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 static PetscErrorCode TaoSetUp_ASILS(Tao tao) 56 { 57 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 58 59 PetscFunctionBegin; 60 PetscCall(VecDuplicate(tao->solution,&tao->gradient)); 61 PetscCall(VecDuplicate(tao->solution,&tao->stepdirection)); 62 PetscCall(VecDuplicate(tao->solution,&asls->ff)); 63 PetscCall(VecDuplicate(tao->solution,&asls->dpsi)); 64 PetscCall(VecDuplicate(tao->solution,&asls->da)); 65 PetscCall(VecDuplicate(tao->solution,&asls->db)); 66 PetscCall(VecDuplicate(tao->solution,&asls->t1)); 67 PetscCall(VecDuplicate(tao->solution,&asls->t2)); 68 asls->fixed = NULL; 69 asls->free = NULL; 70 asls->J_sub = NULL; 71 asls->Jpre_sub = NULL; 72 asls->w = NULL; 73 asls->r1 = NULL; 74 asls->r2 = NULL; 75 asls->r3 = NULL; 76 asls->dxfree = NULL; 77 PetscFunctionReturn(0); 78 } 79 80 static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr) 81 { 82 Tao tao = (Tao)ptr; 83 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 84 85 PetscFunctionBegin; 86 PetscCall(TaoComputeConstraints(tao, X, tao->constraints)); 87 PetscCall(VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff)); 88 PetscCall(VecNorm(asls->ff,NORM_2,&asls->merit)); 89 *fcn = 0.5*asls->merit*asls->merit; 90 91 PetscCall(TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre)); 92 PetscCall(MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db)); 93 PetscCall(VecPointwiseMult(asls->t1, asls->ff, asls->db)); 94 PetscCall(MatMultTranspose(tao->jacobian,asls->t1,G)); 95 PetscCall(VecPointwiseMult(asls->t1, asls->ff, asls->da)); 96 PetscCall(VecAXPY(G,1.0,asls->t1)); 97 PetscFunctionReturn(0); 98 } 99 100 static PetscErrorCode TaoDestroy_ASILS(Tao tao) 101 { 102 TAO_SSLS *ssls = (TAO_SSLS *)tao->data; 103 104 PetscFunctionBegin; 105 PetscCall(VecDestroy(&ssls->ff)); 106 PetscCall(VecDestroy(&ssls->dpsi)); 107 PetscCall(VecDestroy(&ssls->da)); 108 PetscCall(VecDestroy(&ssls->db)); 109 PetscCall(VecDestroy(&ssls->w)); 110 PetscCall(VecDestroy(&ssls->t1)); 111 PetscCall(VecDestroy(&ssls->t2)); 112 PetscCall(VecDestroy(&ssls->r1)); 113 PetscCall(VecDestroy(&ssls->r2)); 114 PetscCall(VecDestroy(&ssls->r3)); 115 PetscCall(VecDestroy(&ssls->dxfree)); 116 PetscCall(MatDestroy(&ssls->J_sub)); 117 PetscCall(MatDestroy(&ssls->Jpre_sub)); 118 PetscCall(ISDestroy(&ssls->fixed)); 119 PetscCall(ISDestroy(&ssls->free)); 120 PetscCall(KSPDestroy(&tao->ksp)); 121 PetscCall(PetscFree(tao->data)); 122 PetscFunctionReturn(0); 123 } 124 125 static PetscErrorCode TaoSolve_ASILS(Tao tao) 126 { 127 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 128 PetscReal psi,ndpsi, normd, innerd, t=0; 129 PetscInt nf; 130 TaoLineSearchConvergedReason ls_reason; 131 132 PetscFunctionBegin; 133 /* Assume that Setup has been called! 134 Set the structure for the Jacobian and create a linear solver. */ 135 136 PetscCall(TaoComputeVariableBounds(tao)); 137 PetscCall(TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao)); 138 PetscCall(TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao)); 139 140 /* Calculate the function value and fischer function value at the 141 current iterate */ 142 PetscCall(TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi)); 143 PetscCall(VecNorm(asls->dpsi,NORM_2,&ndpsi)); 144 145 tao->reason = TAO_CONTINUE_ITERATING; 146 while (1) { 147 /* Check the termination criteria */ 148 PetscCall(PetscInfo(tao,"iter %" PetscInt_FMT ", merit: %g, ||dpsi||: %g\n",tao->niter, (double)asls->merit, (double)ndpsi)); 149 PetscCall(TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its)); 150 PetscCall(TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t)); 151 PetscCall((*tao->ops->convergencetest)(tao,tao->cnvP)); 152 if (TAO_CONTINUE_ITERATING != tao->reason) break; 153 154 /* Call general purpose update function */ 155 if (tao->ops->update) { 156 PetscCall((*tao->ops->update)(tao, tao->niter, tao->user_update)); 157 } 158 tao->niter++; 159 160 /* We are going to solve a linear system of equations. We need to 161 set the tolerances for the solve so that we maintain an asymptotic 162 rate of convergence that is superlinear. 163 Note: these tolerances are for the reduced system. We really need 164 to make sure that the full system satisfies the full-space conditions. 165 166 This rule gives superlinear asymptotic convergence 167 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 168 asls->rtol = 0.0; 169 170 This rule gives quadratic asymptotic convergence 171 asls->atol = min(0.5, asls->merit*asls->merit); 172 asls->rtol = 0.0; 173 174 Calculate a free and fixed set of variables. The fixed set of 175 variables are those for the d_b is approximately equal to zero. 176 The definition of approximately changes as we approach the solution 177 to the problem. 178 179 No one rule is guaranteed to work in all cases. The following 180 definition is based on the norm of the Jacobian matrix. If the 181 norm is large, the tolerance becomes smaller. */ 182 PetscCall(MatNorm(tao->jacobian,NORM_1,&asls->identifier)); 183 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 184 185 PetscCall(VecSet(asls->t1,-asls->identifier)); 186 PetscCall(VecSet(asls->t2, asls->identifier)); 187 188 PetscCall(ISDestroy(&asls->fixed)); 189 PetscCall(ISDestroy(&asls->free)); 190 PetscCall(VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed)); 191 PetscCall(ISComplementVec(asls->fixed,asls->t1, &asls->free)); 192 193 PetscCall(ISGetSize(asls->fixed,&nf)); 194 PetscCall(PetscInfo(tao,"Number of fixed variables: %" PetscInt_FMT "\n", nf)); 195 196 /* We now have our partition. Now calculate the direction in the 197 fixed variable space. */ 198 PetscCall(TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1)); 199 PetscCall(TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2)); 200 PetscCall(VecPointwiseDivide(asls->r1,asls->r1,asls->r2)); 201 PetscCall(VecSet(tao->stepdirection,0.0)); 202 PetscCall(VecISAXPY(tao->stepdirection, asls->fixed,1.0,asls->r1)); 203 204 /* Our direction in the Fixed Variable Set is fixed. Calculate the 205 information needed for the step in the Free Variable Set. To 206 do this, we need to know the diagonal perturbation and the 207 right hand side. */ 208 209 PetscCall(TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1)); 210 PetscCall(TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2)); 211 PetscCall(TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3)); 212 PetscCall(VecPointwiseDivide(asls->r1,asls->r1, asls->r3)); 213 PetscCall(VecPointwiseDivide(asls->r2,asls->r2, asls->r3)); 214 215 /* r1 is the diagonal perturbation 216 r2 is the right hand side 217 r3 is no longer needed 218 219 Now need to modify r2 for our direction choice in the fixed 220 variable set: calculate t1 = J*d, take the reduced vector 221 of t1 and modify r2. */ 222 223 PetscCall(MatMult(tao->jacobian, tao->stepdirection, asls->t1)); 224 PetscCall(TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3)); 225 PetscCall(VecAXPY(asls->r2, -1.0, asls->r3)); 226 227 /* Calculate the reduced problem matrix and the direction */ 228 if (!asls->w && (tao->subset_type == TAO_SUBSET_MASK || tao->subset_type == TAO_SUBSET_MATRIXFREE)) { 229 PetscCall(VecDuplicate(tao->solution, &asls->w)); 230 } 231 PetscCall(TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub)); 232 if (tao->jacobian != tao->jacobian_pre) { 233 PetscCall(TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub)); 234 } else { 235 PetscCall(MatDestroy(&asls->Jpre_sub)); 236 asls->Jpre_sub = asls->J_sub; 237 PetscCall(PetscObjectReference((PetscObject)(asls->Jpre_sub))); 238 } 239 PetscCall(MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES)); 240 PetscCall(TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree)); 241 PetscCall(VecSet(asls->dxfree, 0.0)); 242 243 /* Calculate the reduced direction. (Really negative of Newton 244 direction. Therefore, rest of the code uses -d.) */ 245 PetscCall(KSPReset(tao->ksp)); 246 PetscCall(KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub)); 247 PetscCall(KSPSolve(tao->ksp, asls->r2, asls->dxfree)); 248 PetscCall(KSPGetIterationNumber(tao->ksp,&tao->ksp_its)); 249 tao->ksp_tot_its+=tao->ksp_its; 250 251 /* Add the direction in the free variables back into the real direction. */ 252 PetscCall(VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree)); 253 254 /* Check the real direction for descent and if not, use the negative 255 gradient direction. */ 256 PetscCall(VecNorm(tao->stepdirection, NORM_2, &normd)); 257 PetscCall(VecDot(tao->stepdirection, asls->dpsi, &innerd)); 258 259 if (innerd <= asls->delta*PetscPowReal(normd, asls->rho)) { 260 PetscCall(PetscInfo(tao,"Gradient direction: %5.4e.\n", (double)innerd)); 261 PetscCall(PetscInfo(tao, "Iteration %" PetscInt_FMT ": newton direction not descent\n", tao->niter)); 262 PetscCall(VecCopy(asls->dpsi, tao->stepdirection)); 263 PetscCall(VecDot(asls->dpsi, tao->stepdirection, &innerd)); 264 } 265 266 PetscCall(VecScale(tao->stepdirection, -1.0)); 267 innerd = -innerd; 268 269 /* We now have a correct descent direction. Apply a linesearch to 270 find the new iterate. */ 271 PetscCall(TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0)); 272 PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason)); 273 PetscCall(VecNorm(asls->dpsi, NORM_2, &ndpsi)); 274 } 275 PetscFunctionReturn(0); 276 } 277 278 /* ---------------------------------------------------------- */ 279 /*MC 280 TAOASILS - Active-set infeasible linesearch algorithm for solving 281 complementarity constraints 282 283 Options Database Keys: 284 + -tao_ssls_delta - descent test fraction 285 - -tao_ssls_rho - descent test power 286 287 Level: beginner 288 M*/ 289 PETSC_EXTERN PetscErrorCode TaoCreate_ASILS(Tao tao) 290 { 291 TAO_SSLS *asls; 292 const char *armijo_type = TAOLINESEARCHARMIJO; 293 294 PetscFunctionBegin; 295 PetscCall(PetscNewLog(tao,&asls)); 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 PetscCall(TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch)); 320 PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1)); 321 PetscCall(TaoLineSearchSetType(tao->linesearch, armijo_type)); 322 PetscCall(TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix)); 323 PetscCall(TaoLineSearchSetFromOptions(tao->linesearch)); 324 325 PetscCall(KSPCreate(((PetscObject)tao)->comm, &tao->ksp)); 326 PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1)); 327 PetscCall(KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix)); 328 PetscCall(KSPSetFromOptions(tao->ksp)); 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