Managing Director / CTO NuTech Solutions GmbH / Inc. Martin-Schmeißer-Weg 15 D – 44227 Dortmund

1. Evolution Strategies: A Different Type of EC and its Applications Thomas Bäck October 11, 2004 Natural Computing Leiden Institute for Advanced Computer Science (LIACS) Niels Bohrweg 1 NL-2333 CA Leiden baeck@liacs.nl Tel.: +31 (0) 71 527 7108 Fax: +31 (0) 71 527 6985 Managing Director / CTO NuTech Solutions GmbH / Inc. Martin-Schmeißer-Weg 15 D – 44227 Dortmund baeck@nutechsolutions.de Tel.: +49 (0) 231 / 72 54 63-10 Fax: +49 (0) 231 / 72 54 63-29

2. Background I Daniel Dannett: Biology = Engineering

3. Background II Realistic Scenario ....

4. Background III Phenotypic evolution ... and genotypic

5. Optimization • E.g. costs (min), quality (max), error (min), stability (max), profit (max), ... • Difficulties: • High-dimensional • Nonlinear, non-quadratic: Multimodal • Noisy, dynamic, discontinuous • Evolutionary landscapes are like that !

6. Overview • Evolutionary Algorithm Applications: Examples • Evolutionary Algorithms: Some Algorithmic Details • Genetic Algorithms • Evolution Strategies • Some Theory of EAs • Convergence Velocity Issues • Other Examples • Drug Design • Inverse Design of Cas • Summary

7. Modeling – Simulation - Optimization Modeling / Data Mining ??? !!! !!! Simulation !!! ??? !!! Optimization ??? !!! !!!

8. Model from Data Simulation Function Function(s) Subjective Experiment Evaluation Business Process Model EA-Optimizer General Aspects

9. Examples I: Inflatable Knee Bolster Optimization MAZDA Picture Low Cost ES: 0.677 GA (Ford): 0.72 Hooke Jeeves DoE: 0.88 Initial position of knee bag model deployed knee bag (unit only) Tether FEM #5 Tether Volume of 14L Load distribution plate FEM #3 Support plate FEM #4 Support plate Knee bag FEM #2 Load distribution plate Straps are defined in knee bag(FEM #2) Vent hole

10. IKB: Previous Designs

11. IKB: Problem Statement Objective: Min Ptotal Subject to: Left Femur load <= 7000 Right Femur load <= 7000

12. IKB Results I: Hooke-Jeeves Quality: 8.888 Simulations: 160

13. IKB Results II: (1+1)-ES Quality: 7.142 Simulations: 122

14. Optical Coatings: Design Optimization • Nonlinear mixed-integer problem, variable dimensionality. • Minimize deviation from desired reflection behaviour. • Excellent synthesis method; robust and reliable results.

15. Dielectric Filter Design Problem • Dielectric filter design. • n=40 layers assumed. • Layer thicknesses xi in [0.01, 10.0]. • Quality function: Sum of quadratic penalty terms. • Penalty terms = 0 iff constraints satisfied. Client: Corning, Inc., Corning, NY

16. Results: Overview of Runs • Factor 2 in quality. • Factor 10 in effort. • Reliable, repeatable results. Benchmark

17. Problem Topology Analysis: An Attempt • Grid evaluation for 2 variables. • Close to the optimum (from vector of quality 0.0199). • Global view (left), vs. Local view (right).

18. Examples II: Bridgman Casting Process 18 Speed Variables (continuous) for Casting Schedule Turbine Blade after Casting · FE mesh of 1/3 geometry: 98.610 nodes, 357.300 tetrahedrons, 92.830 radiation surfaces large problem: · run time varies: 16 h 30 min to 32 h (SGI, Origin, R12000, 400 MHz) · at each run: 38,3 GB of view factors (49 positions) are treated!

19. Examples II: Bridgman Casting Process Global Quality Turbine Blade after Casting GCM(Commercial Gradient Based Method) Initial (DoE) Evolution Strategy Quality Comparison of the Initial and Optimized Configurations

20. Examples IV: Traffic Light Control Client: Dutch Ministry of Traffic Rotterdam, NL • Generates green times for next switching schedule. • Minimization of total delay / number of stops. • Better results (3 – 5%) / higher flexibility than with traditional controllers. • Dynamic optimization, depending on actual traffic (measured by control loops).

21. Examples V: Elevator Control Client: Fujitec Co. Ltd., Osaka, Japan • Minimization of passenger waiting times. • Better results (3 – 5%) / higher flexibility than with traditional controllers. • Dynamic optimization, depending on actual traffic.

22. Examples VI: Metal Stamping Process Client: AutoForm Engineering GmbH, Dortmund • Minimization of defects in the produced parts. • Optimization on geometric parameters and forces. • Fast algorithm; finds very good results.

23. Examples VII: Network Routing Client: SIEMENS AG, München • Minimization of end-to-end-blockings under service constraints. • Optimization of routing tables for existing, hard-wired networks. • 10%-1000% improvement.

24. Examples VIII: Nuclear Reactor Refueling Client: SIEMENS AG, München • Minimization of total costs. • Creates new fuel assembly reload patterns. • Clear improvements (1%-5%) of existing expert solutions. • Huge cost saving.

25. Two-Phase Nozzle Design (Experimental) Experimental design optimisation: Optimise efficieny. ... evolves... Initial design Final design: 32% improvement in efficieny.

26. Multipoint Airfoil Optimization (1) Client: Cruise Low Drag! High Lift! Start 22 design parameters.

27. Find pressure profiles that are a compromise between two given target pressure distributions under two given flow conditions! Multipoint Airfoil Optimization (2) Three compromise wing designs Pareto set after 1000 Simulations

28. Evolutionary Algorithms: Some Algorithmic Details

29. Unifying Evolutionary Algorithm t := 0; initialize(P(t)); evaluate(P(t)); while notterminatedo P‘(t) := mating_selection(P(t)); P‘‘(t) := variation(P‘(t)); evaluate(P‘‘(t)); P(t+1) := environmental_selection(P‘‘(t) u Q); t := t+1; od

30. Evolutionary Algorithm Taxonomy Evolution Strategies Genetic Algorithms Classifier Systems Evolutionary Programming Genetic Programming Many mixed forms; agent-based systems, swarm systems, A-life systems, ...

31. Genetic Algorithms vs. Evolution Strategies Genetic Algorithm Evolution Strategies • Binary representation • Fixed mutation rate pm (= 1/n) • Fixed crossover rate pc • Probabilistic selection • Identical population size • No self-adaptation • Real-valued representation • Normally distributed mutations • Fixed recombination rate (= 1) • Deterministic selection • Creation of offspring surplus • Self-adaptation of strategy parameters: Variance(s), Covariances

32. Genetic Algorithms • Often binary representation. • Mutation by bit inversion with probability pm. • Various types of crossover, with probability pc. • k-point crossover. • Uniform crossover. • Probabilistic selection operators. • Proportional selection. • Tournament selection. • Parent and offspring population size identical. • Constant strategy parameters.

33. 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 Mutation 0 1 1 1 0 0 0 1 0 1 0 0 0 1 0 • Mutation by bit inversion with probability pm. • pm identical for all bits. • pm small (e.g., pm = 1/l).

34. Crossover • Crossover applied with probability pc. • pc identical for all individuals. • k-point crossover: k points chosen randomly. • Example: 2-point crossover.

35. Selection • Fitness proportional: • f fitness • l population size • Tournament selection: • Randomly select q << l individuals. • Copy best of these q into next generation. • Repeat l times. • q is the tournament size (often: q = 2).

36. Evolution Strategies • Real-valued representation. • Normally distributed mutations. • Various types of recombination. • Discrete (exchange of variables). • Intermediate (averaging). • Involving two or more parents. • Deterministic selection, offspring surplus l >> m. • Elitist: (m+l) • Non-elitist: (m,l) • Self-Adaptation of strategy parameters.

37. Mutation Creation of a new solution: s-adaptation by means of • 1/5-success rule. • Self-adaptation. More complex / powerful strategies: • Individual step sizes si. • Covariances. • Convergence speed: •  Ca. 10  n down to 5  n is possible.

38. Self-Adaptation • Learning while searching: Intelligent Method. • Different algorithmic approaches, e.g: • Pure self-adaptation: • Mutational step size control MSC: • Derandomized step size adaptation • Covariance adaptation

39. Self-Adaptive Mutation n = 2, ns = 1, na = 0 n = 2, ns = 2, na = 0 n = 2, ns = 2, na = 1

40. Self-Adaptation: • Motivation: General search algorithm • Geometric convergence: Arbitrarily slow, if s wrongly controlled ! • No deterministic / adaptive scheme for arbitrary functions exists. • Self-adaptation: On-line evolution of strategy parameters. • Various schemes: • Schwefel one s, n s, covariances; Rechenberg MSA. • Ostermeier, Hansen: Derandomized, Covariance Matrix Adaptation. • EP variants (meta EP, Rmeta EP). • Bäck: Application to p in GAs. Step size Direction

41. Self-Adaptation: Dynamic Sphere • Optimum s: • Transition time proportionate to n. • Optimum s learned by self-adaptation.

42. Selection (m,l) (m+l)

43. Possible Selection Operators • (1+1)-strategy: one parent, one offspring. • (1,l)-strategies: one parent, l offspring. • Example: (1,10)-strategy. • Derandomized / self-adaptive / mutative step size control. • (m,l)-strategies:m>1 parents, l>m offspring • Example: (2,15)-strategy. • Includes recombination. • Can overcome local optima. • (m+l)-strategies: elitist strategies.

44. Advantages of Evolution Strategies • Self-Adaptation of strategy parameters. • Direct, global optimizers ! • Faster than GAs ! • Extremely good in solution quality. • Very small number of function evaluations. • Dynamical optimization problems. • Design optimization problems. • Discrete or mixed-integer problems. • Experimental design optimisation. • Combination with Meta-Modeling techniques.

45. Some Theory of EAs

46. Robust vs. Fast: • Global convergence with probability one: General, but for practical purposes useless. • Convergence velocity: Local analysis only, specific functions only.

47. Convergence Velocity Analysis, ES: • A convex function („sphere model“). • Simplest case: (1,l)-ES • Illustration: (1,4)-ES

48. Convergence Velocity Analysis, ES: • Order statistics: • pn:l(z) denotes the p.d.f. of Zn:l • Idea: Best offspring has smallest r / largest z‘. • The following holds from geometric considerations: • One gets:

49. Convergence Velocity Analysis, ES: • Using one finally gets • One gets: (dimensionless) l

50. Convergence Velocity Analysis, ES: • Convergence velocity, (1,l)-ES and (1+l)-ES: