Many real-world search and optimization problems involve inequality and/or equality constraints and are thus posed as constrained optimization problems. In trying to solve constrained optimization problems using genetic algorithms (GAs) or classical optimization methods, penalty function methods have been the most popular approach, because of their simplicity and ease of implementation. However, since the penalty function approach is generic and applicable to any type of constraint (linear or nonlinear), their performance is not always satisfactory. Thus, researchers have developed sophisticated penalty functions specific to the problem at hand and the search algorithm used for optimization. However, the most difficult aspect of the penalty function approach is to find appropriate penalty parameters needed to guide the search towards the constrained optimum. In this paper, GA's population-based approach and ability to make pair-wise comparison in tournament selection operator are explo...

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.

Penalty functions are often used in constrained optimization. However, it is very difficult to strike the right balance between objective and penalty functions. This paper introduces a novel approach to balance objective and penalty functions stochastically, i.e., stochastic ranking, and presents a new view on penalty function methods in terms of the dominance of penalty and objective functions. Some of the pitfalls of naive penalty methods are discussed in these terms. The new ranking method is tested using a (µ, ) evolution strategy on 13 benchmark problems. Our results show that suitable ranking alone (i.e., selection), without the introduction of complicated and specialized variation operators, is capable of improving the search performance significantly.

One of the major components of any evolutionary system is the eval� uation function. Evaluation functions are used to assign a quality measure for individuals in a population. Whereas evolutionary com� putation techniques assume the existence of an �e�cient � evaluation function for feasible individuals � there is no uniform methodology for handling �i.e. � evaluating � unfeasible ones. The simplest approach� incorporated by evolution strategies and a version of evolutionary programming �for numerical optimization problems� � is to reject un� feasible solutions. But several other methods for handling unfeasible individuals have emerged recently. This paper reviews such methods �using a domain of nonlinear programming problems � and discusses their merits and drawbacks. 1

During the last two years several methods have been proposed for handling nonlinear constraints by genetic algorithms for numerical optimization problems; most of them were based on penalty functions. However, the performance of these methods is highly problem-dependent; moreover, many methods require additional tunning of several parameters. In this paper we present a new optimization system (Genocop III), which is based on concepts of co-evolution and repair algorithms. We present the resulfs of the system on a few selected test problems and discuss some directions for furlher research.

During the last two years several methods have been proposed for handling constraints by genetic algorithms for numerical optimization problems. In this paper we review these methods, test them on five selected problems, and discuss their strengths and weaknesses.

During the last ve years, several methods have been proposed for handling nonlinear constraints by evolutionary algorithms (EAs) for numerical optimization problems. Recent survey papers classify them into four categories (preservation of feasibility, penalty functions, searching for feasibility, and other hybrids).

Abstract � Adaptation of parameters and operators is one of the most important and promising areas of research in evolutionary computation � it tunes the algorithm to the problem while solving the problem. In this paper we develop a classi�cation of adaptation on the basis of the mechanisms used � and the level at which adaptation operates within the evolutionary algorithm. The classi�cation covers all forms of adaptation in evolutionary computation and suggests fur� ther research. I.

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 presents a new approach to handle constraints using evolutionary algorithms. The new technique treats constraints as objectives, and uses a multiobjective optimization approach to solve the re-stated single-objective optimization problem. The new approach is compared against other numerical and evolutionary optimization techniques in several engineering optimization problems with different kinds of constraints. The results obtained show that the new approach can consistently outperform the other techniques using relatively small sub-populations, and without a significant sacrifice in terms of performance.