What Is Optimization Toolbox?

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lsqnonlinfunThe function whose sum of squares is minimized. fun is afunction that accepts a vector x and returns a vector F, theobjective functions evaluated at x. The function fun can bespecified as a function handle for an M-file functionx = lsqnonlin(@myfun,x0)where myfun is a MATLAB function such asfunction F = myfun(x)F = ...% Compute function values at xfun can also be a function handle for an anonymousfunction.x = lsqnonlin(@(x)sin(x.*x),x0);If the user-defined values for x and F are matrices, they areconverted to a vector using linear indexing.If the Jacobian can also be computed and the Jacobianoption is 'on', setbyoptions = optimset('Jacobian','on')then the function fun must return, in a second outputargument, the Jacobian value J, amatrix,atx. Notethat by checking the value of nargout the function canavoid computing J when fun is called with only one outputargument (in the case where the optimization algorithmonly needs the value of F but not J).function [F,J] = myfun(x)F = ...% Objective function values at xif nargout > 1 % Two output argumentsJ = ... % Jacobian of the function evaluated at xend8-168optionsIf fun returns a vector (matrix) of m components and x haslength n, wheren is the length of x0, then the Jacobian J isan m-by-n matrix where J(i,j) is the partial derivative ofF(i) with respect to x(j). (Note that the Jacobian J is thetranspose of the gradient of F.)“Options” on page 8-170 provides the function-specificdetails for the options values.
lsqnonlinOutputArguments“Function Arguments” on page 6-2 contains general descriptionsof arguments returned by lsqnonlin. This section providesfunction-specific details for exitflag, lambda, andoutput:exitflaglambdaInteger identifying the reason the algorithmterminated. The following lists the values ofexitflag and the corresponding reasons thealgorithm terminated:1 Function converged to a solution x.2 Change in x waslessthanthespecified tolerance.3 Change in the residual was lessthan the specified tolerance.4 Magnitude of search directionwas smaller than the specifiedtolerance.0 Number of iterations exceededoptions.MaxIter or number offunction evaluations exceededoptions.FunEvals.-1 Algorithm was terminated by theoutput function.-2 Problem is infeasible: the boundslb and ub are inconsistent.-4 Line search could not sufficientlydecrease the residual along thecurrent search direction.Structure containing the Lagrange multipliers atthe solution x (separated by constraint type). Thefields arelowerLower bounds lb8-169
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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
Page 401 and 402: fsolve[x,fval] = fsolve(@myfun,x0,o
Page 403 and 404: fsolveYoucanformulateandsolvethepro
Page 405 and 406: fsolveLimitationsThe function to be
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: lsqnonlinreturn a vector of values
Page 455 and 456: lsqnonlinalgorithm. See “Optimiza
Page 457 and 458: lsqnonlinJacobMultFunction handle f
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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