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Journal Article•DOI•

An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point-Based Nondominated Sorting Approach, Part I: Solving Problems With Box Constraints

Kalyanmoy Deb1, Himanshu Jain2•
Michigan State University1, Indian Institute of Technology Delhi2
01 Aug 2014-IEEE Transactions on Evolutionary Computation (IEEE)-Vol. 18, Iss: 4, pp 577-601
TL;DR: A reference-point-based many-objective evolutionary algorithm that emphasizes population members that are nondominated, yet close to a set of supplied reference points is suggested that is found to produce satisfactory results on all problems considered in this paper.
Abstract: Having developed multiobjective optimization algorithms using evolutionary optimization methods and demonstrated their niche on various practical problems involving mostly two and three objectives, there is now a growing need for developing evolutionary multiobjective optimization (EMO) algorithms for handling many-objective (having four or more objectives) optimization problems. In this paper, we recognize a few recent efforts and discuss a number of viable directions for developing a potential EMO algorithm for solving many-objective optimization problems. Thereafter, we suggest a reference-point-based many-objective evolutionary algorithm following NSGA-II framework (we call it NSGA-III) that emphasizes population members that are nondominated, yet close to a set of supplied reference points. The proposed NSGA-III is applied to a number of many-objective test problems with three to 15 objectives and compared with two versions of a recently suggested EMO algorithm (MOEA/D). While each of the two MOEA/D methods works well on different classes of problems, the proposed NSGA-III is found to produce satisfactory results on all problems considered in this paper. This paper presents results on unconstrained problems, and the sequel paper considers constrained and other specialties in handling many-objective optimization problems.

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Citations
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Journal Article•DOI•
An Evolutionary Many-Objective Optimization Algorithm Using Reference-Point Based Nondominated Sorting Approach, Part II: Handling Constraints and Extending to an Adaptive Approach

[...]

Himanshu Jain1, Kalyanmoy Deb2•
Indian Institute of Technology Delhi1, Michigan State University2
01 Aug 2014-IEEE Transactions on Evolutionary Computation
TL;DR: This paper extends NSGA-III to solve generic constrained many-objective optimization problems and suggests three types of constrained test problems that are scalable to any number of objectives and provide different types of challenges to a many- objective optimizer.
Abstract: In the precursor paper, a many-objective optimization method (NSGA-III), based on the NSGA-II framework, was suggested and applied to a number of unconstrained test and practical problems with box constraints alone. In this paper, we extend NSGA-III to solve generic constrained many-objective optimization problems. In the process, we also suggest three types of constrained test problems that are scalable to any number of objectives and provide different types of challenges to a many-objective optimizer. A previously suggested MOEA/D algorithm is also extended to solve constrained problems. Results using constrained NSGA-III and constrained MOEA/D show an edge of the former, particularly in solving problems with a large number of objectives. Furthermore, the NSGA-III algorithm is made adaptive in updating and including new reference points on the fly. The resulting adaptive NSGA-III is shown to provide a denser representation of the Pareto-optimal front, compared to the original NSGA-III with an identical computational effort. This, and the original NSGA-III paper, together suggest and amply test a viable evolutionary many-objective optimization algorithm for handling constrained and unconstrained problems. These studies should encourage researchers to use and pay further attention in evolutionary many-objective optimization.

1,247 citations

Cites background or methods from "An Evolutionary Many-Objective Opti..."

  • ...We also propose the use of PBI metric (instead of the Tchebycheff metric), as PBI metric was found to work better in the original study [1]....

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  • ...The number of reference points, population size, and other parameters are kept in agreement with the original study [1] and are tabulated in Tables I and II....

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  • ...Another aspect of the extension is that if all population members are feasible or an unconstrained problem is supplied, the constrained NSGA-III reduces to the original unconstrained NSGA-III algorithm [1]....

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  • ...The rest of the NSGA-III procedure described in the original paper [1] remains the same....

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  • ...Notice how the procedure becomes similar to the unconstrained NSGA-III selection operator (described in the original study [1]) when there is no infeasible population member or when there are no equality or inequality constraints specified in the optimization problem formulation....

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Book Chapter•DOI•
Multi-objective Optimization

[...]

Kalyanmoy Deb1•
Michigan State University1
01 Jan 2014
TL;DR: This chapter discusses the fundamental principles of multi-objective optimization, the differences between multi-Objective optimization and single-objectives optimization, and describes a few well-known classical and evolutionary algorithms for multi- objective optimization.
Abstract: Multi-objective optimization is an integral part of optimization activities and has a tremendous practical importance, since almost all real-world optimization problems are ideally suited to be modeled using multiple conflicting objectives. The classical means of solving such problems were primarily focused on scalarizing multiple objectives into a single objective, whereas the evolutionary means have been to solve a multi-objective optimization problem as it is. In this chapter, we discuss the fundamental principles of multi-objective optimization, the differences between multi-objective optimization and single-objective optimization, and describe a few well-known classical and evolutionary algorithms for multi-objective optimization. Two application case studies reveal the importance of multi-objective optimization in practice. A number of research challenges are then highlighted. The chapter concludes by suggesting a few tricks of the trade and mentioning some key resources to the field of multi-objective optimization.

1,072 citations

Journal Article•DOI•
A Reference Vector Guided Evolutionary Algorithm for Many-Objective Optimization

[...]

Ran Cheng1, Yaochu Jin1, Markus Olhofer2, Bernhard Sendhoff2•
University of Surrey1, Honda2
19 Jan 2016-IEEE Transactions on Evolutionary Computation
TL;DR: In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space, and reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization.
Abstract: In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems.

1,020 citations

Cites background or methods from "An Evolutionary Many-Objective Opti..."

  • ...For example, in NSGA-III [39], a set of reference pointsor r ference linesare used for niche preservation to manage diversity in each subspace for many-objective optimizatio n, which effectively enhances convergence by giving priority to solutions closer to the reference points....

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  • ...As already shown in [39], reference points can also be used to generate a subset of preferred Pareto optimal solutions, although NSGA-III can be seen as a decomposition based approach if the reference points are evenly distributed in t he whole objective space....

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  • ...Since weight vectorsare typically used to denote the importance of objectives in weighted aggregation, differe nt t rminologies have been coined in the second type of decomposition based approaches to refer to vectors that decompos e he original objective space, including direction vectors[29], reference lines[39], and reference vectors[40]....

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  • ..., the one adopted in NSGAIII [39], the proposed angle normalization approach has two major differences: (1) normalizing the angles (instead of t he objectives) will not change the actual positions of the cand idate solutions, which is important convergence information for the proposed RVEA; (2) angle normalization, which is independently carried out inside each subspace, does not influence t he distribution of the candidate solutions in other subspaces ....

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  • ...The scaling approach is recommended in [39], where each objective is multiplied by a coefficient pi−1, where p is a parameter that controls the scaling size andi = 1, ....

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Journal Article•DOI•
PlatEMO: A MATLAB Platform for Evolutionary Multi-Objective Optimization [Educational Forum]

[...]

Ye Tian1, Ran Cheng2, Xingyi Zhang1, Yaochu Jin3•
Anhui University1, University of Birmingham2, University of Surrey3
11 Oct 2017-IEEE Computational Intelligence Magazine
TL;DR: PlatEMO as discussed by the authors is a MATLAB platform for evolutionary multi-objective optimization, which includes more than 50 multiobjective evolutionary algorithms and more than 100 multobjective test problems, along with several widely used performance indicators.
Abstract: Over the last three decades, a large number of evolutionary algorithms have been developed for solving multi-objective optimization problems. However, there lacks an upto-date and comprehensive software platform for researchers to properly benchmark existing algorithms and for practitioners to apply selected algorithms to solve their real-world problems. The demand of such a common tool becomes even more urgent, when the source code of many proposed algorithms has not been made publicly available. To address these issues, we have developed a MATLAB platform for evolutionary multi-objective optimization in this paper, called PlatEMO, which includes more than 50 multiobjective evolutionary algorithms and more than 100 multi-objective test problems, along with several widely used performance indicators. With a user-friendly graphical user interface, PlatEMO enables users to easily compare several evolutionary algorithms at one time and collect statistical results in Excel or LaTeX files. More importantly, PlatEMO is completely open source, such that users are able to develop new algorithms on the basis of it. This paper introduces the main features of PlatEMO and illustrates how to use it for performing comparative experiments, embedding new algorithms, creating new test problems, and developing performance indicators. Source code of PlatEMO is now available at: http://bimk.ahu.edu.cn/index.php?s=/Index/Software/index.html.

915 citations

Journal Article•DOI•
An Evolutionary Many-Objective Optimization Algorithm Based on Dominance and Decomposition

[...]

Ke Li1, Kalyanmoy Deb2, Qingfu Zhang1, Sam Kwong1•
City University of Hong Kong1, Michigan State University2
01 Oct 2015-IEEE Transactions on Evolutionary Computation
TL;DR: A unified paradigm, which combines dominance- and decomposition-based approaches, for many-objective optimization, is suggested, which shows highly competitive performance on all the constrained optimization problems.
Abstract: Achieving balance between convergence and diversity is a key issue in evolutionary multiobjective optimization. Most existing methodologies, which have demonstrated their niche on various practical problems involving two and three objectives, face significant challenges in many-objective optimization. This paper suggests a unified paradigm, which combines dominance- and decomposition-based approaches, for many-objective optimization. Our major purpose is to exploit the merits of both dominance- and decomposition-based approaches to balance the convergence and diversity of the evolutionary process. The performance of our proposed method is validated and compared with four state-of-the-art algorithms on a number of unconstrained benchmark problems with up to 15 objectives. Empirical results fully demonstrate the superiority of our proposed method on all considered test instances. In addition, we extend this method to solve constrained problems having a large number of objectives. Compared to two other recently proposed constrained optimizers, our proposed method shows highly competitive performance on all the constrained optimization problems.

900 citations

Cites background or methods from "An Evolutionary Many-Objective Opti..."

  • ...Furthermore, the recently proposed NSGA-III [49] also employs a decomposition-based idea to...

    [...]

  • ...binary crossover (SBX) [61] and polynomial mutation [62] as in [49]....

    [...]

  • ...In this paper, we use the penalty-based boundary intersection (PBI) approach [41], due to its promising performance for many-objective optimization reported in [49]....

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  • ...As discussed in [49], in order to have intermediate weight vectors within the simplex, we should set H ≥ m....

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  • ...Most existing studies of MOEA/D in many-objective scenario mainly concentrate on investigations of its search behavior (see [45]–[49])....

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References
More filters
Journal Article•DOI•
A fast and elitist multiobjective genetic algorithm: NSGA-II

[...]

Kalyanmoy Deb1, Amrit Pratap1, Sameer Agarwal1, T. Meyarivan1•
Indian Institute of Technology Kanpur1
01 Apr 2002-IEEE Transactions on Evolutionary Computation
TL;DR: This paper suggests a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties, and modify the definition of dominance in order to solve constrained multi-objective problems efficiently.
Abstract: Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN/sup 3/) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN/sup 2/) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed.

37,111 citations

"An Evolutionary Many-Objective Opti..." refers background or methods in this paper

  • ...Thereafter, we propose a new method that uses the framework of NSGA-II procedure [5], but works with a set of supplied or predefined reference points and demonstrates its efficacy in solving two-objective to 15-objective optimization problems....

    [...]

  • ...The basic framework of the proposed many-objective NSGA-II (or NSGA-III) is similar to the original NSGA-II algorithm [5] with significant changes in its selection operator....

    [...]

  • ...Second, implementation of a diversity-preservation operator (such as the crowding distance operator [5] or clustering operator [6]) becomes a computationally expensive operation....

    [...]

Book•
Multi-Objective Optimization Using Evolutionary Algorithms

[...]

Kalyanmoy Deb, Deb Kalyanmoy
01 Jan 2001
TL;DR: This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.
Abstract: From the Publisher: Evolutionary algorithms are relatively new, but very powerful techniques used to find solutions to many real-world search and optimization problems. Many of these problems have multiple objectives, which leads to the need to obtain a set of optimal solutions, known as effective solutions. It has been found that using evolutionary algorithms is a highly effective way of finding multiple effective solutions in a single simulation run. · Comprehensive coverage of this growing area of research · Carefully introduces each algorithm with examples and in-depth discussion · Includes many applications to real-world problems, including engineering design and scheduling · Includes discussion of advanced topics and future research · Features exercises and solutions, enabling use as a course text or for self-study · Accessible to those with limited knowledge of classical multi-objective optimization and evolutionary algorithms The integrated presentation of theory, algorithms and examples will benefit those working and researching in the areas of optimization, optimal design and evolutionary computing. This text provides an excellent introduction to the use of evolutionary algorithms in multi-objective optimization, allowing use as a graduate course text or for self-study.

12,134 citations

Journal Article•DOI•
Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach

[...]

Eckart Zitzler1, Lothar Thiele1•
ETH Zurich1
01 Nov 1999-IEEE Transactions on Evolutionary Computation
TL;DR: The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface.
Abstract: Evolutionary algorithms (EAs) are often well-suited for optimization problems involving several, often conflicting objectives. Since 1985, various evolutionary approaches to multiobjective optimization have been developed that are capable of searching for multiple solutions concurrently in a single run. However, the few comparative studies of different methods presented up to now remain mostly qualitative and are often restricted to a few approaches. In this paper, four multiobjective EAs are compared quantitatively where an extended 0/1 knapsack problem is taken as a basis. Furthermore, we introduce a new evolutionary approach to multicriteria optimization, the strength Pareto EA (SPEA), that combines several features of previous multiobjective EAs in a unique manner. It is characterized by (a) storing nondominated solutions externally in a second, continuously updated population, (b) evaluating an individual's fitness dependent on the number of external nondominated points that dominate it, (c) preserving population diversity using the Pareto dominance relationship, and (d) incorporating a clustering procedure in order to reduce the nondominated set without destroying its characteristics. The proof-of-principle results obtained on two artificial problems as well as a larger problem, the synthesis of a digital hardware-software multiprocessor system, suggest that SPEA can be very effective in sampling from along the entire Pareto-optimal front and distributing the generated solutions over the tradeoff surface. Moreover, SPEA clearly outperforms the other four multiobjective EAs on the 0/1 knapsack problem.

7,512 citations

"An Evolutionary Many-Objective Opti..." refers background in this paper

  • ...For this purpose, the performance metrics (such as hyper-volume measure [7] or other metrics [3], [8]) are either computationally too expensive or may not be meaningful....

    [...]

Journal Article•DOI•
MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

[...]

Qingfu Zhang1, Hui Li1•
University of Essex1
01 Dec 2007-IEEE Transactions on Evolutionary Computation
TL;DR: Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjectives optimization problems.
Abstract: Decomposition is a basic strategy in traditional multiobjective optimization. However, it has not yet been widely used in multiobjective evolutionary optimization. This paper proposes a multiobjective evolutionary algorithm based on decomposition (MOEA/D). It decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is optimized by only using information from its several neighboring subproblems, which makes MOEA/D have lower computational complexity at each generation than MOGLS and nondominated sorting genetic algorithm II (NSGA-II). Experimental results have demonstrated that MOEA/D with simple decomposition methods outperforms or performs similarly to MOGLS and NSGA-II on multiobjective 0-1 knapsack problems and continuous multiobjective optimization problems. It has been shown that MOEA/D using objective normalization can deal with disparately-scaled objectives, and MOEA/D with an advanced decomposition method can generate a set of very evenly distributed solutions for 3-objective test instances. The ability of MOEA/D with small population, the scalability and sensitivity of MOEA/D have also been experimentally investigated in this paper.

6,657 citations

"An Evolutionary Many-Objective Opti..." refers background or methods or result in this paper

  • ...Such a distribution was also reported in the original MOEA/D study [10]....

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  • ...MOEA/D [10] uses a predefined set of weight vectors to maintain a diverse set of trade-off solutions....

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  • ...A recently proposed MOEA/D procedure [10] uses this concept....

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  • ...This observation is similar to that concluded in the original MOEA/D study [10] based on two-objective and three-objective problems....

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  • ...We compare the performance of the proposed NSGA-III with two versions of an existing many-objective EMO (MOEA/D [10]), as the method is somewhat similar to the proposed...

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Book•
Nonlinear Multiobjective Optimization

[...]

Kaisa Miettinen
26 Sep 2011
TL;DR: This paper is concerned with the development of methods for dealing with the role of symbols in the interpretation of semantics.
Abstract: Preface. Acknowledgements. Notation and Symbols. Part I: Terminology and Theory. 1. Introduction. 2. Concepts. 3. Theoretical Background. Part II: Methods. 1. Introduction. 2. No-Preference Methods. 3. A Posteriori Methods. 4. A Priori Methods. 5. Interactive Methods. Part III: Related Issues. 1. Comparing Methods. 2. Software. 3. Graphical Illustration. 4. Future Directions. 5. Epilogue. References. Index.

4,976 citations

"An Evolutionary Many-Objective Opti..." refers background in this paper

  • ...The use of fixed conedomination [33], [34] or variable cone-domination [35] principles can also be tried....

    [...]

Related Papers (5)
A fast and elitist multiobjective genetic algorithm: NSGA-II

[...]

01 Apr 2002-IEEE Transactions on Evolutionary Computation
Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, T. Meyarivan 
MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

[...]

01 Dec 2007-IEEE Transactions on Evolutionary Computation
Qingfu Zhang, Hui Li
Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach

[...]

01 Nov 1999-IEEE Transactions on Evolutionary Computation
Eckart Zitzler, Lothar Thiele
Multi-Objective Optimization Using Evolutionary Algorithms

[...]

01 Jan 2001
Kalyanmoy Deb, Deb Kalyanmoy
Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems

[...]

01 Mar 1998-Siam Journal on Optimization
Indraneel Das, John E. Dennis