What Is Optimization Toolbox?

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3 Standard AlgorithmsConsider Rosenbrock’s function with an additional nonlinear inequalityconstraint, g(x),(3-26)This was solved by an SQP implementation in 96 iterations compared to 140for the unconstrained case. SQP Method on Nonlinear Linearly ConstrainedRosenbrock’s Function (Eq. 3-2) on page 3-32 shows the path to the solutionpoint starting at .Figure 3-6: SQP Method on Nonlinear Linearly Constrained Rosenbrock’sFunction (Eq. 3-2)SQP ImplementationThe SQP implementation consists of threemainstages,whicharediscussedbriefly in the following subsections:• “Updating the Hessian Matrix” on page 3-33• “Quadratic Programming Solution” on page 3-34• “Line Search and Merit Function” on page 3-383-32
Constrained <strong>Optimization</strong>Updating the Hessian MatrixAt each major iteration a positive definite quasi-Newton approximation of theHessian of the Lagrangian function, H, is calculated using the BFGS method,whereis an estimate of the Lagrange multipliers.where(3-27)Powell [35] recommends keeping the Hessian positive definite even thoughit might be positive indefinite at the solution point. A positive definiteHessian is maintained providing is positive at each update and that H isinitialized with a positive definite matrix. When is not positive, ismodified on an element-by-element basis so that . The general aimof this modification is to distort the elements of , which contribute to apositive definiteupdate,aslittleaspossible. Therefore, in the initial phase ofthe modification, the most negative element of is repeatedly halved.This procedure is continued until is greater than or equal to a smallnegative tolerance. If, after this procedure, is still not positive, modifyby adding a vector v multiplied by a constant scalar w, thatis,where(3-28)ifandotherwiseand increase w systematically untilbecomes positive.3-33
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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
Page 103 and 104: Displaying Iterative Outputbintprog
Page 105 and 106: Displaying Iterative OutputfsolveTh
Page 107 and 108: Displaying Iterative Outputlsqnonli
Page 109 and 110: Calling an Output Function Iterativ
Page 115 and 116: Optimizing Anonymous Functions Inst
Page 117 and 118: Optimizing Anonymous Functions Inst
Page 119 and 120: Typical Problems and How to Deal wi
Page 121 and 122: Typical Problems and How to Deal wi
Page 123 and 124: 3Standard AlgorithmsStandard Algori
Page 125 and 126: Multiobjective Optimization (p. 3-4
Page 127 and 128: Demos of Medium-Scale MethodsDemos
Page 129 and 130: Unconstrained OptimizationFigure 3-
Page 131 and 132: Unconstrained Optimizationvariables
Page 133 and 134: Quasi-Newton ImplementationQuasi-Ne
Page 135 and 136: Quasi-Newton Implementationvalues o
Page 137 and 138: Quasi-Newton ImplementationMixed Cu
Page 139 and 140: Quasi-Newton ImplementationCase 4.3
Page 141 and 142: Least-Squares OptimizationIn proble
Page 143 and 144: Least-Squares OptimizationLevenberg
Page 145 and 146: Least-Squares OptimizationThe linea
Page 147 and 148: Nonlinear Systems of EquationsNonli
Page 149 and 150: Nonlinear Systems of Equationsand s
Page 151 and 152: Constrained OptimizationConstrained
Page 153: Constrained OptimizationGiven the p
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
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Selecting a SolverSelecting a Solve
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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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Defining the ProblemConstraintsBoun
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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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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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