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Quality of Pareto set approximations

This chapter explores the quality of Pareto set approximations in multi-objective optimization. It discusses the motivation behind comparing algorithms and the use of unary and binary measures to evaluate the performance of these approximations. The chapter also includes examples of unary and binary measures and concludes with a discussion on the evaluation of algorithms and the importance of considering preference information.

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Quality of Pareto set approximations

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  1. Quality of Pareto set approximations Eckart Zitzler, Jörg Fliege, Carlos Fonseca, Christian Igel, Andrzej Jaszkiewicz, Joshua Knowles, Alexander Lotov, Serpil Sayin, Andrzej Wierzbicki 4.5 EMO, 4.5 non-EMO

  2. Lessons learned • This chapter is a must • Obvious things are not obvious

  3. Outline • Motivation • Use cases • Testing (true Pareto front is known) • Practice (true Pareto front is not known) • Comparing approximations • Unary measures • Binary measures • Discussion and conclusions

  4. Motivation • Comparison of algorithms • Pictures are valuable but not sufficient • Design of algorithms for vector optimization • To guide the search • Stopping criteria • Learning

  5. Use cases • Testing • True Pareto front is known • Practice • True Pareto front not known in general • Additional useful information: • Lower bounds through relaxation • Upper bounds through random search or other methods • … • Pairwise comparison

  6. Comparing approximations • Unary measures • Evaluation of a single approximation • Binary relations • Purely ordinal • Binary measures • Evaluation of difference between two approximations

  7. Unary measures • Assign a real number to a Pareto set approximation • Relevant properties • Monotonicity (strict or not) • Uniqueness of optimum • Scale invariance • Computational requirements • Required information, e.g. reference set, bounds

  8. Examples of unary measures • Outer diameter • Proportion of Pareto-optimal points found • Cardinality • Hypervolume • ε-dominance • D-measures • D1 - mean value of the Chenycheff distance from the Pareto front • D2 – asymmetric Hausdorff distance with Chebycheff metric • R-measure - expected value of Chebycheff scalarizing function • Uniformity measures • Probability of improvement through random search • Combinations of the above measures

  9. Binary measures • Relevant properties • Monotonicity (strict or not) • Scale invariance • Computational requirements • Required information, e.g. bounds • Transformation of measures into relations • Partial orders stronger than dominance may be constructed • Implicit introduction of preferences

  10. Examples of binary measures • Difference of two unary measure values • Pareto dominance of sets • Binary ε-dominance • Hypervolume of difference between two Edgeworth-Pareto hulls • Coverage

  11. Discussion and conclusions • Evaluation of algorithms – not discussed here • Taking into account preference information - not discussed here • Monotonicity is important in theoretical sense • In practice it may be relaxed • Learning • Practical optimization algorithm may benefit from this relaxation, like constraint relaxation in numerical optimization • Computational complexity also influences the choice of performance indicator(s)

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