What Is Optimization Toolbox?

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How to Contact The MathWorkswww.mathworks.comWebcomp.soft-sys.matlabNewsgroupwww.mathworks.com/contact_TS.html Technical Supportsuggest@mathworks.combugs@mathworks.comdoc@mathworks.comservice@mathworks.cominfo@mathworks.com508-647-7000 (Phone)508-647-7001 (Fax)The MathWorks, Inc.3 Apple Hill DriveNatick, MA 01760-2098Product enhancement suggestionsBug reportsDocumentation error reportsOrder status, license renewals, passcodesSales, pricing, and general informationFor contact information about worldwide offices, see the MathWorks Web site.<strong>Optimization</strong> <strong>Toolbox</strong> User’s Guide© COPYRIGHT 1990–2007 by The MathWorks, Inc.The software described in this document is furnished under a license agreement. The software may be usedor copied only under the terms of the license agreement. No part of this manual may be photocopied orreproduced in any form without prior written consent from The MathWorks, Inc.FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentationby, for, or through the federal government of the United States. By accepting delivery of the Program orDocumentation, the government hereby agrees that this software or documentation qualifies as commercialcomputer software or commercial computer software documentation as such terms are used or definedin FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions ofthis Agreement and only those rights specified in this Agreement, shall pertain to and govern the use,modification, reproduction, release, performance, display, and disclosure of the Program and Documentationby the federal government (or other entity acquiring for or through the federal government) and shallsupersede any conflicting contractual terms or conditions. If this License fails to meet the government’sneeds or is inconsistent in any respect with federal procurement law, the government agrees to return theProgram and Documentation, unused, to The MathWorks, Inc.TrademarksMATLAB, Simulink, Stateflow, Handle Graphics, Real-Time Workshop, and xPC TargetBoxare registered trademarks, and SimBiology, SimEvents, and SimHydraulics are trademarks ofThe MathWorks, Inc.Other product or brand names are trademarks or registered trademarks of their respectiveholders.PatentsThe MathWorks products are protected by one or more U.S. patents. Please seewww.mathworks.com/patents for more information.
Revision HistoryNovember 1990 First printingDecember 1996 Second printing For MATLAB 5January 1999 Third printing For Version 2 (Release 11)September 2000 Fourth printing For Version 2.1 (Release 12)June 2001 Online only Revised for Version 2.1.1 (Release 12.1)September 2003 Online only Revised for Version 2.3 (Release 13SP1)June 2004 Fifth printing Revised for Version 3.0 (Release 14)October 2004 Online only Revised for Version 3.0.1 (Release 14SP1)March 2005 Online only Revised for Version 3.0.2 (Release 14SP2)September 2005 Online only Revised for Version 3.0.3 (Release 14SP3)March 2006 Online only Revised for Version 3.0.4 (Release 2006a)September 2006 Sixth printing Revised for Version 3.1 (Release 2006b)March 2007 Seventh printing Revised for Version 3.1.1 (Release 2007a)
Page 1: Optimization Toolbox 3User’s Guid
Page 6 and 7: Acknowledgments
Page 8 and 9: Nonlinear Equations with Finite-Dif
Page 10 and 11: Quadratic Programming (QP) Subprobl
Page 12 and 13: Specifying the Options ............
Page 14 and 15: xivContents
Page 16 and 17: 1 Getting StartedWhat Is Optimizati
Page 18 and 19: 1 Getting StartedOptimization Examp
Page 20 and 21: 1 Getting Started[x, fval] =lsqlin(
Page 22 and 23: 2 TutorialLarge-Scale Examples (p.
Page 24 and 25: 2 TutorialMinimization (Continued)T
Page 26 and 27: 2 TutorialUsing the Optimization Fu
Page 28 and 29: 2 TutorialA choice of line search s
Page 30 and 31: 2 TutorialThe tutorial uses the fun
Page 32 and 33: 2 Tutorialfunction evaluations. See
Page 34 and 35: 2 TutorialTo restrict x inEquation2
Page 36 and 37: 2 Tutorialceq=[];DCeq = [ ];G conta
Page 38 and 39: 2 TutorialEquality Constrained Exam
Page 40 and 41: 2 Tutorialfunction y = findzero(b,
Page 42 and 43: 2 Tutorial3.7081Sharing Variables U
Page 44 and 45: 2 Tutorialcomponents.');end% Evalua
Page 46 and 47: 2 TutorialThe example produces the
Page 48 and 49: 2 TutorialClosed-Loop ResponseThe p
Page 50 and 51: 2 Tutorialfunction [Kp,Ki,Kd] = run
Page 52 and 53:
2 TutorialThe resulting closed-loop
Page 54 and 55:
2 Tutorialcalling the simulation tw
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2 TutorialThe last value shown in t
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2 TutorialStep 1: Write an M-file f
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2 TutorialLarge-Scale Examples•
Page 62 and 63:
2 TutorialNote The following table
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2 TutorialLarge-Scale Problem Cover
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2 Tutorialoptimset('Display','iter'
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2 Tutorialeither) then, in this pro
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2 TutorialNonlinear Least-Squares w
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2 TutorialThe problem is to find x
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Large-Scale Examplesto zero (for fm
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Large-Scale Examples024681012141618
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Large-Scale Examplesfval =270.4790o
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Large-Scale Examplesans =1.1885e-01
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Large-Scale ExamplesW = Hinfo*Y - V
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Large-Scale Exampleswere not the sa
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Large-Scale Examplestradeoff is ben
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Large-Scale Examplesfunction W = qp
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Large-Scale Examples% RUNQPBOX4PREC
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Large-Scale Examplesalgorithm: 'lar
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Large-Scale Examplescgiterations: 0
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Large-Scale Examplesdoes not give a
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Default Options SettingsDetermining
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Displaying Iterative OutputDisplayi
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Displaying Iterative Outputbintprog
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Displaying Iterative OutputfsolveTh
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Displaying Iterative Outputlsqnonli
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Calling an Output Function Iterativ
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Optimizing Anonymous Functions Inst
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Optimizing Anonymous Functions Inst
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Typical Problems and How to Deal wi
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Typical Problems and How to Deal wi
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3Standard AlgorithmsStandard Algori
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Multiobjective Optimization (p. 3-4
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Demos of Medium-Scale MethodsDemos
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Unconstrained OptimizationFigure 3-
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Unconstrained Optimizationvariables
Page 133 and 134:
Quasi-Newton ImplementationQuasi-Ne
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Quasi-Newton Implementationvalues o
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Quasi-Newton ImplementationMixed Cu
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Quasi-Newton ImplementationCase 4.3
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Least-Squares OptimizationIn proble
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Least-Squares OptimizationLevenberg
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Least-Squares OptimizationThe linea
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Nonlinear Systems of EquationsNonli
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Nonlinear Systems of Equationsand s
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Constrained OptimizationConstrained
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Constrained OptimizationGiven the p
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Constrained OptimizationUpdating th
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Constrained Optimizationis updated
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Constrained OptimizationInitializat
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Constrained OptimizationSimplex Alg
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Constrained Optimization3 Updates t
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Multiobjective OptimizationMultiobj
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Multiobjective OptimizationIn the t
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Multiobjective OptimizationThe afor
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Multiobjective OptimizationWhat is
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Multiobjective Optimization(3-51)wh
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Selected BibliographySelected Bibli
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Selected Bibliography[24] Han, S.P.
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4Large-Scale AlgorithmsLarge-Scale
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Trust-Region Methods for Nonlinear
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Trust-Region Methods for Nonlinear
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Preconditioned Conjugate GradientsP
Page 187 and 188:
Linearly Constrained ProblemsLinear
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Linearly Constrained ProblemsThe sc
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Quadratic ProgrammingQuadratic Prog
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Large-Scale Linear ProgrammingLarge
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Large-Scale Linear ProgrammingThe a
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Large-Scale Linear ProgrammingWhile
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Selected Bibliography[11] Zhang, Y.
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5Optimization ToolGetting Started w
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Getting Started with the Optimizati
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Selecting a SolverSelecting a Solve
Page 207 and 208:
Defining the ProblemDefining the Pr
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Defining the ProblemFunction to Min
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Defining the ProblemConstraintsLine
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Defining the ProblemM-file, or as a
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Defining the ProblemBounds are lowe
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Defining the Problemfseminf Problem
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Defining the ProblemLinear System o
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Page 223 and 224:
Defining the ProblemConstraintsBoun
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Page 227 and 228:
Defining the ProblemConstraintsWith
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Running a Problem in the Optimizati
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Specifying the Options• Xtoleranc
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Specifying the Optionsoption to a d
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Specifying the OptionsThe following
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Specifying the OptionsMultiobjectiv
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Specifying the OptionsPlot Function
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Specifying the Options• final —
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Importing and Exporting Your WorkIm
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Importing and Exporting Your WorkYo
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Optimization Tool ExamplesOptimizat
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Optimization Tool Examples6 In the
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Optimization Tool Examplesmax Line
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Optimization Tool ExamplesThe Aeq a
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6Argument and OptionsReferenceThis
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Function ArgumentsInput Arguments (
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Function ArgumentsInput Arguments (
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Function ArgumentsOutput Arguments
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Optimization OptionsOptimization Op
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Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Optimization Optionsspecifies Outpu
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Optimization OptionsoptimValues Fie
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Page 277 and 278:
Page 279 and 280:
Optimization OptionsStopping an Opt
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7Functions — By CategoryMinimizat
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Least Squares (Curve Fitting)Least
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Functions — AlphabeticalList8
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intprogx = bintprog(f,A,b,Aeq,Beq,x
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intprogBranchStrategyStrategy the a
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intprog• Verifies that no better
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intprogExampleTo minimize the funct
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colorPurposeSyntaxDescriptionColumn
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fgoalattainx = fgoalattain(fun,x0,g
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fgoalattainfunThefunctiontobeminimi
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fgoalattainfunction [c,ceq,GC,GCeq]
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fgoalattainattainfactorexitflaglamb
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fgoalattainFunValCheckGoalsExactAch
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fgoalattainExamplesConsider a linea
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fgoalattainof overattainment is met
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fgoalattainLimitationsReferencesThe
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fminbndInputArguments“Function Ar
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fminbndPlotFcnsPlots various measur
Page 317 and 318:
fminbndLimitationsReferencesThe fun
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fminconx = fmincon(fun,x0,A,b) star
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fminconfunThe function to be minimi
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fminconthen the function nonlcon mu
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fmincongradhessianlambdaoutputGradi
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fminconthe values of these fields i
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fminconHessianHessMultIf 'on', fmin
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fminconPrecondBandWidth Upper bandw
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fminconSince both constraints are l
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fmincon• A dense (or fairly dense
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fminconReferences[1] Coleman, T.F.
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fminimaxx = fminimax(fun,x,A,b,Aeq,
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fminimaxfunThe function to be minim
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fminimaxIf nonlcon returns a vector
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fminimaxlambdamaxfvaloutputStructur
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fminimaxMeritFunctionMinAbsMaxOutpu
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fminimaxx0 = [0.1; 0.1]; % Make a s
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fminimax[3] Han, S.P., “A Globall
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fminsearchInputArguments“Function
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fminsearchOutputFcnPlotFcnsTolFunSp
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fminsearcha = sqrt(2);banana = @(x)
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fminuncPurposeEquationFind minimum
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fminuncfunThefunctiontobeminimized.
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fminuncexitflaggradhessianoutputInt
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fminuncLarge-Scale and Medium-Scale
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fminuncHessianHessMultIf 'on', fmin
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fminuncPrecondBandWidthTolPCGUpper
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fminuncx0 = [1,1];[x,fval] = fminun
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fminunc“Trust-Region Methods for
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fseminfPurposeEquationFind minimum
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fseminf“Avoiding Global Variables
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fseminfoptions“Options” on page
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fseminflambdaoutput5 Magnitude of d
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fseminfOutputFcnPlotFcnsRelLineSrch
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fseminfSecond, write an M-file, myc
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fseminfThe plot command inside 'myc
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fseminfThe goal was to minimize the
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fsolvePurposeEquationSolve system o
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fsolvefunThe nonlinear system of eq
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fsolvefuncCountalgorithmcgiteration
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fsolvePlotFcnsTolFunPlots various m
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fsolveJacobPatternMaxPCGIterPrecond
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fsolve[x,fval] = fsolve(@myfun,x0,o
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fsolveYoucanformulateandsolvethepro
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fsolveLimitationsThe function to be
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fzeroPurposeSyntaxDescriptionFind r
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fzeroDisplayFunValCheckOutputFcnLev
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fzerowrite an M-file called f.m.fun
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fzmultPurposeSyntaxMultiplication w
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linprogPurposeEquationSolve linear
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linproglambdaoutput-2 No feasible p
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linprogsubject toFirst, enter the c
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linprogDiagnosticsLarge-Scale Optim
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linprogthe primal objective < -1e+1
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lsqcurvefitPurposeEquationSolve non
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lsqcurvefitfunThe function you want
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lsqcurvefitoutputupperUpper bounds
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lsqcurvefitJacobianMaxFunEvalsMaxIt
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lsqcurvefitJacobPatternMaxPCGIterSp
Page 435 and 436:
lsqcurvefitNote that at the time th
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lsqcurvefitof J with many nonzeros,
Page 439 and 440:
lsqlinPurposeEquationSolve constrai
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lsqlinlambdaoutput3 Change in the r
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lsqlinDiagnosticsDisplayMaxIterTypi
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lsqlinPrecondBandWidthUpper bandwid
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lsqlinNotesFor problems with no con
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lsqlinReferences[1] Coleman, T.F. a
Page 451 and 452:
lsqnonlinreturn a vector of values
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lsqnonlinOutputArguments“Function
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lsqnonlinalgorithm. See “Optimiza
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lsqnonlinJacobMultFunction handle f
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lsqnonlinfor(that is, F should have
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lsqnonlinand Requirements on page 2
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lsqnonnegPurposeEquationSolve nonne
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lsqnonneglambdaoutputVector contain
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optimgetPurposeSyntaxDescriptionExa
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optimsetIn the following lists, val
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optimsetLineSearchType'cubicpoly' |
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optimtoolPurposeSyntaxDescriptionTo
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quadprogPurposeEquationSolve quadra
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quadproglambdaoutput3 Change in the
Page 479 and 480:
quadprogLargeScaleUse large-scale a
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quadprogTolPCGTermination tolerance
Page 483 and 484:
quadprogNotesIn general quadprog lo
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quadprogWhen the equality constrain
Page 487 and 488:
IndexIndex ε-Constraint method 3-4
Page 489 and 490:
Indexinfeasible solution warninglin
Page 491 and 492:
Indexdescriptions 6-8possible value
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