What Is Optimization Toolbox?

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lsqnonlinJacobianMaxFunEvalsMaxIterOutputFcnPlotFcnsTolFunIf 'on', lsqnonlin uses a user-defined Jacobian(defined in fun), or Jacobian information (whenusing JacobMult), for the objective function. If'off', lsqnonlin approximates the Jacobianusing finite differences.Maximum number of function evaluations allowed.Maximum number of iterations allowed.Specify one or more user-defined functions that anoptimization function calls at each iteration. See“Output Function” on page 6-16.Plots various measures of progress while thealgorithm executes, select from predefined plotsor write your own. Specifying @optimplotx plotsthe current point; @optimplotfunccount plots thefunction count; @optimplotfval plots the functionvalue; @optimplotresnorm plots the norm of theresiduals; @optimplotstepsize plots the step size;@optimplotfirstorderopt plots the first-orderof optimality.Termination tolerance on the function value.TolX Termination tolerance on x.TypicalXTypical x values.Large-Scale Algorithm OnlyThese options are used only by the large-scale algorithm:8-172
lsqnonlinJacobMultFunction handle for Jacobian multiply function.For large-scale structured problems, this functioncomputes the Jacobian matrix product J*Y, J'*Y,or J'*(J*Y) without actually forming J. Thefunction is of the formW = jmfun(Jinfo,Y,flag,p1,p2,...)where Jinfo and the additional parametersp1,p2,... contain the matrices used to computeJ*Y (or J'*Y, orJ'*(J*Y)). The first argumentJinfo must be the same as the second argumentreturned by the objective function fun, forexample by[F,Jinfo] = fun(x)Y is a matrix that has the same number of rowsastherearedimensionsintheproblem. flagdetermines which product to compute:• If flag == 0 then W = J'*(J*Y).• If flag > 0 then W = J*Y.• If flag < 0 then W = J'*Y.In each case, J is not formed explicitly.lsqnonlin uses Jinfo to compute thepreconditioner. The optional parameters p1,p2, ... can be any additional parametersneeded by jmfun. See “Avoiding GlobalVariables via Anonymous and NestedFunctions” on page 2-20 for information onhow to supply values for these parameters.Note 'Jacobian' must be set to 'on' for Jinfoto be passed from fun to jmfun.JacobPatternSee “Nonlinear Minimization with a Dense butStructured Hessian and Equality Constraints” onpage2-61forasimilarexample.Sparsity pattern of the Jacobian for finitedifferencing. If it is not convenient to computethe Jacobian matrix J in fun, lsqnonlin can8-173
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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
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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
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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
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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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Page 407 and 408: fzeroPurposeSyntaxDescriptionFind r
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Page 411 and 412: fzerowrite an M-file called f.m.fun
Page 413 and 414: fzmultPurposeSyntaxMultiplication w
Page 415 and 416: linprogPurposeEquationSolve linear
Page 417 and 418: linproglambdaoutput-2 No feasible p
Page 419 and 420: linprogsubject toFirst, enter the c
Page 421 and 422: linprogDiagnosticsLarge-Scale Optim
Page 423 and 424: linprogthe primal objective < -1e+1
Page 425 and 426: lsqcurvefitPurposeEquationSolve non
Page 427 and 428: lsqcurvefitfunThe function you want
Page 429 and 430: lsqcurvefitoutputupperUpper bounds
Page 431 and 432: lsqcurvefitJacobianMaxFunEvalsMaxIt
Page 433 and 434: lsqcurvefitJacobPatternMaxPCGIterSp
Page 435 and 436: lsqcurvefitNote that at the time th
Page 437 and 438: lsqcurvefitof J with many nonzeros,
Page 439 and 440: lsqlinPurposeEquationSolve constrai
Page 441 and 442: lsqlinlambdaoutput3 Change in the r
Page 443 and 444: lsqlinDiagnosticsDisplayMaxIterTypi
Page 445 and 446: lsqlinPrecondBandWidthUpper bandwid
Page 447 and 448: lsqlinNotesFor problems with no con
Page 449 and 450: lsqlinReferences[1] Coleman, T.F. a
Page 451 and 452: lsqnonlinreturn a vector of values
Page 453 and 454: lsqnonlinOutputArguments“Function
Page 455: lsqnonlinalgorithm. See “Optimiza
Page 459 and 460: lsqnonlinfor(that is, F should have
Page 461 and 462: lsqnonlinand Requirements on page 2
Page 463 and 464: lsqnonnegPurposeEquationSolve nonne
Page 465 and 466: lsqnonneglambdaoutputVector contain
Page 467 and 468: optimgetPurposeSyntaxDescriptionExa
Page 469 and 470: optimsetIn the following lists, val
Page 471 and 472: optimsetLineSearchType'cubicpoly' |
Page 473 and 474: optimtoolPurposeSyntaxDescriptionTo
Page 475 and 476: quadprogPurposeEquationSolve quadra
Page 477 and 478: quadproglambdaoutput3 Change in the
Page 479 and 480: quadprogLargeScaleUse large-scale a
Page 481 and 482: quadprogTolPCGTermination tolerance
Page 483 and 484: quadprogNotesIn general quadprog lo
Page 485 and 486: 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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