Decision Support
A multiobjective evolutionary algorithm for approximating the efficient set

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Abstract

In this article, a new framework for evolutionary algorithms for approximating the efficient set of a multiobjective optimization (MOO) problem with continuous variables is presented. The algorithm is based on populations of variable size and exploits new elite preserving rules for selecting alternatives generated by mutation and recombination. Together with additional assumptions on the considered MOO problem and further specifications on the algorithm, theoretical results on the approximation quality such as convergence in probability and almost sure convergence are derived.

Introduction

For multiobjective decision making (MODM) problems (see, e.g., Hanne, 2001a, Steuer, 1986, Vincke, 1992, Zeleny, 1982), a significant number of algorithms based on evolutionary approaches has been proposed during the last 15 years. Today, there are various survey articles of this research field available (see Fonseca and Fleming, 1995, Horn, 1997, Tamaki et al., 1996), specialized international conferences on evolutionary multi-criterion optimization take place (see the proceedings edited by Zitzler et al., 2001, Fonseca et al., 2003), and comprehensive monographs have been published (see Coello Coello et al., 2002, Deb, 2001). Theoretical results on evolutionary algorithms for multiobjective optimization such as, for instance, approximation proofs are, however, scarcely available. In this paper we introduce a new framework for evolutionary multiobjective algorithms which allows for an analysis of approximation. Approximation, in that sense, means that, providing that sufficient computation time and memory is available, the algorithm is capable of reaching with an arbitrary exactness those alternatives which constitute in a mathematical sense the solution set which is usually denoted as the set of efficient or Pareto-optimal alternatives. The subsequent paper is organized as follows: In the next section some notation on multiobjective optimization problems is specified. In Section 3, an algorithm (or: algorithmic framework) capable of approximating the efficient set is introduced. In Section 4, further theoretical concepts for an analysis of approximation and additional assumptions are introduced and used for a corresponding proof. In Section 5, some conclusions are given.

Section snippets

Some notations

Originally, evolutionary algorithms (EAs) have been developed for scalar, or ordinary, optimization problems (see, e.g., Bäck et al., 1991, Bäck et al., 1997), i.e. problems with a mathematical formulation as follows:maxf(x)s.t.xAwith A  Rn called the feasible set and f : Rn  R being the objective function to be maximized. Instead of maximization a minimization can be assumed as well. The feasible set is usually defined by constraint functions,A={xRn:gj(x)0,j{1,,m}}.Elements a  A are usually

General framework

In the following, we sketch a new framework for multiobjective evolutionary algorithms suitable for approximating the efficient set of a multiobjective optimization (MOO) problem (2.4), (2.5).

The basic idea of EAs is that from a set (population) of intermediary solutions (feasible alternatives) a subsequent set of solutions is generated by imitating concepts of natural evolution such as mutation, recombination, and selection. From a set of ‘parents’, Mt, in generation t  N, a set of ‘offspring’,

Theoretical results on approximation

In Hanne (1999) we have introduced a new concept of ε-efficient solutions used for analyzing the approximation features of our algorithm. This concept is based on ε-neighborhoods and does not require scalarization as some other definitions of ε-efficient solutions do (cf. Helbig and Pateva, 1994). Quite frequently (see, e.g. Laumanns et al., 2002, Ehrgott and Gandibleux, 2004) ε-efficiency is defined component-wise for the objective functions which can be considered as using a specific metrics

Conclusions

In this article we have presented a new framework for multiobjective evolutionary algorithms for approximating the efficient set, denoted as AMOEA. Roughly speaking, this is an extension of our earlier MOEA with a variable and usually increasing population size such that enough alternatives can be generated and kept in population for representing the efficient set with a given exactness ε. The general framework of the algorithm is presented and implementation details of the single steps are

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