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, 1996. 45 Ferris, Kanzow, Munson, "Feasible Descent Algorithms for Mixed 46 Complementarity Problems," Mathematical Programming, 86, 47 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_ASILS(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 asls->fixed = NULL; 71 asls->free = NULL; 72 asls->J_sub = NULL; 73 asls->Jpre_sub = NULL; 74 asls->w = 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 94 ierr = TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre);CHKERRQ(ierr); 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_ASILS(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 PetscFunctionReturn(0); 126 } 127 128 static PetscErrorCode TaoSolve_ASILS(Tao tao) 129 { 130 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 131 PetscReal psi,ndpsi, normd, innerd, t=0; 132 PetscInt nf; 133 PetscErrorCode ierr; 134 TaoLineSearchConvergedReason ls_reason; 135 136 PetscFunctionBegin; 137 /* Assume that Setup has been called! 138 Set the structure for the Jacobian and create a linear solver. */ 139 140 ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr); 141 ierr = TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao);CHKERRQ(ierr); 142 ierr = TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao);CHKERRQ(ierr); 143 144 /* Calculate the function value and fischer function value at the 145 current iterate */ 146 ierr = TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi);CHKERRQ(ierr); 147 ierr = VecNorm(asls->dpsi,NORM_2,&ndpsi);CHKERRQ(ierr); 148 149 tao->reason = TAO_CONTINUE_ITERATING; 150 while (1) { 151 /* Check the termination criteria */ 152 ierr = PetscInfo3(tao,"iter %D, merit: %g, ||dpsi||: %g\n",tao->niter, (double)asls->merit, (double)ndpsi);CHKERRQ(ierr); 153 ierr = TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its);CHKERRQ(ierr); 154 ierr = TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t);CHKERRQ(ierr); 155 ierr = (*tao->ops->convergencetest)(tao,tao->cnvP);CHKERRQ(ierr); 156 if (TAO_CONTINUE_ITERATING != tao->reason) break; 157 tao->niter++; 158 159 /* We are going to solve a linear system of equations. We need to 160 set the tolerances for the solve so that we maintain an asymptotic 161 rate of convergence that is superlinear. 162 Note: these tolerances are for the reduced system. We really need 163 to make sure that the full system satisfies the full-space conditions. 164 165 This rule gives superlinear asymptotic convergence 166 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 167 asls->rtol = 0.0; 168 169 This rule gives quadratic asymptotic convergence 170 asls->atol = min(0.5, asls->merit*asls->merit); 171 asls->rtol = 0.0; 172 173 Calculate a free and fixed set of variables. The fixed set of 174 variables are those for the d_b is approximately equal to zero. 175 The definition of approximately changes as we approach the solution 176 to the problem. 177 178 No one rule is guaranteed to work in all cases. The following 179 definition is based on the norm of the Jacobian matrix. If the 180 norm is large, the tolerance becomes smaller. */ 181 ierr = MatNorm(tao->jacobian,NORM_1,&asls->identifier);CHKERRQ(ierr); 182 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 183 184 ierr = VecSet(asls->t1,-asls->identifier);CHKERRQ(ierr); 185 ierr = VecSet(asls->t2, asls->identifier);CHKERRQ(ierr); 186 187 ierr = ISDestroy(&asls->fixed);CHKERRQ(ierr); 188 ierr = ISDestroy(&asls->free);CHKERRQ(ierr); 189 ierr = VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);CHKERRQ(ierr); 190 ierr = ISComplementVec(asls->fixed,asls->t1, &asls->free);CHKERRQ(ierr); 191 192 ierr = ISGetSize(asls->fixed,&nf);CHKERRQ(ierr); 193 ierr = PetscInfo1(tao,"Number of fixed variables: %D\n", nf);CHKERRQ(ierr); 194 195 /* We now have our partition. Now calculate the direction in the 196 fixed variable space. */ 197 ierr = TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 198 ierr = TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);CHKERRQ(ierr); 199 ierr = VecPointwiseDivide(asls->r1,asls->r1,asls->r2);CHKERRQ(ierr); 200 ierr = VecSet(tao->stepdirection,0.0);CHKERRQ(ierr); 201 ierr = VecISAXPY(tao->stepdirection, asls->fixed,1.0,asls->r1);CHKERRQ(ierr); 202 203 /* Our direction in the Fixed Variable Set is fixed. Calculate the 204 information needed for the step in the Free Variable Set. To 205 do this, we need to know the diagonal perturbation and the 206 right hand side. */ 207 208 ierr = TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 209 ierr = TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);CHKERRQ(ierr); 210 ierr = TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);CHKERRQ(ierr); 211 ierr = VecPointwiseDivide(asls->r1,asls->r1, asls->r3);CHKERRQ(ierr); 212 ierr = VecPointwiseDivide(asls->r2,asls->r2, asls->r3);CHKERRQ(ierr); 213 214 /* r1 is the diagonal perturbation 215 r2 is the right hand side 216 r3 is no longer needed 217 218 Now need to modify r2 for our direction choice in the fixed 219 variable set: calculate t1 = J*d, take the reduced vector 220 of t1 and modify r2. */ 221 222 ierr = MatMult(tao->jacobian, tao->stepdirection, asls->t1);CHKERRQ(ierr); 223 ierr = TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);CHKERRQ(ierr); 224 ierr = VecAXPY(asls->r2, -1.0, asls->r3);CHKERRQ(ierr); 225 226 /* Calculate the reduced problem matrix and the direction */ 227 if (!asls->w && (tao->subset_type == TAO_SUBSET_MASK || tao->subset_type == TAO_SUBSET_MATRIXFREE)) { 228 ierr = VecDuplicate(tao->solution, &asls->w);CHKERRQ(ierr); 229 } 230 ierr = TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);CHKERRQ(ierr); 231 if (tao->jacobian != tao->jacobian_pre) { 232 ierr = TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);CHKERRQ(ierr); 233 } else { 234 ierr = MatDestroy(&asls->Jpre_sub);CHKERRQ(ierr); 235 asls->Jpre_sub = asls->J_sub; 236 ierr = PetscObjectReference((PetscObject)(asls->Jpre_sub));CHKERRQ(ierr); 237 } 238 ierr = MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);CHKERRQ(ierr); 239 ierr = TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);CHKERRQ(ierr); 240 ierr = VecSet(asls->dxfree, 0.0);CHKERRQ(ierr); 241 242 /* Calculate the reduced direction. (Really negative of Newton 243 direction. Therefore, rest of the code uses -d.) */ 244 ierr = KSPReset(tao->ksp);CHKERRQ(ierr); 245 ierr = KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);CHKERRQ(ierr); 246 ierr = KSPSolve(tao->ksp, asls->r2, asls->dxfree);CHKERRQ(ierr); 247 ierr = KSPGetIterationNumber(tao->ksp,&tao->ksp_its);CHKERRQ(ierr); 248 tao->ksp_tot_its+=tao->ksp_its; 249 250 /* Add the direction in the free variables back into the real direction. */ 251 ierr = VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);CHKERRQ(ierr); 252 253 /* Check the real direction for descent and if not, use the negative 254 gradient direction. */ 255 ierr = VecNorm(tao->stepdirection, NORM_2, &normd);CHKERRQ(ierr); 256 ierr = VecDot(tao->stepdirection, asls->dpsi, &innerd);CHKERRQ(ierr); 257 258 if (innerd <= asls->delta*PetscPowReal(normd, asls->rho)) { 259 ierr = PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);CHKERRQ(ierr); 260 ierr = PetscInfo1(tao, "Iteration %D: newton direction not descent\n", tao->niter);CHKERRQ(ierr); 261 ierr = VecCopy(asls->dpsi, tao->stepdirection);CHKERRQ(ierr); 262 ierr = VecDot(asls->dpsi, tao->stepdirection, &innerd);CHKERRQ(ierr); 263 } 264 265 ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr); 266 innerd = -innerd; 267 268 /* We now have a correct descent direction. Apply a linesearch to 269 find the new iterate. */ 270 ierr = TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);CHKERRQ(ierr); 271 ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);CHKERRQ(ierr); 272 ierr = VecNorm(asls->dpsi, NORM_2, &ndpsi);CHKERRQ(ierr); 273 } 274 PetscFunctionReturn(0); 275 } 276 277 /* ---------------------------------------------------------- */ 278 /*MC 279 TAOASILS - Active-set infeasible linesearch algorithm for solving 280 complementarity constraints 281 282 Options Database Keys: 283 + -tao_ssls_delta - descent test fraction 284 - -tao_ssls_rho - descent test power 285 286 Level: beginner 287 M*/ 288 PETSC_EXTERN PetscErrorCode TaoCreate_ASILS(Tao tao) 289 { 290 TAO_SSLS *asls; 291 PetscErrorCode ierr; 292 const char *armijo_type = TAOLINESEARCHARMIJO; 293 294 PetscFunctionBegin; 295 ierr = PetscNewLog(tao,&asls);CHKERRQ(ierr); 296 tao->data = (void*)asls; 297 tao->ops->solve = TaoSolve_ASILS; 298 tao->ops->setup = TaoSetUp_ASILS; 299 tao->ops->view = TaoView_SSLS; 300 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 301 tao->ops->destroy = TaoDestroy_ASILS; 302 tao->subset_type = TAO_SUBSET_SUBVEC; 303 asls->delta = 1e-10; 304 asls->rho = 2.1; 305 asls->fixed = NULL; 306 asls->free = NULL; 307 asls->J_sub = NULL; 308 asls->Jpre_sub = NULL; 309 asls->w = NULL; 310 asls->r1 = NULL; 311 asls->r2 = NULL; 312 asls->r3 = NULL; 313 asls->t1 = NULL; 314 asls->t2 = NULL; 315 asls->dxfree = NULL; 316 317 asls->identifier = 1e-5; 318 319 ierr = TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);CHKERRQ(ierr); 320 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);CHKERRQ(ierr); 321 ierr = TaoLineSearchSetType(tao->linesearch, armijo_type);CHKERRQ(ierr); 322 ierr = TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);CHKERRQ(ierr); 323 ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr); 324 325 ierr = KSPCreate(((PetscObject)tao)->comm, &tao->ksp);CHKERRQ(ierr); 326 ierr = PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);CHKERRQ(ierr); 327 ierr = KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);CHKERRQ(ierr); 328 ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr); 329 330 /* Override default settings (unless already changed) */ 331 if (!tao->max_it_changed) tao->max_it = 2000; 332 if (!tao->max_funcs_changed) tao->max_funcs = 4000; 333 if (!tao->gttol_changed) tao->gttol = 0; 334 if (!tao->grtol_changed) tao->grtol = 0; 335 #if defined(PETSC_USE_REAL_SINGLE) 336 if (!tao->gatol_changed) tao->gatol = 1.0e-6; 337 if (!tao->fmin_changed) tao->fmin = 1.0e-4; 338 #else 339 if (!tao->gatol_changed) tao->gatol = 1.0e-16; 340 if (!tao->fmin_changed) tao->fmin = 1.0e-8; 341 #endif 342 PetscFunctionReturn(0); 343 } 344