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;

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

 79: static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr)
 80: {
 81:   Tao       tao  = (Tao)ptr;
 82:   TAO_SSLS *asls = (TAO_SSLS *)tao->data;

 84:   TaoComputeConstraints(tao, X, tao->constraints);
 85:   VecFischer(X, tao->constraints, tao->XL, tao->XU, asls->ff);
 86:   VecNorm(asls->ff, NORM_2, &asls->merit);
 87:   *fcn = 0.5 * asls->merit * asls->merit;
 88:   TaoComputeJacobian(tao, tao->solution, tao->jacobian, tao->jacobian_pre);

 90:   MatDFischer(tao->jacobian, tao->solution, tao->constraints, tao->XL, tao->XU, asls->t1, asls->t2, asls->da, asls->db);
 91:   VecPointwiseMult(asls->t1, asls->ff, asls->db);
 92:   MatMultTranspose(tao->jacobian, asls->t1, G);
 93:   VecPointwiseMult(asls->t1, asls->ff, asls->da);
 94:   VecAXPY(G, 1.0, asls->t1);
 95:   return 0;
 96: }

 98: static PetscErrorCode TaoDestroy_ASFLS(Tao tao)
 99: {
100:   TAO_SSLS *ssls = (TAO_SSLS *)tao->data;

102:   VecDestroy(&ssls->ff);
103:   VecDestroy(&ssls->dpsi);
104:   VecDestroy(&ssls->da);
105:   VecDestroy(&ssls->db);
106:   VecDestroy(&ssls->w);
107:   VecDestroy(&ssls->t1);
108:   VecDestroy(&ssls->t2);
109:   VecDestroy(&ssls->r1);
110:   VecDestroy(&ssls->r2);
111:   VecDestroy(&ssls->r3);
112:   VecDestroy(&ssls->dxfree);
113:   MatDestroy(&ssls->J_sub);
114:   MatDestroy(&ssls->Jpre_sub);
115:   ISDestroy(&ssls->fixed);
116:   ISDestroy(&ssls->free);
117:   KSPDestroy(&tao->ksp);
118:   PetscFree(tao->data);
119:   return 0;
120: }

122: static PetscErrorCode TaoSolve_ASFLS(Tao tao)
123: {
124:   TAO_SSLS                    *asls = (TAO_SSLS *)tao->data;
125:   PetscReal                    psi, ndpsi, normd, innerd, t = 0;
126:   PetscInt                     nf;
127:   TaoLineSearchConvergedReason ls_reason;

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

132:   TaoComputeVariableBounds(tao);
133:   TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch, Tao_ASLS_FunctionGradient, tao);
134:   TaoLineSearchSetObjectiveRoutine(tao->linesearch, Tao_SSLS_Function, tao);
135:   TaoLineSearchSetVariableBounds(tao->linesearch, tao->XL, tao->XU);

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

139:   /* Calculate the function value and fischer function value at the
140:      current iterate */
141:   TaoLineSearchComputeObjectiveAndGradient(tao->linesearch, tao->solution, &psi, asls->dpsi);
142:   VecNorm(asls->dpsi, NORM_2, &ndpsi);

144:   tao->reason = TAO_CONTINUE_ITERATING;
145:   while (1) {
146:     /* Check the converged criteria */
147:     PetscInfo(tao, "iter %" PetscInt_FMT ", merit: %g, ||dpsi||: %g\n", tao->niter, (double)asls->merit, (double)ndpsi);
148:     TaoLogConvergenceHistory(tao, asls->merit, ndpsi, 0.0, tao->ksp_its);
149:     TaoMonitor(tao, tao->niter, asls->merit, ndpsi, 0.0, t);
150:     PetscUseTypeMethod(tao, convergencetest, tao->cnvP);
151:     if (TAO_CONTINUE_ITERATING != tao->reason) break;

153:     /* Call general purpose update function */
154:     PetscTryTypeMethod(tao, update, tao->niter, tao->user_update);
155:     tao->niter++;

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

163:        This rule gives superlinear asymptotic convergence
164:        asls->atol = min(0.5, asls->merit*sqrt(asls->merit));
165:        asls->rtol = 0.0;

167:        This rule gives quadratic asymptotic convergence
168:        asls->atol = min(0.5, asls->merit*asls->merit);
169:        asls->rtol = 0.0;

171:        Calculate a free and fixed set of variables.  The fixed set of
172:        variables are those for the d_b is approximately equal to zero.
173:        The definition of approximately changes as we approach the solution
174:        to the problem.

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

182:     VecSet(asls->t1, -asls->identifier);
183:     VecSet(asls->t2, asls->identifier);

185:     ISDestroy(&asls->fixed);
186:     ISDestroy(&asls->free);
187:     VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);
188:     ISComplementVec(asls->fixed, asls->t1, &asls->free);

190:     ISGetSize(asls->fixed, &nf);
191:     PetscInfo(tao, "Number of fixed variables: %" PetscInt_FMT "\n", nf);

193:     /* We now have our partition.  Now calculate the direction in the
194:        fixed variable space. */
195:     TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);
196:     TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);
197:     VecPointwiseDivide(asls->r1, asls->r1, asls->r2);
198:     VecSet(tao->stepdirection, 0.0);
199:     VecISAXPY(tao->stepdirection, asls->fixed, 1.0, asls->r1);

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

206:     TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);
207:     TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);
208:     TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);
209:     VecPointwiseDivide(asls->r1, asls->r1, asls->r3);
210:     VecPointwiseDivide(asls->r2, asls->r2, asls->r3);

212:     /* r1 is the diagonal perturbation
213:        r2 is the right hand side
214:        r3 is no longer needed

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

220:     MatMult(tao->jacobian, tao->stepdirection, asls->t1);
221:     TaoVecGetSubVec(asls->t1, asls->free, tao->subset_type, 0.0, &asls->r3);
222:     VecAXPY(asls->r2, -1.0, asls->r3);

224:     /* Calculate the reduced problem matrix and the direction */
225:     TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type, &asls->J_sub);
226:     if (tao->jacobian != tao->jacobian_pre) {
227:       TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);
228:     } else {
229:       MatDestroy(&asls->Jpre_sub);
230:       asls->Jpre_sub = asls->J_sub;
231:       PetscObjectReference((PetscObject)(asls->Jpre_sub));
232:     }
233:     MatDiagonalSet(asls->J_sub, asls->r1, ADD_VALUES);
234:     TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);
235:     VecSet(asls->dxfree, 0.0);

237:     /* Calculate the reduced direction.  (Really negative of Newton
238:        direction.  Therefore, rest of the code uses -d.) */
239:     KSPReset(tao->ksp);
240:     KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);
241:     KSPSolve(tao->ksp, asls->r2, asls->dxfree);
242:     KSPGetIterationNumber(tao->ksp, &tao->ksp_its);
243:     tao->ksp_tot_its += tao->ksp_its;

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

248:     /* Check the projected real direction for descent and if not, use the negative
249:        gradient direction. */
250:     VecCopy(tao->stepdirection, asls->w);
251:     VecScale(asls->w, -1.0);
252:     VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w);
253:     VecNorm(asls->w, NORM_2, &normd);
254:     VecDot(asls->w, asls->dpsi, &innerd);

256:     if (innerd >= -asls->delta * PetscPowReal(normd, asls->rho)) {
257:       PetscInfo(tao, "Gradient direction: %5.4e.\n", (double)innerd);
258:       PetscInfo(tao, "Iteration %" PetscInt_FMT ": newton direction not descent\n", tao->niter);
259:       VecCopy(asls->dpsi, tao->stepdirection);
260:       VecDot(asls->dpsi, tao->stepdirection, &innerd);
261:     }

263:     VecScale(tao->stepdirection, -1.0);
264:     innerd = -innerd;

266:     /* We now have a correct descent direction.  Apply a linesearch to
267:        find the new iterate. */
268:     TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);
269:     TaoLineSearchApply(tao->linesearch, tao->solution, &psi, asls->dpsi, tao->stepdirection, &t, &ls_reason);
270:     VecNorm(asls->dpsi, NORM_2, &ndpsi);
271:   }
272:   return 0;
273: }

275: /* ---------------------------------------------------------- */
276: /*MC
277:    TAOASFLS - Active-set feasible linesearch algorithm for solving
278:        complementarity constraints

280:    Options Database Keys:
281: + -tao_ssls_delta - descent test fraction
282: - -tao_ssls_rho - descent test power

284:    Level: beginner
285: M*/
286: PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao)
287: {
288:   TAO_SSLS   *asls;
289:   const char *armijo_type = TAOLINESEARCHARMIJO;

291:   PetscNew(&asls);
292:   tao->data                = (void *)asls;
293:   tao->ops->solve          = TaoSolve_ASFLS;
294:   tao->ops->setup          = TaoSetUp_ASFLS;
295:   tao->ops->view           = TaoView_SSLS;
296:   tao->ops->setfromoptions = TaoSetFromOptions_SSLS;
297:   tao->ops->destroy        = TaoDestroy_ASFLS;
298:   tao->subset_type         = TAO_SUBSET_SUBVEC;
299:   asls->delta              = 1e-10;
300:   asls->rho                = 2.1;
301:   asls->fixed              = NULL;
302:   asls->free               = NULL;
303:   asls->J_sub              = NULL;
304:   asls->Jpre_sub           = NULL;
305:   asls->w                  = NULL;
306:   asls->r1                 = NULL;
307:   asls->r2                 = NULL;
308:   asls->r3                 = NULL;
309:   asls->t1                 = NULL;
310:   asls->t2                 = NULL;
311:   asls->dxfree             = NULL;
312:   asls->identifier         = 1e-5;

314:   TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);
315:   PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);
316:   TaoLineSearchSetType(tao->linesearch, armijo_type);
317:   TaoLineSearchSetOptionsPrefix(tao->linesearch, tao->hdr.prefix);
318:   TaoLineSearchSetFromOptions(tao->linesearch);

320:   KSPCreate(((PetscObject)tao)->comm, &tao->ksp);
321:   PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);
322:   KSPSetOptionsPrefix(tao->ksp, tao->hdr.prefix);
323:   KSPSetFromOptions(tao->ksp);

325:   /* Override default settings (unless already changed) */
326:   if (!tao->max_it_changed) tao->max_it = 2000;
327:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
328:   if (!tao->gttol_changed) tao->gttol = 0;
329:   if (!tao->grtol_changed) tao->grtol = 0;
330: #if defined(PETSC_USE_REAL_SINGLE)
331:   if (!tao->gatol_changed) tao->gatol = 1.0e-6;
332:   if (!tao->fmin_changed) tao->fmin = 1.0e-4;
333: #else
334:   if (!tao->gatol_changed) tao->gatol = 1.0e-16;
335:   if (!tao->fmin_changed) tao->fmin = 1.0e-8;
336: #endif
337:   return 0;
338: }