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.

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A fast and elitist multiobjective genetic algorithm: NSGA-II

List of Figures. List of Tables. Preface. Foreword. 1. Basic Concepts. 2. Evolutionary Algorithm MOP Approaches. 3. MOEA Test Suites. 4. MOEA Testing and Analysis. 5. MOEA Theory and Issues. 3. MOEA Theoretical Issues. 6. Applications. 7. MOEA Parallelization. 8. Multi-Criteria Decision Making. 9. Special Topics. 10. Epilog. Appendix A: MOEA Classification and Technique Analysis. Appendix B: MOPs in the Literature. Appendix C: Ptrue & PFtrue for Selected Numeric MOPs. Appendix D: Ptrue & PFtrue for Side-Constrained MOPs. Appendix E: MOEA Software Availability. Appendix F: MOEA-Related Information. Index. References.

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Evolutionary algorithms for solving multi-objective problems

The Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele 1999) is a relatively recent technique for finding or approximating the Pareto-optimal set for multiobjective optimization problems. In different studies (Zitzler and Thiele 1999; Zitzler, Deb, and Thiele 2000) 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, e.g., (Corne, Knowles, and Oates 2000). Furthermore, it has been used in different applications, e.g., (Lahanas, Milickovic, Baltas, and Zamboglou 2001). In this paper, an improved version, namely SPEA2, is proposed, which incorporates in contrast to its predecessor a fine-grained fitness assignment strategy, a density estimation technique, and an enhanced archive truncation method. The comparison of SPEA2 with SPEA and two other modern elitist methods, PESA and NSGA-II, on different test problems yields promising results.

SPEA2: Improving the strength pareto evolutionary algorithm

Introduction to Optimization The Binary Genetic Algorithm The Continuous Parameter Genetic Algorithm Applications An Added Level of Sophistication Advanced Applications Evolutionary Trends Appendix Glossary Index.

https://cdn.preterhuman.net/texts/science_and_technology/nature_and_biology/Bioinformatics/Practical%20Genetic%20Algorithms%20-%20Randy%20L.%20Haupt,%20Sue%20Ellen%20Haupt.pdf

Practical Genetic Algorithms

An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques.

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Performance assessment of multiobjective optimizers: an analysis and review

Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN3) computational complexity (where M is the number of objectives and N is the population size), (ii) non-elitism approach, and (iii) the need for specifying a sharing parameter. In this paper, we suggest a non-dominated sorting based multi-objective evolutionary algorithm (we called it the Non-dominated Sorting GA-II or NSGA-II) which alleviates all the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN2) computational complexity is presented. Second, a selection operator is presented which creates a mating pool by combining the parent and child populations and selecting the best (with respect to fitness and spread) N solutions. Simulation results on five difficult test problems show that the proposed NSGA-II, in most problems, is able to find much better spread of solutions and better convergence near the true Pareto-optimal front compared to PAES and SPEA--two other elitist multi-objective EAs which pay special attention towards creating a diverse Pareto-optimal front. Because of NSGA-II's low computational requirements, elitist approach, and parameter-less sharing approach, NSGA-II should find increasing applications in the years to come.

/pdf/a-fast-elitist-non-dominated-sorting-genetic-algorithm-for-3htzhb12sx.pdf

A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II

Genetic algorithms (GAs) are multi-dimensional and stochastic search methods, involving complex interactions among their parameters For last two decades, researchers have been trying to understand the mechanics of GA parameter interactions by using various techniques The methods include careful 'functional' decomposition of parameter interactions, empirical studies, and Markov chain analysis Although the complex knot of these interactions are getting loose with such analyses, it still remains an open question in the mind of a new-comer to the field or to a GA-practitioner as to what values of GA parameters (such as population size, choice of GA operators, operator probabilities, and others) to use in an arbitrary problem In this paper, we investigate the performance of simple tripartite GAs on a number of simple to complex test problems from a practical standpoint Since function evaluations are most time-consuming in a real-world problem, we compare different GAs for a fixed number of function evaluations Based on probability calculations and simulation results, it is observed that for solving simple problems (unimodal or small modality problems) mutation operator plays an important role, although crossover operator can also solve these problems However, two operators (when applied alone) have two different working zones for population size For complex problems involving massive multimodality and misleadingness (deception), crossover operator is the key search operator and performs reliably with an adequate population size Based on these studies, it is recommended that when in doubt, the use of the crossover operator with an adequate population size is a reliable approach

https://www.egr.msu.edu/~kdeb/papers/k99003.pdf

Understanding Interactions among Genetic Algorithm Parameters.

Most applications of genetic algorithms (GAs) in handling constraints use a straightforward penalty function method. Such techniques involve penalty parameters which must be set right in order for GAs to work. Although many researchers use adaptive variation of penalty parameters and penalty functions, the general conclusion is that these variations are specific to a problem and cannot be generalized. In this paper, we propose a niched-penalty approach which does not require any penalty parameter. The penalty function creates a selective pressure towards the feasible region and a niching maintains diversity among feasible solutions for the genetic recombination operator to find new feasible solutions. The approach is only ap plicable to population-based approaches, thereby giving GAs (or other evolutionary algorithms) a niche in exploiting this penalty-parameter-less penalty approach.Simulation results on a number of constrained optimization problems suggest the efficacy of the proposed method.

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Samir Agrawal

Papers

A Fast Elitist Non-dominated Sorting Genetic Algorithm for Multi-objective Optimisation: NSGA-II

Understanding Interactions among Genetic Algorithm Parameters.

A Niched-Penalty Approach for Constraint Handling in Genetic Algorithms