Nonlinear Programming..Concepts, Algorithms, and Applications to Chemical Processes

416 Pages·2011·5.67 MB·English

by  Lorenz T. Biegler

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MOS-SIAM Series on Optimization
This series is published jointly by the Mathematical Optimization Society and the Society for
Industrial and Applied Mathematics. It includes research monographs, books on applications,
textbooks at all levels, and tutorials. Besides being of high scientific quality, books in the series
must advance the understanding and practice of optimization. They must also be written clearly
and at an appropriate level.
Editor-in-Chief
Thomas Liebling
École Polytechnique Fédérale de Lausanne

Editorial Board
William Cook, Georgia Tech
Gérard Cornuejols, Carnegie Mellon University
Oktay Gunluk, IBM T.J. Watson Research Center
Michael Jünger, Universität zu Köln
C.T. Kelley, North Carolina State University
Adrian S. Lewis, Cornell University
Pablo Parrilo, Massachusetts Institute of Technology
Daniel Ralph, University of Cambridge
Éva Tardos, Cornell University
Mike Todd, Cornell University
Laurence Wolsey, Université Catholique de Louvain

Series Volumes
Biegler, Lorenz T., Nonlinear Programming: Concepts, Algorithms, and Applications to
Chemical Processes
Shapiro, Alexander, Dentcheva, Darinka, and Ruszczy´nski, Andrzej, Lectures on Stochastic
Programming: Modeling and Theory
Conn, Andrew R., Scheinberg, Katya, and Vicente, Luis N., Introduction to Derivative-Free
Optimization
Ferris, Michael C., Mangasarian, Olvi L., and Wright, Stephen J., Linear Programming with MATLAB
Attouch, Hedy, Buttazzo, Giuseppe, and Michaille, Gérard, Variational Analysis in Sobolev
and BV Spaces: Applications to PDEs and Optimization
Wallace, Stein W. and Ziemba, William T., editors, Applications of Stochastic Programming
Grötschel, Martin, editor, The Sharpest Cut: The Impact of Manfred Padberg and His Work
Renegar, James, A Mathematical View of Interior-Point Methods in Convex Optimization
Ben-Tal, Aharon and Nemirovski, Arkadi, Lectures on Modern Convex Optimization: Analysis,
Algorithms, and Engineering Applications
Conn, Andrew R., Gould, Nicholas I. M., and Toint, Phillippe L., Trust-Region Methods
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Concepts, Algorithms, and
Applications to Chemical Processes

Lorenz T. Biegler
Carnegie Mellon University
Pittsburgh, Pennsylvania
Society for Industrial and Applied Mathematics Mathematical Optimization Society
Philadelphia Philadelphia
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Copyright © 2010 by the Society for Industrial and Applied Mathematics and the Mathematical
Optimization Society
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may be reproduced,
stored, or transmitted in any manner without the written permission of the publisher. For information,
write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia,
PA 19104-2688.
Trademarked names may be used in this book without the inclusion of a trademark symbol. These
names are used in an editorial context only; no infringement of trademark is intended.
AIMMS is a registered trademark of Paragon Decision Technology B.V.
AMPL is a trademark of AMPL Optimization LLC.
Excel is a trademark of Microsoft Corporation in the United States and/or other countries.
GAMS is a trademark of Gams Development Corp.
gPROMS is a trademark of Process Systems Enterprise, Ltd.
MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please
contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000,
Fax: [phone] [email protected], www.mathworks.com.
TOMLAB is a registered trademark of Tomlab Optimization.
Library of Congress Cataloging-in-Publication Data
Biegler, Lorenz T.
Nonlinear programming : concepts, algorithms, and applications to chemical processes / Lorenz T.
Biegler.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-898717-02-0
1. Chemical processes. 2. Nonlinear programming. I. Title.
TP155.75.B54 2010
519.7’6--dc22 2010013645
is a registered trademark.
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In memory of my father
To my mother
To Lynne and to Matthew
To all my students

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Contents
Preface xiii
1 IntroductiontoProcessOptimization 1
1.1 ScopeofOptimizationProblems . . . . . . . . . . . . . . . . . . . . 1
1.2 ClassificationofOptimizationProblems . . . . . . . . . . . . . . . . 3
1.3 OptimizationApplicationsinChemicalEngineering . . . . . . . . . . 5
1.4 NonlinearProgrammingExamplesinChemicalEngineering. . . . . . 6
1.4.1 DesignofaSmallHeatExchangerNetwork . . . . . . . 7
1.4.2 Real-TimeOptimizationofaDistillationColumn . . . . 9
1.4.3 ModelPredictiveControl . . . . . . . . . . . . . . . . . 11
1.5 AMotivatingApplication . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 SummaryandNotesforFurtherReading . . . . . . . . . . . . . . . . 15
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 ConceptsofUnconstrainedOptimization 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 BasicConcepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 VectorsandMatrices . . . . . . . . . . . . . . . . . . . . 19
2.2.2 QuadraticForms . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 ClassificationofFunctions. . . . . . . . . . . . . . . . . 25
2.3 OptimalityConditions . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 DirectSearchMethods. . . . . . . . . . . . . . . . . . . 30
2.4.2 MethodsThatRequireDerivatives . . . . . . . . . . . . 33
2.5 SummaryandNotesforFurtherReading . . . . . . . . . . . . . . . . 37
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Newton-TypeMethodsforUnconstrainedOptimization 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 ModificationoftheHessianMatrix . . . . . . . . . . . . . . . . . . . 40
3.3 Quasi-NewtonMethods . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 LineSearchMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 TrustRegionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1 ConvexModelProblems . . . . . . . . . . . . . . . . . 53
3.5.2 NonconvexModelProblems. . . . . . . . . . . . . . . . 56
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viii Contents
3.6 SummaryandNotesforFurtherReading . . . . . . . . . . . . . . . . 60
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 ConceptsofConstrainedOptimization 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 ConstrainedConvexProblems. . . . . . . . . . . . . . . 64
4.2 LocalOptimalityConditions—AKinematicInterpretation . . . . . . . 68
4.3 AnalysisofKKTConditions . . . . . . . . . . . . . . . . . . . . . . 72
4.3.1 LinearlyConstrainedProblems . . . . . . . . . . . . . . 75
4.3.2 NonlinearlyConstrainedProblems . . . . . . . . . . . . 76
4.3.3 SecondOrderConditions . . . . . . . . . . . . . . . . . 79
4.4 SpecialCases:LinearandQuadraticPrograms . . . . . . . . . . . . . 84
4.4.1 DescriptionofLinearProgramming . . . . . . . . . . . . 84
4.4.2 DescriptionofQuadraticProgramming . . . . . . . . . . 85
4.4.3 PortfolioPlanningCaseStudy . . . . . . . . . . . . . . . 86
4.5 SummaryandNotesforFurtherReading . . . . . . . . . . . . . . . . 89
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5 NewtonMethodsforEqualityConstrainedOptimization 91
5.1 IntroductiontoEqualityConstrainedOptimization . . . . . . . . . . . 91
5.2 Newton’sMethodwiththeKKTMatrix . . . . . . . . . . . . . . . . 92
5.2.1 NonsingularityofKKTMatrix . . . . . . . . . . . . . . 94
5.2.2 InertiaofKKTMatrix . . . . . . . . . . . . . . . . . . . 95
5.3 TakingNewtonSteps . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Full-SpaceNewtonSteps . . . . . . . . . . . . . . . . . 96
5.3.2 Reduced-SpaceNewtonSteps . . . . . . . . . . . . . . . 99
5.4 Quasi-NewtonMethods . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4.1 AQuasi-NewtonFull-SpaceMethod . . . . . . . . . . . 103
5.4.2 AQuasi-NewtonReduced-SpaceMethod . . . . . . . . . 105
5.5 GlobalizationforConstrainedOptimization . . . . . . . . . . . . . . 109
5.5.1 ConceptsofMeritFunctions. . . . . . . . . . . . . . . . 109
5.5.2 FilterMethodConcepts . . . . . . . . . . . . . . . . . . 112
5.5.3 FilterversusMeritFunctionStrategies . . . . . . . . . . 113
5.6 LineSearchMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6.1 LineSearchwithMeritFunctions . . . . . . . . . . . . . 115
5.6.2 LineSearchFilterMethod . . . . . . . . . . . . . . . . . 119
5.7 TrustRegionMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.7.1 TrustRegionswithMeritFunctions . . . . . . . . . . . . 123
5.7.2 FilterTrustRegionMethods . . . . . . . . . . . . . . . . 126
5.8 CombiningLocalandGlobalProperties . . . . . . . . . . . . . . . . 128
5.8.1 TheMaratosEffect. . . . . . . . . . . . . . . . . . . . . 128
5.9 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . 130
5.10 NotesforFurtherReading . . . . . . . . . . . . . . . . . . . . . . . . 131
5.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6 NumericalAlgorithmsforConstrainedOptimization 133
6.1 ConstrainedNLPFormulations . . . . . . . . . . . . . . . . . . . . . 133
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6.2 SQPMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.1 TheBasic,Full-SpaceSQPAlgorithm . . . . . . . . . . 137
6.2.2 Large-ScaleSQP . . . . . . . . . . . . . . . . . . . . . . 144
6.2.3 ExtensionsofSQPMethods . . . . . . . . . . . . . . . . 148
6.3 InteriorPointMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.3.1 SolutionofthePrimal-DualEquations . . . . . . . . . . 154
6.3.2 ALineSearchFilterMethod . . . . . . . . . . . . . . . . 155
6.3.3 GlobalizationwithTrustRegionMethods . . . . . . . . . 158
6.4 NestedStrategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.4.1 GradientProjectionMethodsforBoundConstrained
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.4.2 LinearlyConstrainedAugmentedLagrangian . . . . . . . 167
6.5 NonlinearProgrammingCodes . . . . . . . . . . . . . . . . . . . . . 168
6.5.1 SQPCodes . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.5.2 InteriorPointNLPCodes . . . . . . . . . . . . . . . . . 170
6.5.3 NestedandGradientProjectionNLPCodes . . . . . . . . 171
6.5.4 PerformanceTrendsforNLPCodes . . . . . . . . . . . . 171
6.6 SummaryandConclusions . . . . . . . . . . . . . . . . . . . . . . . 175
6.7 NotesforFurtherReading . . . . . . . . . . . . . . . . . . . . . . . . 176
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7 SteadyStateProcessOptimization 181
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.2 OptimizationofProcessFlowsheets. . . . . . . . . . . . . . . . . . . 183
7.2.1 ImportanceofAccurateDerivatives . . . . . . . . . . . 188
7.2.2 AmmoniaProcessOptimization . . . . . . . . . . . . . . 191
7.3 Equation-OrientedFormulationofOptimizationModels . . . . . . . . 193
7.3.1 ReformulationoftheWilliams–OttoOptimization
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.4 Real-TimeOptimization . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.4.1 Equation-OrientedRTOModels . . . . . . . . . . . . . . 201
7.4.2 CaseStudyofHydrocrackerFractionationPlant . . . . . 203
7.5 Equation-OrientedModelswithManyDegreesofFreedom . . . . . . 206
7.6 SummaryandNotesforFurtherReading . . . . . . . . . . . . . . . . 209
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8 IntroductiontoDynamicProcessOptimization 213
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2 DynamicSystemsandOptimizationProblems . . . . . . . . . . . . . 214
8.3 OptimalityConditionsforOptimalControlProblems . . . . . . . . . 220
8.3.1 OptimalControlwithoutInequalities . . . . . . . . . . . 223
8.3.2 OptimalControlwithInequalityConstraints . . . . . . . 225
8.4 HandlingPathConstraints . . . . . . . . . . . . . . . . . . . . . . . . 232
8.4.1 TreatmentofEqualityPathConstraints . . . . . . . . . . 232
8.4.2 TreatmentofStatePathInequalities . . . . . . . . . . . . 237
8.5 SingularControlProblems . . . . . . . . . . . . . . . . . . . . . . . 239
8.6 NumericalMethodsBasedonNLPSolvers . . . . . . . . . . . . . . . 243
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