On the Influence of the Number of Objectives on the Hardness of a Multiobjective Optimization Problem

In this paper, we study the influence of the number of objectives of a continuous multiobjective optimization problem on its hardness for evolution strategies which is of particular interest for many-objective optimization problems. To be more precise, we measure the hardness in terms of the evoluti...

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Bibliographic Details
Published in:IEEE transactions on evolutionary computation Vol. 15; no. 4; pp. 444 - 455
Main Authors: Schutze, O., Lara, A., Coello, C. A. C.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.08.2011
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithm design
Algorithmics. Computability. Computer arithmetics
Applied sciences
Approximation methods
Computational modeling
Computer science; control theory; systems
Computer systems and distributed systems. User interface
Convergence
Decision theory. Utility theory
Descent
Evolution
Evolutionary computation
Exact sciences and technology
Hardness
many-objective optimization
Measurement
multiobjective optimization
Operational research and scientific management
Operational research. Management science
Optimization
Pareto optimality
Remedies
Scalability
Software
Strategy
Studies
Theoretical computing
Cone
Modal analysis
Probabilistic approach
Scalability
Pareto optimum
Evolutionary algorithm
Empirical method
Multiobjective programming
many-objective optimization
Vibration test
Hardness
Convex programming
Optimization
Dynamic test
multiobjective optimization
evolutionary computation
Non differentiable programming
Descent method
Problem solving
Algorithm design
Mathematical programming
ISSN:1089-778X, 1941-0026
Online Access:Get full text
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Summary:In this paper, we study the influence of the number of objectives of a continuous multiobjective optimization problem on its hardness for evolution strategies which is of particular interest for many-objective optimization problems. To be more precise, we measure the hardness in terms of the evolution (or convergence) of the population toward the set of interest, the Pareto set. Previous related studies consider mainly the number of nondominated individuals within a population which greatly improved the understanding of the problem and has led to possible remedies. However, in certain cases this ansatz is not sophisticated enough to understand all phenomena, and can even be misleading. In this paper, we suggest alternatively to consider the probability to improve the situation of the population which can, to a certain extent, be measured by the sizes of the descent cones. As an example, we make some qualitative considerations on a general class of uni-modal test problems and conjecture that these problems get harder by adding an objective, but that this difference is practically not significant, and we support this by some empirical studies. Further, we address the scalability in the number of objectives observed in the literature. That is, we try to extract the challenges for the treatment of many-objective problems for evolution strategies based on our observations and use them to explain recent advances in this field.
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ISSN:1089-778X
1941-0026
DOI:10.1109/TEVC.2010.2064321