Actual source code: asfls.c

petsc-3.12.0 2019-09-29
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  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: */


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

 62:   VecDuplicate(tao->solution,&tao->gradient);
 63:   VecDuplicate(tao->solution,&tao->stepdirection);
 64:   VecDuplicate(tao->solution,&asls->ff);
 65:   VecDuplicate(tao->solution,&asls->dpsi);
 66:   VecDuplicate(tao->solution,&asls->da);
 67:   VecDuplicate(tao->solution,&asls->db);
 68:   VecDuplicate(tao->solution,&asls->t1);
 69:   VecDuplicate(tao->solution,&asls->t2);
 70:   VecDuplicate(tao->solution, &asls->w);
 71:   asls->fixed = NULL;
 72:   asls->free = NULL;
 73:   asls->J_sub = NULL;
 74:   asls->Jpre_sub = NULL;
 75:   asls->r1 = NULL;
 76:   asls->r2 = NULL;
 77:   asls->r3 = NULL;
 78:   asls->dxfree = NULL;
 79:   return(0);
 80: }

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

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

 95:   MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db);
 96:   VecPointwiseMult(asls->t1, asls->ff, asls->db);
 97:   MatMultTranspose(tao->jacobian,asls->t1,G);
 98:   VecPointwiseMult(asls->t1, asls->ff, asls->da);
 99:   VecAXPY(G,1.0,asls->t1);
100:   return(0);
101: }

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

109:   VecDestroy(&ssls->ff);
110:   VecDestroy(&ssls->dpsi);
111:   VecDestroy(&ssls->da);
112:   VecDestroy(&ssls->db);
113:   VecDestroy(&ssls->w);
114:   VecDestroy(&ssls->t1);
115:   VecDestroy(&ssls->t2);
116:   VecDestroy(&ssls->r1);
117:   VecDestroy(&ssls->r2);
118:   VecDestroy(&ssls->r3);
119:   VecDestroy(&ssls->dxfree);
120:   MatDestroy(&ssls->J_sub);
121:   MatDestroy(&ssls->Jpre_sub);
122:   ISDestroy(&ssls->fixed);
123:   ISDestroy(&ssls->free);
124:   PetscFree(tao->data);
125:   tao->data = NULL;
126:   return(0);
127: }

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

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

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

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

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

153:   tao->reason = TAO_CONTINUE_ITERATING;
154:   while (1) {
155:     /* Check the converged criteria */
156:     PetscInfo3(tao,"iter %D, merit: %g, ||dpsi||: %g\n",tao->niter,(double)asls->merit,(double)ndpsi);
157:     TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its);
158:     TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t);
159:     (*tao->ops->convergencetest)(tao,tao->cnvP);
160:     if (TAO_CONTINUE_ITERATING != tao->reason) break;
161: 
162:     /* Call general purpose update function */
163:     if (tao->ops->update) {
164:       (*tao->ops->update)(tao, tao->niter, tao->user_update);
165:     }
166:     tao->niter++;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

268:     if (innerd >= -asls->delta*PetscPowReal(normd, asls->rho)) {
269:       PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);
270:       PetscInfo1(tao, "Iteration %D: newton direction not descent\n", tao->niter);
271:       VecCopy(asls->dpsi, tao->stepdirection);
272:       VecDot(asls->dpsi, tao->stepdirection, &innerd);
273:     }

275:     VecScale(tao->stepdirection, -1.0);
276:     innerd = -innerd;

278:     /* We now have a correct descent direction.  Apply a linesearch to
279:        find the new iterate. */
280:     TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);
281:     TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);
282:     VecNorm(asls->dpsi, NORM_2, &ndpsi);
283:   }
284:   return(0);
285: }

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

292:    Options Database Keys:
293: + -tao_ssls_delta - descent test fraction
294: - -tao_ssls_rho - descent test power

296:    Level: beginner
297: M*/
298: PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao)
299: {
300:   TAO_SSLS       *asls;
302:   const char     *armijo_type = TAOLINESEARCHARMIJO;

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

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

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

339:   /* Override default settings (unless already changed) */
340:   if (!tao->max_it_changed) tao->max_it = 2000;
341:   if (!tao->max_funcs_changed) tao->max_funcs = 4000;
342:   if (!tao->gttol_changed) tao->gttol = 0;
343:   if (!tao->grtol_changed) tao->grtol = 0;
344: #if defined(PETSC_USE_REAL_SINGLE)
345:   if (!tao->gatol_changed) tao->gatol = 1.0e-6;
346:   if (!tao->fmin_changed)  tao->fmin = 1.0e-4;
347: #else
348:   if (!tao->gatol_changed) tao->gatol = 1.0e-16;
349:   if (!tao->fmin_changed)  tao->fmin = 1.0e-8;
350: #endif
351:   return(0);
352: }