Actual source code: asils.c

  1: #include "src/complementarity/impls/ssls/ssls.h"
  2: // Context for ASXLS
  3: //   -- active-set        - reduced matrices formed
  4: //                          - inherit properties of original system
  5: //   -- semismooth (S)  - function not differentiable
  6: //                      - merit function continuously differentiable
  7: //                      - Fischer-Burmeister reformulation of complementarity
  8: //                        - Billups composition for two finite bounds
  9: //   -- infeasible (I)  - iterates not guaranteed to remain within bounds
 10: //   -- feasible (F)    - iterates guaranteed to remain within bounds
 11: //   -- linesearch (LS) - Armijo rule on direction
 12: //
 13: // Many other reformulations are possible and combinations of
 14: // feasible/infeasible and linesearch/trust region are possible.
 15: //
 16: // Basic theory
 17: //   Fischer-Burmeister reformulation is semismooth with a continuously
 18: //   differentiable merit function and strongly semismooth if the F has
 19: //   lipschitz continuous derivatives.
 20: //
 21: //   Every accumulation point generated by the algorithm is a stationary
 22: //   point for the merit function.  Stationary points of the merit function
 23: //   are solutions of the complementarity problem if
 24: //     a.  the stationary point has a BD-regular subdifferential, or
 25: //     b.  the Schur complement F'/F'_ff is a P_0-matrix where ff is the
 26: //         index set corresponding to the free variables.
 27: //
 28: //   If one of the accumulation points has a BD-regular subdifferential then
 29: //     a.  the entire sequence converges to this accumulation point at
 30: //         a local q-superlinear rate
 31: //     b.  if in addition the reformulation is strongly semismooth near
 32: //         this accumulation point, then the algorithm converges at a
 33: //         local q-quadratic rate.
 34: //
 35: // The theory for the feasible version follows from the feasible descent
 36: // algorithm framework.
 37: //
 38: // References:
 39: //   Billups, "Algorithms for Complementarity Problems and Generalized
 40: //     Equations," Ph.D thesis, University of Wisconsin - Madison, 1995.
 41: //   De Luca, Facchinei, Kanzow, "A Semismooth Equation Approach to the
 42: //     Solution of Nonlinear Complementarity Problems," Mathematical
 43: //     Programming, 75, pages 407-439, 1996.
 44: //   Ferris, Kanzow, Munson, "Feasible Descent Algorithms for Mixed
 45: //     Complementarity Problems," Mathematical Programming, 86,
 46: //     pages 475-497, 1999.
 47: //   Fischer, "A Special Newton-type Optimization Method," Optimization,
 48: //     24, pages 269-284, 1992
 49: //   Munson, Facchinei, Ferris, Fischer, Kanzow, "The Semismooth Algorithm
 50: //     for Large Scale Complementarity Problems," Technical Report 99-06,
 51: //     University of Wisconsin - Madison, 1999.


 54: /*------------------------------------------------------------*/
 57: static int TaoSolve_ASILS(TAO_SOLVER tao, void *solver)
 58: {
 59:   TAO_SSLS *asls = (TAO_SSLS *)solver;
 60:   TaoVec *x, *l, *u, *ff, *dpsi, *d, *w, *t1, *t2, *da, *db;
 61:   TaoMat *J,*Jsub;
 62:   TaoVec *dxfree,*r1, *r2, *r3;
 63:   TaoIndexSet *FreeVariableSet,*FixedVariableSet;
 64:   //  TaoLinearSolver *lsolver;
 65:   double psi, ndpsi, normd, innerd, t=0;
 66:   double delta, rho;
 67:   int iter=0, info, nn, nf;
 68:   TaoTerminateReason reason;
 69:   TaoTruth flag;

 71:   TaoFunctionBegin;

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

 76:   delta = asls->delta;
 77:   rho = asls->rho;

 79:   info = TaoGetSolution(tao, &x); CHKERRQ(info);
 80:   info = TaoGetVariableBounds(tao, &l, &u); CHKERRQ(info);
 81:   info = TaoEvaluateVariableBounds(tao,l,u); CHKERRQ(info);
 82:   info = TaoGetJacobian(tao, &J); CHKERRQ(info);

 84:   info = x->GetDimension(&nn); CHKERRQ(info);

 86:   info = x->CreateIndexSet(&FreeVariableSet); CHKERRQ(info);
 87:   info = x->CreateIndexSet(&FixedVariableSet); CHKERRQ(info);
 88:   info = J->CreateReducedMatrix(FreeVariableSet,FreeVariableSet,&Jsub); CHKERRQ(info);
 89:   info = x->Clone(&dxfree); CHKERRQ(info);
 90:   info = x->Clone(&r1); CHKERRQ(info);
 91:   info = x->Clone(&r2); CHKERRQ(info);
 92:   info = x->Clone(&r3); CHKERRQ(info);

 94:   ff = asls->ff;
 95:   dpsi = asls->dpsi;
 96:   d = asls->d;
 97:   w = asls->w;
 98:   t1 = asls->t1;
 99:   t2 = asls->t2;
100:   da = asls->da;
101:   db = asls->db;

103:   info = TaoSetMeritFunction(tao, Tao_SSLS_Function, Tao_ASLS_FunctionGradient,
104:                              TAO_NULL, TAO_NULL, asls); CHKERRQ(info);

106:   // Calculate the function value and fischer function value at the
107:   // current iterate

109:   info = TaoComputeMeritFunctionGradient(tao, x, &psi, dpsi); CHKERRQ(info);
110:   info = dpsi->Norm2(&ndpsi); CHKERRQ(info);

112:   while (1) {

114:     // Check the termination criteria

116:     info = TaoMonitor(tao, iter++, asls->merit, ndpsi, 0.0, t, &reason); CHKERRQ(info);
117:     if (TAO_CONTINUE_ITERATING != reason) break;

119:     // We are going to solve a linear system of equations.  We need to
120:     // set the tolerances for the solve so that we maintain an asymptotic
121:     // rate of convergence that is superlinear.
122:     // Note: these tolerances are for the reduced system.  We really need
123:     // to make sure that the full system satisfies the full-space conditions.
124: 
125:     // This rule gives superlinear asymptotic convergence
126:     // asls->atol = TaoMin(0.5, asls->merit*sqrt(asls->merit));
127:     // asls->rtol = 0.0;

129:     // This rule gives quadratic asymptotic convergence
130:     // asls->atol = TaoMin(0.5, asls->merit*asls->merit);
131:     // asls->rtol = 0.0;

133:     // Calculate a free and fixed set of variables.  The fixed set of
134:     // variables are those for the d_b is approximately equal to zero.
135:     // The definition of approximately changes as we approach the solution
136:     // to the problem.

138:     // No one rule is guaranteed to work in all cases.  The following
139:     // definition is based on the norm of the Jacobian matrix.  If the
140:     // norm is large, the tolerance becomes smaller.

142:     info = J->Norm1(&(asls->identifier)); CHKERRQ(info);
143:     // asls->identifier = asls->atol / (1 + asls->identifier);
144:     asls->identifier = TaoMin(asls->merit, 1e-2) / (1 + asls->identifier);
145:     // asls->identifier = 1e-4 / (1 + asls->identifier);
146:     // asls->identifier = 1e-4;
147:     // asls->identifier = 1e-8;

149:     info = t1->SetToConstant(-asls->identifier); CHKERRQ(info);
150:     info = t2->SetToConstant( asls->identifier); CHKERRQ(info);

152:     info = FixedVariableSet->WhichBetweenOrEqual(t1, db, t2); CHKERRQ(info);
153:     info = FreeVariableSet->ComplementOf(FixedVariableSet); CHKERRQ(info);

155:     info = FixedVariableSet->GetSize(&nf);
156:     printf("Fixed size: %d\n", nf);

158:     // We now have our partition.  Now calculate the direction in the
159:     // fixed variable space.

161:     info = r1->SetReducedVec(ff, FixedVariableSet); CHKERRQ(info);
162:     info = r2->SetReducedVec(da, FixedVariableSet); CHKERRQ(info);
163:     info = r1->PointwiseDivide(r1, r2); CHKERRQ(info);

165:     info = d->SetToZero(); CHKERRQ(info);
166:     info = d->ReducedXPY(r1, FixedVariableSet); CHKERRQ(info);

168:     // Our direction in the FixedVariableSet is fixed.  Calculate the
169:     // information needed for the step in the FreeVariableSet.  To
170:     // do this, we need to know the diagonal perturbation and the
171:     // right hand side.

173:     info = r1->SetReducedVec(da, FreeVariableSet); CHKERRQ(info);
174:     info = r2->SetReducedVec(ff, FreeVariableSet); CHKERRQ(info);
175:     info = r3->SetReducedVec(db, FreeVariableSet); CHKERRQ(info);

177:     info = r1->PointwiseDivide(r1, r3); CHKERRQ(info);
178:     info = r2->PointwiseDivide(r2, r3); CHKERRQ(info);

180:     // r1 is the diagonal perturbation
181:     // r2 is the right hand side
182:     // r3 is no longer needed

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

188:     info = J->Multiply(d, t1); CHKERRQ(info);
189:     info = r3->SetReducedVec(t1, FreeVariableSet); CHKERRQ(info);
190:     info = r2->Axpy(-1.0, r3);

192:     // Calculate the reduced problem matrix and the direction

194:     info = Jsub->SetReducedMatrix(J, FreeVariableSet, FreeVariableSet); CHKERRQ(info);
195:     info = Jsub->AddDiagonal(r1); CHKERRQ(info);
196:     info = dxfree->SetReducedVec(d, FreeVariableSet);CHKERRQ(info);
197:     info = dxfree->SetToZero(); CHKERRQ(info);

199:     // Set the convergence for the reduced problem solver.
200:     // info = TaoGetLinearSolver(tao, &lsolver); CHKERRQ(info);
201:     // info = lsolver->SetTolerances(0.1*asls->rtol, 0.1*asls->atol, 1e30, 10*nn); CHKERRQ(info);

203:     // Calculate the reduced direction.  (Really negative of Newton
204:     // direction.  Therefore, rest of the code uses -d.)

206:     info = TaoPreLinearSolve(tao, Jsub); CHKERRQ(info);
207:     info = TaoLinearSolve(tao, Jsub, r2, dxfree, &flag); CHKERRQ(info);

209:     // Add the direction in the free variables back into the real direction.

211:     info = d->ReducedXPY(dxfree, FreeVariableSet);CHKERRQ(info);

213:     // Calculate the norm of the relative residual in the real space

215:     // info = t1->PointwiseMultiply(da, d); CHKERRQ(info);
216:     // info = J->Multiply(d, t2); CHKERRQ(info);
217:     // info = t2->PointwiseMultiply(db, t2); CHKERRQ(info);
218:     // info = t1->Axpy( 1.0, t2); CHKERRQ(info);
219:     // info = t1->Axpy(-1.0, ff); CHKERRQ(info);
220:     // info = t1->Norm2(&two_norm); CHKERRQ(info);

222:     // if (two_norm >= asls->atol) {
223:     //   printf("Bad absolute residual: actual: %5.4e needed: %5.4e\n",
224:     //          two_norm, asls->atol);
225:     // }

227:     // Check the real direction for descent and if not, use the negative
228:     // gradient direction.

230:     info = d->Norm2(&normd); CHKERRQ(info);
231:     info = d->Dot(dpsi, &innerd); CHKERRQ(info);

233:     if (innerd <= delta*pow(normd, rho)) {
234:        printf("Gradient direction: %5.4e.\n", innerd);
235:        info = PetscLogInfo((tao, "TaoSolve_ASILS: %d: newton direction not descent\n", iter)); CHKERRQ(info);

237:        info = d->CopyFrom(dpsi); CHKERRQ(info);
238:        info = d->Dot(dpsi, &innerd); CHKERRQ(info);
239:     }

241:     info = d->Negate(); CHKERRQ(info);
242:     innerd = -innerd;

244:     // We now have a correct descent direction.  Apply a linesearch to
245:     // find the new iterate.

247:     t = 1;
248:     info = TaoLineSearchApply(tao, x, dpsi, d, w,
249:                               &psi, &t, &tao->lsflag); CHKERRQ(info);
250:     info = dpsi->Norm2(&ndpsi); CHKERRQ(info);
251:   }

253:   info = TaoMatDestroy(Jsub);CHKERRQ(info);
254:   info = TaoVecDestroy(dxfree);CHKERRQ(info);
255:   info = TaoVecDestroy(r1);CHKERRQ(info);
256:   info = TaoVecDestroy(r2);CHKERRQ(info);
257:   info = TaoVecDestroy(r3);CHKERRQ(info);
258:   info = TaoIndexSetDestroy(FreeVariableSet);CHKERRQ(info);
259:   info = TaoIndexSetDestroy(FixedVariableSet);CHKERRQ(info);

261:   TaoFunctionReturn(0);
262: }

264: /* ---------------------------------------------------------- */
268: int TaoCreate_ASILS(TAO_SOLVER tao)
269: {
270:   TAO_SSLS *asls;
271:   int        info;

273:   TaoFunctionBegin;

275:   info = TaoNew(TAO_SSLS,&asls); CHKERRQ(info);
276:   info = PetscLogObjectMemory(tao, sizeof(TAO_SSLS)); CHKERRQ(info);

278:   asls->delta = 1e-10;
279:   asls->rho = 2.1;

281:   asls->identifier = 1e-5;

283:   info=TaoSetTaoSolveRoutine(tao,TaoSolve_ASILS,(void*)asls); CHKERRQ(info);
284:   info=TaoSetTaoSetUpDownRoutines(tao,TaoSetUp_SSLS,TaoSetDown_SSLS); CHKERRQ(info);
285:   info=TaoSetTaoOptionsRoutine(tao,TaoSetOptions_SSLS); CHKERRQ(info);
286:   info=TaoSetTaoViewRoutine(tao,TaoView_SSLS); CHKERRQ(info);

288:   info = TaoCreateArmijoLineSearch(tao); CHKERRQ(info);

290:   info = TaoSetMaximumIterates(tao,2000); CHKERRQ(info);
291:   info = TaoSetMaximumFunctionEvaluations(tao,4000); CHKERRQ(info);

293:   info = TaoSetTolerances(tao,0,0,0,0); CHKERRQ(info);
294:   info = TaoSetGradientTolerances(tao,1.0e-16,0.0,0.0); CHKERRQ(info);
295:   info = TaoSetFunctionLowerBound(tao,1.0e-8); CHKERRQ(info);

297:   TaoFunctionReturn(0);
298: }