Date: 2022-03-04T17:39:54+00:00

Actual source code: asfls.c

      1: #include <../src/tao/complementarity/impls/ssls/ssls.h>
      2: /*
      3:    Context for ASXLS
      4:      -- active-set      - reduced matrices formed
      5:                           - inherit properties of original system
      6:      -- semismooth (S)  - function not differentiable
      7:                         - merit function continuously differentiable
      8:                         - Fischer-Burmeister reformulation of complementarity
      9:                           - Billups composition for two finite bounds
     10:      -- infeasible (I)  - iterates not guaranteed to remain within bounds
     11:      -- feasible (F)    - iterates guaranteed to remain within bounds
     12:      -- linesearch (LS) - Armijo rule on direction

     14:    Many other reformulations are possible and combinations of
     15:    feasible/infeasible and linesearch/trust region are possible.

     17:    Basic theory
     18:      Fischer-Burmeister reformulation is semismooth with a continuously
     19:      differentiable merit function and strongly semismooth if the F has
     20:      lipschitz continuous derivatives.

     22:      Every accumulation point generated by the algorithm is a stationary
     23:      point for the merit function.  Stationary points of the merit function
     24:      are solutions of the complementarity problem if
     25:        a.  the stationary point has a BD-regular subdifferential, or
     26:        b.  the Schur complement F'/F'_ff is a P_0-matrix where ff is the
     27:            index set corresponding to the free variables.

     29:      If one of the accumulation points has a BD-regular subdifferential then
     30:        a.  the entire sequence converges to this accumulation point at
     31:            a local q-superlinear rate
     32:        b.  if in addition the reformulation is strongly semismooth near
     33:            this accumulation point, then the algorithm converges at a
     34:            local q-quadratic rate.

     36:    The theory for the feasible version follows from the feasible descent
     37:    algorithm framework.

     39:    References:
     40:      Billups, "Algorithms for Complementarity Problems and Generalized
     41:        Equations," Ph.D thesis, University of Wisconsin  Madison, 1995.
     42:      De Luca, Facchinei, Kanzow, "A Semismooth Equation Approach to the
     43:        Solution of Nonlinear Complementarity Problems," Mathematical
     44:        Programming, 75, pages 407439, 1996.
     45:      Ferris, Kanzow, Munson, "Feasible Descent Algorithms for Mixed
     46:        Complementarity Problems," Mathematical Programming, 86,
     47:        pages 475497, 1999.
     48:      Fischer, "A Special Newton type Optimization Method," Optimization,
     49:        24, 1992
     50:      Munson, Facchinei, Ferris, Fischer, Kanzow, "The Semismooth Algorithm
     51:        for Large Scale Complementarity Problems," Technical Report,
     52:        University of Wisconsin  Madison, 1999.
     53: */

     55: static PetscErrorCode TaoSetUp_ASFLS(Tao tao)
     56: {
     57:   TAO_SSLS       *asls = (TAO_SSLS *)tao->data;

     61:   VecDuplicate(tao->solution,&tao->gradient);
     62:   VecDuplicate(tao->solution,&tao->stepdirection);
     63:   VecDuplicate(tao->solution,&asls->ff);
     64:   VecDuplicate(tao->solution,&asls->dpsi);
     65:   VecDuplicate(tao->solution,&asls->da);
     66:   VecDuplicate(tao->solution,&asls->db);
     67:   VecDuplicate(tao->solution,&asls->t1);
     68:   VecDuplicate(tao->solution,&asls->t2);
     69:   VecDuplicate(tao->solution, &asls->w);
     70:   asls->fixed = NULL;
     71:   asls->free = NULL;
     72:   asls->J_sub = NULL;
     73:   asls->Jpre_sub = NULL;
     74:   asls->r1 = NULL;
     75:   asls->r2 = NULL;
     76:   asls->r3 = NULL;
     77:   asls->dxfree = NULL;
     78:   return(0);
     79: }

     81: static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn,  Vec G, void *ptr)
     82: {
     83:   Tao            tao = (Tao)ptr;
     84:   TAO_SSLS       *asls = (TAO_SSLS *)tao->data;

     88:   TaoComputeConstraints(tao, X, tao->constraints);
     89:   VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff);
     90:   VecNorm(asls->ff,NORM_2,&asls->merit);
     91:   *fcn = 0.5*asls->merit*asls->merit;
     92:   TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre);

     94:   MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db);
     95:   VecPointwiseMult(asls->t1, asls->ff, asls->db);
     96:   MatMultTranspose(tao->jacobian,asls->t1,G);
     97:   VecPointwiseMult(asls->t1, asls->ff, asls->da);
     98:   VecAXPY(G,1.0,asls->t1);
     99:   return(0);
    100: }

    102: static PetscErrorCode TaoDestroy_ASFLS(Tao tao)
    103: {
    104:   TAO_SSLS       *ssls = (TAO_SSLS *)tao->data;

    108:   VecDestroy(&ssls->ff);
    109:   VecDestroy(&ssls->dpsi);
    110:   VecDestroy(&ssls->da);
    111:   VecDestroy(&ssls->db);
    112:   VecDestroy(&ssls->w);
    113:   VecDestroy(&ssls->t1);
    114:   VecDestroy(&ssls->t2);
    115:   VecDestroy(&ssls->r1);
    116:   VecDestroy(&ssls->r2);
    117:   VecDestroy(&ssls->r3);
    118:   VecDestroy(&ssls->dxfree);
    119:   MatDestroy(&ssls->J_sub);
    120:   MatDestroy(&ssls->Jpre_sub);
    121:   ISDestroy(&ssls->fixed);
    122:   ISDestroy(&ssls->free);
    123:   PetscFree(tao->data);
    124:   tao->data = NULL;
    125:   return(0);
    126: }

    128: static PetscErrorCode TaoSolve_ASFLS(Tao tao)
    129: {
    130:   TAO_SSLS                     *asls = (TAO_SSLS *)tao->data;
    131:   PetscReal                    psi,ndpsi, normd, innerd, t=0;
    132:   PetscInt                     nf;
    133:   PetscErrorCode               ierr;
    134:   TaoLineSearchConvergedReason ls_reason;

    137:   /* Assume that Setup has been called!
    138:      Set the structure for the Jacobian and create a linear solver. */

    140:   TaoComputeVariableBounds(tao);
    141:   TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao);
    142:   TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao);
    143:   TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);

    145:   VecMedian(tao->XL, tao->solution, tao->XU, tao->solution);

    147:   /* Calculate the function value and fischer function value at the
    148:      current iterate */
    149:   TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi);
    150:   VecNorm(asls->dpsi,NORM_2,&ndpsi);

    152:   tao->reason = TAO_CONTINUE_ITERATING;
    153:   while (1) {
    154:     /* Check the converged criteria */
    155:     PetscInfo3(tao,"iter %D, merit: %g, ||dpsi||: %g\n",tao->niter,(double)asls->merit,(double)ndpsi);
    156:     TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its);
    157:     TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t);
    158:     (*tao->ops->convergencetest)(tao,tao->cnvP);
    159:     if (TAO_CONTINUE_ITERATING != tao->reason) break;

    161:     /* Call general purpose update function */
    162:     if (tao->ops->update) {
    163:       (*tao->ops->update)(tao, tao->niter, tao->user_update);
    164:     }
    165:     tao->niter++;

    167:     /* We are going to solve a linear system of equations.  We need to
    168:        set the tolerances for the solve so that we maintain an asymptotic
    169:        rate of convergence that is superlinear.
    170:        Note: these tolerances are for the reduced system.  We really need
    171:        to make sure that the full system satisfies the full-space conditions.

    173:        This rule gives superlinear asymptotic convergence
    174:        asls->atol = min(0.5, asls->merit*sqrt(asls->merit));
    175:        asls->rtol = 0.0;

    177:        This rule gives quadratic asymptotic convergence
    178:        asls->atol = min(0.5, asls->merit*asls->merit);
    179:        asls->rtol = 0.0;

    181:        Calculate a free and fixed set of variables.  The fixed set of
    182:        variables are those for the d_b is approximately equal to zero.
    183:        The definition of approximately changes as we approach the solution
    184:        to the problem.

    186:        No one rule is guaranteed to work in all cases.  The following
    187:        definition is based on the norm of the Jacobian matrix.  If the
    188:        norm is large, the tolerance becomes smaller. */
    189:     MatNorm(tao->jacobian,NORM_1,&asls->identifier);
    190:     asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier);

    192:     VecSet(asls->t1,-asls->identifier);
    193:     VecSet(asls->t2, asls->identifier);

    195:     ISDestroy(&asls->fixed);
    196:     ISDestroy(&asls->free);
    197:     VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);
    198:     ISComplementVec(asls->fixed,asls->t1, &asls->free);

    200:     ISGetSize(asls->fixed,&nf);
    201:     PetscInfo1(tao,"Number of fixed variables: %D\n", nf);

    203:     /* We now have our partition.  Now calculate the direction in the
    204:        fixed variable space. */
    205:     TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);
    206:     TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);
    207:     VecPointwiseDivide(asls->r1,asls->r1,asls->r2);
    208:     VecSet(tao->stepdirection,0.0);
    209:     VecISAXPY(tao->stepdirection, asls->fixed, 1.0,asls->r1);

    211:     /* Our direction in the Fixed Variable Set is fixed.  Calculate the
    212:        information needed for the step in the Free Variable Set.  To
    213:        do this, we need to know the diagonal perturbation and the
    214:        right hand side. */

    216:     TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);
    217:     TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);
    218:     TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);
    219:     VecPointwiseDivide(asls->r1,asls->r1, asls->r3);
    220:     VecPointwiseDivide(asls->r2,asls->r2, asls->r3);

    222:     /* r1 is the diagonal perturbation
    223:        r2 is the right hand side
    224:        r3 is no longer needed

    226:        Now need to modify r2 for our direction choice in the fixed
    227:        variable set:  calculate t1 = J*d, take the reduced vector
    228:        of t1 and modify r2. */

    230:     MatMult(tao->jacobian, tao->stepdirection, asls->t1);
    231:     TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);
    232:     VecAXPY(asls->r2, -1.0, asls->r3);

    234:     /* Calculate the reduced problem matrix and the direction */
    235:     TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);
    236:     if (tao->jacobian != tao->jacobian_pre) {
    237:       TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);
    238:     } else {
    239:       MatDestroy(&asls->Jpre_sub);
    240:       asls->Jpre_sub = asls->J_sub;
    241:       PetscObjectReference((PetscObject)(asls->Jpre_sub));
    242:     }
    243:     MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);
    244:     TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);
    245:     VecSet(asls->dxfree, 0.0);

    247:     /* Calculate the reduced direction.  (Really negative of Newton
    248:        direction.  Therefore, rest of the code uses -d.) */
    249:     KSPReset(tao->ksp);
    250:     KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);
    251:     KSPSolve(tao->ksp, asls->r2, asls->dxfree);
    252:     KSPGetIterationNumber(tao->ksp,&tao->ksp_its);
    253:     tao->ksp_tot_its+=tao->ksp_its;

    255:     /* Add the direction in the free variables back into the real direction. */
    256:     VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);

    258:     /* Check the projected real direction for descent and if not, use the negative
    259:        gradient direction. */
    260:     VecCopy(tao->stepdirection, asls->w);
    261:     VecScale(asls->w, -1.0);
    262:     VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w);
    263:     VecNorm(asls->w, NORM_2, &normd);
    264:     VecDot(asls->w, asls->dpsi, &innerd);

    266:     if (innerd >= -asls->delta*PetscPowReal(normd, asls->rho)) {
    267:       PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);
    268:       PetscInfo1(tao, "Iteration %D: newton direction not descent\n", tao->niter);
    269:       VecCopy(asls->dpsi, tao->stepdirection);
    270:       VecDot(asls->dpsi, tao->stepdirection, &innerd);
    271:     }

    273:     VecScale(tao->stepdirection, -1.0);
    274:     innerd = -innerd;

    276:     /* We now have a correct descent direction.  Apply a linesearch to
    277:        find the new iterate. */
    278:     TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);
    279:     TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);
    280:     VecNorm(asls->dpsi, NORM_2, &ndpsi);
    281:   }
    282:   return(0);
    283: }

    285: /* ---------------------------------------------------------- */
    286: /*MC
    287:    TAOASFLS - Active-set feasible linesearch algorithm for solving
    288:        complementarity constraints

    290:    Options Database Keys:
    291: + -tao_ssls_delta - descent test fraction
    292: - -tao_ssls_rho - descent test power

    294:    Level: beginner
    295: M*/
    296: PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao)
    297: {
    298:   TAO_SSLS       *asls;
    300:   const char     *armijo_type = TAOLINESEARCHARMIJO;

    303:   PetscNewLog(tao,&asls);
    304:   tao->data = (void*)asls;
    305:   tao->ops->solve = TaoSolve_ASFLS;
    306:   tao->ops->setup = TaoSetUp_ASFLS;
    307:   tao->ops->view = TaoView_SSLS;
    308:   tao->ops->setfromoptions = TaoSetFromOptions_SSLS;
    309:   tao->ops->destroy = TaoDestroy_ASFLS;
    310:   tao->subset_type = TAO_SUBSET_SUBVEC;
    311:   asls->delta = 1e-10;
    312:   asls->rho = 2.1;
    313:   asls->fixed = NULL;
    314:   asls->free = NULL;
    315:   asls->J_sub = NULL;
    316:   asls->Jpre_sub = NULL;
    317:   asls->w = NULL;
    318:   asls->r1 = NULL;
    319:   asls->r2 = NULL;
    320:   asls->r3 = NULL;
    321:   asls->t1 = NULL;
    322:   asls->t2 = NULL;
    323:   asls->dxfree = NULL;
    324:   asls->identifier = 1e-5;

    326:   TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);
    327:   PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);
    328:   TaoLineSearchSetType(tao->linesearch, armijo_type);
    329:   TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);
    330:   TaoLineSearchSetFromOptions(tao->linesearch);

    332:   KSPCreate(((PetscObject)tao)->comm, &tao->ksp);
    333:   PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);
    334:   KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);
    335:   KSPSetFromOptions(tao->ksp);

    337:   /* Override default settings (unless already changed) */
    338:   if (!tao->max_it_changed) tao->max_it = 2000;
    339:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
    340:   if (!tao->gttol_changed) tao->gttol = 0;
    341:   if (!tao->grtol_changed) tao->grtol = 0;
    342: #if defined(PETSC_USE_REAL_SINGLE)
    343:   if (!tao->gatol_changed) tao->gatol = 1.0e-6;
    344:   if (!tao->fmin_changed)  tao->fmin = 1.0e-4;
    345: #else
    346:   if (!tao->gatol_changed) tao->gatol = 1.0e-16;
    347:   if (!tao->fmin_changed)  tao->fmin = 1.0e-8;
    348: #endif
    349:   return(0);
    350: }