What Is Optimization Toolbox?

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5 <strong>Optimization</strong> ToolSolverfminuncfseminffsolvefzerolinproglsqcurvefitlsqlinlsqnonlinlsqnonnegquadprogDescriptionFind a minimum of an unconstrained nonlinearmultivariable functionFind the minimum of a semi-infinitely constrainednonlinear multivariable functionSolve a system of nonlinear equationsFind a root (zero) of a nonlinear equationSolve a linear programming problemSolve a nonlinear curve-fitting (data-fitting) problemin the least-squares senseSolve a constrained linear least-squares problemSolve a nonlinear least-squares problemFind the minimum of a nonnegative least-squaresproblemSolve a quadratic programming problem5-6
Defining the ProblemDefining the Problem• “Introduction” on page 5-7• “bintprog Problem Setup” on page 5-8• “fgoalattain Problem Setup” on page 5-10• “fminbnd Problem Setup” on page 5-11• “fmincon Problem Setup” on page 5-12• “fminimax Problem Setup” on page 5-14• “fminsearch Problem Setup” on page 5-15• “fminunc Problem Setup” on page 5-16• “fseminf Problem Setup” on page 5-17• “fsolve Problem Setup” on page 5-18• “fzero Problem Setup” on page 5-19• “linprog Problem Setup” on page 5-20• “lsqcurvefit Problem Setup” on page 5-22• “lsqlin Problem Setup” on page 5-23• “lsqnonlin Problem Setup” on page 5-24• “lsqnonneg Problem Setup” on page 5-25• “quadprog Problem Setup” on page 5-26IntroductionAfter selecting a solver, define the problem by specifying the function tominimize and other problem fields, such as the start point and constraints,that are specific to the solver selected. Select a solver from the following listfor a description of the problem definition relevant to that solver.Problem Setupbintprogfgoalattain5-7
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Optimization Toolbox 3User’s Guid
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Revision HistoryNovember 1990 First
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Acknowledgments
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Nonlinear Equations with Finite-Dif
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Quadratic Programming (QP) Subprobl
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Specifying the Options ............
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xivContents
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1 Getting StartedWhat Is Optimizati
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1 Getting StartedOptimization Examp
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1 Getting Started[x, fval] =lsqlin(
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2 TutorialLarge-Scale Examples (p.
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2 TutorialMinimization (Continued)T
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2 TutorialUsing the Optimization Fu
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2 TutorialA choice of line search s
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2 TutorialThe tutorial uses the fun
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2 Tutorialfunction evaluations. See
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2 TutorialTo restrict x inEquation2
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2 Tutorialceq=[];DCeq = [ ];G conta
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2 TutorialEquality Constrained Exam
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2 Tutorialfunction y = findzero(b,
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2 Tutorial3.7081Sharing Variables U
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2 Tutorialcomponents.');end% Evalua
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2 TutorialThe example produces the
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2 TutorialClosed-Loop ResponseThe p
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2 Tutorialfunction [Kp,Ki,Kd] = run
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2 TutorialThe resulting closed-loop
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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•
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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
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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
Page 155 and 156: Constrained OptimizationUpdating th
Page 157 and 158: Constrained Optimizationis updated
Page 159 and 160: Constrained OptimizationInitializat
Page 161 and 162: Constrained OptimizationSimplex Alg
Page 163 and 164: Constrained Optimization3 Updates t
Page 165 and 166: Multiobjective OptimizationMultiobj
Page 167 and 168: Multiobjective OptimizationIn the t
Page 169 and 170: Multiobjective OptimizationThe afor
Page 171 and 172: Multiobjective OptimizationWhat is
Page 173 and 174: Multiobjective Optimization(3-51)wh
Page 175 and 176: Selected BibliographySelected Bibli
Page 177 and 178: Selected Bibliography[24] Han, S.P.
Page 179 and 180: 4Large-Scale AlgorithmsLarge-Scale
Page 181 and 182: Trust-Region Methods for Nonlinear
Page 183 and 184: Trust-Region Methods for Nonlinear
Page 185 and 186: Preconditioned Conjugate GradientsP
Page 187 and 188: Linearly Constrained ProblemsLinear
Page 189 and 190: Linearly Constrained ProblemsThe sc
Page 191 and 192: Quadratic ProgrammingQuadratic Prog
Page 193 and 194: Large-Scale Linear ProgrammingLarge
Page 195 and 196: Large-Scale Linear ProgrammingThe a
Page 197 and 198: Large-Scale Linear ProgrammingWhile
Page 199 and 200: Selected Bibliography[11] Zhang, Y.
Page 201 and 202: 5Optimization ToolGetting Started w
Page 203 and 204: Getting Started with the Optimizati
Page 205: Selecting a SolverSelecting a Solve
Page 209 and 210: Defining the ProblemFunction to Min
Page 211 and 212: Defining the ProblemConstraintsLine
Page 213 and 214: Defining the ProblemM-file, or as a
Page 215 and 216: Defining the ProblemBounds are lowe
Page 217 and 218: Defining the Problemfseminf Problem
Page 219 and 220: Defining the ProblemLinear System o
Page 223 and 224: Defining the ProblemConstraintsBoun
Page 227 and 228: Defining the ProblemConstraintsWith
Page 229 and 230: Running a Problem in the Optimizati
Page 231 and 232: Specifying the Options• Xtoleranc
Page 233 and 234: Specifying the Optionsoption to a d
Page 235 and 236: Specifying the OptionsThe following
Page 237 and 238: Specifying the OptionsMultiobjectiv
Page 239 and 240: Specifying the OptionsPlot Function
Page 241 and 242: Specifying the Options• final —
Page 243 and 244: Importing and Exporting Your WorkIm
Page 245 and 246: Importing and Exporting Your WorkYo
Page 247 and 248: Optimization Tool ExamplesOptimizat
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Page 251 and 252: Optimization Tool Examplesmax Line
Page 253 and 254: Optimization Tool ExamplesThe Aeq a
Page 255 and 256: 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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Optimization Optionsspecifies Outpu
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Optimization OptionsoptimValues Fie
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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
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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
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lsqcurvefitNote that at the time th
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lsqcurvefitof J with many nonzeros,
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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
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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
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quadprogLargeScaleUse large-scale a
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quadprogTolPCGTermination tolerance
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quadprogNotesIn general quadprog lo
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quadprogWhen the equality constrain
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IndexIndex ε-Constraint method 3-4
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Indexinfeasible solution warninglin
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Indexdescriptions 6-8possible value
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