What Is Optimization Toolbox?

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fminuncY is a matrix that has the same number of rowsas there are dimensions in the problem. W =H*Y although H is not formed explicitly. fminuncuses Hinfo to compute the preconditioner. Theoptional parameters p1, p2, ... can be anyadditional parameters needed by hmfun. See“Avoiding Global Variables via Anonymous andNested Functions” on page 2-20 for informationon how to supply values for the parameters.Note 'Hessian' must be set to 'on' for Hinfoto be passed from fun to hmfun.HessPatternMaxPCGIterSee “Nonlinear Minimization with a Dense butStructured Hessian and Equality Constraints”on page 2-61 for an example.Sparsity pattern of the Hessian for finitedifferencing. If it is not convenient to computethesparseHessianmatrixH in fun, thelarge-scale method in fminunc can approximateH via sparse finite differences (of the gradient)provided the sparsity structure of H —i.e.,locations of the nonzeros—is supplied as thevalue for HessPattern. Intheworstcase,ifthestructure is unknown, you can set HessPatternto be a dense matrix and a full finite-differenceapproximation is computed at each iteration(this is the default). This can be very expensivefor large problems, so it is usually worth theeffort to determine the sparsity structure.Maximum number of PCG (preconditionedconjugate gradient) iterations (see “Algorithms”on page 8-88).8-84
fminuncPrecondBandWidthTolPCGUpper bandwidth of preconditioner for PCG. Bydefault, diagonal preconditioning is used (upperbandwidth of 0). For some problems, increasingthe bandwidth reduces the number of PCGiterations. Setting PrecondBandWidth to 'Inf'uses a direct factorization (Cholesky) ratherthan the conjugate gradients (CG). The directfactorization is computationally more expensivethan CG, but produces a better quality steptowards the solution.Termination tolerance on the PCG iteration.Medium-Scale Algorithm OnlyThese options are used only by the medium-scale algorithm:8-85
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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
Page 317 and 318: fminbndLimitationsReferencesThe fun
Page 319 and 320: fminconx = fmincon(fun,x0,A,b) star
Page 321 and 322: fminconfunThe function to be minimi
Page 323 and 324: fminconthen the function nonlcon mu
Page 325 and 326: fmincongradhessianlambdaoutputGradi
Page 327 and 328: 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.
Page 363 and 364: fminuncexitflaggradhessianoutputInt
Page 365 and 366: fminuncLarge-Scale and Medium-Scale
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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 and 380: fseminfoptions“Options” on page
Page 381 and 382: fseminflambdaoutput5 Magnitude of d
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
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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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