Niching in Derandomized Evolution Strategies and its Applications in Quantum Control A Journey from Organic Diversity to Conceptual Quantum Designs Ofer Michael Shir

Niching in Derandomized Evolution Strategies and its Applications in Quantum Control A Journey from Organic Diversity to Conceptual Quantum Designs PROEFSCHRIFT ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van Rector Magnificus prof. mr. P.F. van der Heijden, volgens besluit van het College voor Promoties te verdedigen op woensdag 25 juni 2008 klokke 16:15 uur door Ofer Michael Shir geboren te Jeruzalem, Israël in 1978

Promotiecommissie: Prof. dr. Thomas Bäck Prof. dr. Marc Vrakking (Amolf-Amsterdam) Dr. Michael Emmerich Prof. dr. Darrell Whitley (Colorado State University) Prof. dr. Farhad Arbab Prof. dr. Joost Kok Promotor Promotor Co-promotor Referent This work is part of the research programme of the 'Stichting voor Fundamenteel Onderzoek der Materie (FOM)', which is financially supported by the 'Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)'. Niching in Derandomized Evolution Strategies and its Applications in Quantum Control. Ofer Michael Shir. Thesis Universiteit Leiden. ISBN: 978-90-6464-256-2 Printed in the Netherlands.

To my parents, Mira and Yos'ke

Contents Introduction 1 I Niching in Derandomized Evolution Strategies 5 1 Evolution Strategies 7 1.1 Background............................ 7 1.1.1 The Framework: Global Optimization......... 7 1.1.2 EvolutionaryAlgorithms................. 9 1.2 The Standard Evolution Strategy................ 11 1.2.1 Notation and Terminology................ 11 1.2.2 Motivation: The (1 + 1) Evolution Strategy...... 12 1.2.3 The Self-Adaptation Principle.............. 13 1.2.4 The Canonical ( μ/ν +,λ ) -ES Algorithm........ 14 1.3 Derandomized Evolution Strategies (DES)........... 20 1.3.1 (1,λ) Derandomized ES Variants............ 21 1.3.2 First Level of Derandomization............. 22 1.4 The Covariance Matrix Adaptation ES............. 24 1.4.1 Preliminary........................ 24 1.4.2 The (1,λ) Rank-One CMA............... 26 1.4.3 The (μ W,λ) Rank-μ CMA................ 28 1.4.4 The (1 + λ) CMA.................... 29 1.4.5 Constraints Handling................... 30 1.4.6 Discussion......................... 31 2 Introduction to Niching 33 2.1 Motivation: Speciation Theoryvs. Conceptual Designs.... 33 2.2 From DNA to Organic Diversity................. 34 2.2.1 Genetic Drift....................... 34 2.2.2 Organic Diversity..................... 35 2.3 "Ecological Optima": Basins of Attraction........... 38 2.3.1 Classification of Optima: The Practical Perspective.. 39 2.4 Population Diversitywithin Evolutionary Algorithms..... 39 i

ii Contents 2.4.1 Diversity LossinEvolution Strategies......... 41 2.4.2 Point of Reference: Diversity Loss within GAs..... 44 2.4.3 Neutrality inesvariations: Mutation Drift...... 45 2.5 Classical Niching Techniques.................. 47 2.5.1 Fitness Sharing...................... 48 2.5.2 Dynamic Fitness Sharing................ 49 2.5.3 Clearing.......................... 49 2.5.4 Crowding......................... 50 2.5.5 Clustering......................... 51 2.5.6 The Sequential Niche Technique............ 52 2.5.7 The Islands Model.................... 53 2.5.8 Other GA-Based Methods................ 53 2.5.9 Miscellaneous: Mating Schemes............. 54 2.6 Niching in Evolution Strategies................. 55 2.7 Discussion and Mission Statement............... 55 3 Niching with Derandomized Evolution Strategies 57 3.1 General.............................. 57 3.2 The Proposed Algorithm..................... 57 3.2.1 Niching with ( 1 +,λ ) DES Kernels........... 58 3.3 Niche Radius Calculation.................... 60 3.4 Experimental Procedure..................... 60 3.4.1 Multi-Modal Test Functions............... 61 3.4.2 Performance Criteria................... 63 3.4.3 New Perspective: MPR vs. Time............ 65 3.4.4 MPR Analysis: Previous Observation......... 65 3.5 Numerical Observation...................... 66 3.5.1 Modus Operandi..................... 66 3.5.2 Numerical Results.................... 66 3.5.3 Discussion......................... 70 4 Self-Adaptive Niche-Shape Approaches 71 4.1 General.............................. 71 4.1.1 Related Work....................... 71 4.1.2 Our Approach...................... 73 4.2 New Proposed Approaches.................... 73 4.2.1 Self-Adaptive Radius: Step-Size Coupling....... 74 4.2.2 Mahalanobis Metric: Covariance Exploitation..... 77 4.3 Experimental Procedure..................... 79 4.3.1 Numerical Observation.................. 79 4.3.2 General Behavior..................... 85 4.4 Discussion............................. 85

Contents iii 5 Niching-CMA as EMOA 87 5.1 Multi-Objective Optimization.................. 87 5.1.1 Formulation........................ 87 5.1.2 The NSGA-II Algorithm................. 89 5.2 On Diversity in Multi-Objective Optimization......... 91 5.2.1 Related Work....................... 92 5.3 Multi-Parent Niching with (μ W,λ)-CMA............ 95 5.4 Niching-CMAas EMOA..................... 96 5.4.1 The Niching Distance Metric.............. 97 5.4.2 Selection: Non-dominating Ranking.......... 97 5.4.3 Estimation of the Niche Radius............. 97 5.5 Numerical Simulations...................... 99 5.5.1 Test Functions: Artificial Landscapes.......... 99 5.5.2 Modus Operandi..................... 100 5.5.3 Numerical Observation.................. 101 II Quantum Control 105 6 Introduction to Quantum Control 107 6.1 Optimal Control Theory..................... 108 6.1.1 The Quantum Control Framework........... 108 6.1.2 Controllability...................... 112 6.1.3 Control Level Sets.................... 113 6.1.4 Computational Complexity............... 115 6.2 Optimal Control Experiments.................. 117 6.2.1 Femtosecond Laser Pulse Shaping............ 117 6.2.2 Laboratory Realization: Constraints.......... 119 6.3 Experimental Procedure..................... 122 6.3.1 Numerical Simulations.................. 122 6.3.2 Laboratory Experiments................. 123 7 Two Photon Processes 125 7.1 Introduction............................ 125 7.2 Second Harmonic Generation.................. 125 7.2.1 Total SHG........................ 126 7.2.2 Filtered SHG....................... 128 7.3 Numerical Simulations...................... 130 7.3.1 Preliminary ES Failure: Stretched Phases....... 130 7.3.2 Numerical Observation.................. 131 7.4 Laboratory Experiments..................... 132 7.4.1 Performance Evaluations................. 133 7.4.2 Discussion......................... 138

iv Contents 8 The Rotational Framework 139 8.1 Numerical Modeling....................... 139 8.1.1 Preliminary: Two Electronic States Systems...... 139 8.1.2 Rotational Levels..................... 140 8.2 Population Transfer: Optimization............... 141 8.2.1 Experimental Procedure................. 142 8.2.2 Numerical Observation: J =0 J =4........ 143 8.2.3 Intermediate Discussion................. 144 8.3 Application of Niching...................... 146 8.3.1 Preliminary: Distance Measure............. 146 8.3.2 Numerical Observation.................. 147 9 Dynamic Molecular Alignment 151 9.1 Numerical Modeling....................... 152 9.1.1 Numerical Simulations: Technical Details....... 153 9.2 Experimental Procedure..................... 154 9.2.1 First Numerical Results: Comparison of the Algorithms 155 9.2.2 The Complete-Basis-Functions Parameterization... 155 9.2.3 Further Investigation................... 164 9.3 The Zero Kelvin Case Study................... 165 9.3.1 Conceptual Quantum Structures............ 168 9.3.2 Maximally Attained Yield................ 169 9.3.3 Another Perspective to Optimality: Phasing-Up.... 170 9.4 Evolution of Pulses under Dynamic Intensity......... 173 9.4.1 Evolutionary Algorithms in Dynamic Environments.. 173 9.4.2 Dynamic Intensity Environment: Procedure...... 174 9.5 Scalability: Control Discretization............... 180 9.5.1 Numerical Observation.................. 181 9.6 Intermediate Discussion..................... 184 9.7 Multi-Objective Optimization.................. 185 9.7.1 Choice of Methods.................... 185 9.7.2 Numerical Observation.................. 189 9.8 Application of Niching...................... 191 9.8.1 Numerical Observation.................. 191 Summary and Outlook 197 A Additional Figures 203 B Complete-Basis Functions 221 Bibliography 225 Samenvatting (Dutch) 243