What Is Optimization Toolbox?

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3 Standard AlgorithmsOptimization OverviewOptimization techniques are used to find a set of design parameters,, that can in some way be defined as optimal. In a simplecase this might be the minimization or maximization of some systemcharacteristic that is dependent on x. In a more advanced formulation theobjective function, f(x), to be minimized or maximized, might be subjectto constraints in the form of equality constraints, ;inequality constraints,; and/or parameter bounds,.A General Problem (GP) description is stated assubject to(3-1)where x is the vector of length n design parameters, f(x) istheobjectivefunction, which returns a scalar value, and the vector function G(x) returnsavectorof length m containing the values of the equality and inequalityconstraints evaluated at x.An efficient and accurate solution to this problem depends not only on the sizeof the problem in terms of the number of constraints and design variables butalso on characteristics of the objective function and constraints. When boththe objective function and the constraints are linear functions of the designvariable, the problem is known as a Linear Programming (LP) problem.Quadratic Programming (QP) concerns the minimization or maximization of aquadratic objective function that is linearly constrained. For both the LP andQP problems, reliable solution procedures are readily available. More difficultto solve is the Nonlinear Programming (NP) problem in which the objectivefunction and constraints can be nonlinear functions of the design variables.Asolution of the NP problem generally requires an iterative procedureto establish a direction of search at each major iteration. Thisisusuallyachieved by the solution of an LP, a QP, or an unconstrained subproblem.3-4
Demos of Medium-Scale MethodsDemos of Medium-Scale MethodsFrom the MATLAB Help browser or the MathWorks Web site documentation,click the demo name to display the demo.bandemdatdemodfildemogoaldemoofficeassigntutdemoMinimization of banana functionFitting data to a curveFinite-precision filter design (requires SignalProcessing Toolbox)Goal attainment exampleBinary integer programming to solve the officeassignment problemScript for medium-scale algorithms, followingChapter 2, “Tutorial”3-5
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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
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 and 94: Large-Scale Examplesalgorithm: 'lar
Page 95 and 96: Large-Scale Examplescgiterations: 0
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: Multiobjective Optimization (p. 3-4
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
Page 145 and 146: Least-Squares OptimizationThe linea
Page 147 and 148: Nonlinear Systems of EquationsNonli
Page 149 and 150: Nonlinear Systems of Equationsand s
Page 151 and 152: Constrained OptimizationConstrained
Page 153 and 154: Constrained OptimizationGiven the p
Page 155 and 156: Constrained OptimizationUpdating th
Page 157 and 158: Constrained Optimizationis updated
Page 159 and 160: Constrained OptimizationInitializat
Page 161 and 162: Constrained OptimizationSimplex Alg
Page 163 and 164: Constrained Optimization3 Updates t
Page 165 and 166: Multiobjective OptimizationMultiobj
Page 167 and 168: Multiobjective OptimizationIn the t
Page 169 and 170: Multiobjective OptimizationThe afor
Page 171 and 172: Multiobjective OptimizationWhat is
Page 173 and 174: Multiobjective Optimization(3-51)wh
Page 175 and 176: 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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