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 static PetscErrorCode TaoSetUp_ASFLS(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 PetscCall(VecDuplicate(tao->solution, &asls->w)); 69 asls->fixed = NULL; 70 asls->free = NULL; 71 asls->J_sub = NULL; 72 asls->Jpre_sub = NULL; 73 asls->r1 = NULL; 74 asls->r2 = NULL; 75 asls->r3 = NULL; 76 asls->dxfree = NULL; 77 PetscFunctionReturn(PETSC_SUCCESS); 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 PetscCall(TaoComputeJacobian(tao, tao->solution, tao->jacobian, tao->jacobian_pre)); 91 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(PETSC_SUCCESS); 98 } 99 100 static PetscErrorCode TaoDestroy_ASFLS(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(PETSC_SUCCESS); 123 } 124 125 static PetscErrorCode TaoSolve_ASFLS(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 PetscCall(TaoLineSearchSetVariableBounds(tao->linesearch, tao->XL, tao->XU)); 140 141 PetscCall(VecMedian(tao->XL, tao->solution, tao->XU, tao->solution)); 142 143 /* Calculate the function value and fischer function value at the 144 current iterate */ 145 PetscCall(TaoLineSearchComputeObjectiveAndGradient(tao->linesearch, tao->solution, &psi, asls->dpsi)); 146 PetscCall(VecNorm(asls->dpsi, NORM_2, &ndpsi)); 147 148 tao->reason = TAO_CONTINUE_ITERATING; 149 while (1) { 150 /* Check the converged criteria */ 151 PetscCall(PetscInfo(tao, "iter %" PetscInt_FMT ", merit: %g, ||dpsi||: %g\n", tao->niter, (double)asls->merit, (double)ndpsi)); 152 PetscCall(TaoLogConvergenceHistory(tao, asls->merit, ndpsi, 0.0, tao->ksp_its)); 153 PetscCall(TaoMonitor(tao, tao->niter, asls->merit, ndpsi, 0.0, t)); 154 PetscUseTypeMethod(tao, convergencetest, tao->cnvP); 155 if (TAO_CONTINUE_ITERATING != tao->reason) break; 156 157 /* Call general purpose update function */ 158 PetscTryTypeMethod(tao, update, tao->niter, tao->user_update); 159 tao->niter++; 160 161 /* We are going to solve a linear system of equations. We need to 162 set the tolerances for the solve so that we maintain an asymptotic 163 rate of convergence that is superlinear. 164 Note: these tolerances are for the reduced system. We really need 165 to make sure that the full system satisfies the full-space conditions. 166 167 This rule gives superlinear asymptotic convergence 168 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 169 asls->rtol = 0.0; 170 171 This rule gives quadratic asymptotic convergence 172 asls->atol = min(0.5, asls->merit*asls->merit); 173 asls->rtol = 0.0; 174 175 Calculate a free and fixed set of variables. The fixed set of 176 variables are those for the d_b is approximately equal to zero. 177 The definition of approximately changes as we approach the solution 178 to the problem. 179 180 No one rule is guaranteed to work in all cases. The following 181 definition is based on the norm of the Jacobian matrix. If the 182 norm is large, the tolerance becomes smaller. */ 183 PetscCall(MatNorm(tao->jacobian, NORM_1, &asls->identifier)); 184 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 185 186 PetscCall(VecSet(asls->t1, -asls->identifier)); 187 PetscCall(VecSet(asls->t2, asls->identifier)); 188 189 PetscCall(ISDestroy(&asls->fixed)); 190 PetscCall(ISDestroy(&asls->free)); 191 PetscCall(VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed)); 192 PetscCall(ISComplementVec(asls->fixed, asls->t1, &asls->free)); 193 194 PetscCall(ISGetSize(asls->fixed, &nf)); 195 PetscCall(PetscInfo(tao, "Number of fixed variables: %" PetscInt_FMT "\n", nf)); 196 197 /* We now have our partition. Now calculate the direction in the 198 fixed variable space. */ 199 PetscCall(TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1)); 200 PetscCall(TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2)); 201 PetscCall(VecPointwiseDivide(asls->r1, asls->r1, asls->r2)); 202 PetscCall(VecSet(tao->stepdirection, 0.0)); 203 PetscCall(VecISAXPY(tao->stepdirection, asls->fixed, 1.0, asls->r1)); 204 205 /* Our direction in the Fixed Variable Set is fixed. Calculate the 206 information needed for the step in the Free Variable Set. To 207 do this, we need to know the diagonal perturbation and the 208 right hand side. */ 209 210 PetscCall(TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1)); 211 PetscCall(TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2)); 212 PetscCall(TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3)); 213 PetscCall(VecPointwiseDivide(asls->r1, asls->r1, asls->r3)); 214 PetscCall(VecPointwiseDivide(asls->r2, asls->r2, asls->r3)); 215 216 /* r1 is the diagonal perturbation 217 r2 is the right hand side 218 r3 is no longer needed 219 220 Now need to modify r2 for our direction choice in the fixed 221 variable set: calculate t1 = J*d, take the reduced vector 222 of t1 and modify r2. */ 223 224 PetscCall(MatMult(tao->jacobian, tao->stepdirection, asls->t1)); 225 PetscCall(TaoVecGetSubVec(asls->t1, asls->free, tao->subset_type, 0.0, &asls->r3)); 226 PetscCall(VecAXPY(asls->r2, -1.0, asls->r3)); 227 228 /* Calculate the reduced problem matrix and the direction */ 229 PetscCall(TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type, &asls->J_sub)); 230 if (tao->jacobian != tao->jacobian_pre) { 231 PetscCall(TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub)); 232 } else { 233 PetscCall(MatDestroy(&asls->Jpre_sub)); 234 asls->Jpre_sub = asls->J_sub; 235 PetscCall(PetscObjectReference((PetscObject)(asls->Jpre_sub))); 236 } 237 PetscCall(MatDiagonalSet(asls->J_sub, asls->r1, ADD_VALUES)); 238 PetscCall(TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree)); 239 PetscCall(VecSet(asls->dxfree, 0.0)); 240 241 /* Calculate the reduced direction. (Really negative of Newton 242 direction. Therefore, rest of the code uses -d.) */ 243 PetscCall(KSPReset(tao->ksp)); 244 PetscCall(KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub)); 245 PetscCall(KSPSolve(tao->ksp, asls->r2, asls->dxfree)); 246 PetscCall(KSPGetIterationNumber(tao->ksp, &tao->ksp_its)); 247 tao->ksp_tot_its += tao->ksp_its; 248 249 /* Add the direction in the free variables back into the real direction. */ 250 PetscCall(VecISAXPY(tao->stepdirection, asls->free, 1.0, asls->dxfree)); 251 252 /* Check the projected real direction for descent and if not, use the negative 253 gradient direction. */ 254 PetscCall(VecCopy(tao->stepdirection, asls->w)); 255 PetscCall(VecScale(asls->w, -1.0)); 256 PetscCall(VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w)); 257 PetscCall(VecNorm(asls->w, NORM_2, &normd)); 258 PetscCall(VecDot(asls->w, asls->dpsi, &innerd)); 259 260 if (innerd >= -asls->delta * PetscPowReal(normd, asls->rho)) { 261 PetscCall(PetscInfo(tao, "Gradient direction: %5.4e.\n", (double)innerd)); 262 PetscCall(PetscInfo(tao, "Iteration %" PetscInt_FMT ": newton direction not descent\n", tao->niter)); 263 PetscCall(VecCopy(asls->dpsi, tao->stepdirection)); 264 PetscCall(VecDot(asls->dpsi, tao->stepdirection, &innerd)); 265 } 266 267 PetscCall(VecScale(tao->stepdirection, -1.0)); 268 innerd = -innerd; 269 270 /* We now have a correct descent direction. Apply a linesearch to 271 find the new iterate. */ 272 PetscCall(TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0)); 273 PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &psi, asls->dpsi, tao->stepdirection, &t, &ls_reason)); 274 PetscCall(VecNorm(asls->dpsi, NORM_2, &ndpsi)); 275 } 276 PetscFunctionReturn(PETSC_SUCCESS); 277 } 278 279 /* ---------------------------------------------------------- */ 280 /*MC 281 TAOASFLS - Active-set feasible linesearch algorithm for solving 282 complementarity constraints 283 284 Options Database Keys: 285 + -tao_ssls_delta - descent test fraction 286 - -tao_ssls_rho - descent test power 287 288 Level: beginner 289 M*/ 290 PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao) 291 { 292 TAO_SSLS *asls; 293 const char *armijo_type = TAOLINESEARCHARMIJO; 294 295 PetscFunctionBegin; 296 PetscCall(PetscNew(&asls)); 297 tao->data = (void *)asls; 298 tao->ops->solve = TaoSolve_ASFLS; 299 tao->ops->setup = TaoSetUp_ASFLS; 300 tao->ops->view = TaoView_SSLS; 301 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 302 tao->ops->destroy = TaoDestroy_ASFLS; 303 tao->subset_type = TAO_SUBSET_SUBVEC; 304 asls->delta = 1e-10; 305 asls->rho = 2.1; 306 asls->fixed = NULL; 307 asls->free = NULL; 308 asls->J_sub = NULL; 309 asls->Jpre_sub = NULL; 310 asls->w = NULL; 311 asls->r1 = NULL; 312 asls->r2 = NULL; 313 asls->r3 = NULL; 314 asls->t1 = NULL; 315 asls->t2 = NULL; 316 asls->dxfree = NULL; 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(PETSC_SUCCESS); 343 } 344