What Is Optimization Toolbox?

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6 Argument and Options ReferenceoptimValues Fields (Continued)OptimValues Field(optimValues.field)residualresnormDescriptionThe residual vector.For fsolve, residualmeans the 2-norm of theresidual squared.2-norm of the residualsquared.Returned byFunctionslsqcurvefit (B),lsqnonlin (B), fsolve(B)lsqcurvefit (B),lsqnonlin (B)searchdirection Search direction. fgoalattain (M),fmincon (M), fminimax(M), fminunc (M),fseminf (M), fsolve(M, NonlEqn-Algorithm='lm','gn'),lsqcurvefit (M),lsqnonlin (M)stepacceptStatus of the currenttrust-region step.Returns true if thecurrent trust-regionstep was successful, andfalse if the trust-regionstep was unsuccessful.fsolve (M, NonlEqn-Algorithm='dogleg')Command-LineDisplayResidualSee “DisplayingIterativeOutput” on page2-81.ResnormSee “DisplayingIterativeOutput” on page2-81.NoneNone6-22
<strong>Optimization</strong> OptionsoptimValues Fields (Continued)OptimValues Field(optimValues.field)stepsizeDescriptionCurrent step size(displacement inx). Final valueequals optimizationfunction outputoutput.stepsize.Returned byFunctionsfgoalattain (M),fmincon (B), fminimax(M), fminunc (B),fseminf (M), fsolve(B), lsqcurvefit (B),lsqnonlin (B)trustregionradius Radius of trust region. fmincon (L), fsolve(L,M,NonlEqn-Algorithm='lm','gn'),lsqcurvefit, fsolve(L), lsqnonlin (L)Command-LineDisplayStep-size orNorm of StepSee “DisplayingIterativeOutput” on page2-81.Trust-regionradiusSee “DisplayingIterativeOutput” on page2-81.Degeneracy. Thevalueofthefielddegenerate, which measures thedegeneracy of the current optimization point x, is defined as follows. First,define a vector r, of the same size as x,forwhichr(i) is the minimum distancefrom x(i) to the ith entries of the lower and upper bounds, lb and ub. Thatis,r = min(abs(ub-x, x-lb))Then the value of degenerate is the minimum entry of the vectorr + abs(grad), wheregrad is the gradient of the objective function. Thevalue of degenerate is 0 if there is an index i for which both of the followingare true:• grad(i) = 0• x(i) equals the ith entry of either the lower or upper bound.6-23
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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 ProblemFunction to Min
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Defining the ProblemConstraintsBoun
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Page 229 and 230: Running a Problem in the Optimizati
Page 231 and 232: Specifying the Options• Xtoleranc
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Page 235 and 236: Specifying the OptionsThe following
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Page 241 and 242: Specifying the Options• final —
Page 243 and 244: Importing and Exporting Your WorkIm
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Page 247 and 248: Optimization Tool ExamplesOptimizat
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Page 251 and 252: Optimization Tool Examplesmax Line
Page 253 and 254: Optimization Tool ExamplesThe Aeq a
Page 255 and 256: 6Argument and OptionsReferenceThis
Page 257 and 258: Function ArgumentsInput Arguments (
Page 259 and 260: Function ArgumentsInput Arguments (
Page 261 and 262: Function ArgumentsOutput Arguments
Page 263 and 264: Optimization OptionsOptimization Op
Page 271 and 272: Optimization Optionsspecifies Outpu
Page 273 and 274: Optimization OptionsoptimValues Fie
Page 275: Optimization OptionsoptimValues Fie
Page 279 and 280: Optimization OptionsStopping an Opt
Page 281 and 282: 7Functions — By CategoryMinimizat
Page 283 and 284: Least Squares (Curve Fitting)Least
Page 285 and 286: Functions — AlphabeticalList8
Page 287 and 288: intprogx = bintprog(f,A,b,Aeq,Beq,x
Page 289 and 290: intprogBranchStrategyStrategy the a
Page 291 and 292: intprog• Verifies that no better
Page 293 and 294: intprogExampleTo minimize the funct
Page 295 and 296: colorPurposeSyntaxDescriptionColumn
Page 297 and 298: fgoalattainx = fgoalattain(fun,x0,g
Page 299 and 300: fgoalattainfunThefunctiontobeminimi
Page 301 and 302: fgoalattainfunction [c,ceq,GC,GCeq]
Page 303 and 304: fgoalattainattainfactorexitflaglamb
Page 305 and 306: fgoalattainFunValCheckGoalsExactAch
Page 307 and 308: fgoalattainExamplesConsider a linea
Page 309 and 310: fgoalattainof overattainment is met
Page 311 and 312: fgoalattainLimitationsReferencesThe
Page 313 and 314: fminbndInputArguments“Function Ar
Page 315 and 316: 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
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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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