Actual source code: ntl.c
petsc-3.7.1 2016-05-15
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      1: #include <../src/tao/matrix/lmvmmat.h>
      2: #include <../src/tao/unconstrained/impls/ntl/ntl.h>

      4: #include <petscksp.h>
      5: #include <petscpc.h>
      6: #include <petsc/private/kspimpl.h>
      7: #include <petsc/private/pcimpl.h>

      9: #define NTL_KSP_NASH    0
     10: #define NTL_KSP_STCG    1
     11: #define NTL_KSP_GLTR    2
     12: #define NTL_KSP_TYPES   3

     14: #define NTL_PC_NONE     0
     15: #define NTL_PC_AHESS    1
     16: #define NTL_PC_BFGS     2
     17: #define NTL_PC_PETSC    3
     18: #define NTL_PC_TYPES    4

     20: #define BFGS_SCALE_AHESS        0
     21: #define BFGS_SCALE_BFGS         1
     22: #define BFGS_SCALE_TYPES        2

     24: #define NTL_INIT_CONSTANT         0
     25: #define NTL_INIT_DIRECTION        1
     26: #define NTL_INIT_INTERPOLATION    2
     27: #define NTL_INIT_TYPES            3

     29: #define NTL_UPDATE_REDUCTION      0
     30: #define NTL_UPDATE_INTERPOLATION  1
     31: #define NTL_UPDATE_TYPES          2

     33: static const char *NTL_KSP[64] = {"nash", "stcg", "gltr"};

     35: static const char *NTL_PC[64] = {"none", "ahess", "bfgs", "petsc"};

     37: static const char *BFGS_SCALE[64] = {"ahess", "bfgs"};

     39: static const char *NTL_INIT[64] = {"constant", "direction", "interpolation"};

     41: static const char *NTL_UPDATE[64] = {"reduction", "interpolation"};

     43: /* Routine for BFGS preconditioner */

     47: static PetscErrorCode MatLMVMSolveShell(PC pc, Vec b, Vec x)
     48: {
     50:   Mat            M;

     56:   PCShellGetContext(pc,(void**)&M);
     57:   MatLMVMSolve(M, b, x);
     58:   return(0);
     59: }

     61: /* Implements Newton's Method with a trust-region, line-search approach for
     62:    solving unconstrained minimization problems.  A More'-Thuente line search
     63:    is used to guarantee that the bfgs preconditioner remains positive
     64:    definite. */

     66: #define NTL_NEWTON              0
     67: #define NTL_BFGS                1
     68: #define NTL_SCALED_GRADIENT     2
     69: #define NTL_GRADIENT            3

     73: static PetscErrorCode TaoSolve_NTL(Tao tao)
     74: {
     75:   TAO_NTL                      *tl = (TAO_NTL *)tao->data;
     76:   PC                           pc;
     77:   KSPConvergedReason           ksp_reason;
     78:   TaoConvergedReason           reason;
     79:   TaoLineSearchConvergedReason ls_reason;

     81:   PetscReal                    fmin, ftrial, prered, actred, kappa, sigma;
     82:   PetscReal                    tau, tau_1, tau_2, tau_max, tau_min, max_radius;
     83:   PetscReal                    f, fold, gdx, gnorm;
     84:   PetscReal                    step = 1.0;

     86:   PetscReal                    delta;
     87:   PetscReal                    norm_d = 0.0;
     88:   PetscErrorCode               ierr;
     89:   PetscInt                     stepType;
     90:   PetscInt                     its;

     92:   PetscInt                     bfgsUpdates = 0;
     93:   PetscInt                     needH;

     95:   PetscInt                     i_max = 5;
     96:   PetscInt                     j_max = 1;
     97:   PetscInt                     i, j, n, N;

     99:   PetscInt                     tr_reject;

    102:   if (tao->XL || tao->XU || tao->ops->computebounds) {
    103:     PetscPrintf(((PetscObject)tao)->comm,"WARNING: Variable bounds have been set but will be ignored by ntl algorithm\n");
    104:   }

    106:   /* Initialize trust-region radius */
    107:   tao->trust = tao->trust0;

    109:   /* Modify the radius if it is too large or small */
    110:   tao->trust = PetscMax(tao->trust, tl->min_radius);
    111:   tao->trust = PetscMin(tao->trust, tl->max_radius);

    113:   if (NTL_PC_BFGS == tl->pc_type && !tl->M) {
    114:     VecGetLocalSize(tao->solution,&n);
    115:     VecGetSize(tao->solution,&N);
    116:     MatCreateLMVM(((PetscObject)tao)->comm,n,N,&tl->M);
    117:     MatLMVMAllocateVectors(tl->M,tao->solution);
    118:   }

    120:   /* Check convergence criteria */
    121:   TaoComputeObjectiveAndGradient(tao, tao->solution, &f, tao->gradient);
    122:   VecNorm(tao->gradient, NORM_2, &gnorm);
    123:   if (PetscIsInfOrNanReal(f) || PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
    124:   needH = 1;

    126:   TaoMonitor(tao, tao->niter, f, gnorm, 0.0, 1.0, &reason);
    127:   if (reason != TAO_CONTINUE_ITERATING) return(0);

    129:   /* Create vectors for the limited memory preconditioner */
    130:   if ((NTL_PC_BFGS == tl->pc_type) && (BFGS_SCALE_BFGS != tl->bfgs_scale_type)) {
    131:     if (!tl->Diag) {
    132:       VecDuplicate(tao->solution, &tl->Diag);
    133:     }
    134:   }

    136:   /* Modify the linear solver to a conjugate gradient method */
    137:   switch(tl->ksp_type) {
    138:   case NTL_KSP_NASH:
    139:     KSPSetType(tao->ksp, KSPNASH);
    140:     KSPSetFromOptions(tao->ksp);
    141:     break;

    143:   case NTL_KSP_STCG:
    144:     KSPSetType(tao->ksp, KSPSTCG);
    145:     KSPSetFromOptions(tao->ksp);
    146:     break;

    148:   default:
    149:     KSPSetType(tao->ksp, KSPGLTR);
    150:     KSPSetFromOptions(tao->ksp);
    151:     break;
    152:   }

    154:   /* Modify the preconditioner to use the bfgs approximation */
    155:   KSPGetPC(tao->ksp, &pc);
    156:   switch(tl->pc_type) {
    157:   case NTL_PC_NONE:
    158:     PCSetType(pc, PCNONE);
    159:     PCSetFromOptions(pc);
    160:     break;

    162:   case NTL_PC_AHESS:
    163:     PCSetType(pc, PCJACOBI);
    164:     PCSetFromOptions(pc);
    165:     PCJacobiSetUseAbs(pc,PETSC_TRUE);
    166:     break;

    168:   case NTL_PC_BFGS:
    169:     PCSetType(pc, PCSHELL);
    170:     PCSetFromOptions(pc);
    171:     PCShellSetName(pc, "bfgs");
    172:     PCShellSetContext(pc, tl->M);
    173:     PCShellSetApply(pc, MatLMVMSolveShell);
    174:     break;

    176:   default:
    177:     /* Use the pc method set by pc_type */
    178:     break;
    179:   }

    181:   /* Initialize trust-region radius.  The initialization is only performed
    182:      when we are using Steihaug-Toint or the Generalized Lanczos method. */
    183:   switch(tl->init_type) {
    184:   case NTL_INIT_CONSTANT:
    185:     /* Use the initial radius specified */
    186:     break;

    188:   case NTL_INIT_INTERPOLATION:
    189:     /* Use the initial radius specified */
    190:     max_radius = 0.0;

    192:     for (j = 0; j < j_max; ++j) {
    193:       fmin = f;
    194:       sigma = 0.0;

    196:       if (needH) {
    197:         TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);
    198:         needH = 0;
    199:       }

    201:       for (i = 0; i < i_max; ++i) {
    202:         VecCopy(tao->solution, tl->W);
    203:         VecAXPY(tl->W, -tao->trust/gnorm, tao->gradient);

    205:         TaoComputeObjective(tao, tl->W, &ftrial);
    206:         if (PetscIsInfOrNanReal(ftrial)) {
    207:           tau = tl->gamma1_i;
    208:         } else {
    209:           if (ftrial < fmin) {
    210:             fmin = ftrial;
    211:             sigma = -tao->trust / gnorm;
    212:           }

    214:           MatMult(tao->hessian, tao->gradient, tao->stepdirection);
    215:           VecDot(tao->gradient, tao->stepdirection, &prered);

    217:           prered = tao->trust * (gnorm - 0.5 * tao->trust * prered / (gnorm * gnorm));
    218:           actred = f - ftrial;
    219:           if ((PetscAbsScalar(actred) <= tl->epsilon) && (PetscAbsScalar(prered) <= tl->epsilon)) {
    220:             kappa = 1.0;
    221:           } else {
    222:             kappa = actred / prered;
    223:           }

    225:           tau_1 = tl->theta_i * gnorm * tao->trust / (tl->theta_i * gnorm * tao->trust + (1.0 - tl->theta_i) * prered - actred);
    226:           tau_2 = tl->theta_i * gnorm * tao->trust / (tl->theta_i * gnorm * tao->trust - (1.0 + tl->theta_i) * prered + actred);
    227:           tau_min = PetscMin(tau_1, tau_2);
    228:           tau_max = PetscMax(tau_1, tau_2);

    230:           if (PetscAbsScalar(kappa - 1.0) <= tl->mu1_i) {
    231:             /* Great agreement */
    232:             max_radius = PetscMax(max_radius, tao->trust);

    234:             if (tau_max < 1.0) {
    235:               tau = tl->gamma3_i;
    236:             } else if (tau_max > tl->gamma4_i) {
    237:               tau = tl->gamma4_i;
    238:             } else if (tau_1 >= 1.0 && tau_1 <= tl->gamma4_i && tau_2 < 1.0) {
    239:               tau = tau_1;
    240:             } else if (tau_2 >= 1.0 && tau_2 <= tl->gamma4_i && tau_1 < 1.0) {
    241:               tau = tau_2;
    242:             } else {
    243:               tau = tau_max;
    244:             }
    245:           } else if (PetscAbsScalar(kappa - 1.0) <= tl->mu2_i) {
    246:             /* Good agreement */
    247:             max_radius = PetscMax(max_radius, tao->trust);

    249:             if (tau_max < tl->gamma2_i) {
    250:               tau = tl->gamma2_i;
    251:             } else if (tau_max > tl->gamma3_i) {
    252:               tau = tl->gamma3_i;
    253:             } else {
    254:               tau = tau_max;
    255:             }
    256:           } else {
    257:             /* Not good agreement */
    258:             if (tau_min > 1.0) {
    259:               tau = tl->gamma2_i;
    260:             } else if (tau_max < tl->gamma1_i) {
    261:               tau = tl->gamma1_i;
    262:             } else if ((tau_min < tl->gamma1_i) && (tau_max >= 1.0)) {
    263:               tau = tl->gamma1_i;
    264:             } else if ((tau_1 >= tl->gamma1_i) && (tau_1 < 1.0) &&  ((tau_2 < tl->gamma1_i) || (tau_2 >= 1.0))) {
    265:               tau = tau_1;
    266:             } else if ((tau_2 >= tl->gamma1_i) && (tau_2 < 1.0) &&  ((tau_1 < tl->gamma1_i) || (tau_2 >= 1.0))) {
    267:               tau = tau_2;
    268:             } else {
    269:               tau = tau_max;
    270:             }
    271:           }
    272:         }
    273:         tao->trust = tau * tao->trust;
    274:       }

    276:       if (fmin < f) {
    277:         f = fmin;
    278:         VecAXPY(tao->solution, sigma, tao->gradient);
    279:         TaoComputeGradient(tao, tao->solution, tao->gradient);

    281:         VecNorm(tao->gradient, NORM_2, &gnorm);
    282:         if (PetscIsInfOrNanReal(f) || PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1, "User provided compute function generated Inf or NaN");
    283:         needH = 1;

    285:         TaoMonitor(tao, tao->niter, f, gnorm, 0.0, 1.0, &reason);
    286:         if (reason != TAO_CONTINUE_ITERATING) return(0);
    287:       }
    288:     }
    289:     tao->trust = PetscMax(tao->trust, max_radius);

    291:     /* Modify the radius if it is too large or small */
    292:     tao->trust = PetscMax(tao->trust, tl->min_radius);
    293:     tao->trust = PetscMin(tao->trust, tl->max_radius);
    294:     break;

    296:   default:
    297:     /* Norm of the first direction will initialize radius */
    298:     tao->trust = 0.0;
    299:     break;
    300:   }

    302:   /* Set initial scaling for the BFGS preconditioner
    303:      This step is done after computing the initial trust-region radius
    304:      since the function value may have decreased */
    305:   if (NTL_PC_BFGS == tl->pc_type) {
    306:     if (f != 0.0) {
    307:       delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    308:     } else {
    309:       delta = 2.0 / (gnorm*gnorm);
    310:     }
    311:     MatLMVMSetDelta(tl->M, delta);
    312:   }

    314:   /* Set counter for gradient/reset steps */
    315:   tl->ntrust = 0;
    316:   tl->newt = 0;
    317:   tl->bfgs = 0;
    318:   tl->sgrad = 0;
    319:   tl->grad = 0;

    321:   /* Have not converged; continue with Newton method */
    322:   while (reason == TAO_CONTINUE_ITERATING) {
    323:     ++tao->niter;
    324:     tao->ksp_its=0;
    325:     /* Compute the Hessian */
    326:     if (needH) {
    327:       TaoComputeHessian(tao,tao->solution,tao->hessian,tao->hessian_pre);
    328:     }

    330:     if (NTL_PC_BFGS == tl->pc_type) {
    331:       if (BFGS_SCALE_AHESS == tl->bfgs_scale_type) {
    332:         /* Obtain diagonal for the bfgs preconditioner */
    333:         MatGetDiagonal(tao->hessian, tl->Diag);
    334:         VecAbs(tl->Diag);
    335:         VecReciprocal(tl->Diag);
    336:         MatLMVMSetScale(tl->M, tl->Diag);
    337:       }

    339:       /* Update the limited memory preconditioner */
    340:       MatLMVMUpdate(tl->M,tao->solution, tao->gradient);
    341:       ++bfgsUpdates;
    342:     }
    343:     KSPSetOperators(tao->ksp, tao->hessian, tao->hessian_pre);
    344:     /* Solve the Newton system of equations */
    345:     if (NTL_KSP_NASH == tl->ksp_type) {
    346:       KSPNASHSetRadius(tao->ksp,tl->max_radius);
    347:       KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    348:       KSPGetIterationNumber(tao->ksp,&its);
    349:       tao->ksp_its+=its;
    350:       tao->ksp_tot_its+=its;
    351:       KSPNASHGetNormD(tao->ksp, &norm_d);
    352:     } else if (NTL_KSP_STCG == tl->ksp_type) {
    353:       KSPSTCGSetRadius(tao->ksp,tl->max_radius);
    354:       KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    355:       KSPGetIterationNumber(tao->ksp,&its);
    356:       tao->ksp_its+=its;
    357:       tao->ksp_tot_its+=its;
    358:       KSPSTCGGetNormD(tao->ksp, &norm_d);
    359:     } else { /* NTL_KSP_GLTR */
    360:       KSPGLTRSetRadius(tao->ksp,tl->max_radius);
    361:       KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    362:       KSPGetIterationNumber(tao->ksp,&its);
    363:       tao->ksp_its+=its;
    364:       tao->ksp_tot_its+=its;
    365:       KSPGLTRGetNormD(tao->ksp, &norm_d);
    366:     }

    368:     if (0.0 == tao->trust) {
    369:       /* Radius was uninitialized; use the norm of the direction */
    370:       if (norm_d > 0.0) {
    371:         tao->trust = norm_d;

    373:         /* Modify the radius if it is too large or small */
    374:         tao->trust = PetscMax(tao->trust, tl->min_radius);
    375:         tao->trust = PetscMin(tao->trust, tl->max_radius);
    376:       } else {
    377:         /* The direction was bad; set radius to default value and re-solve
    378:            the trust-region subproblem to get a direction */
    379:         tao->trust = tao->trust0;

    381:         /* Modify the radius if it is too large or small */
    382:         tao->trust = PetscMax(tao->trust, tl->min_radius);
    383:         tao->trust = PetscMin(tao->trust, tl->max_radius);

    385:         if (NTL_KSP_NASH == tl->ksp_type) {
    386:           KSPNASHSetRadius(tao->ksp,tl->max_radius);
    387:           KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    388:           KSPGetIterationNumber(tao->ksp,&its);
    389:           tao->ksp_its+=its;
    390:           tao->ksp_tot_its+=its;
    391:           KSPNASHGetNormD(tao->ksp, &norm_d);
    392:         } else if (NTL_KSP_STCG == tl->ksp_type) {
    393:           KSPSTCGSetRadius(tao->ksp,tl->max_radius);
    394:           KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    395:           KSPGetIterationNumber(tao->ksp,&its);
    396:           tao->ksp_its+=its;
    397:           tao->ksp_tot_its+=its;
    398:           KSPSTCGGetNormD(tao->ksp, &norm_d);
    399:         } else { /* NTL_KSP_GLTR */
    400:           KSPGLTRSetRadius(tao->ksp,tl->max_radius);
    401:           KSPSolve(tao->ksp, tao->gradient, tao->stepdirection);
    402:           KSPGetIterationNumber(tao->ksp,&its);
    403:           tao->ksp_its+=its;
    404:           tao->ksp_tot_its+=its;
    405:           KSPGLTRGetNormD(tao->ksp, &norm_d);
    406:         }

    409:         if (norm_d == 0.0) SETERRQ(PETSC_COMM_SELF,1, "Initial direction zero");
    410:       }
    411:     }

    413:     VecScale(tao->stepdirection, -1.0);
    414:     KSPGetConvergedReason(tao->ksp, &ksp_reason);
    415:     if ((KSP_DIVERGED_INDEFINITE_PC == ksp_reason) && (NTL_PC_BFGS == tl->pc_type) && (bfgsUpdates > 1)) {
    416:       /* Preconditioner is numerically indefinite; reset the
    417:          approximate if using BFGS preconditioning. */

    419:       if (f != 0.0) {
    420:         delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    421:       } else {
    422:         delta = 2.0 / (gnorm*gnorm);
    423:       }
    424:       MatLMVMSetDelta(tl->M, delta);
    425:       MatLMVMReset(tl->M);
    426:       MatLMVMUpdate(tl->M, tao->solution, tao->gradient);
    427:       bfgsUpdates = 1;
    428:     }

    430:     /* Check trust-region reduction conditions */
    431:     tr_reject = 0;
    432:     if (NTL_UPDATE_REDUCTION == tl->update_type) {
    433:       /* Get predicted reduction */
    434:       if (NTL_KSP_NASH == tl->ksp_type) {
    435:         KSPNASHGetObjFcn(tao->ksp,&prered);
    436:       } else if (NTL_KSP_STCG == tl->ksp_type) {
    437:         KSPSTCGGetObjFcn(tao->ksp,&prered);
    438:       } else { /* gltr */
    439:         KSPGLTRGetObjFcn(tao->ksp,&prered);
    440:       }

    442:       if (prered >= 0.0) {
    443:         /* The predicted reduction has the wrong sign.  This cannot
    444:            happen in infinite precision arithmetic.  Step should
    445:            be rejected! */
    446:         tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
    447:         tr_reject = 1;
    448:       } else {
    449:         /* Compute trial step and function value */
    450:         VecCopy(tao->solution, tl->W);
    451:         VecAXPY(tl->W, 1.0, tao->stepdirection);
    452:         TaoComputeObjective(tao, tl->W, &ftrial);

    454:         if (PetscIsInfOrNanReal(ftrial)) {
    455:           tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
    456:           tr_reject = 1;
    457:         } else {
    458:           /* Compute and actual reduction */
    459:           actred = f - ftrial;
    460:           prered = -prered;
    461:           if ((PetscAbsScalar(actred) <= tl->epsilon) &&
    462:               (PetscAbsScalar(prered) <= tl->epsilon)) {
    463:             kappa = 1.0;
    464:           } else {
    465:             kappa = actred / prered;
    466:           }

    468:           /* Accept of reject the step and update radius */
    469:           if (kappa < tl->eta1) {
    470:             /* Reject the step */
    471:             tao->trust = tl->alpha1 * PetscMin(tao->trust, norm_d);
    472:             tr_reject = 1;
    473:           } else {
    474:             /* Accept the step */
    475:             if (kappa < tl->eta2) {
    476:               /* Marginal bad step */
    477:               tao->trust = tl->alpha2 * PetscMin(tao->trust, norm_d);
    478:             } else if (kappa < tl->eta3) {
    479:               /* Reasonable step */
    480:               tao->trust = tl->alpha3 * tao->trust;
    481:             } else if (kappa < tl->eta4) {
    482:               /* Good step */
    483:               tao->trust = PetscMax(tl->alpha4 * norm_d, tao->trust);
    484:             } else {
    485:               /* Very good step */
    486:               tao->trust = PetscMax(tl->alpha5 * norm_d, tao->trust);
    487:             }
    488:           }
    489:         }
    490:       }
    491:     } else {
    492:       /* Get predicted reduction */
    493:       if (NTL_KSP_NASH == tl->ksp_type) {
    494:         KSPNASHGetObjFcn(tao->ksp,&prered);
    495:       } else if (NTL_KSP_STCG == tl->ksp_type) {
    496:         KSPSTCGGetObjFcn(tao->ksp,&prered);
    497:       } else { /* gltr */
    498:         KSPGLTRGetObjFcn(tao->ksp,&prered);
    499:       }

    501:       if (prered >= 0.0) {
    502:         /* The predicted reduction has the wrong sign.  This cannot
    503:            happen in infinite precision arithmetic.  Step should
    504:            be rejected! */
    505:         tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
    506:         tr_reject = 1;
    507:       } else {
    508:         VecCopy(tao->solution, tl->W);
    509:         VecAXPY(tl->W, 1.0, tao->stepdirection);
    510:         TaoComputeObjective(tao, tl->W, &ftrial);
    511:         if (PetscIsInfOrNanReal(ftrial)) {
    512:           tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
    513:           tr_reject = 1;
    514:         } else {
    515:           VecDot(tao->gradient, tao->stepdirection, &gdx);

    517:           actred = f - ftrial;
    518:           prered = -prered;
    519:           if ((PetscAbsScalar(actred) <= tl->epsilon) &&
    520:               (PetscAbsScalar(prered) <= tl->epsilon)) {
    521:             kappa = 1.0;
    522:           } else {
    523:             kappa = actred / prered;
    524:           }

    526:           tau_1 = tl->theta * gdx / (tl->theta * gdx - (1.0 - tl->theta) * prered + actred);
    527:           tau_2 = tl->theta * gdx / (tl->theta * gdx + (1.0 + tl->theta) * prered - actred);
    528:           tau_min = PetscMin(tau_1, tau_2);
    529:           tau_max = PetscMax(tau_1, tau_2);

    531:           if (kappa >= 1.0 - tl->mu1) {
    532:             /* Great agreement; accept step and update radius */
    533:             if (tau_max < 1.0) {
    534:               tao->trust = PetscMax(tao->trust, tl->gamma3 * norm_d);
    535:             } else if (tau_max > tl->gamma4) {
    536:               tao->trust = PetscMax(tao->trust, tl->gamma4 * norm_d);
    537:             } else {
    538:               tao->trust = PetscMax(tao->trust, tau_max * norm_d);
    539:             }
    540:           } else if (kappa >= 1.0 - tl->mu2) {
    541:             /* Good agreement */

    543:             if (tau_max < tl->gamma2) {
    544:               tao->trust = tl->gamma2 * PetscMin(tao->trust, norm_d);
    545:             } else if (tau_max > tl->gamma3) {
    546:               tao->trust = PetscMax(tao->trust, tl->gamma3 * norm_d);
    547:             } else if (tau_max < 1.0) {
    548:               tao->trust = tau_max * PetscMin(tao->trust, norm_d);
    549:             } else {
    550:               tao->trust = PetscMax(tao->trust, tau_max * norm_d);
    551:             }
    552:           } else {
    553:             /* Not good agreement */
    554:             if (tau_min > 1.0) {
    555:               tao->trust = tl->gamma2 * PetscMin(tao->trust, norm_d);
    556:             } else if (tau_max < tl->gamma1) {
    557:               tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
    558:             } else if ((tau_min < tl->gamma1) && (tau_max >= 1.0)) {
    559:               tao->trust = tl->gamma1 * PetscMin(tao->trust, norm_d);
    560:             } else if ((tau_1 >= tl->gamma1) && (tau_1 < 1.0) && ((tau_2 < tl->gamma1) || (tau_2 >= 1.0))) {
    561:               tao->trust = tau_1 * PetscMin(tao->trust, norm_d);
    562:             } else if ((tau_2 >= tl->gamma1) && (tau_2 < 1.0) && ((tau_1 < tl->gamma1) || (tau_2 >= 1.0))) {
    563:               tao->trust = tau_2 * PetscMin(tao->trust, norm_d);
    564:             } else {
    565:               tao->trust = tau_max * PetscMin(tao->trust, norm_d);
    566:             }
    567:             tr_reject = 1;
    568:           }
    569:         }
    570:       }
    571:     }

    573:     if (tr_reject) {
    574:       /* The trust-region constraints rejected the step.  Apply a linesearch.
    575:          Check for descent direction. */
    576:       VecDot(tao->stepdirection, tao->gradient, &gdx);
    577:       if ((gdx >= 0.0) || PetscIsInfOrNanReal(gdx)) {
    578:         /* Newton step is not descent or direction produced Inf or NaN */

    580:         if (NTL_PC_BFGS != tl->pc_type) {
    581:           /* We don't have the bfgs matrix around and updated
    582:              Must use gradient direction in this case */
    583:           VecCopy(tao->gradient, tao->stepdirection);
    584:           VecScale(tao->stepdirection, -1.0);
    585:           ++tl->grad;
    586:           stepType = NTL_GRADIENT;
    587:         } else {
    588:           /* Attempt to use the BFGS direction */
    589:           MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);
    590:           VecScale(tao->stepdirection, -1.0);

    592:           /* Check for success (descent direction) */
    593:           VecDot(tao->stepdirection, tao->gradient, &gdx);
    594:           if ((gdx >= 0) || PetscIsInfOrNanReal(gdx)) {
    595:             /* BFGS direction is not descent or direction produced not a number
    596:                We can assert bfgsUpdates > 1 in this case because
    597:                the first solve produces the scaled gradient direction,
    598:                which is guaranteed to be descent */

    600:             /* Use steepest descent direction (scaled) */
    601:             if (f != 0.0) {
    602:               delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    603:             } else {
    604:               delta = 2.0 / (gnorm*gnorm);
    605:             }
    606:             MatLMVMSetDelta(tl->M, delta);
    607:             MatLMVMReset(tl->M);
    608:             MatLMVMUpdate(tl->M, tao->solution, tao->gradient);
    609:             MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);
    610:             VecScale(tao->stepdirection, -1.0);

    612:             bfgsUpdates = 1;
    613:             ++tl->sgrad;
    614:             stepType = NTL_SCALED_GRADIENT;
    615:           } else {
    616:             if (1 == bfgsUpdates) {
    617:               /* The first BFGS direction is always the scaled gradient */
    618:               ++tl->sgrad;
    619:               stepType = NTL_SCALED_GRADIENT;
    620:             } else {
    621:               ++tl->bfgs;
    622:               stepType = NTL_BFGS;
    623:             }
    624:           }
    625:         }
    626:       } else {
    627:         /* Computed Newton step is descent */
    628:         ++tl->newt;
    629:         stepType = NTL_NEWTON;
    630:       }

    632:       /* Perform the linesearch */
    633:       fold = f;
    634:       VecCopy(tao->solution, tl->Xold);
    635:       VecCopy(tao->gradient, tl->Gold);

    637:       step = 1.0;
    638:       TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, tao->stepdirection, &step, &ls_reason);
    639:       TaoAddLineSearchCounts(tao);

    641:       while (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER && stepType != NTL_GRADIENT) {      /* Linesearch failed */
    642:         /* Linesearch failed */
    643:         f = fold;
    644:         VecCopy(tl->Xold, tao->solution);
    645:         VecCopy(tl->Gold, tao->gradient);

    647:         switch(stepType) {
    648:         case NTL_NEWTON:
    649:           /* Failed to obtain acceptable iterate with Newton step */

    651:           if (NTL_PC_BFGS != tl->pc_type) {
    652:             /* We don't have the bfgs matrix around and being updated
    653:                Must use gradient direction in this case */
    654:             VecCopy(tao->gradient, tao->stepdirection);
    655:             ++tl->grad;
    656:             stepType = NTL_GRADIENT;
    657:           } else {
    658:             /* Attempt to use the BFGS direction */
    659:             MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);

    662:             /* Check for success (descent direction) */
    663:             VecDot(tao->stepdirection, tao->gradient, &gdx);
    664:             if ((gdx <= 0) || PetscIsInfOrNanReal(gdx)) {
    665:               /* BFGS direction is not descent or direction produced
    666:                  not a number.  We can assert bfgsUpdates > 1 in this case
    667:                  Use steepest descent direction (scaled) */

    669:               if (f != 0.0) {
    670:                 delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    671:               } else {
    672:                 delta = 2.0 / (gnorm*gnorm);
    673:               }
    674:               MatLMVMSetDelta(tl->M, delta);
    675:               MatLMVMReset(tl->M);
    676:               MatLMVMUpdate(tl->M, tao->solution, tao->gradient);
    677:               MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);

    679:               bfgsUpdates = 1;
    680:               ++tl->sgrad;
    681:               stepType = NTL_SCALED_GRADIENT;
    682:             } else {
    683:               if (1 == bfgsUpdates) {
    684:                 /* The first BFGS direction is always the scaled gradient */
    685:                 ++tl->sgrad;
    686:                 stepType = NTL_SCALED_GRADIENT;
    687:               } else {
    688:                 ++tl->bfgs;
    689:                 stepType = NTL_BFGS;
    690:               }
    691:             }
    692:           }
    693:           break;

    695:         case NTL_BFGS:
    696:           /* Can only enter if pc_type == NTL_PC_BFGS
    697:              Failed to obtain acceptable iterate with BFGS step
    698:              Attempt to use the scaled gradient direction */

    700:           if (f != 0.0) {
    701:             delta = 2.0 * PetscAbsScalar(f) / (gnorm*gnorm);
    702:           } else {
    703:             delta = 2.0 / (gnorm*gnorm);
    704:           }
    705:           MatLMVMSetDelta(tl->M, delta);
    706:           MatLMVMReset(tl->M);
    707:           MatLMVMUpdate(tl->M, tao->solution, tao->gradient);
    708:           MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);

    710:           bfgsUpdates = 1;
    711:           ++tl->sgrad;
    712:           stepType = NTL_SCALED_GRADIENT;
    713:           break;

    715:         case NTL_SCALED_GRADIENT:
    716:           /* Can only enter if pc_type == NTL_PC_BFGS
    717:              The scaled gradient step did not produce a new iterate;
    718:              attemp to use the gradient direction.
    719:              Need to make sure we are not using a different diagonal scaling */
    720:           MatLMVMSetScale(tl->M, tl->Diag);
    721:           MatLMVMSetDelta(tl->M, 1.0);
    722:           MatLMVMReset(tl->M);
    723:           MatLMVMUpdate(tl->M, tao->solution, tao->gradient);
    724:           MatLMVMSolve(tl->M, tao->gradient, tao->stepdirection);

    726:           bfgsUpdates = 1;
    727:           ++tl->grad;
    728:           stepType = NTL_GRADIENT;
    729:           break;
    730:         }
    731:         VecScale(tao->stepdirection, -1.0);

    733:         /* This may be incorrect; linesearch has values for stepmax and stepmin
    734:            that should be reset. */
    735:         step = 1.0;
    736:         TaoLineSearchApply(tao->linesearch, tao->solution, &f, tao->gradient, tao->stepdirection, &step, &ls_reason);
    737:         TaoAddLineSearchCounts(tao);
    738:       }

    740:       if (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER) {
    741:         /* Failed to find an improving point */
    742:         f = fold;
    743:         VecCopy(tl->Xold, tao->solution);
    744:         VecCopy(tl->Gold, tao->gradient);
    745:         tao->trust = 0.0;
    746:         step = 0.0;
    747:         reason = TAO_DIVERGED_LS_FAILURE;
    748:         tao->reason = TAO_DIVERGED_LS_FAILURE;
    749:         break;
    750:       } else if (stepType == NTL_NEWTON) {
    751:         if (step < tl->nu1) {
    752:           /* Very bad step taken; reduce radius */
    753:           tao->trust = tl->omega1 * PetscMin(norm_d, tao->trust);
    754:         } else if (step < tl->nu2) {
    755:           /* Reasonably bad step taken; reduce radius */
    756:           tao->trust = tl->omega2 * PetscMin(norm_d, tao->trust);
    757:         } else if (step < tl->nu3) {
    758:           /* Reasonable step was taken; leave radius alone */
    759:           if (tl->omega3 < 1.0) {
    760:             tao->trust = tl->omega3 * PetscMin(norm_d, tao->trust);
    761:           } else if (tl->omega3 > 1.0) {
    762:             tao->trust = PetscMax(tl->omega3 * norm_d, tao->trust);
    763:           }
    764:         } else if (step < tl->nu4) {
    765:           /* Full step taken; increase the radius */
    766:           tao->trust = PetscMax(tl->omega4 * norm_d, tao->trust);
    767:         } else {
    768:           /* More than full step taken; increase the radius */
    769:           tao->trust = PetscMax(tl->omega5 * norm_d, tao->trust);
    770:         }
    771:       } else {
    772:         /* Newton step was not good; reduce the radius */
    773:         tao->trust = tl->omega1 * PetscMin(norm_d, tao->trust);
    774:       }
    775:     } else {
    776:       /* Trust-region step is accepted */
    777:       VecCopy(tl->W, tao->solution);
    778:       f = ftrial;
    779:       TaoComputeGradient(tao, tao->solution, tao->gradient);
    780:       ++tl->ntrust;
    781:     }

    783:     /* The radius may have been increased; modify if it is too large */
    784:     tao->trust = PetscMin(tao->trust, tl->max_radius);

    786:     /* Check for converged */
    787:     VecNorm(tao->gradient, NORM_2, &gnorm);
    788:     if (PetscIsInfOrNanReal(f) || PetscIsInfOrNanReal(gnorm)) SETERRQ(PETSC_COMM_SELF,1,"User provided compute function generated Not-a-Number");
    789:     needH = 1;

    791:     TaoMonitor(tao, tao->niter, f, gnorm, 0.0, tao->trust, &reason);
    792:   }
    793:   return(0);
    794: }

    796: /* ---------------------------------------------------------- */
    799: static PetscErrorCode TaoSetUp_NTL(Tao tao)
    800: {
    801:   TAO_NTL        *tl = (TAO_NTL *)tao->data;

    805:   if (!tao->gradient) {VecDuplicate(tao->solution, &tao->gradient); }
    806:   if (!tao->stepdirection) {VecDuplicate(tao->solution, &tao->stepdirection);}
    807:   if (!tl->W) { VecDuplicate(tao->solution, &tl->W);}
    808:   if (!tl->Xold) { VecDuplicate(tao->solution, &tl->Xold);}
    809:   if (!tl->Gold) { VecDuplicate(tao->solution, &tl->Gold);}
    810:   tl->Diag = 0;
    811:   tl->M = 0;
    812:   return(0);
    813: }

    815: /*------------------------------------------------------------*/
    818: static PetscErrorCode TaoDestroy_NTL(Tao tao)
    819: {
    820:   TAO_NTL        *tl = (TAO_NTL *)tao->data;

    824:   if (tao->setupcalled) {
    825:     VecDestroy(&tl->W);
    826:     VecDestroy(&tl->Xold);
    827:     VecDestroy(&tl->Gold);
    828:   }
    829:   VecDestroy(&tl->Diag);
    830:   MatDestroy(&tl->M);
    831:   PetscFree(tao->data);
    832:   return(0);
    833: }

    835: /*------------------------------------------------------------*/
    838: static PetscErrorCode TaoSetFromOptions_NTL(PetscOptionItems *PetscOptionsObject,Tao tao)
    839: {
    840:   TAO_NTL        *tl = (TAO_NTL *)tao->data;

    844:   PetscOptionsHead(PetscOptionsObject,"Newton trust region with line search method for unconstrained optimization");
    845:   PetscOptionsEList("-tao_ntl_ksp_type", "ksp type", "", NTL_KSP, NTL_KSP_TYPES, NTL_KSP[tl->ksp_type], &tl->ksp_type,NULL);
    846:   PetscOptionsEList("-tao_ntl_pc_type", "pc type", "", NTL_PC, NTL_PC_TYPES, NTL_PC[tl->pc_type], &tl->pc_type,NULL);
    847:   PetscOptionsEList("-tao_ntl_bfgs_scale_type", "bfgs scale type", "", BFGS_SCALE, BFGS_SCALE_TYPES, BFGS_SCALE[tl->bfgs_scale_type], &tl->bfgs_scale_type,NULL);
    848:   PetscOptionsEList("-tao_ntl_init_type", "radius initialization type", "", NTL_INIT, NTL_INIT_TYPES, NTL_INIT[tl->init_type], &tl->init_type,NULL);
    849:   PetscOptionsEList("-tao_ntl_update_type", "radius update type", "", NTL_UPDATE, NTL_UPDATE_TYPES, NTL_UPDATE[tl->update_type], &tl->update_type,NULL);
    850:   PetscOptionsReal("-tao_ntl_eta1", "poor steplength; reduce radius", "", tl->eta1, &tl->eta1,NULL);
    851:   PetscOptionsReal("-tao_ntl_eta2", "reasonable steplength; leave radius alone", "", tl->eta2, &tl->eta2,NULL);
    852:   PetscOptionsReal("-tao_ntl_eta3", "good steplength; increase radius", "", tl->eta3, &tl->eta3,NULL);
    853:   PetscOptionsReal("-tao_ntl_eta4", "excellent steplength; greatly increase radius", "", tl->eta4, &tl->eta4,NULL);
    854:   PetscOptionsReal("-tao_ntl_alpha1", "", "", tl->alpha1, &tl->alpha1,NULL);
    855:   PetscOptionsReal("-tao_ntl_alpha2", "", "", tl->alpha2, &tl->alpha2,NULL);
    856:   PetscOptionsReal("-tao_ntl_alpha3", "", "", tl->alpha3, &tl->alpha3,NULL);
    857:   PetscOptionsReal("-tao_ntl_alpha4", "", "", tl->alpha4, &tl->alpha4,NULL);
    858:   PetscOptionsReal("-tao_ntl_alpha5", "", "", tl->alpha5, &tl->alpha5,NULL);
    859:   PetscOptionsReal("-tao_ntl_nu1", "poor steplength; reduce radius", "", tl->nu1, &tl->nu1,NULL);
    860:   PetscOptionsReal("-tao_ntl_nu2", "reasonable steplength; leave radius alone", "", tl->nu2, &tl->nu2,NULL);
    861:   PetscOptionsReal("-tao_ntl_nu3", "good steplength; increase radius", "", tl->nu3, &tl->nu3,NULL);
    862:   PetscOptionsReal("-tao_ntl_nu4", "excellent steplength; greatly increase radius", "", tl->nu4, &tl->nu4,NULL);
    863:   PetscOptionsReal("-tao_ntl_omega1", "", "", tl->omega1, &tl->omega1,NULL);
    864:   PetscOptionsReal("-tao_ntl_omega2", "", "", tl->omega2, &tl->omega2,NULL);
    865:   PetscOptionsReal("-tao_ntl_omega3", "", "", tl->omega3, &tl->omega3,NULL);
    866:   PetscOptionsReal("-tao_ntl_omega4", "", "", tl->omega4, &tl->omega4,NULL);
    867:   PetscOptionsReal("-tao_ntl_omega5", "", "", tl->omega5, &tl->omega5,NULL);
    868:   PetscOptionsReal("-tao_ntl_mu1_i", "", "", tl->mu1_i, &tl->mu1_i,NULL);
    869:   PetscOptionsReal("-tao_ntl_mu2_i", "", "", tl->mu2_i, &tl->mu2_i,NULL);
    870:   PetscOptionsReal("-tao_ntl_gamma1_i", "", "", tl->gamma1_i, &tl->gamma1_i,NULL);
    871:   PetscOptionsReal("-tao_ntl_gamma2_i", "", "", tl->gamma2_i, &tl->gamma2_i,NULL);
    872:   PetscOptionsReal("-tao_ntl_gamma3_i", "", "", tl->gamma3_i, &tl->gamma3_i,NULL);
    873:   PetscOptionsReal("-tao_ntl_gamma4_i", "", "", tl->gamma4_i, &tl->gamma4_i,NULL);
    874:   PetscOptionsReal("-tao_ntl_theta_i", "", "", tl->theta_i, &tl->theta_i,NULL);
    875:   PetscOptionsReal("-tao_ntl_mu1", "", "", tl->mu1, &tl->mu1,NULL);
    876:   PetscOptionsReal("-tao_ntl_mu2", "", "", tl->mu2, &tl->mu2,NULL);
    877:   PetscOptionsReal("-tao_ntl_gamma1", "", "", tl->gamma1, &tl->gamma1,NULL);
    878:   PetscOptionsReal("-tao_ntl_gamma2", "", "", tl->gamma2, &tl->gamma2,NULL);
    879:   PetscOptionsReal("-tao_ntl_gamma3", "", "", tl->gamma3, &tl->gamma3,NULL);
    880:   PetscOptionsReal("-tao_ntl_gamma4", "", "", tl->gamma4, &tl->gamma4,NULL);
    881:   PetscOptionsReal("-tao_ntl_theta", "", "", tl->theta, &tl->theta,NULL);
    882:   PetscOptionsReal("-tao_ntl_min_radius", "lower bound on initial radius", "", tl->min_radius, &tl->min_radius,NULL);
    883:   PetscOptionsReal("-tao_ntl_max_radius", "upper bound on radius", "", tl->max_radius, &tl->max_radius,NULL);
    884:   PetscOptionsReal("-tao_ntl_epsilon", "tolerance used when computing actual and predicted reduction", "", tl->epsilon, &tl->epsilon,NULL);
    885:   PetscOptionsTail();
    886:   TaoLineSearchSetFromOptions(tao->linesearch);
    887:   KSPSetFromOptions(tao->ksp);
    888:   return(0);
    889: }

    891: /*------------------------------------------------------------*/
    894: static PetscErrorCode TaoView_NTL(Tao tao, PetscViewer viewer)
    895: {
    896:   TAO_NTL        *tl = (TAO_NTL *)tao->data;
    897:   PetscInt       nrejects;
    898:   PetscBool      isascii;

    902:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    903:   if (isascii) {
    904:     PetscViewerASCIIPushTab(viewer);
    905:     if (NTL_PC_BFGS == tl->pc_type && tl->M) {
    906:       MatLMVMGetRejects(tl->M, &nrejects);
    907:       PetscViewerASCIIPrintf(viewer, "Rejected matrix updates: %D\n", nrejects);
    908:     }
    909:     PetscViewerASCIIPrintf(viewer, "Trust-region steps: %D\n", tl->ntrust);
    910:     PetscViewerASCIIPrintf(viewer, "Newton search steps: %D\n", tl->newt);
    911:     PetscViewerASCIIPrintf(viewer, "BFGS search steps: %D\n", tl->bfgs);
    912:     PetscViewerASCIIPrintf(viewer, "Scaled gradient search steps: %D\n", tl->sgrad);
    913:     PetscViewerASCIIPrintf(viewer, "Gradient search steps: %D\n", tl->grad);
    914:     PetscViewerASCIIPopTab(viewer);
    915:   }
    916:   return(0);
    917: }

    919: /* ---------------------------------------------------------- */
    920: /*MC
    921:   TAONTR - Newton's method with trust region and linesearch
    922:   for unconstrained minimization.
    923:   At each iteration, the Newton trust region method solves the system for d
    924:   and performs a line search in the d direction:

    926:             min_d  .5 dT Hk d + gkT d,  s.t.   ||d|| < Delta_k

    928:   Options Database Keys:
    929: + -tao_ntl_ksp_type - "nash","stcg","gltr"
    930: . -tao_ntl_pc_type - "none","ahess","bfgs","petsc"
    931: . -tao_ntl_bfgs_scale_type - type of scaling with bfgs pc, "ahess" or "bfgs"
    932: . -tao_ntl_init_type - "constant","direction","interpolation"
    933: . -tao_ntl_update_type - "reduction","interpolation"
    934: . -tao_ntl_min_radius - lower bound on trust region radius
    935: . -tao_ntl_max_radius - upper bound on trust region radius
    936: . -tao_ntl_epsilon - tolerance for accepting actual / predicted reduction
    937: . -tao_ntl_mu1_i - mu1 interpolation init factor
    938: . -tao_ntl_mu2_i - mu2 interpolation init factor
    939: . -tao_ntl_gamma1_i - gamma1 interpolation init factor
    940: . -tao_ntl_gamma2_i - gamma2 interpolation init factor
    941: . -tao_ntl_gamma3_i - gamma3 interpolation init factor
    942: . -tao_ntl_gamma4_i - gamma4 interpolation init factor
    943: . -tao_ntl_theta_i - thetha1 interpolation init factor
    944: . -tao_ntl_eta1 - eta1 reduction update factor
    945: . -tao_ntl_eta2 - eta2 reduction update factor
    946: . -tao_ntl_eta3 - eta3 reduction update factor
    947: . -tao_ntl_eta4 - eta4 reduction update factor
    948: . -tao_ntl_alpha1 - alpha1 reduction update factor
    949: . -tao_ntl_alpha2 - alpha2 reduction update factor
    950: . -tao_ntl_alpha3 - alpha3 reduction update factor
    951: . -tao_ntl_alpha4 - alpha4 reduction update factor
    952: . -tao_ntl_alpha4 - alpha4 reduction update factor
    953: . -tao_ntl_mu1 - mu1 interpolation update
    954: . -tao_ntl_mu2 - mu2 interpolation update
    955: . -tao_ntl_gamma1 - gamma1 interpolcation update
    956: . -tao_ntl_gamma2 - gamma2 interpolcation update
    957: . -tao_ntl_gamma3 - gamma3 interpolcation update
    958: . -tao_ntl_gamma4 - gamma4 interpolation update
    959: - -tao_ntl_theta - theta1 interpolation update

    961:   Level: beginner
    962: M*/

    966: PETSC_EXTERN PetscErrorCode TaoCreate_NTL(Tao tao)
    967: {
    968:   TAO_NTL        *tl;
    970:   const char     *morethuente_type = TAOLINESEARCHMT;

    973:   PetscNewLog(tao,&tl);
    974:   tao->ops->setup = TaoSetUp_NTL;
    975:   tao->ops->solve = TaoSolve_NTL;
    976:   tao->ops->view = TaoView_NTL;
    977:   tao->ops->setfromoptions = TaoSetFromOptions_NTL;
    978:   tao->ops->destroy = TaoDestroy_NTL;

    980:   /* Override default settings (unless already changed) */
    981:   if (!tao->max_it_changed) tao->max_it = 50;
    982:   if (!tao->trust0_changed) tao->trust0 = 100.0;

    984:   tao->data = (void*)tl;

    986:   /* Default values for trust-region radius update based on steplength */
    987:   tl->nu1 = 0.25;
    988:   tl->nu2 = 0.50;
    989:   tl->nu3 = 1.00;
    990:   tl->nu4 = 1.25;

    992:   tl->omega1 = 0.25;
    993:   tl->omega2 = 0.50;
    994:   tl->omega3 = 1.00;
    995:   tl->omega4 = 2.00;
    996:   tl->omega5 = 4.00;

    998:   /* Default values for trust-region radius update based on reduction */
    999:   tl->eta1 = 1.0e-4;
    1000:   tl->eta2 = 0.25;
    1001:   tl->eta3 = 0.50;
    1002:   tl->eta4 = 0.90;

    1004:   tl->alpha1 = 0.25;
    1005:   tl->alpha2 = 0.50;
    1006:   tl->alpha3 = 1.00;
    1007:   tl->alpha4 = 2.00;
    1008:   tl->alpha5 = 4.00;

    1010:   /* Default values for trust-region radius update based on interpolation */
    1011:   tl->mu1 = 0.10;
    1012:   tl->mu2 = 0.50;

    1014:   tl->gamma1 = 0.25;
    1015:   tl->gamma2 = 0.50;
    1016:   tl->gamma3 = 2.00;
    1017:   tl->gamma4 = 4.00;

    1019:   tl->theta = 0.05;

    1021:   /* Default values for trust region initialization based on interpolation */
    1022:   tl->mu1_i = 0.35;
    1023:   tl->mu2_i = 0.50;

    1025:   tl->gamma1_i = 0.0625;
    1026:   tl->gamma2_i = 0.5;
    1027:   tl->gamma3_i = 2.0;
    1028:   tl->gamma4_i = 5.0;

    1030:   tl->theta_i = 0.25;

    1032:   /* Remaining parameters */
    1033:   tl->min_radius = 1.0e-10;
    1034:   tl->max_radius = 1.0e10;
    1035:   tl->epsilon = 1.0e-6;

    1037:   tl->ksp_type        = NTL_KSP_STCG;
    1038:   tl->pc_type         = NTL_PC_BFGS;
    1039:   tl->bfgs_scale_type = BFGS_SCALE_AHESS;
    1040:   tl->init_type       = NTL_INIT_INTERPOLATION;
    1041:   tl->update_type     = NTL_UPDATE_REDUCTION;

    1043:   TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);
    1044:   TaoLineSearchSetType(tao->linesearch, morethuente_type);
    1045:   TaoLineSearchUseTaoRoutines(tao->linesearch, tao);
    1046:   TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);
    1047:   KSPCreate(((PetscObject)tao)->comm, &tao->ksp);
    1048:   KSPSetOptionsPrefix(tao->ksp, tao->hdr.prefix);
    1049:   return(0);
    1050: }