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Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems" (2004)

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Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 9-2004 Pattern Search Ranking and Selection Algorithms for Mixed- Variable Optimization of Stochastic Systems Todd A. Sriver Follow this and additional works at: https://scholar.afit.edu/etd Part of the Theory and Algorithms Commons Recommended Citation Sriver, Todd A., "Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems" (2004). Theses and Dissertations. 3903. https://scholar.afit.edu/etd/3903 This Dissertation is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. Pattern Search Ranking and Selection Algorithms for Mixed-Variable Optimization of Stochastic Systems DISSERTATION Todd A. Sriver B.S., M.S. Major, USAF AFIT/DS/ENS/04-02 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio Approved for public release; distribution unlimited The views expressed in this dissertation are those of the author and do not reflect the official policy or position of the United States Air Force, the Department of Defense, or the United States Government. AFIT/DS/ENS/04-02 PatternSearchRankingandSelectionAlgorithmsfor Mixed-Variable Optimization of Stochastic Systems DISSERTATION Presented to the Faculty of the Graduate School of Engineering and Management Air Force Institute of Technology Air University In Partial Fulfillment for the Degree of Doctor of Philosophy Specialization in: Operations Research Todd A. Sriver B.S., M.S. Major, USAF September, 2004 Sponsored by the Air Force Office of ScientificResearch Approved for public release; distribution unlimited AFIT/DS/ENS/04-02 Abstract A new class of algorithms is introduced and analyzed for bound and linearly con- strained optimization problems with stochastic objective functions and a mixture of design variable types. The generalized pattern search (GPS) class of algorithms is extended to a new problem setting in which objective function evaluations require sampling from a model of a stochastic system. The approach combines GPS with ranking and selection (R&S) statistical procedures to select new iterates. The derivative-free algorithms require only black-box simulation responses and are applicable over domains with mixed variables (con- tinuous, discrete numeric, and discrete categorical) to include bound and linear constraints on the continuous variables. A convergence analysis for the general class of algorithms establishes almost sure convergence of an iteration subsequence to stationary points appro- priately defined in the mixed-variable domain. Additionally, specific algorithm instances are implemented that provide computational enhancements to the basic algorithm. Im- plementation alternatives include the use of modern R&S procedures designed to provide efficient sampling strategies and the use of surrogate functions that augment the search by approximating the unknown objective function with nonparametric response surfaces. In a computational evaluation, six variants of the algorithm are tested along with four com- peting methods on 26 standardized test problems. The numerical results validate the use of advanced implementations as a means to improve algorithm performance. iv Acknowledgments I express my heartfelt gratitude to many individuals who directly or indirectly sup- ported me in the challenging, but rewarding, endeavor of completing a Ph.D. program. I begin with the most important person — my wife — whose strength and encouragement were essential to my success. I am also grateful for our three children (who make me very proud) for the ability to make me smile during those times when the road ahead seemed difficult. I owe a deep thanks my research advisor, Professor Jim Chrissis. His expertise, guid- ance, and friendship kept me on the right track while enabling me to enjoy the ride. I am also grateful to the remainder of my research committee, Lt Col Mark Abramson, Professor Dick Deckro, and Professor J. O. Miller for all of the positive support they provided me. In particular, Lt Col Abramson’s help on some of the more theoretical issues was crucial to the development of the material presented in Chapter 3. I also thank the Air Force Office of Scientific Research for sponsoring my work. I would be negligent if I did not acknowledge my friends and colleagues in the B.A.R.F. cubicle farm. The synergy amongst the Ph.D. students in that small building led to some novel ideas incorporated in my research (Major Trevor Laine gets credit for introducing me to kernel regression); but, perhaps more importantly, the moments of levity kept things in the proper perspective. Finally, I offer a special thanks to both my mother and father. Their love and guidance throughout my life has inspired me to seek achievement. Todd A. Sriver v Table of Contents Page Abstract............................................................... iv Acknowledgments........................................................v ListofFigures...........................................................x ListofTables...........................................................xii Chapter1. INTRODUCTION.............................................1 1.1 ProblemSetting...............................................1 1.2 PurposeoftheResearch........................................5 1.2.1 ProblemStatement.........................................6 1.2.2 ResearchObjectives........................................6 1.3 Overview....................................................7 Chapter2. LITERATUREREVIEW.......................................8 2.1 MethodsforStochasticOptimization..............................8 2.1.1 StochasticApproximation ...................................9 2.1.2 RandomSearch...........................................14 2.1.3 RankingandSelection.....................................19 2.1.4 DirectSearch.............................................22 2.1.5 ResponseSurfaceMethods..................................28 2.1.6 OtherMethods...........................................34 2.1.7 SummaryofMethods......................................35 vi Page 2.2 GeneralizedPatternSearch.....................................36 2.2.1 PatternSearchforContinuousVariables.......................36 2.2.2 PatternSearchforMixedVariables...........................39 2.2.3 PatternSearchforRandomResponseFunctions.................39 2.3 Summary...................................................40 Chapter 3. ALGORITHMIC FRAMEWORK AND CONVERGENCE THEORY...................................................41 3.1 MixedVariables..............................................41 3.2 PositiveSpanningSetsandMeshConstruction.....................43 3.3 BoundandLinearConstraintHandling...........................46 3.4 TheMGPSAlgorithmforDeterministicOptimization...............48 3.5 IterateSelectionforNoisyResponseFunctions.....................51 3.6 TheMGPS-RSAlgorithmforStochasticOptimization...............53 3.7 ConvergenceAnalysis.........................................57 3.7.1 ControllingIncorrectSelections..............................59 3.7.2 MeshSizeBehavior........................................61 3.7.3 MainResults.............................................65 3.8 IllustrativeExample..........................................69 Chapter4. ALGORITHMIMPLEMENTATIONS............................77 4.1 Specific Ranking and Selection (R&S) Procedures . .................77 4.2 UseofSurrogateModels.......................................83 vii Page 4.3 TerminationCriteria..........................................90 4.4 AlgorithmDesign ............................................94 4.4.1 BuildingtheSurrogateDuringInitialization....................95 4.4.2 AlgorithmSearchStepsandTermination ......................99 4.4.3 AlgorithmParameters .................................... 102 4.5 Summary.................................................. 104 Chapter5. COMPUTATIONALEVALUATION............................ 105 5.1 TestScenario............................................... 105 5.2 CompetingAlgorithms....................................... 106 5.3 TestProblems.............................................. 112 5.3.1 Continuous-VariableProblems.............................. 113 5.3.2 Mixed-VariableProblems.................................. 115 5.4 ExperimentalDesign......................................... 117 5.4.1 PerformanceMeasuresandStatisticalModel .................. 118 5.4.2 SelectionofParameterSettings............................. 119 5.5 ResultsandAnalysis......................................... 122 5.5.1 AnalysisofMGPS-RSVariantImplementations................ 123 5.5.2 ComparativeAnalysisofAllAlgorithmImplementations......... 133 5.5.3 TerminationCriteriaAnalysis.............................. 137 5.5.4 SummaryoftheAnalysis.................................. 140 viii Page Chapter6. CONCLUSIONSANDRECOMMENDATIONS................... 142 6.1 Contributions............................................... 142 6.2 FutureResearch............................................. 144 6.2.1 ModificationstoExistingSearchFramework................... 144 6.2.2 ExtensionstoBroaderProblemClasses....................... 147 AppendixA.TESTPROBLEMDETAILS.................................. 150 AppendixB.TESTRESULTDATA....................................... 166 B.1 IterationHistoryCharts.....................................
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