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 tao->niter++; 162 163 /* We are going to solve a linear system of equations. We need to 164 set the tolerances for the solve so that we maintain an asymptotic 165 rate of convergence that is superlinear. 166 Note: these tolerances are for the reduced system. We really need 167 to make sure that the full system satisfies the full-space conditions. 168 169 This rule gives superlinear asymptotic convergence 170 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 171 asls->rtol = 0.0; 172 173 This rule gives quadratic asymptotic convergence 174 asls->atol = min(0.5, asls->merit*asls->merit); 175 asls->rtol = 0.0; 176 177 Calculate a free and fixed set of variables. The fixed set of 178 variables are those for the d_b is approximately equal to zero. 179 The definition of approximately changes as we approach the solution 180 to the problem. 181 182 No one rule is guaranteed to work in all cases. The following 183 definition is based on the norm of the Jacobian matrix. If the 184 norm is large, the tolerance becomes smaller. */ 185 ierr = MatNorm(tao->jacobian,NORM_1,&asls->identifier);CHKERRQ(ierr); 186 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 187 188 ierr = VecSet(asls->t1,-asls->identifier);CHKERRQ(ierr); 189 ierr = VecSet(asls->t2, asls->identifier);CHKERRQ(ierr); 190 191 ierr = ISDestroy(&asls->fixed);CHKERRQ(ierr); 192 ierr = ISDestroy(&asls->free);CHKERRQ(ierr); 193 ierr = VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);CHKERRQ(ierr); 194 ierr = ISComplementVec(asls->fixed,asls->t1, &asls->free);CHKERRQ(ierr); 195 196 ierr = ISGetSize(asls->fixed,&nf);CHKERRQ(ierr); 197 ierr = PetscInfo1(tao,"Number of fixed variables: %D\n", nf);CHKERRQ(ierr); 198 199 /* We now have our partition. Now calculate the direction in the 200 fixed variable space. */ 201 ierr = TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 202 ierr = TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);CHKERRQ(ierr); 203 ierr = VecPointwiseDivide(asls->r1,asls->r1,asls->r2);CHKERRQ(ierr); 204 ierr = VecSet(tao->stepdirection,0.0);CHKERRQ(ierr); 205 ierr = VecISAXPY(tao->stepdirection, asls->fixed, 1.0,asls->r1);CHKERRQ(ierr); 206 207 /* Our direction in the Fixed Variable Set is fixed. Calculate the 208 information needed for the step in the Free Variable Set. To 209 do this, we need to know the diagonal perturbation and the 210 right hand side. */ 211 212 ierr = TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 213 ierr = TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);CHKERRQ(ierr); 214 ierr = TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);CHKERRQ(ierr); 215 ierr = VecPointwiseDivide(asls->r1,asls->r1, asls->r3);CHKERRQ(ierr); 216 ierr = VecPointwiseDivide(asls->r2,asls->r2, asls->r3);CHKERRQ(ierr); 217 218 /* r1 is the diagonal perturbation 219 r2 is the right hand side 220 r3 is no longer needed 221 222 Now need to modify r2 for our direction choice in the fixed 223 variable set: calculate t1 = J*d, take the reduced vector 224 of t1 and modify r2. */ 225 226 ierr = MatMult(tao->jacobian, tao->stepdirection, asls->t1);CHKERRQ(ierr); 227 ierr = TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);CHKERRQ(ierr); 228 ierr = VecAXPY(asls->r2, -1.0, asls->r3);CHKERRQ(ierr); 229 230 /* Calculate the reduced problem matrix and the direction */ 231 ierr = TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);CHKERRQ(ierr); 232 if (tao->jacobian != tao->jacobian_pre) { 233 ierr = TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);CHKERRQ(ierr); 234 } else { 235 ierr = MatDestroy(&asls->Jpre_sub);CHKERRQ(ierr); 236 asls->Jpre_sub = asls->J_sub; 237 ierr = PetscObjectReference((PetscObject)(asls->Jpre_sub));CHKERRQ(ierr); 238 } 239 ierr = MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);CHKERRQ(ierr); 240 ierr = TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);CHKERRQ(ierr); 241 ierr = VecSet(asls->dxfree, 0.0);CHKERRQ(ierr); 242 243 /* Calculate the reduced direction. (Really negative of Newton 244 direction. Therefore, rest of the code uses -d.) */ 245 ierr = KSPReset(tao->ksp);CHKERRQ(ierr); 246 ierr = KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);CHKERRQ(ierr); 247 ierr = KSPSolve(tao->ksp, asls->r2, asls->dxfree);CHKERRQ(ierr); 248 ierr = KSPGetIterationNumber(tao->ksp,&tao->ksp_its);CHKERRQ(ierr); 249 tao->ksp_tot_its+=tao->ksp_its; 250 251 /* Add the direction in the free variables back into the real direction. */ 252 ierr = VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);CHKERRQ(ierr); 253 254 255 /* Check the projected real direction for descent and if not, use the negative 256 gradient direction. */ 257 ierr = VecCopy(tao->stepdirection, asls->w);CHKERRQ(ierr); 258 ierr = VecScale(asls->w, -1.0);CHKERRQ(ierr); 259 ierr = VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w);CHKERRQ(ierr); 260 ierr = VecNorm(asls->w, NORM_2, &normd);CHKERRQ(ierr); 261 ierr = VecDot(asls->w, asls->dpsi, &innerd);CHKERRQ(ierr); 262 263 if (innerd >= -asls->delta*PetscPowReal(normd, asls->rho)) { 264 ierr = PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);CHKERRQ(ierr); 265 ierr = PetscInfo1(tao, "Iteration %D: newton direction not descent\n", tao->niter);CHKERRQ(ierr); 266 ierr = VecCopy(asls->dpsi, tao->stepdirection);CHKERRQ(ierr); 267 ierr = VecDot(asls->dpsi, tao->stepdirection, &innerd);CHKERRQ(ierr); 268 } 269 270 ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr); 271 innerd = -innerd; 272 273 /* We now have a correct descent direction. Apply a linesearch to 274 find the new iterate. */ 275 ierr = TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);CHKERRQ(ierr); 276 ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);CHKERRQ(ierr); 277 ierr = VecNorm(asls->dpsi, NORM_2, &ndpsi);CHKERRQ(ierr); 278 } 279 PetscFunctionReturn(0); 280 } 281 282 /* ---------------------------------------------------------- */ 283 /*MC 284 TAOASFLS - Active-set feasible linesearch algorithm for solving 285 complementarity constraints 286 287 Options Database Keys: 288 + -tao_ssls_delta - descent test fraction 289 - -tao_ssls_rho - descent test power 290 291 Level: beginner 292 M*/ 293 PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao) 294 { 295 TAO_SSLS *asls; 296 PetscErrorCode ierr; 297 const char *armijo_type = TAOLINESEARCHARMIJO; 298 299 PetscFunctionBegin; 300 ierr = PetscNewLog(tao,&asls);CHKERRQ(ierr); 301 tao->data = (void*)asls; 302 tao->ops->solve = TaoSolve_ASFLS; 303 tao->ops->setup = TaoSetUp_ASFLS; 304 tao->ops->view = TaoView_SSLS; 305 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 306 tao->ops->destroy = TaoDestroy_ASFLS; 307 tao->subset_type = TAO_SUBSET_SUBVEC; 308 asls->delta = 1e-10; 309 asls->rho = 2.1; 310 asls->fixed = NULL; 311 asls->free = NULL; 312 asls->J_sub = NULL; 313 asls->Jpre_sub = NULL; 314 asls->w = NULL; 315 asls->r1 = NULL; 316 asls->r2 = NULL; 317 asls->r3 = NULL; 318 asls->t1 = NULL; 319 asls->t2 = NULL; 320 asls->dxfree = NULL; 321 asls->identifier = 1e-5; 322 323 ierr = TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);CHKERRQ(ierr); 324 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr); 325 ierr = TaoLineSearchSetType(tao->linesearch, armijo_type);CHKERRQ(ierr); 326 ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr); 327 ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr); 328 329 ierr = KSPCreate(((PetscObject)tao)->comm, &tao->ksp);CHKERRQ(ierr); 330 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr); 331 ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr); 332 ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr); 333 334 /* Override default settings (unless already changed) */ 335 if (!tao->max_it_changed) tao->max_it = 2000; 336 if (!tao->max_funcs_changed) tao->max_funcs = 4000; 337 if (!tao->gttol_changed) tao->gttol = 0; 338 if (!tao->grtol_changed) tao->grtol = 0; 339 #if defined(PETSC_USE_REAL_SINGLE) 340 if (!tao->gatol_changed) tao->gatol = 1.0e-6; 341 if (!tao->fmin_changed) tao->fmin = 1.0e-4; 342 #else 343 if (!tao->gatol_changed) tao->gatol = 1.0e-16; 344 if (!tao->fmin_changed) tao->fmin = 1.0e-8; 345 #endif 346 PetscFunctionReturn(0); 347 } 348