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Multiobjective Optimization, Autumn 2005: Michael Emmerich 1

Historical milestones in MCDMand LIACS Projects

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 2

Georg Cantor and Felix Haussdorff

• The origins of the mathematical foundations of multiobjective optimization can be traced back to the period that goes from 1895 to 1906. During that period, Georg Cantor and Felix Hausdorff laid the foundations of infinite dimensional ordered spaces.

• Haussdorff gave the first example for complete orderings

• Cantor introduced equivalence classes and gave the first example of a utility function

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 3

Francis Ysidro Edgeworth and Wilfredo Pareto

Having several objective functions, the notion of “optimum” changes, because in MOPs, we are really trying to find good compromises (or “trade-offs”) rather than a single solution as in global optimization. The notion of “optimum” that is most commonly adopted is that originally proposed by Francis Ysidro Edgeworth in 1881.

This notion was later generalized by Vilfredo Pareto (in 1896). Although some authors call Edgeworth-Pareto optimum to this notion, the most commonly accepted term remains Pareto optimum.

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 4

Albert W.Tucker Leonid Hurwitz

• Nevertheless, multiobjective optimization theory remained relatively undeveloped during the 1950s. It was until the 1960s that the foundations of multiobjective optimization were consolidated and taken seriously by pure mathematicians when Leonid Hurwicz generalized the results of Kuhn & Tucker to topological vector spaces.

• Albert Tucker was the first who systematically worked on vector optimization

• Kuhn and Tucker analyzed local Pareto optimality and stated conditions for local optimal based on differential geometry of the vector valued function

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 5

John Nash/ John von Neumann

Foundations of game theory: Many players in conflicting and cooperative gamesGame theory is today an important field that is closely related multiobjective optimization, as interests of different players can be viewed as multiple objectives.

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 6

TC. Koopmans and S.A. Marglin

The application of multiobjectiveoptimization to domains outsideeconomics began with the work byKoopmans (1951) in productiontheory and with the work of Marglin(1967) in water resources planning.

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 7

Kaiza Miettinen and Matthias Ehrgott

Ehrgott generalized many combinatorial Algorithms to multiobjective versions.

Kaiza Miettinen summed up state-of-the-art on seterministic methods inMCDM and mathematicallprogramming

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 8

Kalyanmoy Deb and Carlos Coello Coello (C3)

Deb Introduced Large Scale Multi-objective Optimization algorithmsBased on EvolutionaryAlgorithms

Web RepositoryOn Metaheuristics,Thesis and Test-Problems

http://www.lania.mx/~ccoello/EMOO/

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 9

Many more researchers …

• International Society of MCDM

Awards:• MCDM Gold Medal, • Edgeworth-Pareto Award, and • Georg Cantor Award

http://www.terry.uga.edu/mcdm/index.html

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 10

LIACS Contributions to MCDM

• Evolutionary Strategies for MCO (Bäck, Willmes, Emmerich 2002, 2004)

• SMS-EMOA: Algorithm proposed first on EMO 2005 conference, now is one of the state-of-the-art techniques used in the field.

• Recently: CMA-SMS-EMOA and S-Gradient Method have been developed an published in 2007

• Pareto Optimization for Optimization with time consuming evaluation functions (Emmerich, Naujoks)

• Niching Methods in EMOA (Emmerich, Shir, Preuss)

• 2006: Rigorous analysis of spherical target problems using superellipsoid theory (Deutz, Emmerich)

• 2004: Interval-based orders in Pareto Optimization (Emmerich, Naujoks) – University of Dortmund

• Various Applications: Airfoil optimization, Turbine Blade Optimization, Laser Pulse Shaping, High Purity Silicon Production, and many more, Grid Computing, Building Design …(TU Eindhoven, TU Aachen, TU Dortmund, Fraunhofer UMSICHT, Bayer, DEGUSSA)

• Gradient-Based S-Metric Maximization

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 11

Current Research:S-Gradient (Hybrid Metaheuristics 2007)

Idea: Gradient-Based Maximization of the S-MetricFirst Paper: HM 2007, Hybrid-EMO/Gradient

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 12

Current Research:Exploring the Chemical Universe

• LACDR Cooperation for Medical Drug Design• Multi-objective Search for Drug-like Molecules

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 13

Current Research:Robust Design Optimization

• Noisy Objective Functions, Fuzzy Constraints => Interval Ordered Spaces

• Applications in Building Performance Design

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 14

Current Research:Grid-Scheduling

• Finding optimal Scheduling Strategies• Each User-Group defines Objective

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Multiobjective Optimization, Autumn 2005: Michael Emmerich 15

Some Recent Research Fields

• Optimization of Interval-Ordered Sets• Complexity/Reliability Theory for Algorithms• S-Metric and Indicator Based Approaches• Geometrical Analysis of Partially-Ordered Landscapes• Many-Objective Optimization• Combination of Mathematical Programming and Metaheuristics• Multi-objective Robust Design Optimization• Multi-objectivization in Combinatorial Optimization• Geometrical Classification of Conflicts

Long term goal: • Reliable, flexible, efficient methods for Pareto Optimization• Optimal support of human decision making