What Is Optimization Toolbox?

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2 Tutorialload sc50bThis problem in sc50b.mat has 48 variables, 30 inequalities, and 20 equalities.You can use linprog to solve the problem:[x,fval,exitflag,output] = ...linprog(f,A,b,Aeq,beq,lb,[],[],optimset('Display','iter'));Because the iterative display was set using optimset, theresultsdisplayedareResiduals: Primal Dual Duality TotalInfeas Infeas Gap RelA*x-b A'*y+z-f x'*z Error---------------------------------------------------Iter 0: 1.50e+003 2.19e+001 1.91e+004 1.00e+002Iter 1: 1.15e+002 3.18e-015 3.62e+003 9.90e-001Iter 2: 8.32e-013 1.96e-015 4.32e+002 9.48e-001Iter 3: 3.47e-012 7.49e-015 7.78e+001 6.88e-001Iter 4: 5.66e-011 1.16e-015 2.38e+001 2.69e-001Iter 5: 1.13e-010 3.67e-015 5.05e+000 6.89e-002Iter 6: 5.03e-011 1.21e-016 1.64e-001 2.34e-003Iter 7: 5.75e-012 1.12e-016 1.09e-005 1.55e-007Iter 8: 8.08e-014 5.67e-013 1.09e-011 3.82e-012<strong>Optimization</strong> terminated.For this problem, the large-scale linear programming algorithm quicklyreduces the scaled residuals below the default tolerance of 1e-08.The exitflag value is positive, telling you linprog converged. You canalso get the final function value in fval and the number of iterations inoutput.iterations:exitflag =1fval =-70.0000output =iterations: 8algorithm: 'large-scale: interior point'2-74
Large-Scale Examplescgiterations: 0message: '<strong>Optimization</strong> terminated.'Linear Programming with Dense Columns in theEqualitiesThe problem isand you can load the matrices and vectors Aeq, beq, f, lb, andub into theMATLAB workspace withload densecolumnsThe problem in densecolumns.mat has 1677 variables and 627 equalities withlower bounds on all the variables, and upper bounds on 399 of the variables.The equality matrix Aeq has dense columns among its first 25 columns, whichis easy to see with a spy plot:spy(Aeq)You can use linprog to solve the problem:[x,fval,exitflag,output] = ...linprog(f,[],[],Aeq,beq,lb,ub,[],optimset('Display','iter'));2-75
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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
Page 44 and 45: 2 Tutorialcomponents.');end% Evalua
Page 46 and 47: 2 TutorialThe example produces the
Page 48 and 49: 2 TutorialClosed-Loop ResponseThe p
Page 50 and 51: 2 Tutorialfunction [Kp,Ki,Kd] = run
Page 52 and 53: 2 TutorialThe resulting closed-loop
Page 54 and 55: 2 Tutorialcalling the simulation tw
Page 56 and 57: 2 TutorialThe last value shown in t
Page 58 and 59: 2 TutorialStep 1: Write an M-file f
Page 60 and 61: 2 TutorialLarge-Scale Examples•
Page 62 and 63: 2 TutorialNote The following table
Page 64 and 65: 2 TutorialLarge-Scale Problem Cover
Page 66 and 67: 2 Tutorialoptimset('Display','iter'
Page 68 and 69: 2 Tutorialeither) then, in this pro
Page 70 and 71: 2 TutorialNonlinear Least-Squares w
Page 72: 2 TutorialThe problem is to find x
Page 75 and 76: Large-Scale Examplesto zero (for fm
Page 77 and 78: Large-Scale Examples024681012141618
Page 79 and 80: Large-Scale Examplesfval =270.4790o
Page 81 and 82: Large-Scale Examplesans =1.1885e-01
Page 83 and 84: Large-Scale ExamplesW = Hinfo*Y - V
Page 85 and 86: Large-Scale Exampleswere not the sa
Page 87 and 88: Large-Scale Examplestradeoff is ben
Page 89 and 90: Large-Scale Examplesfunction W = qp
Page 91 and 92: Large-Scale Examples% RUNQPBOX4PREC
Page 93: Large-Scale Examplesalgorithm: 'lar
Page 97 and 98: Large-Scale Examplesdoes not give a
Page 99 and 100: Default Options SettingsDetermining
Page 101 and 102: Displaying Iterative OutputDisplayi
Page 103 and 104: Displaying Iterative Outputbintprog
Page 105 and 106: Displaying Iterative OutputfsolveTh
Page 107 and 108: Displaying Iterative Outputlsqnonli
Page 109 and 110: Calling an Output Function Iterativ
Page 115 and 116: Optimizing Anonymous Functions Inst
Page 117 and 118: Optimizing Anonymous Functions Inst
Page 119 and 120: Typical Problems and How to Deal wi
Page 121 and 122: Typical Problems and How to Deal wi
Page 123 and 124: 3Standard AlgorithmsStandard Algori
Page 125 and 126: Multiobjective Optimization (p. 3-4
Page 127 and 128: Demos of Medium-Scale MethodsDemos
Page 129 and 130: Unconstrained OptimizationFigure 3-
Page 131 and 132: Unconstrained Optimizationvariables
Page 133 and 134: Quasi-Newton ImplementationQuasi-Ne
Page 135 and 136: Quasi-Newton Implementationvalues o
Page 137 and 138: Quasi-Newton ImplementationMixed Cu
Page 139 and 140: Quasi-Newton ImplementationCase 4.3
Page 141 and 142: Least-Squares OptimizationIn proble
Page 143 and 144: 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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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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