEGO, and expected inverted penalty boundary intersection improvement (EIPBII) to understand their capabilities on handling multi-objective aerodynamic optimization problems. Numerical tests were performed on a set consisting of six generalized Schaffer problems (GSP), five low-fidelity, and one high-fidelity airfoil design problems. Results suggest that EHVI is the only method which consistently performed well on artificial and aerodynamic problems. EEI yields the worst performance and is not suitable to deal with various problem complexities. ParEGO, although it performs modestly on GSP, surprisingly works well on the low- and high-fidelity problems. On the other hand, EIPBII encounters the opposite case, where it is one of the best performer on GSP but yields modest performance on the aerodynamic problems. In light of the results, we suggest that EHVI is a highly potential MOBO method to be applied for multi-objective aerodynamic design optimization.

An adaptive decomposition-based evolutionary algorithm for many-objective optimization

Penalty boundary intersection (PBI) is one popular method in decomposition based evolutionary multi-objective algorithms, where the penalty factor is crucial for striking a balance between convergence and diversity in a high-dimensional objective space. Meanwhile, the distribution of the obtained solutions highly depends on the setting of the weight vectors. This paper proposes an adaptive decomposition-based evolutionary algorithm for many-objective optimization, which introduces one adaptation mechanism for PBI-based decomposition and the other for adjusting the weight vector. The former assigns a specific penalty factor for each subproblem by using the distribution information of both population and the weight vectors, while the latter adjusts the weight vectors based on the objective ranges to handle problems with different scales on the objectives. We have compared the proposed algorithm with seven state-of-the-art many-objective evolutionary algorithms on a number of benchmark problems. The empirical results demonstrate the superiority of the proposed algorithm.

Evolutionary Many-Objective Algorithm Using Decomposition-Based Dominance Relationship.

Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm.

Improving the multiobjective evolutionary algorithm based on decomposition with new penalty schemes

It has been increasingly reported that the multiobjective optimization evolutionary algorithm based on decomposition (MOEA/D) is promising for handling multiobjective optimization problems (MOPs). MOEA/D employs scalarizing functions to convert an MOP into a number of single-objective subproblems. Among them, penalty boundary intersection (PBI) is one of the most popular decomposition approaches and has been widely adopted for dealing with MOPs. However, the original PBI uses a constant penalty value for all subproblems and has difficulties in achieving a good distribution and coverage of the Pareto front for some problems. In this paper, we investigate the influence of the penalty factor on PBI, and suggest two new penalty schemes, i.e., adaptive penalty scheme and subproblem-based penalty scheme (SPS), to enhance the spread of Pareto-optimal solutions. The new penalty schemes are examined on several complex MOPs, showing that PBI with the use of them is able to provide a better approximation of the Pareto front than the original one. The SPS is further integrated into two recently developed MOEA/D variants to help balance the population diversity and convergence. Experimental results show that it can significantly enhance the algorithm’s performance.

An alternative hypervolume-based selection mechanism for multi-objective evolutionary algorithms

In this paper, we are interested in selection mechanisms based on the hypervolume indicator with a particular emphasis on the mechanism used in an improved version of the S metric selection evolutionary multi-objective algorithm (SMS-EMOA) called iSMS-EMOA, which exploits the locality property of the hypervolume. Here, we propose a new selection scheme which approximates the contribution of solutions to the hypervolume and it is designed to preserve the locality property exploited by iSMS-EMOA. This approach is proposed as an alternative to the use of exact hypervolume calculations and is aimed for solving many-objective optimization problems. The proposed approach is called "approximate version of the improved SMS-EMOA (aviSMS-EMOA)" and is validated using standard test problems (with three or more objectives) and performance indicators taken from the specialized literature. Our preliminary results indicate that our proposed approach is a good alternative to solve many-objective optimization problems, if we consider both quality in the solutions and running time required to obtain them because it outperforms two versions of the original SMS-EMOA that approximate the contributions to the hypervolume, it outperforms MOEA/D using penalty boundary intersection and it is competitive with respect to the original SMS-EMOA in several of the test problems adopted. Also, its computational cost is reasonable (it is slower than MOEA/D, but faster than SMS-EMOA).