What Is Optimization Toolbox?

More documents

Recommendations

Info

$Producing Beautiful Documents With TEX and LATEX$

2 TutorialNorm of First-orderIteration f(x) step optimality CG-iterations1 1997.07 1 555 02 1072.56 6.31716 377 13 480.232 8.19554 159 24 136.861 10.3015 59.5 25 44.3708 9.04697 16.3 26 44.3708 100 16.3 27 44.3708 25 16.3 08 -8.90967 6.25 28.5 09 -318.486 12.5 107 110 -318.486 12.5 107 111 -415.445 3.125 73.9 012 -561.688 3.125 47.4 213 -785.326 6.25 126 314 -785.326 4.30584 126 515 -804.414 1.07646 26.9 016 -822.399 2.16965 2.8 317 -823.173 0.40754 1.34 318 -823.241 0.154885 0.555 319 -823.246 0.0518407 0.214 520 -823.246 0.00977601 0.00724 6<strong>Optimization</strong> terminated successfully:Relative function value changing by less than OPTIONS.TolFunConvergence is rapid for a problem of this size with the PCG iteration costincreasing modestly as the optimization progresses. Feasibility of the equalityconstraints is maintained at the solutionnorm(Aeq*x-beq)ans =1.9854e-013PreconditioningIn this example, fmincon cannot use H to compute a preconditioner because Honly exists implicitly. Instead of H, fmincon uses Hinfo, the third argumentreturned by brownvv, to compute a preconditioner. Hinfo is a good choicebecause it is the same size as H and approximates H to some degree. If Hinfo2-64
Large-Scale Exampleswere not the same size as H, fmincon would compute a preconditioner basedon some diagonal scaling matrices determined from the algorithm. Typically,this would not perform as well.Quadratic Minimization with Bound ConstraintsTo minimize a large-scale quadratic with upper and lower bounds, you canuse the quadprog function.The problem stored in the MAT-file qpbox1.mat is a positive definitequadratic, and the Hessian matrix H is tridiagonal, subject to upper (ub) andlower (lb) bounds.Step 1: Load the Hessian and define f, lb, and ub.load qpbox1 % Get Hlb = zeros(400,1); lb(400) = -inf;ub = 0.9*ones(400,1); ub(400) = inf;f = zeros(400,1); f([1 400]) = -2;Step 2: Call a quadratic minimization routine with a startingpoint xstart.xstart = 0.5*ones(400,1);[x,fval,exitflag,output] = ...quadprog(H,f,[],[],[],[],lb,ub,xstart);Looking at the resulting values of exitflag and output,exitflag =3output =iterations: 20algorithm: 'large-scale: reflective trust-region'firstorderopt: 9.6206e-006cgiterations: 1834message: [1x206 char]you can see that while convergence occurred in 20 iterations, the high numberof CG iterations indicates that the cost of the linear system solve is high. In2-65
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: Large-Scale ExamplesW = Hinfo*Y - V
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 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
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?