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Part II.2 A-Posteriori Methods and Evolutionary Multiobjective Optimization

Part II.2 A-Posteriori Methods and Evolutionary Multiobjective Optimization. Scalar solution methods A-posteriori methods Evaluation of algorithms. A posteriori-methods. A priori-method:

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Part II.2 A-Posteriori Methods and Evolutionary Multiobjective Optimization

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  1. Part II.2 A-Posteriori Methods and Evolutionary Multiobjective Optimization • Scalar solution methods • A-posteriori methods • Evaluation of algorithms

  2. A posteriori-methods A priori-method: First define preferences, including decisions in case of conflicts (e.g. by specifying a utility function) Let the algorithm search for a single best solution User selects the obtained solution => Single objective optimization can be used A posteriori-method Specify general, possibly conflicting, goals Let algorithm find Pareto front User selects best solution among solution on Pareto front => Algorithms for obtaining Pareto fronts needed!

  3. A-Posteriori methods (Pareto Optimization) • A finite set of non-dominated solutions = Approximation set • Strive for good coverage and convergence to the pareto front !

  4. Overview Approaches Continuation methods Starting from a Karush Kuhn Tucker point gradually extend Pareto front by including neighboring Karush Kuhn Tucker points Problem: Connected Pareto front is required and differentiability Epsilon-Constraint method Obtain all points on the Pareto front by solving constraint optimization problems.;All but one objective is set to a constraint value The constraint values are changed gradually until the whole Pareto front is sampled; density/position of points can be easily controlled; Problem: Effort growth exponentially. Population-based Metaheuristics (evolutionary, particle swarm, …) Use selection/variation scheme to gradually move a population of search points to the Pareto front Very flexible method, easy to apply in different search spaces Problem: Cannot guarantee optimality of result

  5. Overview Approaches • Indicator-based method • Approximation set A to pareto front of q=|A| points (each one of dimension n) is viewed as n*m dimensional vector • A quality measure is defined for a set of points; e.g. the dominated hyper-volume and functions as surrogate objective function • Problem: Reference point is required; Dimension of problem may be to large if q is to large An example of indicator for the quality of an approximation set is the S-metric, Measuring the area between the points in A and the reference point to be maximized.

  6. Evolutionary Multiobjective Optimization (EMO)

  7. Overview of the EMO field

  8. State of the literature in EMOA

  9. Biological term Individual Fitness Population Generation Mutation Operator Recombination Parents, Offspring Biologically inspired terminology Mathematical term Element of search space S Objective function value (+penalty) Multi-set of elements of S Iteration of main loop Operator generating a new solution by adding a small perturbation to a given solution Operator generating a new solution by combining information of at least two given solutions Given a set of variations generated from an original set, the original set is called parents, and the set of variations offspring The concepts are used slightly differently, depending on authors. This will be the way we use them.

  10. General schema of evolutionary search Application of variation operators parents offspring Population of individuals Evaluation of fitness

  11. Example: Steady state evolutionary algorithm

  12. Variation Operators

  13. Initialization

  14. Stochastic variation operators

  15. Representation Independence Variation Operators Representation Operators Population Model and Selection Operator The concept of PISA (ETH Zuerich) Algorithms such as NSGA, SPEA2, PAES are widely independent of represenations (search spaces and variation operators)!

  16. Mutation

  17. Recombination: Crossover Operators

  18. Advanced operators

  19. Selection operator in EMOA

  20. First generation EMOA

  21. NSGA-II Algorithm

  22. Non-dominated sorting genetic algorithm (NSGA-II)

  23. (m+l)- Evolutionary algorithm

  24. A-Posteriori methods (Pareto Optimization) • A finite set of solutions • Strive for good coverage and convergence to the pareto front !

  25. A: Non-dominated sorting

  26. A: Non-dominated sorting

  27. B: Crowding distance sorting

  28. B: Crowding distance sorting Objective space (NSGA-II) Variable space (NSGA)

  29. B: Crowding distance sorting

  30. NSGA-II Complete procedure Download NSGA-II: http://www.iitk.ac.in/kangal/soft.htm

  31. NSGA-II ZDT1

  32. NSGA-II ZDT2

  33. Results ZDTL4

  34. Non-dominated sorting genetic algorithm

  35. SMS-EMOA

  36. Remarks Introduced by Beume, Emmerich, Naujoks, 2005 Outperforms standard approaches on common Benchmarks for continuous multiobjective optimization ZDT and DTLZ Especially well suited for small approximation set Paradigm shift: Indicator-based Pareto optimization Can be hybridized with S-Gradient method (Deutz, Beume, Emmerich 2007)

  37. SMS-EMOA – Basic Algorithm

  38. Replacement The following invariant holds:

  39. SMS-EMOA

  40. Hypervolume in 2-D

  41. Computation of hypervolume in 3-D O(m q3) algorithm was proposed by Emmerich

  42. SMS-EMOA (20.000 Auswertungen)

  43. Pareto front in higher dimensons Points demark finite Set approximation of Pareto fronts

  44. Pareto front in higher dimensons

  45. SMS-EMOA Conclusions SMS EMOA tries to maximize the dominated hypervolume Increment in hypervolume are used as a selection criterion in an Evolutionary Algorithm The evolutionary algorithm gradually improves the dominated hypervolume of the population by variation-selection scheme The SMS EMOA is the best performing MCO techniques on standard benchmarks from literature (cf. EJOR 2006 Preprint, EMO Conference 2005) A bottleneck is the high computational cost of computing hypervolume increments, if no, of objectives is high

  46. Summary Large number of EMOA algorithms available today Most popular variants are NSGA-II, SPEA-II In EMOA field also other population-based algorithms (particle swarm optimization, simulated annealing) are discussed New generation of EMOA is developed (IBEA, SMS-EMOA) that directly addresses optimization of performance measure Statistical performance measuring on test problems crucial technique to engineer and select EMOA technique Bi-annual conference: EMOO 2001, 2003, 2005 (Lecture notes in Computer Science) Bi-annual conference MCDM: Operations Research Oriented

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