Laboratory of Information Processing

Jouni Lampinen

Multiobjective Nonlinear Pareto-Optimization

A Pre-Investigation Report

LAPPEENRANTA 2000

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Contents

1 Introduction 2 Major Information Sources 2.1 2.2 2.3 2.4 2.5 Literature surveys, reviews Bibliographies Thesis works Books Some significant articles 13 15 17 18 23 25 27 2 9

3. Basic Problem Statements 4. Classification for Multiobjective Optimization Approaches 5. Usage of Weighted Objective Functions 6. Pareto Optimization – Definitions 7. Evaluation of Multiobjective Evolutionary Algorithms 8. Concluding Remarks References

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1 Introduction

This report surveys briefly multiobjective nonlinear Pareto-optimization. The focus is on the usage of evolutionary algorithms for multiobjective decision making. The term evolutionary algorithms refer to a class of computationally intelligent algorithms, which are based on an artificial model of evolution in nature. When considering multiobjective optimization problems, the most frequently applied evolutionary optimization algorithms are genetic algorithms [Gol90] and evolution strategies [Schw95]. The purpose of this report is to serve as an initial investigation for establishing a future research project. The weight is on literature survey, finding the most important previous works, explaining the problem, methodology, terminology, classifications, concepts etc. In a nutshell, the objective is a brief up-to-date presentation of Status Quo, including the recent advances. Practical optimization problems, especially the engineering design optimization problems, seem to have a multiobjective nature much more frequently than a single objective one. Typically, some structural performance criteria are to be maximized, while the weight of the structure and the implementation costs should be minimized simultaneously. For example, consider designing the appearance of a wing for an advanced military aircraft. The aerodynamic lift of the wing should be maximized, while simultaneously, the aerodynamic drag is to be minimized. In addition, the electromagnetic backscattering of the wing should be minimized in order to prevent detecting the military aircraft easily by radar. Furthermore, the weight of the wing should be minimized in order to maximize the weaponry payload. The production costs should be minimized, too. Unfortunately, these five objectives are all conflicting with each other. Virtually, there exist an infinite number of similar problems. In practice, real-world decision making problems with only one objective are rare. Despite of that, solving single objective optimization problems is far more common than solving multiobjective problems, since there appears to be no generally effective and efficient method available for solving multiobjective problems directly as they are. Typically a multiobjective problem is to be effectively converted to a single objective problem before applying an optimization algorithm. This conversion can be done easily by first deciding the relative importance for each objective a priori. Then, for example, the decision-maker may combine the individual objective functions into a scalar cost function (linear or nonlinear combination), which effectively converts a multiobjective problem into a single objective one. Anyway, single objective problems are only a subclass of multiobjective problems. Thus finding a method for solving the multiobjective problems as multiobjective problems, without any a priori preference decisions, and without first converting the problem into a single objective one, is one of the most important optimization research objectives at the moment. Justifying more practically, the decision-makers (having multiple objectives), are willing to perform unbiased searches in general. They are often unwilling or unable to assign priorities without having further information about the other potential/effective solutions....