Multi-objective evolutionary algorithms which use non-dominated sorting and sharing have been mainly criticized for their (i) O(MN 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(MN ) 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 a number of 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. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems eciently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint non-linear problem, are compared with another constrained multi-objective optimizer and much better performance of NSGA-II is observed. Because of NSGA-II's low computational requirements, elitist approach, parameter-less niching approach, and simple constraint-handling strategy, NSGA-II should find increasing applications in the coming years.

This paper provides a comprehensive survey of the most popular constraint-handling techniques currently used with evolutionary algorithms. We review approaches that go from simple variations of a penalty function, to others, more sophisticated, that are biologically inspired on emulations of the immune system, culture or ant colonies. Besides describing briefly each of these approaches (or groups of techniques), we provide some criticism regarding their highlights and drawbacks. A small comparative study is also conducted, in order to assess the performance of several penalty-based approaches with respect to a dominance-based technique proposed by the author, and with respect to some mathematical programming approaches. Finally, we provide some guidelines regarding how to select the most appropriate constraint-handling technique for a certain application, ad we conclude with some of the the most promising paths of future research in this area.

This paper focuses on the issue of evaluation of constraints handling methods, as the advantages and disadvantages of various methods are not well understood. The general way of dealing with constraints -- whatever the optimization method -- is by penalizing infeasible points. However, there are no guidelines on designing penalty functions. Some suggestions for evolutionary algorithms are given in [37], but they do not generalize. Other techniques that can be used to handle constraints in are more or less problem dependent. For instance, the knowledge about linear constraints can be incorporated into specific operators [24], or a repair operator can be designed that projects infeasible points onto feasible ones [30]

this paper, we briefly outline the principles of multi-objective optimization. Thereafter, we discuss why classical search and optimization methods are not adequate for multi-criterion optimization by discussing the working of two popular methods. We then outline several evolutionary methods for handling multi-criterion optimization problems. Of them, we discuss one implementation (non-dominated sorting GA or NSGA [38]) in somewhat greater details. Thereafter, we demonstrate the working of the evolutionary methods by applying NSGA to three test problems having constraints and discontinuous Pareto-optimal region. We also show the efficacy of evolutionary algorithms in engineering design problems by solving a welded beam design problem. The results show that evolutionary methods can find widely different yet near-Pareto-optimal solutions in such problems. Based on the above studies, this paper also suggests a number of immediate future studies which would make this emerging field more mature and applicable in practice. 1.2 PRINCIPLES OF MULTI-CRITERION OPTIMIZATION

This tutorial will review some of the basic concepts related to evolutionary multiobjective optimization (i.e., the use of evolutionary algorithms to handle more than one objective function at a time). The most commonly used evolutionary multiobjective optimization techniques will be described and criticized, including some of their applications. Theory, test functions and metrics will be also discussed. Finally, we will provide some possible paths of future research in this area.

Despite the extended applicability of evolutionary algorithms to a wide range of domains, the fact that these algorithms are unconstrained optimization techniques leaves open the issue regarding how to incorporate constraints of any kind (linear, non-linear, equality and inequality) into the fitness function as to search efficiently. The main goal of this paper is to provide a detailed and comprehensive survey of the many constraint handling approaches that have been proposed for evolutionary algorithms, analyzing in each case their advantages and disadvantages, and concluding with some of the most promising paths of research.

A hybrid particle swarm with differential evolution operator, termed DEPSO, which provide the bell-shaped mutations with consensus on the population diversity along with the evolution, while keeps the self-organized particle swarm dynamics, is proposed. Then it is applied to a set of benchmark functions, and the experimental results illustrate its efficiency.

Evolutionary computation is emerging as a new engineering computational paradigm, which may significantly change the present structural design practice. For this reason, an extensive study of evolutionary computation in the context of structural design has been conducted in the Information Technology and Engineering School at George Mason University and its results are reported here. First, a general introduction to evolutionary computation is presented and recent developments in this field are briefly described. Next, the field of evolutionary design is introduced and its relevance to structural design is explained. Further, the issue of creativity/novelty is discussed and possible ways of achieving it during a structural design process are suggested. Current research progress in building engineering systems ’ representations, one of the key issues in evolutionary design, is subsequently discussed. Next, recent developments in constraint-handling methods in evolutionary optimization are reported. Further, the rapidly growing field of evolutionary multiobjective optimization is presented and briefly described. An emerging subfield of coevolutionary design is subsequently introduced and its current advancements reported. Next, a comprehensive review of the applications of evolutionary computation in structural design is provided and chronologically classified. Finally, a summary of the current research status and a discussion on the most promising paths of future research are also presented.

In this paper, we introduce the concept of non-dominance (commonly used in multiobjective optimization) as a way to incorporate constraints into the fitness function of a genetic algorithm. Each individual is assigned a rank based on its degree of dominance over the rest of the population. Feasible individuals are always ranked higher than infeasible ones, and the degree of constraint violation determines the rank among infeasible individuals. The proposed technique does not require fine tuning of factors like the traditional penalty function and uses a self-adaptation mechanism that avoids the traditional empirical adjustment of the main genetic operators (i.e., crossover and mutation). Keywords: genetic algorithms, constraint handling, multiobjective optimization, self-adaptation, evolutionary optimization, numerical optimization. 1 Introduction Despite the wide success of genetic algorithms (GAs) in a wide range of applications [25, 3, 36, 34], their use in constrained optimizati...

Goal programming is a technique often used in engineering design activities primarily to find a compromised solution which will simultaneously satisfy a number of design goals. In solving goal programming problems, classical methods reduce the multiple goal-attainment problem into a single objective of minimizing a weighted sum of deviations from goals. Moreover, in tackling non-linear goal programming problems, classical methods use successive linearization techniques, which are sensitive to the chosen starting solution. In this paper, we pose the goal programming problem as a multi-objective optimization problem of minimizing deviations from individual goals. This procedure eliminates the need of having extra constraints needed with classical formulations and also eliminates the need of any user-defined weight factor for each goal. The proposed technique can also solve goal programming problems having nonconvex trade-off region, which are difficult to solve using classical methods. T...