Actual source code: bmrm.c

      1: #include <../src/tao/unconstrained/impls/bmrm/bmrm.h>

      3: static PetscErrorCode init_df_solver(TAO_DF*);
      4: static PetscErrorCode ensure_df_space(PetscInt, TAO_DF*);
      5: static PetscErrorCode destroy_df_solver(TAO_DF*);
      6: static PetscReal phi(PetscReal*,PetscInt,PetscReal,PetscReal*,PetscReal,PetscReal*,PetscReal*,PetscReal*);
      7: static PetscInt project(PetscInt,PetscReal*,PetscReal,PetscReal*,PetscReal*,PetscReal*,PetscReal*,PetscReal*,TAO_DF*);
      8: static PetscErrorCode solve(TAO_DF*);

     11: /*------------------------------------------------------------*/
     12: /* The main solver function

     14:    f = Remp(W)          This is what the user provides us from the application layer
     15:    So the ComputeGradient function for instance should get us back the subgradient of Remp(W)

     17:    Regularizer assumed to be L2 norm = lambda*0.5*W'W ()
     18: */

     20: static PetscErrorCode make_grad_node(Vec X, Vec_Chain **p)
     21: {

     25:   PetscNew(p);
     26:   VecDuplicate(X, &(*p)->V);
     27:   VecCopy(X, (*p)->V);
     28:   (*p)->next = NULL;
     29:   return(0);
     30: }

     32: static PetscErrorCode destroy_grad_list(Vec_Chain *head)
     33: {
     35:   Vec_Chain      *p = head->next, *q;

     38:   while (p) {
     39:     q = p->next;
     40:     VecDestroy(&p->V);
     41:     PetscFree(p);
     42:     p = q;
     43:   }
     44:   head->next = NULL;
     45:   return(0);
     46: }

     49: static PetscErrorCode TaoSolve_BMRM(Tao tao)
     50: {
     51:   PetscErrorCode     ierr;
     52:   TAO_DF             df;
     53:   TAO_BMRM           *bmrm = (TAO_BMRM*)tao->data;

     55:   /* Values and pointers to parts of the optimization problem */
     56:   PetscReal          f = 0.0;
     57:   Vec                W = tao->solution;
     58:   Vec                G = tao->gradient;
     59:   PetscReal          lambda;
     60:   PetscReal          bt;
     61:   Vec_Chain          grad_list, *tail_glist, *pgrad;
     62:   PetscInt           i;
     63:   PetscMPIInt        rank;

     65:   /* Used in converged criteria check */
     66:   PetscReal          reg;
     67:   PetscReal          jtwt = 0.0, max_jtwt, pre_epsilon, epsilon, jw, min_jw;
     68:   PetscReal          innerSolverTol;
     69:   MPI_Comm           comm;

     72:   PetscObjectGetComm((PetscObject)tao,&comm);
     73:   MPI_Comm_rank(comm, &rank);
     74:   lambda = bmrm->lambda;

     76:   /* Check Stopping Condition */
     77:   tao->step = 1.0;
     78:   max_jtwt = -BMRM_INFTY;
     79:   min_jw = BMRM_INFTY;
     80:   innerSolverTol = 1.0;
     81:   epsilon = 0.0;

     83:   if (!rank) {
     84:     init_df_solver(&df);
     85:     grad_list.next = NULL;
     86:     tail_glist = &grad_list;
     87:   }

     89:   df.tol = 1e-6;
     90:   tao->reason = TAO_CONTINUE_ITERATING;

     92:   /*-----------------Algorithm Begins------------------------*/
     93:   /* make the scatter */
     94:   VecScatterCreateToZero(W, &bmrm->scatter, &bmrm->local_w);
     95:   VecAssemblyBegin(bmrm->local_w);
     96:   VecAssemblyEnd(bmrm->local_w);

     98:   /* NOTE: In application pass the sub-gradient of Remp(W) */
     99:   TaoComputeObjectiveAndGradient(tao, W, &f, G);
    100:   TaoLogConvergenceHistory(tao,f,1.0,0.0,tao->ksp_its);
    101:   TaoMonitor(tao,tao->niter,f,1.0,0.0,tao->step);
    102:   (*tao->ops->convergencetest)(tao,tao->cnvP);

    104:   while (tao->reason == TAO_CONTINUE_ITERATING) {
    105:     /* Call general purpose update function */
    106:     if (tao->ops->update) {
    107:       (*tao->ops->update)(tao, tao->niter, tao->user_update);
    108:     }

    110:     /* compute bt = Remp(Wt-1) - <Wt-1, At> */
    111:     VecDot(W, G, &bt);
    112:     bt = f - bt;

    114:     /* First gather the gradient to the rank-0 node */
    115:     VecScatterBegin(bmrm->scatter, G, bmrm->local_w, INSERT_VALUES, SCATTER_FORWARD);
    116:     VecScatterEnd(bmrm->scatter, G, bmrm->local_w, INSERT_VALUES, SCATTER_FORWARD);

    118:     /* Bring up the inner solver */
    119:     if (!rank) {
    120:       ensure_df_space(tao->niter+1, &df);
    121:       make_grad_node(bmrm->local_w, &pgrad);
    122:       tail_glist->next = pgrad;
    123:       tail_glist = pgrad;

    125:       df.a[tao->niter] = 1.0;
    126:       df.f[tao->niter] = -bt;
    127:       df.u[tao->niter] = 1.0;
    128:       df.l[tao->niter] = 0.0;

    130:       /* set up the Q */
    131:       pgrad = grad_list.next;
    132:       for (i=0; i<=tao->niter; i++) {
    133:         if (!pgrad) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Assert that there are at least tao->niter+1 pgrad available");
    134:         VecDot(pgrad->V, bmrm->local_w, &reg);
    135:         df.Q[i][tao->niter] = df.Q[tao->niter][i] = reg / lambda;
    136:         pgrad = pgrad->next;
    137:       }

    139:       if (tao->niter > 0) {
    140:         df.x[tao->niter] = 0.0;
    141:         solve(&df);
    142:       } else
    143:         df.x[0] = 1.0;

    145:       /* now computing Jt*(alpha_t) which should be = Jt(wt) to check convergence */
    146:       jtwt = 0.0;
    147:       VecSet(bmrm->local_w, 0.0);
    148:       pgrad = grad_list.next;
    149:       for (i=0; i<=tao->niter; i++) {
    150:         jtwt -= df.x[i] * df.f[i];
    151:         VecAXPY(bmrm->local_w, -df.x[i] / lambda, pgrad->V);
    152:         pgrad = pgrad->next;
    153:       }

    155:       VecNorm(bmrm->local_w, NORM_2, &reg);
    156:       reg = 0.5*lambda*reg*reg;
    157:       jtwt -= reg;
    158:     } /* end if rank == 0 */

    160:     /* scatter the new W to all nodes */
    161:     VecScatterBegin(bmrm->scatter,bmrm->local_w,W,INSERT_VALUES,SCATTER_REVERSE);
    162:     VecScatterEnd(bmrm->scatter,bmrm->local_w,W,INSERT_VALUES,SCATTER_REVERSE);

    164:     TaoComputeObjectiveAndGradient(tao, W, &f, G);

    166:     MPI_Bcast(&jtwt,1,MPIU_REAL,0,comm);
    167:     MPI_Bcast(&reg,1,MPIU_REAL,0,comm);

    169:     jw = reg + f;                                       /* J(w) = regularizer + Remp(w) */
    170:     if (jw < min_jw) min_jw = jw;
    171:     if (jtwt > max_jtwt) max_jtwt = jtwt;

    173:     pre_epsilon = epsilon;
    174:     epsilon = min_jw - jtwt;

    176:     if (!rank) {
    177:       if (innerSolverTol > epsilon) innerSolverTol = epsilon;
    178:       else if (innerSolverTol < 1e-7) innerSolverTol = 1e-7;

    180:       /* if the annealing doesn't work well, lower the inner solver tolerance */
    181:       if (pre_epsilon < epsilon) innerSolverTol *= 0.2;

    183:       df.tol = innerSolverTol*0.5;
    184:     }

    186:     tao->niter++;
    187:     TaoLogConvergenceHistory(tao,min_jw,epsilon,0.0,tao->ksp_its);
    188:     TaoMonitor(tao,tao->niter,min_jw,epsilon,0.0,tao->step);
    189:     (*tao->ops->convergencetest)(tao,tao->cnvP);
    190:   }

    192:   /* free all the memory */
    193:   if (!rank) {
    194:     destroy_grad_list(&grad_list);
    195:     destroy_df_solver(&df);
    196:   }

    198:   VecDestroy(&bmrm->local_w);
    199:   VecScatterDestroy(&bmrm->scatter);
    200:   return(0);
    201: }

    204: /* ---------------------------------------------------------- */

    206: static PetscErrorCode TaoSetup_BMRM(Tao tao)
    207: {

    212:   /* Allocate some arrays */
    213:   if (!tao->gradient) {
    214:     VecDuplicate(tao->solution, &tao->gradient);
    215:   }
    216:   return(0);
    217: }

    219: /*------------------------------------------------------------*/
    220: static PetscErrorCode TaoDestroy_BMRM(Tao tao)
    221: {

    225:   PetscFree(tao->data);
    226:   return(0);
    227: }

    229: static PetscErrorCode TaoSetFromOptions_BMRM(PetscOptionItems *PetscOptionsObject,Tao tao)
    230: {
    232:   TAO_BMRM*      bmrm = (TAO_BMRM*)tao->data;

    235:   PetscOptionsHead(PetscOptionsObject,"BMRM for regularized risk minimization");
    236:   PetscOptionsReal("-tao_bmrm_lambda", "regulariser weight","", 100,&bmrm->lambda,NULL);
    237:   PetscOptionsTail();
    238:   return(0);
    239: }

    241: /*------------------------------------------------------------*/
    242: static PetscErrorCode TaoView_BMRM(Tao tao, PetscViewer viewer)
    243: {
    244:   PetscBool      isascii;

    248:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    249:   if (isascii) {
    250:     PetscViewerASCIIPushTab(viewer);
    251:     PetscViewerASCIIPopTab(viewer);
    252:   }
    253:   return(0);
    254: }

    256: /*------------------------------------------------------------*/
    257: /*MC
    258:   TAOBMRM - bundle method for regularized risk minimization

    260:   Options Database Keys:
    261: . - tao_bmrm_lambda - regulariser weight

    263:   Level: beginner
    264: M*/

    266: PETSC_EXTERN PetscErrorCode TaoCreate_BMRM(Tao tao)
    267: {
    268:   TAO_BMRM       *bmrm;

    272:   tao->ops->setup = TaoSetup_BMRM;
    273:   tao->ops->solve = TaoSolve_BMRM;
    274:   tao->ops->view  = TaoView_BMRM;
    275:   tao->ops->setfromoptions = TaoSetFromOptions_BMRM;
    276:   tao->ops->destroy = TaoDestroy_BMRM;

    278:   PetscNewLog(tao,&bmrm);
    279:   bmrm->lambda = 1.0;
    280:   tao->data = (void*)bmrm;

    282:   /* Override default settings (unless already changed) */
    283:   if (!tao->max_it_changed) tao->max_it = 2000;
    284:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
    285:   if (!tao->gatol_changed) tao->gatol = 1.0e-12;
    286:   if (!tao->grtol_changed) tao->grtol = 1.0e-12;

    288:   return(0);
    289: }

    291: PetscErrorCode init_df_solver(TAO_DF *df)
    292: {
    293:   PetscInt       i, n = INCRE_DIM;

    297:   /* default values */
    298:   df->maxProjIter = 200;
    299:   df->maxPGMIter = 300000;
    300:   df->b = 1.0;

    302:   /* memory space required by Dai-Fletcher */
    303:   df->cur_num_cp = n;
    304:   PetscMalloc1(n, &df->f);
    305:   PetscMalloc1(n, &df->a);
    306:   PetscMalloc1(n, &df->l);
    307:   PetscMalloc1(n, &df->u);
    308:   PetscMalloc1(n, &df->x);
    309:   PetscMalloc1(n, &df->Q);

    311:   for (i = 0; i < n; i ++) {
    312:     PetscMalloc1(n, &df->Q[i]);
    313:   }

    315:   PetscMalloc1(n, &df->g);
    316:   PetscMalloc1(n, &df->y);
    317:   PetscMalloc1(n, &df->tempv);
    318:   PetscMalloc1(n, &df->d);
    319:   PetscMalloc1(n, &df->Qd);
    320:   PetscMalloc1(n, &df->t);
    321:   PetscMalloc1(n, &df->xplus);
    322:   PetscMalloc1(n, &df->tplus);
    323:   PetscMalloc1(n, &df->sk);
    324:   PetscMalloc1(n, &df->yk);

    326:   PetscMalloc1(n, &df->ipt);
    327:   PetscMalloc1(n, &df->ipt2);
    328:   PetscMalloc1(n, &df->uv);
    329:   return(0);
    330: }

    332: PetscErrorCode ensure_df_space(PetscInt dim, TAO_DF *df)
    333: {
    335:   PetscReal      *tmp, **tmp_Q;
    336:   PetscInt       i, n, old_n;

    339:   df->dim = dim;
    340:   if (dim <= df->cur_num_cp) return(0);

    342:   old_n = df->cur_num_cp;
    343:   df->cur_num_cp += INCRE_DIM;
    344:   n = df->cur_num_cp;

    346:   /* memory space required by dai-fletcher */
    347:   PetscMalloc1(n, &tmp);
    348:   PetscArraycpy(tmp, df->f, old_n);
    349:   PetscFree(df->f);
    350:   df->f = tmp;

    352:   PetscMalloc1(n, &tmp);
    353:   PetscArraycpy(tmp, df->a, old_n);
    354:   PetscFree(df->a);
    355:   df->a = tmp;

    357:   PetscMalloc1(n, &tmp);
    358:   PetscArraycpy(tmp, df->l, old_n);
    359:   PetscFree(df->l);
    360:   df->l = tmp;

    362:   PetscMalloc1(n, &tmp);
    363:   PetscArraycpy(tmp, df->u, old_n);
    364:   PetscFree(df->u);
    365:   df->u = tmp;

    367:   PetscMalloc1(n, &tmp);
    368:   PetscArraycpy(tmp, df->x, old_n);
    369:   PetscFree(df->x);
    370:   df->x = tmp;

    372:   PetscMalloc1(n, &tmp_Q);
    373:   for (i = 0; i < n; i ++) {
    374:     PetscMalloc1(n, &tmp_Q[i]);
    375:     if (i < old_n) {
    376:       PetscArraycpy(tmp_Q[i], df->Q[i], old_n);
    377:       PetscFree(df->Q[i]);
    378:     }
    379:   }

    381:   PetscFree(df->Q);
    382:   df->Q = tmp_Q;

    384:   PetscFree(df->g);
    385:   PetscMalloc1(n, &df->g);

    387:   PetscFree(df->y);
    388:   PetscMalloc1(n, &df->y);

    390:   PetscFree(df->tempv);
    391:   PetscMalloc1(n, &df->tempv);

    393:   PetscFree(df->d);
    394:   PetscMalloc1(n, &df->d);

    396:   PetscFree(df->Qd);
    397:   PetscMalloc1(n, &df->Qd);

    399:   PetscFree(df->t);
    400:   PetscMalloc1(n, &df->t);

    402:   PetscFree(df->xplus);
    403:   PetscMalloc1(n, &df->xplus);

    405:   PetscFree(df->tplus);
    406:   PetscMalloc1(n, &df->tplus);

    408:   PetscFree(df->sk);
    409:   PetscMalloc1(n, &df->sk);

    411:   PetscFree(df->yk);
    412:   PetscMalloc1(n, &df->yk);

    414:   PetscFree(df->ipt);
    415:   PetscMalloc1(n, &df->ipt);

    417:   PetscFree(df->ipt2);
    418:   PetscMalloc1(n, &df->ipt2);

    420:   PetscFree(df->uv);
    421:   PetscMalloc1(n, &df->uv);
    422:   return(0);
    423: }

    425: PetscErrorCode destroy_df_solver(TAO_DF *df)
    426: {
    428:   PetscInt       i;

    431:   PetscFree(df->f);
    432:   PetscFree(df->a);
    433:   PetscFree(df->l);
    434:   PetscFree(df->u);
    435:   PetscFree(df->x);

    437:   for (i = 0; i < df->cur_num_cp; i ++) {
    438:     PetscFree(df->Q[i]);
    439:   }
    440:   PetscFree(df->Q);
    441:   PetscFree(df->ipt);
    442:   PetscFree(df->ipt2);
    443:   PetscFree(df->uv);
    444:   PetscFree(df->g);
    445:   PetscFree(df->y);
    446:   PetscFree(df->tempv);
    447:   PetscFree(df->d);
    448:   PetscFree(df->Qd);
    449:   PetscFree(df->t);
    450:   PetscFree(df->xplus);
    451:   PetscFree(df->tplus);
    452:   PetscFree(df->sk);
    453:   PetscFree(df->yk);
    454:   return(0);
    455: }

    457: /* Piecewise linear monotone target function for the Dai-Fletcher projector */
    458: PetscReal phi(PetscReal *x,PetscInt n,PetscReal lambda,PetscReal *a,PetscReal b,PetscReal *c,PetscReal *l,PetscReal *u)
    459: {
    460:   PetscReal r = 0.0;
    461:   PetscInt  i;

    463:   for (i = 0; i < n; i++){
    464:     x[i] = -c[i] + lambda*a[i];
    465:     if (x[i] > u[i])     x[i] = u[i];
    466:     else if (x[i] < l[i]) x[i] = l[i];
    467:     r += a[i]*x[i];
    468:   }
    469:   return r - b;
    470: }

    472: /** Modified Dai-Fletcher QP projector solves the problem:
    473:  *
    474:  *      minimise  0.5*x'*x - c'*x
    475:  *      subj to   a'*x = b
    476:  *                l \leq x \leq u
    477:  *
    478:  *  \param c The point to be projected onto feasible set
    479:  */
    480: PetscInt project(PetscInt n,PetscReal *a,PetscReal b,PetscReal *c,PetscReal *l,PetscReal *u,PetscReal *x,PetscReal *lam_ext,TAO_DF *df)
    481: {
    482:   PetscReal      lambda, lambdal, lambdau, dlambda, lambda_new;
    483:   PetscReal      r, rl, ru, s;
    484:   PetscInt       innerIter;
    485:   PetscBool      nonNegativeSlack = PETSC_FALSE;

    488:   *lam_ext = 0;
    489:   lambda  = 0;
    490:   dlambda = 0.5;
    491:   innerIter = 1;

    493:   /*  \phi(x;lambda) := 0.5*x'*x + c'*x - lambda*(a'*x-b)
    494:    *
    495:    *  Optimality conditions for \phi:
    496:    *
    497:    *  1. lambda   <= 0
    498:    *  2. r        <= 0
    499:    *  3. r*lambda == 0
    500:    */

    502:   /* Bracketing Phase */
    503:   r = phi(x, n, lambda, a, b, c, l, u);

    505:   if (nonNegativeSlack) {
    506:     /* inequality constraint, i.e., with \xi >= 0 constraint */
    507:     if (r < TOL_R) return 0;
    508:   } else  {
    509:     /* equality constraint ,i.e., without \xi >= 0 constraint */
    510:     if (PetscAbsReal(r) < TOL_R) return 0;
    511:   }

    513:   if (r < 0.0){
    514:     lambdal = lambda;
    515:     rl      = r;
    516:     lambda  = lambda + dlambda;
    517:     r       = phi(x, n, lambda, a, b, c, l, u);
    518:     while (r < 0.0 && dlambda < BMRM_INFTY)  {
    519:       lambdal = lambda;
    520:       s       = rl/r - 1.0;
    521:       if (s < 0.1) s = 0.1;
    522:       dlambda = dlambda + dlambda/s;
    523:       lambda  = lambda + dlambda;
    524:       rl      = r;
    525:       r       = phi(x, n, lambda, a, b, c, l, u);
    526:     }
    527:     lambdau = lambda;
    528:     ru      = r;
    529:   } else {
    530:     lambdau = lambda;
    531:     ru      = r;
    532:     lambda  = lambda - dlambda;
    533:     r       = phi(x, n, lambda, a, b, c, l, u);
    534:     while (r > 0.0 && dlambda > -BMRM_INFTY) {
    535:       lambdau = lambda;
    536:       s       = ru/r - 1.0;
    537:       if (s < 0.1) s = 0.1;
    538:       dlambda = dlambda + dlambda/s;
    539:       lambda  = lambda - dlambda;
    540:       ru      = r;
    541:       r       = phi(x, n, lambda, a, b, c, l, u);
    542:     }
    543:     lambdal = lambda;
    544:     rl      = r;
    545:   }

    547:   if (PetscAbsReal(dlambda) > BMRM_INFTY) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"L2N2_DaiFletcherPGM detected Infeasible QP problem!");

    549:   if (ru == 0){
    550:     return innerIter;
    551:   }

    553:   /* Secant Phase */
    554:   s       = 1.0 - rl/ru;
    555:   dlambda = dlambda/s;
    556:   lambda  = lambdau - dlambda;
    557:   r       = phi(x, n, lambda, a, b, c, l, u);

    559:   while (PetscAbsReal(r) > TOL_R
    560:          && dlambda > TOL_LAM * (1.0 + PetscAbsReal(lambda))
    561:          && innerIter < df->maxProjIter){
    562:     innerIter++;
    563:     if (r > 0.0){
    564:       if (s <= 2.0){
    565:         lambdau = lambda;
    566:         ru      = r;
    567:         s       = 1.0 - rl/ru;
    568:         dlambda = (lambdau - lambdal) / s;
    569:         lambda  = lambdau - dlambda;
    570:       } else {
    571:         s          = ru/r-1.0;
    572:         if (s < 0.1) s = 0.1;
    573:         dlambda    = (lambdau - lambda) / s;
    574:         lambda_new = 0.75*lambdal + 0.25*lambda;
    575:         if (lambda_new < (lambda - dlambda))
    576:           lambda_new = lambda - dlambda;
    577:         lambdau    = lambda;
    578:         ru         = r;
    579:         lambda     = lambda_new;
    580:         s          = (lambdau - lambdal) / (lambdau - lambda);
    581:       }
    582:     } else {
    583:       if (s >= 2.0){
    584:         lambdal = lambda;
    585:         rl      = r;
    586:         s       = 1.0 - rl/ru;
    587:         dlambda = (lambdau - lambdal) / s;
    588:         lambda  = lambdau - dlambda;
    589:       } else {
    590:         s          = rl/r - 1.0;
    591:         if (s < 0.1) s = 0.1;
    592:         dlambda    = (lambda-lambdal) / s;
    593:         lambda_new = 0.75*lambdau + 0.25*lambda;
    594:         if (lambda_new > (lambda + dlambda))
    595:           lambda_new = lambda + dlambda;
    596:         lambdal    = lambda;
    597:         rl         = r;
    598:         lambda     = lambda_new;
    599:         s          = (lambdau - lambdal) / (lambdau-lambda);
    600:       }
    601:     }
    602:     r = phi(x, n, lambda, a, b, c, l, u);
    603:   }

    605:   *lam_ext = lambda;
    606:   if (innerIter >= df->maxProjIter) {
    607:     PetscInfo(NULL,"WARNING: DaiFletcher max iterations\n");
    608:   }
    609:   return innerIter;
    610: }

    613: PetscErrorCode solve(TAO_DF *df)
    614: {
    616:   PetscInt       i, j, innerIter, it, it2, luv, info, lscount = 0, projcount = 0;
    617:   PetscReal      gd, max, ak, bk, akold, bkold, lamnew, alpha, kktlam=0.0, lam_ext;
    618:   PetscReal      DELTAsv, ProdDELTAsv;
    619:   PetscReal      c, *tempQ;
    620:   PetscReal      *x = df->x, *a = df->a, b = df->b, *l = df->l, *u = df->u, tol = df->tol;
    621:   PetscReal      *tempv = df->tempv, *y = df->y, *g = df->g, *d = df->d, *Qd = df->Qd;
    622:   PetscReal      *xplus = df->xplus, *tplus = df->tplus, *sk = df->sk, *yk = df->yk;
    623:   PetscReal      **Q = df->Q, *f = df->f, *t = df->t;
    624:   PetscInt       dim = df->dim, *ipt = df->ipt, *ipt2 = df->ipt2, *uv = df->uv;

    626:   /*** variables for the adaptive nonmonotone linesearch ***/
    627:   PetscInt    L, llast;
    628:   PetscReal   fr, fbest, fv, fc, fv0;

    630:   c = BMRM_INFTY;

    632:   DELTAsv = EPS_SV;
    633:   if (tol <= 1.0e-5 || dim <= 20) ProdDELTAsv = 0.0F;
    634:   else  ProdDELTAsv = EPS_SV;

    636:   for (i = 0; i < dim; i++)  tempv[i] = -x[i];

    638:   lam_ext = 0.0;

    640:   /* Project the initial solution */
    641:   projcount += project(dim, a, b, tempv, l, u, x, &lam_ext, df);

    643:   /* Compute gradient
    644:      g = Q*x + f; */

    646:   it = 0;
    647:   for (i = 0; i < dim; i++) {
    648:     if (PetscAbsReal(x[i]) > ProdDELTAsv) ipt[it++] = i;
    649:   }

    651:   PetscArrayzero(t, dim);
    652:   for (i = 0; i < it; i++){
    653:     tempQ = Q[ipt[i]];
    654:     for (j = 0; j < dim; j++) t[j] += (tempQ[j]*x[ipt[i]]);
    655:   }
    656:   for (i = 0; i < dim; i++){
    657:     g[i] = t[i] + f[i];
    658:   }

    661:   /* y = -(x_{k} - g_{k}) */
    662:   for (i = 0; i < dim; i++){
    663:     y[i] = g[i] - x[i];
    664:   }

    666:   /* Project x_{k} - g_{k} */
    667:   projcount += project(dim, a, b, y, l, u, tempv, &lam_ext, df);

    669:   /* y = P(x_{k} - g_{k}) - x_{k} */
    670:   max = ALPHA_MIN;
    671:   for (i = 0; i < dim; i++){
    672:     y[i] = tempv[i] - x[i];
    673:     if (PetscAbsReal(y[i]) > max) max = PetscAbsReal(y[i]);
    674:   }

    676:   if (max < tol*1e-3){
    677:     return 0;
    678:   }

    680:   alpha = 1.0 / max;

    682:   /* fv0 = f(x_{0}). Recall t = Q x_{k}  */
    683:   fv0   = 0.0;
    684:   for (i = 0; i < dim; i++) fv0 += x[i] * (0.5*t[i] + f[i]);

    686:   /*** adaptive nonmonotone linesearch ***/
    687:   L     = 2;
    688:   fr    = ALPHA_MAX;
    689:   fbest = fv0;
    690:   fc    = fv0;
    691:   llast = 0;
    692:   akold = bkold = 0.0;

    694:   /***      Iterator begins     ***/
    695:   for (innerIter = 1; innerIter <= df->maxPGMIter; innerIter++) {

    697:     /* tempv = -(x_{k} - alpha*g_{k}) */
    698:     for (i = 0; i < dim; i++)  tempv[i] = alpha*g[i] - x[i];

    700:     /* Project x_{k} - alpha*g_{k} */
    701:     projcount += project(dim, a, b, tempv, l, u, y, &lam_ext, df);

    704:     /* gd = \inner{d_{k}}{g_{k}}
    705:         d = P(x_{k} - alpha*g_{k}) - x_{k}
    706:     */
    707:     gd = 0.0;
    708:     for (i = 0; i < dim; i++){
    709:       d[i] = y[i] - x[i];
    710:       gd  += d[i] * g[i];
    711:     }

    713:     /* Gradient computation  */

    715:     /* compute Qd = Q*d  or  Qd = Q*y - t depending on their sparsity */

    717:     it = it2 = 0;
    718:     for (i = 0; i < dim; i++){
    719:       if (PetscAbsReal(d[i]) > (ProdDELTAsv*1.0e-2)) ipt[it++]   = i;
    720:     }
    721:     for (i = 0; i < dim; i++) {
    722:       if (PetscAbsReal(y[i]) > ProdDELTAsv) ipt2[it2++] = i;
    723:     }

    725:     PetscArrayzero(Qd, dim);
    726:     /* compute Qd = Q*d */
    727:     if (it < it2){
    728:       for (i = 0; i < it; i++){
    729:         tempQ = Q[ipt[i]];
    730:         for (j = 0; j < dim; j++) Qd[j] += (tempQ[j] * d[ipt[i]]);
    731:       }
    732:     } else { /* compute Qd = Q*y-t */
    733:       for (i = 0; i < it2; i++){
    734:         tempQ = Q[ipt2[i]];
    735:         for (j = 0; j < dim; j++) Qd[j] += (tempQ[j] * y[ipt2[i]]);
    736:       }
    737:       for (j = 0; j < dim; j++) Qd[j] -= t[j];
    738:     }

    740:     /* ak = inner{d_{k}}{d_{k}} */
    741:     ak = 0.0;
    742:     for (i = 0; i < dim; i++) ak += d[i] * d[i];

    744:     bk = 0.0;
    745:     for (i = 0; i < dim; i++) bk += d[i]*Qd[i];

    747:     if (bk > EPS*ak && gd < 0.0)  lamnew = -gd/bk;
    748:     else lamnew = 1.0;

    750:     /* fv is computing f(x_{k} + d_{k}) */
    751:     fv = 0.0;
    752:     for (i = 0; i < dim; i++){
    753:       xplus[i] = x[i] + d[i];
    754:       tplus[i] = t[i] + Qd[i];
    755:       fv      += xplus[i] * (0.5*tplus[i] + f[i]);
    756:     }

    758:     /* fr is fref */
    759:     if ((innerIter == 1 && fv >= fv0) || (innerIter > 1 && fv >= fr)){
    760:       lscount++;
    761:       fv = 0.0;
    762:       for (i = 0; i < dim; i++){
    763:         xplus[i] = x[i] + lamnew*d[i];
    764:         tplus[i] = t[i] + lamnew*Qd[i];
    765:         fv      += xplus[i] * (0.5*tplus[i] + f[i]);
    766:       }
    767:     }

    769:     for (i = 0; i < dim; i++){
    770:       sk[i] = xplus[i] - x[i];
    771:       yk[i] = tplus[i] - t[i];
    772:       x[i]  = xplus[i];
    773:       t[i]  = tplus[i];
    774:       g[i]  = t[i] + f[i];
    775:     }

    777:     /* update the line search control parameters */
    778:     if (fv < fbest){
    779:       fbest = fv;
    780:       fc    = fv;
    781:       llast = 0;
    782:     } else {
    783:       fc = (fc > fv ? fc : fv);
    784:       llast++;
    785:       if (llast == L){
    786:         fr    = fc;
    787:         fc    = fv;
    788:         llast = 0;
    789:       }
    790:     }

    792:     ak = bk = 0.0;
    793:     for (i = 0; i < dim; i++){
    794:       ak += sk[i] * sk[i];
    795:       bk += sk[i] * yk[i];
    796:     }

    798:     if (bk <= EPS*ak) alpha = ALPHA_MAX;
    799:     else {
    800:       if (bkold < EPS*akold) alpha = ak/bk;
    801:       else alpha = (akold+ak)/(bkold+bk);

    803:       if (alpha > ALPHA_MAX) alpha = ALPHA_MAX;
    804:       else if (alpha < ALPHA_MIN) alpha = ALPHA_MIN;
    805:     }

    807:     akold = ak;
    808:     bkold = bk;

    810:     /*** stopping criterion based on KKT conditions ***/
    811:     /* at optimal, gradient of lagrangian w.r.t. x is zero */

    813:     bk = 0.0;
    814:     for (i = 0; i < dim; i++) bk +=  x[i] * x[i];

    816:     if (PetscSqrtReal(ak) < tol*10 * PetscSqrtReal(bk)){
    817:       it     = 0;
    818:       luv    = 0;
    819:       kktlam = 0.0;
    820:       for (i = 0; i < dim; i++){
    821:         /* x[i] is active hence lagrange multipliers for box constraints
    822:                 are zero. The lagrange multiplier for ineq. const. is then
    823:                 defined as below
    824:         */
    825:         if ((x[i] > DELTAsv) && (x[i] < c-DELTAsv)){
    826:           ipt[it++] = i;
    827:           kktlam    = kktlam - a[i]*g[i];
    828:         } else  uv[luv++] = i;
    829:       }

    831:       if (it == 0 && PetscSqrtReal(ak) < tol*0.5 * PetscSqrtReal(bk)) return 0;
    832:       else {
    833:         kktlam = kktlam/it;
    834:         info   = 1;
    835:         for (i = 0; i < it; i++) {
    836:           if (PetscAbsReal(a[ipt[i]] * g[ipt[i]] + kktlam) > tol) {
    837:             info = 0;
    838:             break;
    839:           }
    840:         }
    841:         if (info == 1)  {
    842:           for (i = 0; i < luv; i++)  {
    843:             if (x[uv[i]] <= DELTAsv){
    844:               /* x[i] == lower bound, hence, lagrange multiplier (say, beta) for lower bound may
    845:                      not be zero. So, the gradient without beta is > 0
    846:               */
    847:               if (g[uv[i]] + kktlam*a[uv[i]] < -tol){
    848:                 info = 0;
    849:                 break;
    850:               }
    851:             } else {
    852:               /* x[i] == upper bound, hence, lagrange multiplier (say, eta) for upper bound may
    853:                      not be zero. So, the gradient without eta is < 0
    854:               */
    855:               if (g[uv[i]] + kktlam*a[uv[i]] > tol) {
    856:                 info = 0;
    857:                 break;
    858:               }
    859:             }
    860:           }
    861:         }

    863:         if (info == 1) return 0;
    864:       }
    865:     }
    866:   }
    867:   return 0;
    868: }