Algorithms and Environments for Complementarity By Todd S. Munson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN – MADISON 2000 i Abstract Complementarity problems arise in a wide variety of disciplines. Prototypical examples include the Wardropian and Walrasian equilibrium models encountered in the engineering and economic disciplines and the first order optimality conditions for nonlinear programs from the optimization community. The main focus of this thesis is algorithms and envi- ronments for solving complementarity problems. Environments, such as AMPL and GAMS, are used by practitioners to easily write large, complex models. Support for these packages is provided by PATH 4.x and SEMI through the customizable solver interface specified in this thesis. The main design feature is the abstraction of core components from the code with implementations tailored to a particular environment supplied either at compile or run time. This solver interface is then used to develop new links to the MATLAB and NEOS tools. Preprocessing techniques are an integral part of linear and mixed integer programming codes and are primarily used to reduce the size and complexity of a model prior to solving it. For example, wasted computation is avoided when an infeasible model is detected. This thesis documents the new techniques for preprocessing complementarity problems contained in the PATH 4.x and SEMI algorithms. PATH 4.x is more reliable than prior versions of the code and has been able to find solutions to previously unsolvable models. The reasons for the improvement are discussed in this thesis and include new theoretical developments for a globalization scheme based on the Fischer-Burmeister merit function, and enhancements made to the linear complementarity solver, nonmonotone linesearch, and restart strategy. SEMI is an alternative to PATH 4.x based on the semismooth algorithm. The main ii computation in PATH 4.x is to solve linear complementarity problems, which can be quite expensive. The new SEMI code described in this thesis only solves linear systems of equations and uses iterative methods to process very large models. The theme of this thesis is “enabling” technologies in complementarity: environments enable practitioners to easily specify models; sophisticated codes enable them to solve the models; and theory assures them that the algorithm is well-defined and efficiently processes models. iii Acknowledgements I would like to thank my family and educators throughout the years who have taught me many important lessons. Without them, I would not have attended university, let alone pursue an advanced degree. Several people have influenced the work in this thesis. The most important are my advisor, Michael Ferris, who first introduced me to complementarity, and Steven Dirkse and Danny Ralph who paved the way for the current version of PATH. Others include Stephen Billups, Arne Drud, Francisco Facchinei, Andreas Fischer, David Gay, Christian Kanzow, Alex Meeraus, Jorge Mor´e, and Steve Wright. I have also benefitted immensely from interaction with other graduate students including Paul Bradley, Qun Chen, William Donaldson, Glenn Fung, Yuh-Jye Lee, Jinho Lim, David Musicant, Krung Sinapiromsaran, and Meta Voelker. Many individuals have tested the software developed in this thesis and have provided feedback. Special thanks to Sherman Robinson, Thomas Rutherford, and Francis Tin-loi who continue to challenge the solvers. The members of my committee, Thomas Cox, Miron Livny, Olvi Mangasarian, and Stephen Robinson, have made many useful suggestions to improve the presentation of this thesis. Finally, I want to give many thanks to the secretarial and support staff for the Com- puter Sciences Department at the University of Wisconsin - Madison. They have kept me on track and provided encouragement throughout my graduate studies. I am especially grateful to Laura Cuccia as she has constantly gone above and beyond the call of duty. When I needed something done on short notice, she always came through. iv The work in this thesis was supported by NSF grants CCR-9972372 and CCR- 9619765, AFOSR grant F49620-98-1-0417, GAMS Development Corporation, and a fel- lowship from Cisco Systems. v Contents Abstract i Acknowledgements iii 1 Introduction 1 1.1 Complementarity ............................... 2 1.2 NonsmoothReformulations . 6 1.2.1 NormalMap.............................. 6 1.2.2 SemismoothFormulation. 7 1.3 ExistenceofSolutions ............................ 10 1.3.1 LinearProblems ........................... 10 1.3.2 NonlinearProblems. .. .. 12 1.4 AlgorithmsandRegularity . 15 1.5 Summary ................................... 17 2 Environments 21 2.1 ExampleApplications ............................ 21 2.1.1 TransportationModel . 22 2.1.2 WalrasianEquilibrium . 26 2.2 AMPL..................................... 31 2.2.1 GeneralizedDeclaration . 40 2.2.2 SideConstraints ........................... 41 2.2.3 Presolve ................................ 42 vi 2.3 GAMS..................................... 43 2.3.1 ListingFile .............................. 50 2.3.2 RedefinedEquations . .. .. 53 2.3.3 Pitfalls................................. 54 2.4 MATLAB ................................... 56 2.4.1 Linear Complementarity Problems . 56 2.4.2 Nonlinear Complementarity Problems . .. 61 2.4.3 MexFunctions ............................ 64 2.4.4 Pitfalls................................. 65 2.5 NEOS ..................................... 66 2.5.1 Pitfalls................................. 70 2.6 Summary ................................... 71 3 Interfaces 73 3.1 Subroutines .................................. 74 3.2 ProblemStructures .............................. 78 3.2.1 MCP Interface ............................ 82 3.2.2 Preprocessing Interface........................ 85 3.3 SolverStructures ............................... 87 3.3.1 Options ................................ 87 3.3.2 WorkspaceAllocation. 89 3.4 ControlStructures .............................. 90 3.4.1 Error.................................. 90 3.4.2 Interrupts ............................... 92 3.4.3 Memory ................................ 92 vii 3.4.4 Output................................. 95 3.4.5 Timer ................................. 96 3.5 Driver ..................................... 97 3.6 Summary ................................... 100 4 Preprocessing 101 4.1 PolyhedralConstraints . 105 4.1.1 Presolve ................................ 105 4.1.2 Postsolve ............................... 112 4.2 StructuralImplications . 113 4.2.1 Intervals................................ 114 4.2.2 Duplicates............................... 114 4.2.3 SpecialStructure . .. .. 115 4.3 ComputationalResults . 118 4.4 Summary ................................... 123 5 Diagnostic Information 124 5.1 MeritFunctions................................ 125 5.2 Ill-DefinedModels............................... 127 5.2.1 FunctionUndefined. 127 5.2.2 JacobianUndefined. 131 5.3 PoorlyScaledModels............................. 132 5.4 SingularModels................................ 134 5.5 Summary ................................... 135 6 PATH 4.x 136 viii 6.1 AlgorithmandTheory ............................ 138 6.1.1 EquationProperties . 140 6.1.2 MeritFunctionProperties . 146 6.1.3 AlgorithmicFramework . 153 6.2 ImplementationFeatures . 161 6.2.1 CrashingMethod ........................... 162 6.2.2 NonmonotoneSearches . 162 6.2.3 Linear Complementarity Subproblems . 164 6.2.4 OtherFeatures ............................ 167 6.3 ComputationalResults . 167 6.4 UserDocumentation ............................. 178 6.4.1 AnatomyoftheLogFile . 179 6.4.2 OptionsSummary. .. .. 186 6.5 Summary ................................... 190 7 SEMI 192 7.1 MathematicalFoundation . 193 7.2 TheLinearSystem .............................. 198 7.2.1 IterativeTechniques . 199 7.2.2 DirectMethods ............................ 204 7.2.3 Summary ............................... 210 7.3 TheNonlinearModel............................. 210 k 7.3.1 Φ(x ) and Hk ............................. 211 7.3.2 Crashing................................ 213 7.3.3 StationaryPoints . .. .. 214 ix 7.4 ComputationalResults . 217 7.5 Summary ................................... 224 8 Conclusion 225 Bibliography 227 x List of Tables 1 AMPLSpecificOptions ........................... 35 2 AMPLReturnCodes............................. 36 3 AMPLReturnCodeMeanings........................ 36 4 SolutionStatusCodesforGAMS . .. .. 52 5 ModelStatusCodesforGAMS ....................... 52 6 ReturnmessagesforNEOS.......................... 69 7 Keywords for NEOS e-mail submissions . 70 8 StatuscodesforSubroutineInterfaces. .... 75 9 Initializationroutines. .. .. 80 10 Jacobianinformationroutines . .. 81 11 Accessfunctions................................ 81 12 Basisinformationcodes ........................... 81 13 Interfacefunctions .............................. 82 14 Required functions for MCP Interface ................... 84 15 Optional functions for MCP Interface .................... 85 16 MCP Termination codes ........................... 87 17 Key fields in Information structure..................... 88 18 Functionsfordealingwithoptions. ... 89 19 ErrorObjectFunctions............................ 91 20 InterruptObjectFunctions. 92 21 MemoryObjectFunctions .......................... 93 22 OutputObjectFunctions........................... 95 xi 23 TimerObjectFunctions ........................... 96 24 RedundantRowsCases...........................