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

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

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.

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Multi-Objective Optimization Using Evolutionary Algorithms

Algorithms are important tools for solving problems computationally. All computation involves algorithms, and the efficiency of an algorithm largely determines its usefulness. This chapter provides an overview of the fundamentals of algorithms and their links to self-organization, exploration, and exploitation. A brief history of recent nature-inspired algorithms for optimization is outlined in this chapter.

Introduction to Algorithms

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.

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MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition

We introduce a simple evolution scheme for multiobjective optimization problems, called the Pareto Archived Evolution Strategy (PAES) We argue that PAES may represent the simplest possible nontrivial algorithm capable of generating diverse solutions in the Pareto optimal set The algorithm, in its simplest form, is a (1 + 1) evolution strategy employing local search but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors (1 + 1)-PAES is intended to be a baseline approach against which more involved methods may be compared It may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods We introduce (1 + λ) and (μ | λ) variants of PAES as extensions to the basic algorithm Six variants of PAES are compared to variants of the Niched Pareto Genetic Algorithm and the Nondominated Sorting Genetic Algorithm over a diverse suite of six test functions Results are analyzed and presented using techniques that reduce the attainment surfaces generated from several optimization runs into a set of univariate distributions This allows standard statistical analysis to be carried out for comparative purposes Our results provide strong evidence that PAES performs consistently well on a range of multiobjective optimization tasks

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Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy

Optimization is a pivotal aspect of software design. The techniques treated in this text represent research as elucidated by the leaders in the field. The optimization methods are applied to real problems, such as hilllclimbing, simulated annealing, and tabu search.

New Ideas In Optimization

Most popular evolutionary algorithms for multiobjective optimisation maintain a population of solutions from which individuals are selected for reproduction. In this paper, we introduce a simpler evolution scheme for multiobjective problems, called the Pareto archived evolution strategy (PAES). We argue that PAES may represent the simplest possible non-trivial algorithm capable of generating diverse solutions in the Pareto optimal set. The algorithm is identified as being a (1+1) evolution strategy, using local search from a population of one but using a reference archive of previously found solutions in order to identify the approximate dominance ranking of the current and candidate solution vectors. PAES is intended as a good baseline approach, against which more involved methods may be compared, and may also serve well in some real-world applications when local search seems superior to or competitive with population-based methods. The performance of the new algorithm is compared with that of a MOEA based on the niched Pareto GA on a real world application from the telecommunications field. In addition, we include results from experiments carried out on a suite of four test functions, to demonstrate the algorithm's general capability.

The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation

We describe a new selection technique for evolutionary multiobjective optimization algorithms in which the unit of selection is a hyperbox in objective space. In this technique, instead of assigning a selective fitness to an individual, selective fitness is assigned to the hyperboxes in objective space which are currently occupied by at least one individual in the current approximation to the Pareto frontier. A hyperbox is thereby selected, and the resulting selected individual is randomly chosen from this hyperbox. This method of selection is shown to be more sensitive to ensuring a good spread of development along the Pareto frontier than individual-based selection. The method is implemented in a modern multiobjective evolutionary algorithm, and performance is tested by using Deb's test suite of `T' functions with varying properties. The new selection technique is found to give significantly superior results to the other methods compared, namely PAES, PESA, and SPEA; each is a modern multi-objective optimization algorithm previously found to outperform earlier approaches on various problems.

PESA-II: region-based selection in evolutionary multiobjective optimization

We introduce a new multiobjective evolutionary algorithm called PESA (the Pareto Envelope-based Selection Algorithm), in which selection and diversity maintenance are controlled via a simple hyper-grid based scheme. PESA's selection method is relatively unusual in comparison with current well known multiobjective evolutionary algorithms, which tend to use counts based on the degree to which solutions dominate others in the population. The diversity maintenance method is similar to that used by certain other methods. The main attraction of PESA is the integration of selection and diversity maintenance, whereby essentially the same technique is used for both tasks. The resulting algorithm is simple to describe, with full pseudocode provided here and real code available from the authors. We compare PESA with two recent strong-performing MOEAs on some multiobjective test problems recently proposed by Deb. We find that PESA emerges as the best method overall on these problems.

David Corne

Papers

Approximating the Nondominated Front Using the Pareto Archived Evolution Strategy

New Ideas In Optimization

The Pareto archived evolution strategy: a new baseline algorithm for Pareto multiobjective optimisation

PESA-II: region-based selection in evolutionary multiobjective optimization

The Pareto Envelope-Based Selection Algorithm for Multi-objective Optimisation