What Is Optimization Toolbox?

More documents

Recommendations

Info

$Producing Beautiful Documents With TEX and LATEX$

3 Standard AlgorithmsThe functions use one of two line search strategies, depending on whethergradient information is readily available or whether it must be calculatedusing a finite difference method:• When gradient information is available, the default is to use a cubicpolynomial method.• When gradient information is not available, the default is to use a mixedcubic and quadratic polynomial method.Cubic Polynomial MethodIn the proposed cubic polynomial method, a gradient and a function evaluationaremadeateveryiterationk. At each iteration an update is performed whena new point is found, , that satisfies the condition(3-12)At each iteration a step,, is attempted to form a new iterate of the form(3-13)If this step does not satisfy the condition (Equation 3-12), then is reducedto form a new step, . The usual method for this reduction is to usebisection, i.e., to continually halve the step length until a reduction is achievedin f(x). However, this procedure is slow when compared to an approachthat involves using gradient and function evaluations together with cubicinterpolation/extrapolation methods to identify estimates of step length.When a point is found that satisfies the condition (Equation 3-12), an updateis performed if is positive. If it is not, then further cubic interpolationsare performed until the univariate gradient termis sufficientlysmall so that is positive.It is usual practice to reset to unity after every iteration. However, notethat the quadratic model (Equation 3-3) is generally only a good one nearto the solution point. Therefore, is modified at each major iterationto compensate for the case when the approximation to the Hessian ismonotonically increasing or decreasing. To ensure that, as approachesthe solution point, the procedure reverts to a value of close to unity, the3-12
Quasi-Newton Implementationvalues of and are used to estimate the closeness to thesolution point and thus to control the variation in .Cubic Polynomial Line Search Procedures. After each update procedure,asteplength is attempted, following which a number of scenarios arepossible. Consideration of all the possible cases is quite complicated and sothey are represented pictorially below.For each case:• The left point on the graph represents the point .• The slope of the line bisecting each point represents the slope of theunivariate gradient, , which is always negative for the left point.• The right point is the point after a step of istakeninthedirectiond.Case 1.Case 2.3-13
Page 1 and 2:
Optimization Toolbox 3User’s Guid
Page 3:
Revision HistoryNovember 1990 First
Page 6 and 7:
Acknowledgments
Page 8 and 9:
Nonlinear Equations with Finite-Dif
Page 10 and 11:
Quadratic Programming (QP) Subprobl
Page 12 and 13:
Specifying the Options ............
Page 14 and 15:
xivContents
Page 16 and 17:
1 Getting StartedWhat Is Optimizati
Page 18 and 19:
1 Getting StartedOptimization Examp
Page 20 and 21:
1 Getting Started[x, fval] =lsqlin(
Page 22 and 23:
2 TutorialLarge-Scale Examples (p.
Page 24 and 25:
2 TutorialMinimization (Continued)T
Page 26 and 27:
2 TutorialUsing the Optimization Fu
Page 28 and 29:
2 TutorialA choice of line search s
Page 30 and 31:
2 TutorialThe tutorial uses the fun
Page 32 and 33:
2 Tutorialfunction evaluations. See
Page 34 and 35:
2 TutorialTo restrict x inEquation2
Page 36 and 37:
2 Tutorialceq=[];DCeq = [ ];G conta
Page 38 and 39:
2 TutorialEquality Constrained Exam
Page 40 and 41:
2 Tutorialfunction y = findzero(b,
Page 42 and 43:
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 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 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: Quasi-Newton ImplementationQuasi-Ne
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
Page 177 and 178: Selected Bibliography[24] Han, S.P.
Page 179 and 180: 4Large-Scale AlgorithmsLarge-Scale
Page 181 and 182: Trust-Region Methods for Nonlinear
Page 183 and 184: Trust-Region Methods for Nonlinear
Page 185 and 186:
Preconditioned Conjugate GradientsP
Page 187 and 188:
Linearly Constrained ProblemsLinear
Page 189 and 190:
Linearly Constrained ProblemsThe sc
Page 191 and 192:
Quadratic ProgrammingQuadratic Prog
Page 193 and 194:
Large-Scale Linear ProgrammingLarge
Page 195 and 196:
Large-Scale Linear ProgrammingThe a
Page 197 and 198:
Large-Scale Linear ProgrammingWhile
Page 199 and 200:
Selected Bibliography[11] Zhang, Y.
Page 201 and 202:
5Optimization ToolGetting Started w
Page 203 and 204:
Getting Started with the Optimizati
Page 205 and 206:
Selecting a SolverSelecting a Solve
Page 207 and 208:
Defining the ProblemDefining the Pr
Page 209 and 210:
Defining the ProblemFunction to Min
Page 211 and 212:
Defining the ProblemConstraintsLine
Page 213 and 214:
Defining the ProblemM-file, or as a
Page 215 and 216:
Defining the ProblemBounds are lowe
Page 217 and 218:
Defining the Problemfseminf Problem
Page 219 and 220:
Defining the ProblemLinear System o
Page 221 and 222:
Page 223 and 224:
Defining the ProblemConstraintsBoun
Page 225 and 226:
Page 227 and 228:
Defining the ProblemConstraintsWith
Page 229 and 230:
Running a Problem in the Optimizati
Page 231 and 232:
Specifying the Options• Xtoleranc
Page 233 and 234:
Specifying the Optionsoption to a d
Page 235 and 236:
Specifying the OptionsThe following
Page 237 and 238:
Specifying the OptionsMultiobjectiv
Page 239 and 240:
Specifying the OptionsPlot Function
Page 241 and 242:
Specifying the Options• final —
Page 243 and 244:
Importing and Exporting Your WorkIm
Page 245 and 246:
Importing and Exporting Your WorkYo
Page 247 and 248:
Optimization Tool ExamplesOptimizat
Page 249 and 250:
Optimization Tool Examples6 In the
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 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Optimization Optionsspecifies Outpu
Page 273 and 274:
Optimization OptionsoptimValues Fie
Page 275 and 276:
Page 277 and 278:
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
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
Page 367 and 368:
fminuncHessianHessMultIf 'on', fmin
Page 369 and 370:
fminuncPrecondBandWidthTolPCGUpper
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
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
Page 431 and 432:
lsqcurvefitJacobianMaxFunEvalsMaxIt
Page 433 and 434:
lsqcurvefitJacobPatternMaxPCGIterSp
Page 435 and 436:
lsqcurvefitNote that at the time th
Page 437 and 438:
lsqcurvefitof J with many nonzeros,
Page 439 and 440:
lsqlinPurposeEquationSolve constrai
Page 441 and 442:
lsqlinlambdaoutput3 Change in the r
Page 443 and 444:
lsqlinDiagnosticsDisplayMaxIterTypi
Page 445 and 446:
lsqlinPrecondBandWidthUpper bandwid
Page 447 and 448:
lsqlinNotesFor problems with no con
Page 449 and 450:
lsqlinReferences[1] Coleman, T.F. a
Page 451 and 452:
lsqnonlinreturn a vector of values
Page 453 and 454:
lsqnonlinOutputArguments“Function
Page 455 and 456:
lsqnonlinalgorithm. See “Optimiza
Page 457 and 458:
lsqnonlinJacobMultFunction handle f
Page 459 and 460:
lsqnonlinfor(that is, F should have
Page 461 and 462:
lsqnonlinand Requirements on page 2
Page 463 and 464:
lsqnonnegPurposeEquationSolve nonne
Page 465 and 466:
lsqnonneglambdaoutputVector contain
Page 467 and 468:
optimgetPurposeSyntaxDescriptionExa
Page 469 and 470:
optimsetIn the following lists, val
Page 471 and 472:
optimsetLineSearchType'cubicpoly' |
Page 473 and 474:
optimtoolPurposeSyntaxDescriptionTo
Page 475 and 476:
quadprogPurposeEquationSolve quadra
Page 477 and 478:
quadproglambdaoutput3 Change in the
Page 479 and 480:
quadprogLargeScaleUse large-scale a
Page 481 and 482:
quadprogTolPCGTermination tolerance
Page 483 and 484:
quadprogNotesIn general quadprog lo
Page 485 and 486:
quadprogWhen the equality constrain
Page 487 and 488:
IndexIndex ε-Constraint method 3-4
Page 489 and 490:
Indexinfeasible solution warninglin
Page 491 and 492:
Indexdescriptions 6-8possible value
show all

What Is Optimization Toolbox?

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?