What Is Optimization Toolbox?

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3 Standard Algorithmsto characterize the objectives. A noninferior solution is one in which animprovement in one objective requires a degradation of another. To definethis concept more precisely, consider a feasible region, ,intheparameterspace x is an element of the n-dimensional real numbers that satisfiesall the constraints, i.e.,(3-45)subject toThis allows definition of the corresponding feasible region for the objectivefunction space .The performance vector, F(x), maps parameter space into objective functionspace, as represented in two dimensions in the figure below.(3-46)Figure 3-7:Mapping from Parameter Space into Objective Function SpaceA noninferior solution point can now be defined.Definition: point is a noninferior solution if for some neighborhood ofthere does not exist a such that and3-44
Multiobjective <strong>Optimization</strong>In the two-dimensional representation of the figure below, the set ofnoninferior solutions lies on the curve between C and D. PointsA and Brepresent specific noninferior points.(3-47)Figure 3-8:Set of Noninferior SolutionsA and B are clearly noninferior solution points because an improvementin one objective, ,requiresadegradationintheotherobjective, , i.e.,.Since any point in that is not a noninferior point represents a point in whichimprovement can be attained in all the objectives, it is clear that such a pointis of no value. Multiobjective optimization is, therefore, concerned with thegeneration and selection of noninferior solution points. The techniques formultiobjective optimization are wide andvariedandallthemethodscannotbe covered within the scope of this toolbox. Subsequent sections describesome of the techniques.3-45
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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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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 and 154: 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: Multiobjective OptimizationMultiobj
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 and 206: Selecting a SolverSelecting a Solve
Page 207 and 208: Defining the ProblemDefining the Pr
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
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Defining the Problemfseminf Problem
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Defining the ProblemLinear System o
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Defining the ProblemFunction to Min
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Defining the ProblemConstraintsBoun
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Defining the ProblemFunction to Min
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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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