Actual source code: bmrm.c
petsc-3.10.5 2019-03-28
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      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);
    103:
    104:   while (tao->reason == TAO_CONTINUE_ITERATING) {
    105:     /* compute bt = Remp(Wt-1) - <Wt-1, At> */
    106:     VecDot(W, G, &bt);
    107:     bt = f - bt;

    109:     /* First gather the gradient to the master node */
    110:     VecScatterBegin(bmrm->scatter, G, bmrm->local_w, INSERT_VALUES, SCATTER_FORWARD);
    111:     VecScatterEnd(bmrm->scatter, G, bmrm->local_w, INSERT_VALUES, SCATTER_FORWARD);

    113:     /* Bring up the inner solver */
    114:     if (!rank) {
    115:       ensure_df_space(tao->niter+1, &df);
    116:       make_grad_node(bmrm->local_w, &pgrad);
    117:       tail_glist->next = pgrad;
    118:       tail_glist = pgrad;

    120:       df.a[tao->niter] = 1.0;
    121:       df.f[tao->niter] = -bt;
    122:       df.u[tao->niter] = 1.0;
    123:       df.l[tao->niter] = 0.0;

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

    134:       if (tao->niter > 0) {
    135:         df.x[tao->niter] = 0.0;
    136:         solve(&df);
    137:       } else
    138:         df.x[0] = 1.0;

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

    150:       VecNorm(bmrm->local_w, NORM_2, &reg);
    151:       reg = 0.5*lambda*reg*reg;
    152:       jtwt -= reg;
    153:     } /* end if rank == 0 */

    155:     /* scatter the new W to all nodes */
    156:     VecScatterBegin(bmrm->scatter,bmrm->local_w,W,INSERT_VALUES,SCATTER_REVERSE);
    157:     VecScatterEnd(bmrm->scatter,bmrm->local_w,W,INSERT_VALUES,SCATTER_REVERSE);

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

    161:     MPI_Bcast(&jtwt,1,MPIU_REAL,0,comm);
    162:     MPI_Bcast(&reg,1,MPIU_REAL,0,comm);

    164:     jw = reg + f;                                       /* J(w) = regularizer + Remp(w) */
    165:     if (jw < min_jw) min_jw = jw;
    166:     if (jtwt > max_jtwt) max_jtwt = jtwt;

    168:     pre_epsilon = epsilon;
    169:     epsilon = min_jw - jtwt;

    171:     if (!rank) {
    172:       if (innerSolverTol > epsilon) innerSolverTol = epsilon;
    173:       else if (innerSolverTol < 1e-7) innerSolverTol = 1e-7;

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

    178:       df.tol = innerSolverTol*0.5;
    179:     }

    181:     tao->niter++;
    182:     TaoLogConvergenceHistory(tao,min_jw,epsilon,0.0,tao->ksp_its);
    183:     TaoMonitor(tao,tao->niter,min_jw,epsilon,0.0,tao->step);
    184:     (*tao->ops->convergencetest)(tao,tao->cnvP);
    185:   }

    187:   /* free all the memory */
    188:   if (!rank) {
    189:     destroy_grad_list(&grad_list);
    190:     destroy_df_solver(&df);
    191:   }

    193:   VecDestroy(&bmrm->local_w);
    194:   VecScatterDestroy(&bmrm->scatter);
    195:   return(0);
    196: }

    199: /* ---------------------------------------------------------- */

    201: static PetscErrorCode TaoSetup_BMRM(Tao tao)
    202: {

    207:   /* Allocate some arrays */
    208:   if (!tao->gradient) {
    209:     VecDuplicate(tao->solution, &tao->gradient);
    210:   }
    211:   return(0);
    212: }

    214: /*------------------------------------------------------------*/
    215: static PetscErrorCode TaoDestroy_BMRM(Tao tao)
    216: {

    220:   PetscFree(tao->data);
    221:   return(0);
    222: }

    224: static PetscErrorCode TaoSetFromOptions_BMRM(PetscOptionItems *PetscOptionsObject,Tao tao)
    225: {
    227:   TAO_BMRM*      bmrm = (TAO_BMRM*)tao->data;

    230:   PetscOptionsHead(PetscOptionsObject,"BMRM for regularized risk minimization");
    231:   PetscOptionsReal("-tao_bmrm_lambda", "regulariser weight","", 100,&bmrm->lambda,NULL);
    232:   PetscOptionsTail();
    233:   return(0);
    234: }

    236: /*------------------------------------------------------------*/
    237: static PetscErrorCode TaoView_BMRM(Tao tao, PetscViewer viewer)
    238: {
    239:   PetscBool      isascii;

    243:   PetscObjectTypeCompare((PetscObject)viewer,PETSCVIEWERASCII,&isascii);
    244:   if (isascii) {
    245:     PetscViewerASCIIPushTab(viewer);
    246:     PetscViewerASCIIPopTab(viewer);
    247:   }
    248:   return(0);
    249: }

    251: /*------------------------------------------------------------*/
    252: /*MC
    253:   TAOBMRM - bundle method for regularized risk minimization

    255:   Options Database Keys:
    256: . - tao_bmrm_lambda - regulariser weight

    258:   Level: beginner
    259: M*/

    261: PETSC_EXTERN PetscErrorCode TaoCreate_BMRM(Tao tao)
    262: {
    263:   TAO_BMRM       *bmrm;

    267:   tao->ops->setup = TaoSetup_BMRM;
    268:   tao->ops->solve = TaoSolve_BMRM;
    269:   tao->ops->view  = TaoView_BMRM;
    270:   tao->ops->setfromoptions = TaoSetFromOptions_BMRM;
    271:   tao->ops->destroy = TaoDestroy_BMRM;

    273:   PetscNewLog(tao,&bmrm);
    274:   bmrm->lambda = 1.0;
    275:   tao->data = (void*)bmrm;

    277:   /* Override default settings (unless already changed) */
    278:   if (!tao->max_it_changed) tao->max_it = 2000;
    279:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
    280:   if (!tao->gatol_changed) tao->gatol = 1.0e-12;
    281:   if (!tao->grtol_changed) tao->grtol = 1.0e-12;

    283:   return(0);
    284: }

    286: PetscErrorCode init_df_solver(TAO_DF *df)
    287: {
    288:   PetscInt       i, n = INCRE_DIM;

    292:   /* default values */
    293:   df->maxProjIter = 200;
    294:   df->maxPGMIter = 300000;
    295:   df->b = 1.0;

    297:   /* memory space required by Dai-Fletcher */
    298:   df->cur_num_cp = n;
    299:   PetscMalloc1(n, &df->f);
    300:   PetscMalloc1(n, &df->a);
    301:   PetscMalloc1(n, &df->l);
    302:   PetscMalloc1(n, &df->u);
    303:   PetscMalloc1(n, &df->x);
    304:   PetscMalloc1(n, &df->Q);

    306:   for (i = 0; i < n; i ++) {
    307:     PetscMalloc1(n, &df->Q[i]);
    308:   }

    310:   PetscMalloc1(n, &df->g);
    311:   PetscMalloc1(n, &df->y);
    312:   PetscMalloc1(n, &df->tempv);
    313:   PetscMalloc1(n, &df->d);
    314:   PetscMalloc1(n, &df->Qd);
    315:   PetscMalloc1(n, &df->t);
    316:   PetscMalloc1(n, &df->xplus);
    317:   PetscMalloc1(n, &df->tplus);
    318:   PetscMalloc1(n, &df->sk);
    319:   PetscMalloc1(n, &df->yk);

    321:   PetscMalloc1(n, &df->ipt);
    322:   PetscMalloc1(n, &df->ipt2);
    323:   PetscMalloc1(n, &df->uv);
    324:   return(0);
    325: }

    327: PetscErrorCode ensure_df_space(PetscInt dim, TAO_DF *df)
    328: {
    330:   PetscReal      *tmp, **tmp_Q;
    331:   PetscInt       i, n, old_n;

    334:   df->dim = dim;
    335:   if (dim <= df->cur_num_cp) return(0);

    337:   old_n = df->cur_num_cp;
    338:   df->cur_num_cp += INCRE_DIM;
    339:   n = df->cur_num_cp;

    341:   /* memory space required by dai-fletcher */
    342:   PetscMalloc1(n, &tmp);
    343:   PetscMemcpy(tmp, df->f, sizeof(PetscReal)*old_n);
    344:   PetscFree(df->f);
    345:   df->f = tmp;

    347:   PetscMalloc1(n, &tmp);
    348:   PetscMemcpy(tmp, df->a, sizeof(PetscReal)*old_n);
    349:   PetscFree(df->a);
    350:   df->a = tmp;

    352:   PetscMalloc1(n, &tmp);
    353:   PetscMemcpy(tmp, df->l, sizeof(PetscReal)*old_n);
    354:   PetscFree(df->l);
    355:   df->l = tmp;

    357:   PetscMalloc1(n, &tmp);
    358:   PetscMemcpy(tmp, df->u, sizeof(PetscReal)*old_n);
    359:   PetscFree(df->u);
    360:   df->u = tmp;

    362:   PetscMalloc1(n, &tmp);
    363:   PetscMemcpy(tmp, df->x, sizeof(PetscReal)*old_n);
    364:   PetscFree(df->x);
    365:   df->x = tmp;

    367:   PetscMalloc1(n, &tmp_Q);
    368:   for (i = 0; i < n; i ++) {
    369:     PetscMalloc1(n, &tmp_Q[i]);
    370:     if (i < old_n) {
    371:       PetscMemcpy(tmp_Q[i], df->Q[i], sizeof(PetscReal)*old_n);
    372:       PetscFree(df->Q[i]);
    373:     }
    374:   }

    376:   PetscFree(df->Q);
    377:   df->Q = tmp_Q;

    379:   PetscFree(df->g);
    380:   PetscMalloc1(n, &df->g);

    382:   PetscFree(df->y);
    383:   PetscMalloc1(n, &df->y);

    385:   PetscFree(df->tempv);
    386:   PetscMalloc1(n, &df->tempv);

    388:   PetscFree(df->d);
    389:   PetscMalloc1(n, &df->d);

    391:   PetscFree(df->Qd);
    392:   PetscMalloc1(n, &df->Qd);

    394:   PetscFree(df->t);
    395:   PetscMalloc1(n, &df->t);

    397:   PetscFree(df->xplus);
    398:   PetscMalloc1(n, &df->xplus);

    400:   PetscFree(df->tplus);
    401:   PetscMalloc1(n, &df->tplus);

    403:   PetscFree(df->sk);
    404:   PetscMalloc1(n, &df->sk);

    406:   PetscFree(df->yk);
    407:   PetscMalloc1(n, &df->yk);

    409:   PetscFree(df->ipt);
    410:   PetscMalloc1(n, &df->ipt);

    412:   PetscFree(df->ipt2);
    413:   PetscMalloc1(n, &df->ipt2);

    415:   PetscFree(df->uv);
    416:   PetscMalloc1(n, &df->uv);
    417:   return(0);
    418: }

    420: PetscErrorCode destroy_df_solver(TAO_DF *df)
    421: {
    423:   PetscInt       i;

    426:   PetscFree(df->f);
    427:   PetscFree(df->a);
    428:   PetscFree(df->l);
    429:   PetscFree(df->u);
    430:   PetscFree(df->x);

    432:   for (i = 0; i < df->cur_num_cp; i ++) {
    433:     PetscFree(df->Q[i]);
    434:   }
    435:   PetscFree(df->Q);
    436:   PetscFree(df->ipt);
    437:   PetscFree(df->ipt2);
    438:   PetscFree(df->uv);
    439:   PetscFree(df->g);
    440:   PetscFree(df->y);
    441:   PetscFree(df->tempv);
    442:   PetscFree(df->d);
    443:   PetscFree(df->Qd);
    444:   PetscFree(df->t);
    445:   PetscFree(df->xplus);
    446:   PetscFree(df->tplus);
    447:   PetscFree(df->sk);
    448:   PetscFree(df->yk);
    449:   return(0);
    450: }

    452: /* Piecewise linear monotone target function for the Dai-Fletcher projector */
    453: PetscReal phi(PetscReal *x,PetscInt n,PetscReal lambda,PetscReal *a,PetscReal b,PetscReal *c,PetscReal *l,PetscReal *u)
    454: {
    455:   PetscReal r = 0.0;
    456:   PetscInt  i;

    458:   for (i = 0; i < n; i++){
    459:     x[i] = -c[i] + lambda*a[i];
    460:     if (x[i] > u[i])     x[i] = u[i];
    461:     else if(x[i] < l[i]) x[i] = l[i];
    462:     r += a[i]*x[i];
    463:   }
    464:   return r - b;
    465: }

    467: /** Modified Dai-Fletcher QP projector solves the problem:
    468:  *
    469:  *      minimise  0.5*x'*x - c'*x
    470:  *      subj to   a'*x = b
    471:  *                l \leq x \leq u
    472:  *
    473:  *  \param c The point to be projected onto feasible set
    474:  */
    475: PetscInt project(PetscInt n,PetscReal *a,PetscReal b,PetscReal *c,PetscReal *l,PetscReal *u,PetscReal *x,PetscReal *lam_ext,TAO_DF *df)
    476: {
    477:   PetscReal      lambda, lambdal, lambdau, dlambda, lambda_new;
    478:   PetscReal      r, rl, ru, s;
    479:   PetscInt       innerIter;
    480:   PetscBool      nonNegativeSlack = PETSC_FALSE;

    483:   *lam_ext = 0;
    484:   lambda  = 0;
    485:   dlambda = 0.5;
    486:   innerIter = 1;

    488:   /*  \phi(x;lambda) := 0.5*x'*x + c'*x - lambda*(a'*x-b)
    489:    *
    490:    *  Optimality conditions for \phi:
    491:    *
    492:    *  1. lambda   <= 0
    493:    *  2. r        <= 0
    494:    *  3. r*lambda == 0
    495:    */

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

    500:   if(nonNegativeSlack) {
    501:     /* inequality constraint, i.e., with \xi >= 0 constraint */
    502:     if (r < TOL_R) return 0;
    503:   } else  {
    504:     /* equality constraint ,i.e., without \xi >= 0 constraint */
    505:     if (PetscAbsReal(r) < TOL_R) return 0;
    506:   }

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

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

    544:   if(ru == 0){
    545:     return innerIter;
    546:   }

    548:   /* Secant Phase */
    549:   s       = 1.0 - rl/ru;
    550:   dlambda = dlambda/s;
    551:   lambda  = lambdau - dlambda;
    552:   r       = phi(x, n, lambda, a, b, c, l, u);

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

    600:   *lam_ext = lambda;
    601:   if(innerIter >= df->maxProjIter) {
    602:     PetscPrintf(PETSC_COMM_SELF, "WARNING: DaiFletcher max iterations\n");
    603:   }
    604:   return innerIter;
    605: }

    608: PetscErrorCode solve(TAO_DF *df)
    609: {
    611:   PetscInt       i, j, innerIter, it, it2, luv, info, lscount = 0, projcount = 0;
    612:   PetscReal      gd, max, ak, bk, akold, bkold, lamnew, alpha, kktlam=0.0, lam_ext;
    613:   PetscReal      DELTAsv, ProdDELTAsv;
    614:   PetscReal      c, *tempQ;
    615:   PetscReal      *x = df->x, *a = df->a, b = df->b, *l = df->l, *u = df->u, tol = df->tol;
    616:   PetscReal      *tempv = df->tempv, *y = df->y, *g = df->g, *d = df->d, *Qd = df->Qd;
    617:   PetscReal      *xplus = df->xplus, *tplus = df->tplus, *sk = df->sk, *yk = df->yk;
    618:   PetscReal      **Q = df->Q, *f = df->f, *t = df->t;
    619:   PetscInt       dim = df->dim, *ipt = df->ipt, *ipt2 = df->ipt2, *uv = df->uv;

    621:   /*** variables for the adaptive nonmonotone linesearch ***/
    622:   PetscInt    L, llast;
    623:   PetscReal   fr, fbest, fv, fc, fv0;

    625:   c = BMRM_INFTY;

    627:   DELTAsv = EPS_SV;
    628:   if (tol <= 1.0e-5 || dim <= 20) ProdDELTAsv = 0.0F;
    629:   else  ProdDELTAsv = EPS_SV;

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

    633:   lam_ext = 0.0;

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

    638:   /* Compute gradient
    639:      g = Q*x + f; */

    641:   it = 0;
    642:   for (i = 0; i < dim; i++) {
    643:     if (PetscAbsReal(x[i]) > ProdDELTAsv) ipt[it++] = i;
    644:   }

    646:   PetscMemzero(t, dim*sizeof(PetscReal));
    647:   for (i = 0; i < it; i++){
    648:     tempQ = Q[ipt[i]];
    649:     for (j = 0; j < dim; j++) t[j] += (tempQ[j]*x[ipt[i]]);
    650:   }
    651:   for (i = 0; i < dim; i++){
    652:     g[i] = t[i] + f[i];
    653:   }

    656:   /* y = -(x_{k} - g_{k}) */
    657:   for (i = 0; i < dim; i++){
    658:     y[i] = g[i] - x[i];
    659:   }

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

    664:   /* y = P(x_{k} - g_{k}) - x_{k} */
    665:   max = ALPHA_MIN;
    666:   for (i = 0; i < dim; i++){
    667:     y[i] = tempv[i] - x[i];
    668:     if (PetscAbsReal(y[i]) > max) max = PetscAbsReal(y[i]);
    669:   }

    671:   if (max < tol*1e-3){
    672:     return 0;
    673:   }

    675:   alpha = 1.0 / max;

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

    681:   /*** adaptive nonmonotone linesearch ***/
    682:   L     = 2;
    683:   fr    = ALPHA_MAX;
    684:   fbest = fv0;
    685:   fc    = fv0;
    686:   llast = 0;
    687:   akold = bkold = 0.0;

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

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

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

    699:     /* gd = \inner{d_{k}}{g_{k}}
    700:         d = P(x_{k} - alpha*g_{k}) - x_{k}
    701:     */
    702:     gd = 0.0;
    703:     for (i = 0; i < dim; i++){
    704:       d[i] = y[i] - x[i];
    705:       gd  += d[i] * g[i];
    706:     }

    708:     /* Gradient computation  */

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

    712:     it = it2 = 0;
    713:     for (i = 0; i < dim; i++){
    714:       if (PetscAbsReal(d[i]) > (ProdDELTAsv*1.0e-2)) ipt[it++]   = i;
    715:     }
    716:     for (i = 0; i < dim; i++) {
    717:       if (PetscAbsReal(y[i]) > ProdDELTAsv) ipt2[it2++] = i;
    718:     }

    720:     PetscMemzero(Qd, dim*sizeof(PetscReal));
    721:     /* compute Qd = Q*d */
    722:     if (it < it2){
    723:       for (i = 0; i < it; i++){
    724:         tempQ = Q[ipt[i]];
    725:         for (j = 0; j < dim; j++) Qd[j] += (tempQ[j] * d[ipt[i]]);
    726:       }
    727:     } else { /* compute Qd = Q*y-t */
    728:       for (i = 0; i < it2; i++){
    729:         tempQ = Q[ipt2[i]];
    730:         for (j = 0; j < dim; j++) Qd[j] += (tempQ[j] * y[ipt2[i]]);
    731:       }
    732:       for (j = 0; j < dim; j++) Qd[j] -= t[j];
    733:     }

    735:     /* ak = inner{d_{k}}{d_{k}} */
    736:     ak = 0.0;
    737:     for (i = 0; i < dim; i++) ak += d[i] * d[i];

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

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

    745:     /* fv is computing f(x_{k} + d_{k}) */
    746:     fv = 0.0;
    747:     for (i = 0; i < dim; i++){
    748:       xplus[i] = x[i] + d[i];
    749:       tplus[i] = t[i] + Qd[i];
    750:       fv      += xplus[i] * (0.5*tplus[i] + f[i]);
    751:     }

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

    764:     for (i = 0; i < dim; i++){
    765:       sk[i] = xplus[i] - x[i];
    766:       yk[i] = tplus[i] - t[i];
    767:       x[i]  = xplus[i];
    768:       t[i]  = tplus[i];
    769:       g[i]  = t[i] + f[i];
    770:     }

    772:     /* update the line search control parameters */
    773:     if (fv < fbest){
    774:       fbest = fv;
    775:       fc    = fv;
    776:       llast = 0;
    777:     } else {
    778:       fc = (fc > fv ? fc : fv);
    779:       llast++;
    780:       if (llast == L){
    781:         fr    = fc;
    782:         fc    = fv;
    783:         llast = 0;
    784:       }
    785:     }

    787:     ak = bk = 0.0;
    788:     for (i = 0; i < dim; i++){
    789:       ak += sk[i] * sk[i];
    790:       bk += sk[i] * yk[i];
    791:     }

    793:     if (bk <= EPS*ak) alpha = ALPHA_MAX;
    794:     else {
    795:       if (bkold < EPS*akold) alpha = ak/bk;
    796:       else alpha = (akold+ak)/(bkold+bk);

    798:       if (alpha > ALPHA_MAX) alpha = ALPHA_MAX;
    799:       else if (alpha < ALPHA_MIN) alpha = ALPHA_MIN;
    800:     }

    802:     akold = ak;
    803:     bkold = bk;

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

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

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

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

    858:         if (info == 1) return 0;
    859:       }
    860:     }
    861:   }
    862:   return 0;
    863: }