Date: 2022-03-31T04:49:02+00:00

Actual source code: bntl.c

      1: #include <../src/tao/bound/impls/bnk/bnk.h>
      2: #include <petscksp.h>

      4: /*
      5:  Implements Newton's Method with a trust region approach for solving
      6:  bound constrained minimization problems.

      8:  In this variant, the trust region failures trigger a line search with
      9:  the existing Newton step instead of re-solving the step with a
     10:  different radius.

     12:  ------------------------------------------------------------

     14:  x_0 = VecMedian(x_0)
     15:  f_0, g_0 = TaoComputeObjectiveAndGradient(x_0)
     16:  pg_0 = project(g_0)
     17:  check convergence at pg_0
     18:  needH = TaoBNKInitialize(default:BNK_INIT_INTERPOLATION)
     19:  niter = 0
     20:  step_accepted = true

     22:  while niter <= max_it
     23:     niter += 1

     25:     if needH
     26:       If max_cg_steps > 0
     27:         x_k, g_k, pg_k = TaoSolve(BNCG)
     28:       end

     30:       H_k = TaoComputeHessian(x_k)
     31:       if pc_type == BNK_PC_BFGS
     32:         add correction to BFGS approx
     33:         if scale_type == BNK_SCALE_AHESS
     34:           D = VecMedian(1e-6, abs(diag(H_k)), 1e6)
     35:           scale BFGS with VecReciprocal(D)
     36:         end
     37:       end
     38:       needH = False
     39:     end

     41:     if pc_type = BNK_PC_BFGS
     42:       B_k = BFGS
     43:     else
     44:       B_k = VecMedian(1e-6, abs(diag(H_k)), 1e6)
     45:       B_k = VecReciprocal(B_k)
     46:     end
     47:     w = x_k - VecMedian(x_k - 0.001*B_k*g_k)
     48:     eps = min(eps, norm2(w))
     49:     determine the active and inactive index sets such that
     50:       L = {i : (x_k)_i <= l_i + eps && (g_k)_i > 0}
     51:       U = {i : (x_k)_i >= u_i - eps && (g_k)_i < 0}
     52:       F = {i : l_i = (x_k)_i = u_i}
     53:       A = {L + U + F}
     54:       IA = {i : i not in A}

     56:     generate the reduced system Hr_k dr_k = -gr_k for variables in IA
     57:     if pc_type == BNK_PC_BFGS && scale_type == BNK_SCALE_PHESS
     58:       D = VecMedian(1e-6, abs(diag(Hr_k)), 1e6)
     59:       scale BFGS with VecReciprocal(D)
     60:     end
     61:     solve Hr_k dr_k = -gr_k
     62:     set d_k to (l - x) for variables in L, (u - x) for variables in U, and 0 for variables in F

     64:     x_{k+1} = VecMedian(x_k + d_k)
     65:     s = x_{k+1} - x_k
     66:     prered = dot(s, 0.5*gr_k - Hr_k*s)
     67:     f_{k+1} = TaoComputeObjective(x_{k+1})
     68:     actred = f_k - f_{k+1}

     70:     oldTrust = trust
     71:     step_accepted, trust = TaoBNKUpdateTrustRadius(default: BNK_UPDATE_REDUCTION)
     72:     if step_accepted
     73:       g_{k+1} = TaoComputeGradient(x_{k+1})
     74:       pg_{k+1} = project(g_{k+1})
     75:       count the accepted Newton step
     76:     else
     77:       if dot(d_k, pg_k)) >= 0 || norm(d_k) == NaN || norm(d_k) == Inf
     78:         dr_k = -BFGS*gr_k for variables in I
     79:         if dot(d_k, pg_k)) >= 0 || norm(d_k) == NaN || norm(d_k) == Inf
     80:           reset the BFGS preconditioner
     81:           calculate scale delta and apply it to BFGS
     82:           dr_k = -BFGS*gr_k for variables in I
     83:           if dot(d_k, pg_k)) >= 0 || norm(d_k) == NaN || norm(d_k) == Inf
     84:             dr_k = -gr_k for variables in I
     85:           end
     86:         end
     87:       end

     89:       x_{k+1}, f_{k+1}, g_{k+1}, ls_failed = TaoBNKPerformLineSearch()
     90:       if ls_failed
     91:         f_{k+1} = f_k
     92:         x_{k+1} = x_k
     93:         g_{k+1} = g_k
     94:         pg_{k+1} = pg_k
     95:         terminate
     96:       else
     97:         pg_{k+1} = project(g_{k+1})
     98:         trust = oldTrust
     99:         trust = TaoBNKUpdateTrustRadius(BNK_UPDATE_STEP)
    100:         count the accepted step type (Newton, BFGS, scaled grad or grad)
    101:       end
    102:     end

    104:     check convergence at pg_{k+1}
    105:  end
    106: */

    108: PetscErrorCode TaoSolve_BNTL(Tao tao)
    109: {
    110:   TAO_BNK                      *bnk = (TAO_BNK *)tao->data;
    111:   KSPConvergedReason           ksp_reason;
    112:   TaoLineSearchConvergedReason ls_reason;

    114:   PetscReal                    oldTrust, prered, actred, steplen, resnorm;
    115:   PetscBool                    cgTerminate, needH = PETSC_TRUE, stepAccepted, shift = PETSC_FALSE;
    116:   PetscInt                     stepType, nDiff;

    118:   /* Initialize the preconditioner, KSP solver and trust radius/line search */
    119:   tao->reason = TAO_CONTINUE_ITERATING;
    120:   TaoBNKInitialize(tao, bnk->init_type, &needH);
    121:   if (tao->reason != TAO_CONTINUE_ITERATING) return 0;

    123:   /* Have not converged; continue with Newton method */
    124:   while (tao->reason == TAO_CONTINUE_ITERATING) {
    125:     /* Call general purpose update function */
    126:     if (tao->ops->update) {
    127:       (*tao->ops->update)(tao, tao->niter, tao->user_update);
    128:     }
    129:     ++tao->niter;

    131:     if (needH && bnk->inactive_idx) {
    132:       /* Take BNCG steps (if enabled) to trade-off Hessian evaluations for more gradient evaluations */
    133:       TaoBNKTakeCGSteps(tao, &cgTerminate);
    134:       if (cgTerminate) {
    135:         tao->reason = bnk->bncg->reason;
    136:         return 0;
    137:       }
    138:       /* Compute the hessian and update the BFGS preconditioner at the new iterate */
    139:       (*bnk->computehessian)(tao);
    140:       needH = PETSC_FALSE;
    141:     }

    143:     /* Use the common BNK kernel to compute the Newton step (for inactive variables only) */
    144:     (*bnk->computestep)(tao, shift, &ksp_reason, &stepType);

    146:     /* Store current solution before it changes */
    147:     oldTrust = tao->trust;
    148:     bnk->fold = bnk->f;
    149:     VecCopy(tao->solution, bnk->Xold);
    150:     VecCopy(tao->gradient, bnk->Gold);
    151:     VecCopy(bnk->unprojected_gradient, bnk->unprojected_gradient_old);

    153:     /* Temporarily accept the step and project it into the bounds */
    154:     VecAXPY(tao->solution, 1.0, tao->stepdirection);
    155:     TaoBoundSolution(tao->solution, tao->XL,tao->XU, 0.0, &nDiff, tao->solution);

    157:     /* Check if the projection changed the step direction */
    158:     if (nDiff > 0) {
    159:       /* Projection changed the step, so we have to recompute the step and
    160:          the predicted reduction. Leave the trust radius unchanged. */
    161:       VecCopy(tao->solution, tao->stepdirection);
    162:       VecAXPY(tao->stepdirection, -1.0, bnk->Xold);
    163:       TaoBNKRecomputePred(tao, tao->stepdirection, &prered);
    164:     } else {
    165:       /* Step did not change, so we can just recover the pre-computed prediction */
    166:       KSPCGGetObjFcn(tao->ksp, &prered);
    167:     }
    168:     prered = -prered;

    170:     /* Compute the actual reduction and update the trust radius */
    171:     TaoComputeObjective(tao, tao->solution, &bnk->f);
    173:     actred = bnk->fold - bnk->f;
    174:     TaoBNKUpdateTrustRadius(tao, prered, actred, bnk->update_type, stepType, &stepAccepted);

    176:     if (stepAccepted) {
    177:       /* Step is good, evaluate the gradient and the hessian */
    178:       steplen = 1.0;
    179:       needH = PETSC_TRUE;
    180:       ++bnk->newt;
    181:       TaoComputeGradient(tao, tao->solution, bnk->unprojected_gradient);
    182:       TaoBNKEstimateActiveSet(tao, bnk->as_type);
    183:       VecCopy(bnk->unprojected_gradient, tao->gradient);
    184:       VecISSet(tao->gradient, bnk->active_idx, 0.0);
    185:       TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);
    186:     } else {
    187:       /* Trust-region rejected the step. Revert the solution. */
    188:       bnk->f = bnk->fold;
    189:       VecCopy(bnk->Xold, tao->solution);
    190:       /* Trigger the line search */
    191:       TaoBNKSafeguardStep(tao, ksp_reason, &stepType);
    192:       TaoBNKPerformLineSearch(tao, &stepType, &steplen, &ls_reason);
    193:       if (ls_reason != TAOLINESEARCH_SUCCESS && ls_reason != TAOLINESEARCH_SUCCESS_USER) {
    194:         /* Line search failed, revert solution and terminate */
    195:         stepAccepted = PETSC_FALSE;
    196:         needH = PETSC_FALSE;
    197:         bnk->f = bnk->fold;
    198:         VecCopy(bnk->Xold, tao->solution);
    199:         VecCopy(bnk->Gold, tao->gradient);
    200:         VecCopy(bnk->unprojected_gradient_old, bnk->unprojected_gradient);
    201:         tao->trust = 0.0;
    202:         tao->reason = TAO_DIVERGED_LS_FAILURE;
    203:       } else {
    204:         /* new iterate so we need to recompute the Hessian */
    205:         needH = PETSC_TRUE;
    206:         /* compute the projected gradient */
    207:         TaoBNKEstimateActiveSet(tao, bnk->as_type);
    208:         VecCopy(bnk->unprojected_gradient, tao->gradient);
    209:         VecISSet(tao->gradient, bnk->active_idx, 0.0);
    210:         TaoGradientNorm(tao, tao->gradient, NORM_2, &bnk->gnorm);
    211:         /* Line search succeeded so we should update the trust radius based on the LS step length */
    212:         tao->trust = oldTrust;
    213:         TaoBNKUpdateTrustRadius(tao, prered, actred, BNK_UPDATE_STEP, stepType, &stepAccepted);
    214:         /* count the accepted step type */
    215:         TaoBNKAddStepCounts(tao, stepType);
    216:       }
    217:     }

    219:     /*  Check for termination */
    220:     VecFischer(tao->solution, bnk->unprojected_gradient, tao->XL, tao->XU, bnk->W);
    221:     VecNorm(bnk->W, NORM_2, &resnorm);
    223:     TaoLogConvergenceHistory(tao, bnk->f, resnorm, 0.0, tao->ksp_its);
    224:     TaoMonitor(tao, tao->niter, bnk->f, resnorm, 0.0, steplen);
    225:     (*tao->ops->convergencetest)(tao, tao->cnvP);
    226:   }
    227:   return 0;
    228: }

    230: /*------------------------------------------------------------*/
    231: static PetscErrorCode TaoSetUp_BNTL(Tao tao)
    232: {
    233:   KSP               ksp;
    234:   PetscVoidFunction valid;

    236:   TaoSetUp_BNK(tao);
    237:   TaoGetKSP(tao,&ksp);
    238:   PetscObjectQueryFunction((PetscObject)ksp,"KSPCGSetRadius_C",&valid);
    240:   return 0;
    241: }

    243: /*------------------------------------------------------------*/
    244: static PetscErrorCode TaoSetFromOptions_BNTL(PetscOptionItems *PetscOptionsObject,Tao tao)
    245: {
    246:   TAO_BNK        *bnk = (TAO_BNK *)tao->data;

    248:   TaoSetFromOptions_BNK(PetscOptionsObject, tao);
    249:   if (bnk->update_type == BNK_UPDATE_STEP) bnk->update_type = BNK_UPDATE_REDUCTION;
    250:   return 0;
    251: }

    253: /*------------------------------------------------------------*/
    254: /*MC
    255:   TAOBNTL - Bounded Newton Trust Region method with line-search fall-back for nonlinear
    256:             minimization with bound constraints.

    258:   Options Database Keys:
    259:   + -tao_bnk_max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
    260:   . -tao_bnk_init_type - trust radius initialization method ("constant", "direction", "interpolation")
    261:   . -tao_bnk_update_type - trust radius update method ("step", "direction", "interpolation")
    262:   - -tao_bnk_as_type - active-set estimation method ("none", "bertsekas")

    264:   Level: beginner
    265: M*/
    266: PETSC_EXTERN PetscErrorCode TaoCreate_BNTL(Tao tao)
    267: {
    268:   TAO_BNK        *bnk;

    270:   TaoCreate_BNK(tao);
    271:   tao->ops->solve=TaoSolve_BNTL;
    272:   tao->ops->setup=TaoSetUp_BNTL;
    273:   tao->ops->setfromoptions=TaoSetFromOptions_BNTL;

    275:   bnk = (TAO_BNK *)tao->data;
    276:   bnk->update_type = BNK_UPDATE_REDUCTION; /* trust region updates based on predicted/actual reduction */
    277:   return 0;
    278: }