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 13 14 Many other reformulations are possible and combinations of 15 feasible/infeasible and linesearch/trust region are possible. 16 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. 21 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. 28 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. 35 36 The theory for the feasible version follows from the feasible descent 37 algorithm framework. 38 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 */ 54 55 56 static PetscErrorCode TaoSetUp_ASFLS(Tao tao) 57 { 58 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 59 PetscErrorCode ierr; 60 61 PetscFunctionBegin; 62 ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr); 63 ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr); 64 ierr = VecDuplicate(tao->solution,&asls->ff);CHKERRQ(ierr); 65 ierr = VecDuplicate(tao->solution,&asls->dpsi);CHKERRQ(ierr); 66 ierr = VecDuplicate(tao->solution,&asls->da);CHKERRQ(ierr); 67 ierr = VecDuplicate(tao->solution,&asls->db);CHKERRQ(ierr); 68 ierr = VecDuplicate(tao->solution,&asls->t1);CHKERRQ(ierr); 69 ierr = VecDuplicate(tao->solution,&asls->t2);CHKERRQ(ierr); 70 ierr = VecDuplicate(tao->solution, &asls->w);CHKERRQ(ierr); 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 PetscFunctionReturn(0); 80 } 81 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; 86 PetscErrorCode ierr; 87 88 PetscFunctionBegin; 89 ierr = TaoComputeConstraints(tao, X, tao->constraints);CHKERRQ(ierr); 90 ierr = VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff);CHKERRQ(ierr); 91 ierr = VecNorm(asls->ff,NORM_2,&asls->merit);CHKERRQ(ierr); 92 *fcn = 0.5*asls->merit*asls->merit; 93 ierr = TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre);CHKERRQ(ierr); 94 95 ierr = MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db);CHKERRQ(ierr); 96 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->db);CHKERRQ(ierr); 97 ierr = MatMultTranspose(tao->jacobian,asls->t1,G);CHKERRQ(ierr); 98 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->da);CHKERRQ(ierr); 99 ierr = VecAXPY(G,1.0,asls->t1);CHKERRQ(ierr); 100 PetscFunctionReturn(0); 101 } 102 103 static PetscErrorCode TaoDestroy_ASFLS(Tao tao) 104 { 105 TAO_SSLS *ssls = (TAO_SSLS *)tao->data; 106 PetscErrorCode ierr; 107 108 PetscFunctionBegin; 109 ierr = VecDestroy(&ssls->ff);CHKERRQ(ierr); 110 ierr = VecDestroy(&ssls->dpsi);CHKERRQ(ierr); 111 ierr = VecDestroy(&ssls->da);CHKERRQ(ierr); 112 ierr = VecDestroy(&ssls->db);CHKERRQ(ierr); 113 ierr = VecDestroy(&ssls->w);CHKERRQ(ierr); 114 ierr = VecDestroy(&ssls->t1);CHKERRQ(ierr); 115 ierr = VecDestroy(&ssls->t2);CHKERRQ(ierr); 116 ierr = VecDestroy(&ssls->r1);CHKERRQ(ierr); 117 ierr = VecDestroy(&ssls->r2);CHKERRQ(ierr); 118 ierr = VecDestroy(&ssls->r3);CHKERRQ(ierr); 119 ierr = VecDestroy(&ssls->dxfree);CHKERRQ(ierr); 120 ierr = MatDestroy(&ssls->J_sub);CHKERRQ(ierr); 121 ierr = MatDestroy(&ssls->Jpre_sub);CHKERRQ(ierr); 122 ierr = ISDestroy(&ssls->fixed);CHKERRQ(ierr); 123 ierr = ISDestroy(&ssls->free);CHKERRQ(ierr); 124 ierr = PetscFree(tao->data);CHKERRQ(ierr); 125 tao->data = NULL; 126 PetscFunctionReturn(0); 127 } 128 129 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; 136 137 PetscFunctionBegin; 138 /* Assume that Setup has been called! 139 Set the structure for the Jacobian and create a linear solver. */ 140 141 ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr); 142 ierr = TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao);CHKERRQ(ierr); 143 ierr = TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao);CHKERRQ(ierr); 144 ierr = TaoLineSearchSetVariableBounds(tao->linesearch,tao->XL,tao->XU);CHKERRQ(ierr); 145 146 ierr = VecMedian(tao->XL, tao->solution, tao->XU, tao->solution);CHKERRQ(ierr); 147 148 /* Calculate the function value and fischer function value at the 149 current iterate */ 150 ierr = TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi);CHKERRQ(ierr); 151 ierr = VecNorm(asls->dpsi,NORM_2,&ndpsi);CHKERRQ(ierr); 152 153 tao->reason = TAO_CONTINUE_ITERATING; 154 while (1) { 155 /* Check the converged criteria */ 156 ierr = PetscInfo3(tao,"iter %D, merit: %g, ||dpsi||: %g\n",tao->niter,(double)asls->merit,(double)ndpsi);CHKERRQ(ierr); 157 ierr = TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its);CHKERRQ(ierr); 158 ierr = TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t);CHKERRQ(ierr); 159 ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr); 160 if (TAO_CONTINUE_ITERATING != tao->reason) break; 161 162 /* Call general purpose update function */ 163 if (tao->ops->update) { 164 ierr = (*tao->ops->update)(tao, tao->niter, tao->user_update);CHKERRQ(ierr); 165 } 166 tao->niter++; 167 168 /* 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. 173 174 This rule gives superlinear asymptotic convergence 175 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 176 asls->rtol = 0.0; 177 178 This rule gives quadratic asymptotic convergence 179 asls->atol = min(0.5, asls->merit*asls->merit); 180 asls->rtol = 0.0; 181 182 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. 186 187 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 ierr = MatNorm(tao->jacobian,NORM_1,&asls->identifier);CHKERRQ(ierr); 191 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 192 193 ierr = VecSet(asls->t1,-asls->identifier);CHKERRQ(ierr); 194 ierr = VecSet(asls->t2, asls->identifier);CHKERRQ(ierr); 195 196 ierr = ISDestroy(&asls->fixed);CHKERRQ(ierr); 197 ierr = ISDestroy(&asls->free);CHKERRQ(ierr); 198 ierr = VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);CHKERRQ(ierr); 199 ierr = ISComplementVec(asls->fixed,asls->t1, &asls->free);CHKERRQ(ierr); 200 201 ierr = ISGetSize(asls->fixed,&nf);CHKERRQ(ierr); 202 ierr = PetscInfo1(tao,"Number of fixed variables: %D\n", nf);CHKERRQ(ierr); 203 204 /* We now have our partition. Now calculate the direction in the 205 fixed variable space. */ 206 ierr = TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 207 ierr = TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);CHKERRQ(ierr); 208 ierr = VecPointwiseDivide(asls->r1,asls->r1,asls->r2);CHKERRQ(ierr); 209 ierr = VecSet(tao->stepdirection,0.0);CHKERRQ(ierr); 210 ierr = VecISAXPY(tao->stepdirection, asls->fixed, 1.0,asls->r1);CHKERRQ(ierr); 211 212 /* 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. */ 216 217 ierr = TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 218 ierr = TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);CHKERRQ(ierr); 219 ierr = TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);CHKERRQ(ierr); 220 ierr = VecPointwiseDivide(asls->r1,asls->r1, asls->r3);CHKERRQ(ierr); 221 ierr = VecPointwiseDivide(asls->r2,asls->r2, asls->r3);CHKERRQ(ierr); 222 223 /* r1 is the diagonal perturbation 224 r2 is the right hand side 225 r3 is no longer needed 226 227 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. */ 230 231 ierr = MatMult(tao->jacobian, tao->stepdirection, asls->t1);CHKERRQ(ierr); 232 ierr = TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);CHKERRQ(ierr); 233 ierr = VecAXPY(asls->r2, -1.0, asls->r3);CHKERRQ(ierr); 234 235 /* Calculate the reduced problem matrix and the direction */ 236 ierr = TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);CHKERRQ(ierr); 237 if (tao->jacobian != tao->jacobian_pre) { 238 ierr = TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);CHKERRQ(ierr); 239 } else { 240 ierr = MatDestroy(&asls->Jpre_sub);CHKERRQ(ierr); 241 asls->Jpre_sub = asls->J_sub; 242 ierr = PetscObjectReference((PetscObject)(asls->Jpre_sub));CHKERRQ(ierr); 243 } 244 ierr = MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);CHKERRQ(ierr); 245 ierr = TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);CHKERRQ(ierr); 246 ierr = VecSet(asls->dxfree, 0.0);CHKERRQ(ierr); 247 248 /* Calculate the reduced direction. (Really negative of Newton 249 direction. Therefore, rest of the code uses -d.) */ 250 ierr = KSPReset(tao->ksp);CHKERRQ(ierr); 251 ierr = KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);CHKERRQ(ierr); 252 ierr = KSPSolve(tao->ksp, asls->r2, asls->dxfree);CHKERRQ(ierr); 253 ierr = KSPGetIterationNumber(tao->ksp,&tao->ksp_its);CHKERRQ(ierr); 254 tao->ksp_tot_its+=tao->ksp_its; 255 256 /* Add the direction in the free variables back into the real direction. */ 257 ierr = VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);CHKERRQ(ierr); 258 259 260 /* Check the projected real direction for descent and if not, use the negative 261 gradient direction. */ 262 ierr = VecCopy(tao->stepdirection, asls->w);CHKERRQ(ierr); 263 ierr = VecScale(asls->w, -1.0);CHKERRQ(ierr); 264 ierr = VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w);CHKERRQ(ierr); 265 ierr = VecNorm(asls->w, NORM_2, &normd);CHKERRQ(ierr); 266 ierr = VecDot(asls->w, asls->dpsi, &innerd);CHKERRQ(ierr); 267 268 if (innerd >= -asls->delta*PetscPowReal(normd, asls->rho)) { 269 ierr = PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);CHKERRQ(ierr); 270 ierr = PetscInfo1(tao, "Iteration %D: newton direction not descent\n", tao->niter);CHKERRQ(ierr); 271 ierr = VecCopy(asls->dpsi, tao->stepdirection);CHKERRQ(ierr); 272 ierr = VecDot(asls->dpsi, tao->stepdirection, &innerd);CHKERRQ(ierr); 273 } 274 275 ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr); 276 innerd = -innerd; 277 278 /* We now have a correct descent direction. Apply a linesearch to 279 find the new iterate. */ 280 ierr = TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);CHKERRQ(ierr); 281 ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);CHKERRQ(ierr); 282 ierr = VecNorm(asls->dpsi, NORM_2, &ndpsi);CHKERRQ(ierr); 283 } 284 PetscFunctionReturn(0); 285 } 286 287 /* ---------------------------------------------------------- */ 288 /*MC 289 TAOASFLS - Active-set feasible linesearch algorithm for solving 290 complementarity constraints 291 292 Options Database Keys: 293 + -tao_ssls_delta - descent test fraction 294 - -tao_ssls_rho - descent test power 295 296 Level: beginner 297 M*/ 298 PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao) 299 { 300 TAO_SSLS *asls; 301 PetscErrorCode ierr; 302 const char *armijo_type = TAOLINESEARCHARMIJO; 303 304 PetscFunctionBegin; 305 ierr = PetscNewLog(tao,&asls);CHKERRQ(ierr); 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; 327 328 ierr = TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);CHKERRQ(ierr); 329 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr); 330 ierr = TaoLineSearchSetType(tao->linesearch, armijo_type);CHKERRQ(ierr); 331 ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr); 332 ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr); 333 334 ierr = KSPCreate(((PetscObject)tao)->comm, &tao->ksp);CHKERRQ(ierr); 335 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr); 336 ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr); 337 ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr); 338 339 /* 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 PetscFunctionReturn(0); 352 } 353