Thesis Title

Limited Memory BFGS method for Sparse and Large-Scale Nonlinear Optimization by Sonja Rauˇski A thesis submitted to University of Bremen for the degree of Dr.rer.nat. May 2014 Date of Defense: 16.09.2014 1. referee: Prof. Dr. Christof Büskens 2. referee: Prof. Dr. Matthias Gerdts Acknowledgements I want to thank my supervisor Prof. Dr. Christof Büskens for his motivation and guidance throughout this thesis. My second supervisor Prof. Dr. Matthias Gerdts for helpful discussions and advice. Moreover, I want to thank Sven Weikert and all my former colleagues from ASTOS Solutions GmbH for their help in the area of aerospace engineering. Further Prof. Dr. Moritz Diehl for giving me the possibility to work in his research group in KU Leuven and for a very helpful exchange period of six months. Thanks to Dr. Matthias Knauer for reviewing my thesis and for all his help and advice during the last three and a half years. To my current and former colleagues from the AG Optimierung und Optimale Steuerung at the University of Bremen for motivating conversations and software related support, especially to Dr.-Ing. Dennis Wassel, Dr. Tim Nikolayzik and Dr. Bodo Blume. Further thanks to Riccardo Parenti for all his patience and support throughout this thesis and his English grammar corrections. I thank my colleague Anne Schattel for her help with administration, paperwork and the German language. This thesis wouldn’t have been possible without the funding from the European Union under the 7th Framework Programme FP7- PEOPLE-2010-ITN Grant agreement number 264735-SADCO. Many thanks to the SADCO’s coordinators for organizing various inspiring conferences, workshops, summer schools and train- ings. Final thanks to all SADCO participants for interesting discussions and a helpful cooperation. Contents List of Figures vii List of Tables ix Abbreviations and Acronyms xi 1 Introduction 1 2 Nonlinear Optimization 5 2.1 Problem formulation and optimality conditions ............ 6 2.2 Nonlinear optimization methods .................... 10 2.2.1 Newton-typemethods..................... 11 2.2.2 Sequential quadratic programming .............. 14 2.2.3 Globalizationstrategies..................... 15 2.2.4 Interiorpointmethod...................... 17 2.3WORHP................................. 19 2.3.1 Mainalgorithm......................... 19 2.3.2 Termination criteria ...................... 20 2.3.3 Filtermethod.......................... 21 2.3.4 Constraintrelaxation...................... 21 2.3.5 Hessianregularization..................... 22 2.3.6 Derivatives........................... 22 2.3.6.1 Firstorderderivatives................ 22 2.3.6.2 Secondorderderivatives............... 24 2.3.7 Hotstart............................. 25 2.3.8 Reverse Communication .................... 26 2.3.9 Recoverystrategy........................ 26 2.3.10Interfaces............................ 26 2.3.11Parameters........................... 27 2.4QPSOL................................. 27 2.4.1 Primal-DualInterior-PointAlgorithm............. 27 2.4.2 Termination criteria ...................... 31 2.4.3 Solving the linear equation system .............. 32 2.4.4 StoringSparseMatrices.................... 34 vi CONTENTS 3 Update methods 37 3.1BFGSmethod.............................. 38 3.1.1 Compact representations of BFGS formula .......... 40 3.2BlockBFGSmethods.......................... 42 3.2.1 Block matrix without intersections .............. 43 3.2.2 Single intersection blocks .................... 44 3.2.3 Multiple intersections blocks .................. 45 3.2.4 SparseBFGS.......................... 46 3.3 Limited memory BFGS ......................... 47 3.3.1 Unconstrained optimization .................. 48 3.3.2 Equality constrained optimization ............... 50 3.4LBFGS................................. 51 3.4.1 Computations .......................... 51 3.4.2 QPformulationforLBFGS.................. 55 3.4.3 LBFGSwithPrimal-DualInterior-PointAlgorithm..... 56 3.4.4 Termination criteria ...................... 60 3.4.5 Solving linear equation system ................. 61 3.4.6 Powell’s trick .......................... 62 3.4.7 LBFGSalgorithm........................ 63 4 Optimal Control Problem 65 4.1Problemformulation.......................... 66 4.2Discretization.............................. 70 4.3Integrationmethods.......................... 71 4.4Multi-phaseoptimalcontrolproblems................. 72 4.5TransWORHP.............................. 74 5 Numerical results 77 5.1 KUKA youBot robot collision avoidance ............... 78 5.2ASTOS................................. 88 5.2.1 Ariane5............................. 89 5.2.1.1 Ariane5intoGTO.................. 90 5.2.1.2 Ariane5dualpayloadintoLEO/GTO....... 91 5.2.2 VEGA.............................. 93 5.3CUTErtestset............................. 96 5.4COPStestset..............................102 6 Conclusions 107 Bibliography 109 List of Figures 2.1 The complementarity condition .................... 9 2.2Newton’smethod............................ 12 2.3 Armijo condition ............................ 16 2.4Theinteriorpointmethodidea.................... 18 2.5Graphcoloring............................. 24 4.1Linkagesforamulti-phaseoptimalcontrolproblem......... 74 4.2 TransWORHP graphical interface screenshot ............. 75 4.3TransWORHP:StructureofJacobianmatrix............. 76 4.4 TransWORHP: Structure of Hessian matrix ............. 76 5.1 KUKA youBot omni-directional mobile platform with arm ..... 78 5.2 KUKA youBot robot: the Jacobian matrix and the Hessian matrix structures................................ 82 5.3KUKAyouBotrobotsimulation.................... 83 5.4 KUKA youBot robot smaller obstacle: Comparison of the Hessian methods by the computational times in seconds for the different numberofdiscretizationpoints..................... 84 5.5 KUKA youBot robot simulation of collision avoidance of a bigger obstacle................................. 85 5.6 KUKA youBot robot problem with a bigger obstacle: Comparison of the Hessian methods by the computational times in seconds for thedifferentnumberofdiscretizationpoints.............. 86 5.7 KUKA youBot robot problem with a bigger obstacle: Screenshot of the TransWORHP graphical interface. ................ 86 5.8 KUKA youBot robot problem with a bigger obstacle: Screenshot of the TransWORHP graphical interface obtained using the Lagrange weight α =0.001. ............................ 87 5.9 KUKA youBot robot problem with a bigger obstacle: Screenshot of the TransWORHP graphical interface obtained using the Lagrange weight α =0.1.............................. 88 5.10Ariane5................................. 89 5.11 Ariane 5 GTO: Progression of the cost function using LBFGS20 method................................. 90 5.12 Ariane 5 GTO: Flight simulation. Altitude, latitude, longitude and timeoftheflightaredisplayed..................... 91 viii LIST OF FIGURES 5.13 Ariane 5 LEO/GTO: Progression of the cost function using LBFGS method................................. 92 5.14 Ariane 5 LEO/GTO: Flight simulation. Altitude, latitude, longi- tudeandtimeoftheflightaredisplayed............... 93 5.15VEGA.................................. 94 5.16 VEGA: Altitude, latitude, longitude and time of the flight are dis- played.................................. 95 5.17 VEGA: Progression of the cost function using LBFGS method . 96 5.18 CUTEr: Comparison of different update-methods by percentage of solved problems according to the time ................. 98 5.19 CUTEr problems with n + m ≤ 500: Comparison of different update-methods by the percentage of solved problems according tothetime............................... 99 5.20 CUTEr problems with 500 <n+m ≤ 1000: Comparison of different update-methods by the percentage of solved problems according to thetime.................................100 5.21 CUTEr problems with 1000 <n+ m ≤ 5000: Comparison of different update-methods by the percentage of solved problems according tothetime...............................101 5.22 CUTEr problems with n + m>5000: Comparison of different update-methods by the percentage of solved problems according to thetime.................................102 5.23 COPS: Comparison of different update-methods by percentage of solved problems according to the time .................104 5.24 COPS problems with n + m ≤ 5000: Comparison of different update-methods by percentage of solved problems according to the time...................................105 5.25 COPS problems with n + m>5000: Comparison of different update-methods by percentage of solved problems according to the time...................................106 List of Tables 5.1KUKAyouBotrobot:Optimalcontrolproblemsize......... 81 5.2KUKAyouBotrobot:DiscretizedNLPproblemsize......... 82 5.3KUKAyouBotrobot:testresults................... 83 5.4 KUKA youBot robot test results of collision avoidance of a bigger obstacle................................. 85 5.5Ariane5GTO:NLPproblemsize................... 90 5.6 Ariane 5 GTO: Comparison of different Hessian-update methods . 90 5.7Ariane5LEO/GTO:NLPproblemsize................ 92 5.8 Ariane 5 LEO/GTO: Comparison of different Hessian-update methods 92 5.9VEGA:NLPproblemsize....................... 94 5.10 VEGA: Comparison of different Hessian-update methods ...... 94 5.11 CUTEr: Comparison of different update-methods by number of solvedproblems............................. 97 5.12CUTErnumberofproblemsbytheproblemsizecategories..... 99 5.13 CUTEr problems with n + m ≤ 500: Comparison of different update-methodsbythenumberofsolvedproblems........

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