What Is Optimization Toolbox?

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fseminfThe vectors or matrices K1, K2, ..., Kntheta contain the semi-infiniteconstraints evaluated for a sampled set of values for the independent variablesw1, w2, ... wntheta, respectively. The two-column matrix, S, containsarecommended sampling interval for values of w1, w2, ..., wntheta, whichareusedtoevaluateK1, K2, ..., Kntheta. The ith row of S contains therecommended sampling interval for evaluating Ki. WhenKi is a vector, useonly S(i,1) (the second column can be all zeros). When Ki is a matrix, S(i,2)is used for the sampling of the rows in Ki, S(i,1) is used for the samplinginterval of the columns of Ki (see “Two-Dimensional Example” on page 8-103).On the first iteration S is NaN, so that some initial sampling interval mustbe determined by seminfcon.Note Because the functions in <strong>Optimization</strong> <strong>Toolbox</strong> only accept inputs oftype double, user-supplied objective and nonlinear constraint functions mustreturn outputs of type double.“Avoiding Global Variables via Anonymous and Nested Functions” on page2-20 explains how to parameterize seminfcon, if necessary.OutputArguments“Function Arguments” on page 6-2 contains general descriptions ofarguments returnedbyfseminf. This section provides function-specificdetails for exitflag, lambda, andoutput:exitflagInteger identifying the reason the algorithmterminated. The following lists the values ofexitflag and the corresponding reasons thealgorithm terminated.1 Function converged to a solution x.4 Magnitude of the search directionwaslessthanthespecifiedtoleranceand constraint violation was less thanoptions.TolCon.8-96
fseminflambdaoutput5 Magnitude of directional derivativewaslessthanthespecifiedtoleranceand constraint violation was less thanoptions.TolCon.0 Number of iterations exceededoptions.MaxIter or number offunction evaluations exceededoptions.FunEvals.-1 Algorithm was terminated by the outputfunction.-2 No feasible point was found.Structure containing the Lagrange multipliers atthe solution x (separated by constraint type). Thefields of the structure arelower Lower bounds lbupper Upper bounds ubineqlin Linear inequalitieseqlin Linear equalitiesineqnonlin Nonlinear inequalitieseqnonlin Nonlinear equalitiesStructure containing information about theoptimization. The fields of the structure areiterations Number of iterations takenfuncCountalgorithmstepsizeNumber of function evaluationsAlgorithm usedFinal step size taken8-97
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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
Page 329 and 330: fminconHessianHessMultIf 'on', fmin
Page 331 and 332: fminconPrecondBandWidth Upper bandw
Page 333 and 334: fminconSince both constraints are l
Page 335 and 336: fmincon• A dense (or fairly dense
Page 337 and 338: fminconReferences[1] Coleman, T.F.
Page 339 and 340: fminimaxx = fminimax(fun,x,A,b,Aeq,
Page 341 and 342: fminimaxfunThe function to be minim
Page 343 and 344: fminimaxIf nonlcon returns a vector
Page 345 and 346: fminimaxlambdamaxfvaloutputStructur
Page 347 and 348: fminimaxMeritFunctionMinAbsMaxOutpu
Page 349 and 350: fminimaxx0 = [0.1; 0.1]; % Make a s
Page 351 and 352: fminimax[3] Han, S.P., “A Globall
Page 353 and 354: fminsearchInputArguments“Function
Page 355 and 356: fminsearchOutputFcnPlotFcnsTolFunSp
Page 357 and 358: fminsearcha = sqrt(2);banana = @(x)
Page 359 and 360: fminuncPurposeEquationFind minimum
Page 361 and 362: fminuncfunThefunctiontobeminimized.
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Page 371 and 372: fminuncx0 = [1,1];[x,fval] = fminun
Page 373 and 374: fminunc“Trust-Region Methods for
Page 375 and 376: fseminfPurposeEquationFind minimum
Page 377 and 378: fseminf“Avoiding Global Variables
Page 379: fseminfoptions“Options” on page
Page 383 and 384: fseminfOutputFcnPlotFcnsRelLineSrch
Page 385 and 386: fseminfSecond, write an M-file, myc
Page 387 and 388: fseminfThe plot command inside 'myc
Page 389 and 390: fseminfThe goal was to minimize the
Page 391 and 392: fsolvePurposeEquationSolve system o
Page 393 and 394: fsolvefunThe nonlinear system of eq
Page 395 and 396: fsolvefuncCountalgorithmcgiteration
Page 397 and 398: fsolvePlotFcnsTolFunPlots various m
Page 399 and 400: 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
Page 409 and 410: fzeroDisplayFunValCheckOutputFcnLev
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
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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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