vi/ss/viss.c

#include <../src/snes/impls/vi/ss/vissimpl.h> /*I "petscsnes.h" I*/

/*@
  SNESVIComputeMeritFunction - Evaluates the merit function for the mixed complementarity problem.

   Input Parameter:
.  phi - the `Vec` holding the evaluation of the semismooth function

   Output Parameters:
+  merit - the merit function 1/2 ||phi||^2
-  phinorm -  the two-norm of the vector, ||phi||

   Level: developer

.seealso: `SNESVINEWTONSSLS`, `SNESVIComputeFunction()`
@*/
PetscErrorCode SNESVIComputeMeritFunction(Vec phi, PetscReal *merit, PetscReal *phinorm) {
  PetscFunctionBegin;
  PetscCall(VecNormBegin(phi, NORM_2, phinorm));
  PetscCall(VecNormEnd(phi, NORM_2, phinorm));
  *merit = 0.5 * (*phinorm) * (*phinorm);
  PetscFunctionReturn(0);
}

static inline PetscScalar Phi(PetscScalar a, PetscScalar b) {
  return a + b - PetscSqrtScalar(a * a + b * b);
}

static inline PetscScalar DPhi(PetscScalar a, PetscScalar b) {
  if ((PetscAbsScalar(a) >= 1.e-6) || (PetscAbsScalar(b) >= 1.e-6)) return 1.0 - a / PetscSqrtScalar(a * a + b * b);
  else return .5;
}

/*@
   SNESVIComputeFunction - Provides the function that reformulates a system of nonlinear equations in mixed complementarity form to a system of nonlinear
   equations in semismooth form.

   Input Parameters:
+  snes - the SNES context
.  X - current iterate
-  functx - user defined function context

   Output Parameter:
.  phi - the evaluation of Semismooth function at X

   Level: developer

.seealso: `SNESVINEWTONSSLS`, `SNESVIComputeMeritFunction()`
@*/
PetscErrorCode SNESVIComputeFunction(SNES snes, Vec X, Vec phi, void *functx) {
  SNES_VINEWTONSSLS *vi = (SNES_VINEWTONSSLS *)snes->data;
  Vec                Xl = snes->xl, Xu = snes->xu, F = snes->vec_func;
  PetscScalar       *phi_arr, *f_arr, *l, *u;
  const PetscScalar *x_arr;
  PetscInt           i, nlocal;

  PetscFunctionBegin;
  PetscCall((*vi->computeuserfunction)(snes, X, F, functx));
  PetscCall(VecGetLocalSize(X, &nlocal));
  PetscCall(VecGetArrayRead(X, &x_arr));
  PetscCall(VecGetArray(F, &f_arr));
  PetscCall(VecGetArray(Xl, &l));
  PetscCall(VecGetArray(Xu, &u));
  PetscCall(VecGetArray(phi, &phi_arr));

  for (i = 0; i < nlocal; i++) {
    if ((PetscRealPart(l[i]) <= PETSC_NINFINITY) && (PetscRealPart(u[i]) >= PETSC_INFINITY)) { /* no constraints on variable */
      phi_arr[i] = f_arr[i];
    } else if (PetscRealPart(l[i]) <= PETSC_NINFINITY) { /* upper bound on variable only */
      phi_arr[i] = -Phi(u[i] - x_arr[i], -f_arr[i]);
    } else if (PetscRealPart(u[i]) >= PETSC_INFINITY) { /* lower bound on variable only */
      phi_arr[i] = Phi(x_arr[i] - l[i], f_arr[i]);
    } else if (l[i] == u[i]) {
      phi_arr[i] = l[i] - x_arr[i];
    } else { /* both bounds on variable */
      phi_arr[i] = Phi(x_arr[i] - l[i], -Phi(u[i] - x_arr[i], -f_arr[i]));
    }
  }

  PetscCall(VecRestoreArrayRead(X, &x_arr));
  PetscCall(VecRestoreArray(F, &f_arr));
  PetscCall(VecRestoreArray(Xl, &l));
  PetscCall(VecRestoreArray(Xu, &u));
  PetscCall(VecRestoreArray(phi, &phi_arr));
  PetscFunctionReturn(0);
}

/*
   SNESVIComputeBsubdifferentialVectors - Computes the diagonal shift (Da) and row scaling (Db) vectors needed for the
                                          the semismooth jacobian.
*/
PetscErrorCode SNESVIComputeBsubdifferentialVectors(SNES snes, Vec X, Vec F, Mat jac, Vec Da, Vec Db) {
  PetscScalar *l, *u, *x, *f, *da, *db, da1, da2, db1, db2;
  PetscInt     i, nlocal;

  PetscFunctionBegin;
  PetscCall(VecGetArray(X, &x));
  PetscCall(VecGetArray(F, &f));
  PetscCall(VecGetArray(snes->xl, &l));
  PetscCall(VecGetArray(snes->xu, &u));
  PetscCall(VecGetArray(Da, &da));
  PetscCall(VecGetArray(Db, &db));
  PetscCall(VecGetLocalSize(X, &nlocal));

  for (i = 0; i < nlocal; i++) {
    if ((PetscRealPart(l[i]) <= PETSC_NINFINITY) && (PetscRealPart(u[i]) >= PETSC_INFINITY)) { /* no constraints on variable */
      da[i] = 0;
      db[i] = 1;
    } else if (PetscRealPart(l[i]) <= PETSC_NINFINITY) { /* upper bound on variable only */
      da[i] = DPhi(u[i] - x[i], -f[i]);
      db[i] = DPhi(-f[i], u[i] - x[i]);
    } else if (PetscRealPart(u[i]) >= PETSC_INFINITY) { /* lower bound on variable only */
      da[i] = DPhi(x[i] - l[i], f[i]);
      db[i] = DPhi(f[i], x[i] - l[i]);
    } else if (l[i] == u[i]) { /* fixed variable */
      da[i] = 1;
      db[i] = 0;
    } else { /* upper and lower bounds on variable */
      da1   = DPhi(x[i] - l[i], -Phi(u[i] - x[i], -f[i]));
      db1   = DPhi(-Phi(u[i] - x[i], -f[i]), x[i] - l[i]);
      da2   = DPhi(u[i] - x[i], -f[i]);
      db2   = DPhi(-f[i], u[i] - x[i]);
      da[i] = da1 + db1 * da2;
      db[i] = db1 * db2;
    }
  }

  PetscCall(VecRestoreArray(X, &x));
  PetscCall(VecRestoreArray(F, &f));
  PetscCall(VecRestoreArray(snes->xl, &l));
  PetscCall(VecRestoreArray(snes->xu, &u));
  PetscCall(VecRestoreArray(Da, &da));
  PetscCall(VecRestoreArray(Db, &db));
  PetscFunctionReturn(0);
}

/*
   SNESVIComputeJacobian - Computes the jacobian of the semismooth function.The Jacobian for the semismooth function is an element of the B-subdifferential of the Fischer-Burmeister function for complementarity problems.

   Input Parameters:
.  Da       - Diagonal shift vector for the semismooth jacobian.
.  Db       - Row scaling vector for the semismooth jacobian.

   Output Parameters:
.  jac      - semismooth jacobian
.  jac_pre  - optional preconditioning matrix

   Note:
   The semismooth jacobian matrix is given by
   jac = Da + Db*jacfun
   where Db is the row scaling matrix stored as a vector,
         Da is the diagonal perturbation matrix stored as a vector
   and   jacfun is the jacobian of the original nonlinear function.
*/
PetscErrorCode SNESVIComputeJacobian(Mat jac, Mat jac_pre, Vec Da, Vec Db) {
  /* Do row scaling  and add diagonal perturbation */
  PetscFunctionBegin;
  PetscCall(MatDiagonalScale(jac, Db, NULL));
  PetscCall(MatDiagonalSet(jac, Da, ADD_VALUES));
  if (jac != jac_pre) { /* If jac and jac_pre are different */
    PetscCall(MatDiagonalScale(jac_pre, Db, NULL));
    PetscCall(MatDiagonalSet(jac_pre, Da, ADD_VALUES));
  }
  PetscFunctionReturn(0);
}

/*
   SNESVIComputeMeritFunctionGradient - Computes the gradient of the merit function psi.

   Input Parameters:
   phi - semismooth function.
   H   - semismooth jacobian

   Output Parameter:
   dpsi - merit function gradient

   Note:
  The merit function gradient is computed as follows
        dpsi = H^T*phi
*/
PetscErrorCode SNESVIComputeMeritFunctionGradient(Mat H, Vec phi, Vec dpsi) {
  PetscFunctionBegin;
  PetscCall(MatMultTranspose(H, phi, dpsi));
  PetscFunctionReturn(0);
}

/*
   SNESSolve_VINEWTONSSLS - Solves the complementarity problem with a semismooth Newton
   method using a line search.

   Input Parameter:
.  snes - the SNES context

   Application Interface Routine: SNESSolve()

   Note:
   This implements essentially a semismooth Newton method with a
   line search. The default line search does not do any line search
   but rather takes a full Newton step.

   Developer Note: the code in this file should be slightly modified so that this routine need not exist and the SNESSolve_NEWTONLS() routine is called directly with the appropriate wrapped function and Jacobian evaluations

*/
PetscErrorCode SNESSolve_VINEWTONSSLS(SNES snes) {
  SNES_VINEWTONSSLS   *vi = (SNES_VINEWTONSSLS *)snes->data;
  PetscInt             maxits, i, lits;
  SNESLineSearchReason lssucceed;
  PetscReal            gnorm, xnorm = 0, ynorm;
  Vec                  Y, X, F;
  KSPConvergedReason   kspreason;
  DM                   dm;
  DMSNES               sdm;

  PetscFunctionBegin;
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(DMGetDMSNES(dm, &sdm));

  vi->computeuserfunction   = sdm->ops->computefunction;
  sdm->ops->computefunction = SNESVIComputeFunction;

  snes->numFailures            = 0;
  snes->numLinearSolveFailures = 0;
  snes->reason                 = SNES_CONVERGED_ITERATING;

  maxits = snes->max_its;  /* maximum number of iterations */
  X      = snes->vec_sol;  /* solution vector */
  F      = snes->vec_func; /* residual vector */
  Y      = snes->work[0];  /* work vectors */

  PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
  snes->iter = 0;
  snes->norm = 0.0;
  PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));

  PetscCall(SNESVIProjectOntoBounds(snes, X));
  PetscCall(SNESComputeFunction(snes, X, vi->phi));
  if (snes->domainerror) {
    snes->reason              = SNES_DIVERGED_FUNCTION_DOMAIN;
    sdm->ops->computefunction = vi->computeuserfunction;
    PetscFunctionReturn(0);
  }
  /* Compute Merit function */
  PetscCall(SNESVIComputeMeritFunction(vi->phi, &vi->merit, &vi->phinorm));

  PetscCall(VecNormBegin(X, NORM_2, &xnorm)); /* xnorm <- ||x||  */
  PetscCall(VecNormEnd(X, NORM_2, &xnorm));
  SNESCheckFunctionNorm(snes, vi->merit);

  PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
  snes->norm = vi->phinorm;
  PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));
  PetscCall(SNESLogConvergenceHistory(snes, vi->phinorm, 0));
  PetscCall(SNESMonitor(snes, 0, vi->phinorm));

  /* test convergence */
  PetscUseTypeMethod(snes, converged, 0, 0.0, 0.0, vi->phinorm, &snes->reason, snes->cnvP);
  if (snes->reason) {
    sdm->ops->computefunction = vi->computeuserfunction;
    PetscFunctionReturn(0);
  }

  for (i = 0; i < maxits; i++) {
    /* Call general purpose update function */
    PetscTryTypeMethod(snes, update, snes->iter);

    /* Solve J Y = Phi, where J is the semismooth jacobian */

    /* Get the jacobian -- note that the function must be the original function for snes_fd and snes_fd_color to work for this*/
    sdm->ops->computefunction = vi->computeuserfunction;
    PetscCall(SNESComputeJacobian(snes, X, snes->jacobian, snes->jacobian_pre));
    SNESCheckJacobianDomainerror(snes);
    sdm->ops->computefunction = SNESVIComputeFunction;

    /* Get the diagonal shift and row scaling vectors */
    PetscCall(SNESVIComputeBsubdifferentialVectors(snes, X, F, snes->jacobian, vi->Da, vi->Db));
    /* Compute the semismooth jacobian */
    PetscCall(SNESVIComputeJacobian(snes->jacobian, snes->jacobian_pre, vi->Da, vi->Db));
    /* Compute the merit function gradient */
    PetscCall(SNESVIComputeMeritFunctionGradient(snes->jacobian, vi->phi, vi->dpsi));
    PetscCall(KSPSetOperators(snes->ksp, snes->jacobian, snes->jacobian_pre));
    PetscCall(KSPSolve(snes->ksp, vi->phi, Y));
    PetscCall(KSPGetConvergedReason(snes->ksp, &kspreason));

    if (kspreason < 0) {
      if (++snes->numLinearSolveFailures >= snes->maxLinearSolveFailures) {
        PetscCall(PetscInfo(snes, "iter=%" PetscInt_FMT ", number linear solve failures %" PetscInt_FMT " greater than current SNES allowed, stopping solve\n", snes->iter, snes->numLinearSolveFailures));
        snes->reason = SNES_DIVERGED_LINEAR_SOLVE;
        break;
      }
    }
    PetscCall(KSPGetIterationNumber(snes->ksp, &lits));
    snes->linear_its += lits;
    PetscCall(PetscInfo(snes, "iter=%" PetscInt_FMT ", linear solve iterations=%" PetscInt_FMT "\n", snes->iter, lits));
    /*
    if (snes->ops->precheck) {
      PetscBool changed_y = PETSC_FALSE;
      PetscUseTypeMethod(snes,precheck ,X,Y,snes->precheck,&changed_y);
    }

    if (PetscLogPrintInfo) PetscCall(SNESVICheckResidual_Private(snes,snes->jacobian,F,Y,G,W));
    */
    /* Compute a (scaled) negative update in the line search routine:
         Y <- X - lambda*Y
       and evaluate G = function(Y) (depends on the line search).
    */
    PetscCall(VecCopy(Y, snes->vec_sol_update));
    ynorm = 1;
    gnorm = vi->phinorm;
    PetscCall(SNESLineSearchApply(snes->linesearch, X, vi->phi, &gnorm, Y));
    PetscCall(SNESLineSearchGetReason(snes->linesearch, &lssucceed));
    PetscCall(SNESLineSearchGetNorms(snes->linesearch, &xnorm, &gnorm, &ynorm));
    PetscCall(PetscInfo(snes, "fnorm=%18.16e, gnorm=%18.16e, ynorm=%18.16e, lssucceed=%d\n", (double)vi->phinorm, (double)gnorm, (double)ynorm, (int)lssucceed));
    if (snes->reason == SNES_DIVERGED_FUNCTION_COUNT) break;
    if (snes->domainerror) {
      snes->reason              = SNES_DIVERGED_FUNCTION_DOMAIN;
      sdm->ops->computefunction = vi->computeuserfunction;
      PetscFunctionReturn(0);
    }
    if (lssucceed) {
      if (++snes->numFailures >= snes->maxFailures) {
        PetscBool ismin;
        snes->reason = SNES_DIVERGED_LINE_SEARCH;
        PetscCall(SNESVICheckLocalMin_Private(snes, snes->jacobian, vi->phi, X, gnorm, &ismin));
        if (ismin) snes->reason = SNES_DIVERGED_LOCAL_MIN;
        break;
      }
    }
    /* Update function and solution vectors */
    vi->phinorm = gnorm;
    vi->merit   = 0.5 * vi->phinorm * vi->phinorm;
    /* Monitor convergence */
    PetscCall(PetscObjectSAWsTakeAccess((PetscObject)snes));
    snes->iter  = i + 1;
    snes->norm  = vi->phinorm;
    snes->xnorm = xnorm;
    snes->ynorm = ynorm;
    PetscCall(PetscObjectSAWsGrantAccess((PetscObject)snes));
    PetscCall(SNESLogConvergenceHistory(snes, snes->norm, lits));
    PetscCall(SNESMonitor(snes, snes->iter, snes->norm));
    /* Test for convergence, xnorm = || X || */
    if (snes->ops->converged != SNESConvergedSkip) PetscCall(VecNorm(X, NORM_2, &xnorm));
    PetscUseTypeMethod(snes, converged, snes->iter, xnorm, ynorm, vi->phinorm, &snes->reason, snes->cnvP);
    if (snes->reason) break;
  }
  if (i == maxits) {
    PetscCall(PetscInfo(snes, "Maximum number of iterations has been reached: %" PetscInt_FMT "\n", maxits));
    if (!snes->reason) snes->reason = SNES_DIVERGED_MAX_IT;
  }
  sdm->ops->computefunction = vi->computeuserfunction;
  PetscFunctionReturn(0);
}

/*
   SNESSetUp_VINEWTONSSLS - Sets up the internal data structures for the later use
   of the SNES nonlinear solver.

   Input Parameter:
.  snes - the SNES context

   Application Interface Routine: SNESSetUp()

   Note:
   For basic use of the SNES solvers, the user need not explicitly call
   SNESSetUp(), since these actions will automatically occur during
   the call to SNESSolve().
 */
PetscErrorCode SNESSetUp_VINEWTONSSLS(SNES snes) {
  SNES_VINEWTONSSLS *vi = (SNES_VINEWTONSSLS *)snes->data;

  PetscFunctionBegin;
  PetscCall(SNESSetUp_VI(snes));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->dpsi));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->phi));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->Da));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->Db));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->z));
  PetscCall(VecDuplicate(snes->vec_sol, &vi->t));
  PetscFunctionReturn(0);
}

PetscErrorCode SNESReset_VINEWTONSSLS(SNES snes) {
  SNES_VINEWTONSSLS *vi = (SNES_VINEWTONSSLS *)snes->data;

  PetscFunctionBegin;
  PetscCall(SNESReset_VI(snes));
  PetscCall(VecDestroy(&vi->dpsi));
  PetscCall(VecDestroy(&vi->phi));
  PetscCall(VecDestroy(&vi->Da));
  PetscCall(VecDestroy(&vi->Db));
  PetscCall(VecDestroy(&vi->z));
  PetscCall(VecDestroy(&vi->t));
  PetscFunctionReturn(0);
}

/*
   SNESSetFromOptions_VINEWTONSSLS - Sets various parameters for the SNESVI method.

   Input Parameter:
.  snes - the SNES context

   Application Interface Routine: SNESSetFromOptions()
*/
static PetscErrorCode SNESSetFromOptions_VINEWTONSSLS(SNES snes, PetscOptionItems *PetscOptionsObject) {
  PetscFunctionBegin;
  PetscCall(SNESSetFromOptions_VI(snes, PetscOptionsObject));
  PetscOptionsHeadBegin(PetscOptionsObject, "SNES semismooth method options");
  PetscOptionsHeadEnd();
  PetscFunctionReturn(0);
}

/*MC
      SNESVINEWTONSSLS - Semi-smooth solver for variational inequalities based on Newton's method

   Options Database Keys:
+   -snes_type <vinewtonssls,vinewtonrsls> a semi-smooth solver, a reduced space active set method
-   -snes_vi_monitor - prints the number of active constraints at each iteration.

   Level: beginner

   References:
+  * -  T. S. Munson, F. Facchinei, M. C. Ferris, A. Fischer, and C. Kanzow. The semismooth
     algorithm for large scale complementarity problems. INFORMS Journal on Computing, 13 (2001).
-  * -  T. S. Munson, and S. Benson. Flexible Complementarity Solvers for Large Scale
     Applications, Optimization Methods and Software, 21 (2006).

   Notes:
   This family of algorithm is much like an interior point method.

   The reduced space active set solvers `SNESVINEWTONRSLS` provide an alternative approach that does not result in extremely ill-conditioned linear systems

.seealso: `SNESVINEWTONRSLS`, `SNESVISetVariableBounds()`, `SNESVISetComputeVariableBounds()`, `SNESCreate()`, `SNES`, `SNESSetType()`, `SNESVINEWTONRSLS`, `SNESNEWTONTR`, `SNESLineSearchSetType()`, `SNESLineSearchSetPostCheck()`, `SNESLineSearchSetPreCheck()`
M*/
PETSC_EXTERN PetscErrorCode SNESCreate_VINEWTONSSLS(SNES snes) {
  SNES_VINEWTONSSLS *vi;
  SNESLineSearch     linesearch;

  PetscFunctionBegin;
  snes->ops->reset          = SNESReset_VINEWTONSSLS;
  snes->ops->setup          = SNESSetUp_VINEWTONSSLS;
  snes->ops->solve          = SNESSolve_VINEWTONSSLS;
  snes->ops->destroy        = SNESDestroy_VI;
  snes->ops->setfromoptions = SNESSetFromOptions_VINEWTONSSLS;
  snes->ops->view           = NULL;

  snes->usesksp = PETSC_TRUE;
  snes->usesnpc = PETSC_FALSE;

  PetscCall(SNESGetLineSearch(snes, &linesearch));
  if (!((PetscObject)linesearch)->type_name) {
    PetscCall(SNESLineSearchSetType(linesearch, SNESLINESEARCHBT));
    PetscCall(SNESLineSearchBTSetAlpha(linesearch, 0.0));
  }

  snes->alwayscomputesfinalresidual = PETSC_FALSE;

  PetscCall(PetscNew(&vi));
  snes->data = (void *)vi;

  PetscCall(PetscObjectComposeFunction((PetscObject)snes, "SNESVISetVariableBounds_C", SNESVISetVariableBounds_VI));
  PetscCall(PetscObjectComposeFunction((PetscObject)snes, "SNESVISetComputeVariableBounds_C", SNESVISetComputeVariableBounds_VI));
  PetscFunctionReturn(0);
}