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Very often real-world applications have several multiple conflicting objectives. Recently there has been a growing interest in evolutionary multiobjective optimization algorithms that combine two major disciplines: evolutionary computation and the theoretical frameworks of multicriteria decision making. In this introductory chapter, some fundamental concepts of multiobjective optimization are introduced, emphasizing the motivation and advantages of using evolutionary algorithms. We then lay out the important contributions of the remaining chapters of this volume.

This chapter presents a brief review of some of the most relevant research currently taking place in evolutionary multiobjective optimization. The main topics covered include algorithms, applications, metrics, test functions, and theory. Some of the most promising future paths of research are also addressed.

This chapter investigates the convergence behavior of simple evolutionary algorithms with different selection strategies on a continuous multiobjective model problem. Special focus is given to the problem of controlling the mutation strength, since an adaptation of the mutation strength is necessary to converge to the optimum with arbitrary precision, and to achieve linear convergence order. Adaptive parameter control represents a major research topic in the field of evolutionary computation, and several methods have been proposed and applied successfully for single-objective optimization problems. We demonstrate that the convergence properties achieved by a self-adaptation of the mutation strength on single-objective problems do not carry over to the multiobjective case, if a simple dominance-based selection scheme is used. As a solution, a combined strategy is proposed using dominance-based selection in the archive and scalarizing functions in the working population.

This chapter describes a Pareto-based approach to evolutionary multiobjective optimization, that avoids most of the time-consuming global calculations typical of other multi-objective evolutionary techniques. The new approach uses a simple uniform selection strategy within a steady-state evolutionary algorithm (EA) and employs a straightforward elitist mechanism for replacing population members with their offspring. Global calculations for fitness and Pareto dominance are not needed. Other state-of-the-art Pareto-based EAs depend heavily on various fitness functions and niche evaluations, mostly based on Pareto dominance, and the calculations involved tend to be rather time consuming (at least O ( N ^2) for a population size, N ). The new approach has performed well on some benchmark combinatorial problems and continuous functions, outperforming the latest state-of-the-art EAs in several cases. In this chapter the new approach will be explained in detail.

In multiobjective evolutionary algorithms (MOEAs) with elitism, the data structures for storing and updating archives may have a great impact on the required computational (CPU) time, especially when optimizing higher-dimensional problems with large Pareto sets. In this chapter, we introduce Quad-trees as an alternative data structure to linear lists for storing Pareto sets. In particular, we investigate several variants of Quad-trees and compare them with conventional linear lists. We also study the influence of population size and number of objectives on the required CPU time. These data structures are evaluated and compared on several multiobjective example problems. The results presented show that typically, linear lists perform better for small population sizes and higher-dimensional Pareto fronts (large archives) whereas Quad-trees perform better for larger population sizes and Pareto sets of small cardinality.

After adequately demonstrating the ability to solve different two-objective optimization problems, multiobjective evolutionary algorithms (MOEAs) must demonstrate their efficacy in handling problems having more than two objectives. In this study, we have suggested three different approaches for systematically designing test problems for this purpose. The simplicity of construction, scalability to any number of decision variables and objectives, knowledge of the shape and the location of the resulting Pareto-optimal front, and introduction of controlled difficulties in both converging to the true Pareto-optimal front and maintaining a widely distributed set of solutions are the main features of the suggested test problems. Because of the above features, they should be found useful in various research activities on MOEAs, such as testing the performance of a new MOEA, comparing different MOEAs, and better understanding of the working principles of MOEAs.

This chapter presents a synergistic combination of particle swarm optimization and evolutionary algorithm for optimization problems. The performance of the hybrid algorithm is bench-marked against conventional genetic algorithm and particle swarm optimization algorithm. Finally, the hybrid algorithm is illustrated as a multiobjective optimization algorithm using the Fonseca 2-objective function.

In this chapter we propose a new evolutionary elitist approach combining a non-standard solution representation and an evolutionary optimization technique. The proposed method permits detection of continuous decision regions. In our approach an individual (a solution) is either a closed interval or a point. The individuals in the final population give a realistic representation of the Pareto-optimal set. Each solution in this population corresponds to a decision region of the Pareto-optimal set. The proposed technique is an elitist one. It uses a unique population. The current population contains non-dominated solutions already computed.

This chapter proposes a Multi-Objective Genetic Algorithm with Distributed Environment Scheme (MOGADES). The performance of MAGADES is compared with SPEA2 and NSGA-II. Further a Distributed Cooperation Model of Multi-Objective Genetic Algorithm (DEMOGA) is introduced. The effectiveness of DCMOGA is illustrated by comparing with SPEA2 and other multiobjective optimization algorithms. Finally MOGADES and DCMOGA are combined into a hybrid algorithm called Distributed Cooperation Model of Multi-Objective Genetic Algorithm with Environmental Scheme (DCMOGADES) and applied to some test problems. Performance of MOGADES is also illustrated by applying to some real world problems.

This chapter describes the general multiobjective optimization concepts that can and have been used to incorporate constraints of any type (linear, nonlinear, equality and inequality) into the fitness function of a genetic algorithm used for global optimization. Several approaches reported in the literature are also described and four of them are compared using several test functions. The results obtained are discussed and further ideas about how to devise new approaches are also briefly analyzed.