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 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 57 58 PetscFunctionBegin; 59 PetscCall(VecDuplicate(tao->solution, &tao->gradient)); 60 PetscCall(VecDuplicate(tao->solution, &tao->stepdirection)); 61 PetscCall(VecDuplicate(tao->solution, &asls->ff)); 62 PetscCall(VecDuplicate(tao->solution, &asls->dpsi)); 63 PetscCall(VecDuplicate(tao->solution, &asls->da)); 64 PetscCall(VecDuplicate(tao->solution, &asls->db)); 65 PetscCall(VecDuplicate(tao->solution, &asls->t1)); 66 PetscCall(VecDuplicate(tao->solution, &asls->t2)); 67 PetscCall(VecDuplicate(tao->solution, &asls->w)); 68 asls->fixed = NULL; 69 asls->free = NULL; 70 asls->J_sub = NULL; 71 asls->Jpre_sub = NULL; 72 asls->r1 = NULL; 73 asls->r2 = NULL; 74 asls->r3 = NULL; 75 asls->dxfree = NULL; 76 PetscFunctionReturn(0); 77 } 78 79 static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr) { 80 Tao tao = (Tao)ptr; 81 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 82 83 PetscFunctionBegin; 84 PetscCall(TaoComputeConstraints(tao, X, tao->constraints)); 85 PetscCall(VecFischer(X, tao->constraints, tao->XL, tao->XU, asls->ff)); 86 PetscCall(VecNorm(asls->ff, NORM_2, &asls->merit)); 87 *fcn = 0.5 * asls->merit * asls->merit; 88 PetscCall(TaoComputeJacobian(tao, tao->solution, tao->jacobian, tao->jacobian_pre)); 89 90 PetscCall(MatDFischer(tao->jacobian, tao->solution, tao->constraints, tao->XL, tao->XU, asls->t1, asls->t2, asls->da, asls->db)); 91 PetscCall(VecPointwiseMult(asls->t1, asls->ff, asls->db)); 92 PetscCall(MatMultTranspose(tao->jacobian, asls->t1, G)); 93 PetscCall(VecPointwiseMult(asls->t1, asls->ff, asls->da)); 94 PetscCall(VecAXPY(G, 1.0, asls->t1)); 95 PetscFunctionReturn(0); 96 } 97 98 static PetscErrorCode TaoDestroy_ASFLS(Tao tao) { 99 TAO_SSLS *ssls = (TAO_SSLS *)tao->data; 100 101 PetscFunctionBegin; 102 PetscCall(VecDestroy(&ssls->ff)); 103 PetscCall(VecDestroy(&ssls->dpsi)); 104 PetscCall(VecDestroy(&ssls->da)); 105 PetscCall(VecDestroy(&ssls->db)); 106 PetscCall(VecDestroy(&ssls->w)); 107 PetscCall(VecDestroy(&ssls->t1)); 108 PetscCall(VecDestroy(&ssls->t2)); 109 PetscCall(VecDestroy(&ssls->r1)); 110 PetscCall(VecDestroy(&ssls->r2)); 111 PetscCall(VecDestroy(&ssls->r3)); 112 PetscCall(VecDestroy(&ssls->dxfree)); 113 PetscCall(MatDestroy(&ssls->J_sub)); 114 PetscCall(MatDestroy(&ssls->Jpre_sub)); 115 PetscCall(ISDestroy(&ssls->fixed)); 116 PetscCall(ISDestroy(&ssls->free)); 117 PetscCall(KSPDestroy(&tao->ksp)); 118 PetscCall(PetscFree(tao->data)); 119 PetscFunctionReturn(0); 120 } 121 122 static PetscErrorCode TaoSolve_ASFLS(Tao tao) { 123 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 124 PetscReal psi, ndpsi, normd, innerd, t = 0; 125 PetscInt nf; 126 TaoLineSearchConvergedReason ls_reason; 127 128 PetscFunctionBegin; 129 /* Assume that Setup has been called! 130 Set the structure for the Jacobian and create a linear solver. */ 131 132 PetscCall(TaoComputeVariableBounds(tao)); 133 PetscCall(TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch, Tao_ASLS_FunctionGradient, tao)); 134 PetscCall(TaoLineSearchSetObjectiveRoutine(tao->linesearch, Tao_SSLS_Function, tao)); 135 PetscCall(TaoLineSearchSetVariableBounds(tao->linesearch, tao->XL, tao->XU)); 136 137 PetscCall(VecMedian(tao->XL, tao->solution, tao->XU, tao->solution)); 138 139 /* Calculate the function value and fischer function value at the 140 current iterate */ 141 PetscCall(TaoLineSearchComputeObjectiveAndGradient(tao->linesearch, tao->solution, &psi, asls->dpsi)); 142 PetscCall(VecNorm(asls->dpsi, NORM_2, &ndpsi)); 143 144 tao->reason = TAO_CONTINUE_ITERATING; 145 while (1) { 146 /* Check the converged criteria */ 147 PetscCall(PetscInfo(tao, "iter %" PetscInt_FMT ", merit: %g, ||dpsi||: %g\n", tao->niter, (double)asls->merit, (double)ndpsi)); 148 PetscCall(TaoLogConvergenceHistory(tao, asls->merit, ndpsi, 0.0, tao->ksp_its)); 149 PetscCall(TaoMonitor(tao, tao->niter, asls->merit, ndpsi, 0.0, t)); 150 PetscUseTypeMethod(tao, convergencetest, tao->cnvP); 151 if (TAO_CONTINUE_ITERATING != tao->reason) break; 152 153 /* Call general purpose update function */ 154 PetscTryTypeMethod(tao, update, tao->niter, tao->user_update); 155 tao->niter++; 156 157 /* We are going to solve a linear system of equations. We need to 158 set the tolerances for the solve so that we maintain an asymptotic 159 rate of convergence that is superlinear. 160 Note: these tolerances are for the reduced system. We really need 161 to make sure that the full system satisfies the full-space conditions. 162 163 This rule gives superlinear asymptotic convergence 164 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 165 asls->rtol = 0.0; 166 167 This rule gives quadratic asymptotic convergence 168 asls->atol = min(0.5, asls->merit*asls->merit); 169 asls->rtol = 0.0; 170 171 Calculate a free and fixed set of variables. The fixed set of 172 variables are those for the d_b is approximately equal to zero. 173 The definition of approximately changes as we approach the solution 174 to the problem. 175 176 No one rule is guaranteed to work in all cases. The following 177 definition is based on the norm of the Jacobian matrix. If the 178 norm is large, the tolerance becomes smaller. */ 179 PetscCall(MatNorm(tao->jacobian, NORM_1, &asls->identifier)); 180 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 181 182 PetscCall(VecSet(asls->t1, -asls->identifier)); 183 PetscCall(VecSet(asls->t2, asls->identifier)); 184 185 PetscCall(ISDestroy(&asls->fixed)); 186 PetscCall(ISDestroy(&asls->free)); 187 PetscCall(VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed)); 188 PetscCall(ISComplementVec(asls->fixed, asls->t1, &asls->free)); 189 190 PetscCall(ISGetSize(asls->fixed, &nf)); 191 PetscCall(PetscInfo(tao, "Number of fixed variables: %" PetscInt_FMT "\n", nf)); 192 193 /* We now have our partition. Now calculate the direction in the 194 fixed variable space. */ 195 PetscCall(TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1)); 196 PetscCall(TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2)); 197 PetscCall(VecPointwiseDivide(asls->r1, asls->r1, asls->r2)); 198 PetscCall(VecSet(tao->stepdirection, 0.0)); 199 PetscCall(VecISAXPY(tao->stepdirection, asls->fixed, 1.0, asls->r1)); 200 201 /* Our direction in the Fixed Variable Set is fixed. Calculate the 202 information needed for the step in the Free Variable Set. To 203 do this, we need to know the diagonal perturbation and the 204 right hand side. */ 205 206 PetscCall(TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1)); 207 PetscCall(TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2)); 208 PetscCall(TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3)); 209 PetscCall(VecPointwiseDivide(asls->r1, asls->r1, asls->r3)); 210 PetscCall(VecPointwiseDivide(asls->r2, asls->r2, asls->r3)); 211 212 /* r1 is the diagonal perturbation 213 r2 is the right hand side 214 r3 is no longer needed 215 216 Now need to modify r2 for our direction choice in the fixed 217 variable set: calculate t1 = J*d, take the reduced vector 218 of t1 and modify r2. */ 219 220 PetscCall(MatMult(tao->jacobian, tao->stepdirection, asls->t1)); 221 PetscCall(TaoVecGetSubVec(asls->t1, asls->free, tao->subset_type, 0.0, &asls->r3)); 222 PetscCall(VecAXPY(asls->r2, -1.0, asls->r3)); 223 224 /* Calculate the reduced problem matrix and the direction */ 225 PetscCall(TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type, &asls->J_sub)); 226 if (tao->jacobian != tao->jacobian_pre) { 227 PetscCall(TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub)); 228 } else { 229 PetscCall(MatDestroy(&asls->Jpre_sub)); 230 asls->Jpre_sub = asls->J_sub; 231 PetscCall(PetscObjectReference((PetscObject)(asls->Jpre_sub))); 232 } 233 PetscCall(MatDiagonalSet(asls->J_sub, asls->r1, ADD_VALUES)); 234 PetscCall(TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree)); 235 PetscCall(VecSet(asls->dxfree, 0.0)); 236 237 /* Calculate the reduced direction. (Really negative of Newton 238 direction. Therefore, rest of the code uses -d.) */ 239 PetscCall(KSPReset(tao->ksp)); 240 PetscCall(KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub)); 241 PetscCall(KSPSolve(tao->ksp, asls->r2, asls->dxfree)); 242 PetscCall(KSPGetIterationNumber(tao->ksp, &tao->ksp_its)); 243 tao->ksp_tot_its += tao->ksp_its; 244 245 /* Add the direction in the free variables back into the real direction. */ 246 PetscCall(VecISAXPY(tao->stepdirection, asls->free, 1.0, asls->dxfree)); 247 248 /* Check the projected real direction for descent and if not, use the negative 249 gradient direction. */ 250 PetscCall(VecCopy(tao->stepdirection, asls->w)); 251 PetscCall(VecScale(asls->w, -1.0)); 252 PetscCall(VecBoundGradientProjection(asls->w, tao->solution, tao->XL, tao->XU, asls->w)); 253 PetscCall(VecNorm(asls->w, NORM_2, &normd)); 254 PetscCall(VecDot(asls->w, asls->dpsi, &innerd)); 255 256 if (innerd >= -asls->delta * PetscPowReal(normd, asls->rho)) { 257 PetscCall(PetscInfo(tao, "Gradient direction: %5.4e.\n", (double)innerd)); 258 PetscCall(PetscInfo(tao, "Iteration %" PetscInt_FMT ": newton direction not descent\n", tao->niter)); 259 PetscCall(VecCopy(asls->dpsi, tao->stepdirection)); 260 PetscCall(VecDot(asls->dpsi, tao->stepdirection, &innerd)); 261 } 262 263 PetscCall(VecScale(tao->stepdirection, -1.0)); 264 innerd = -innerd; 265 266 /* We now have a correct descent direction. Apply a linesearch to 267 find the new iterate. */ 268 PetscCall(TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0)); 269 PetscCall(TaoLineSearchApply(tao->linesearch, tao->solution, &psi, asls->dpsi, tao->stepdirection, &t, &ls_reason)); 270 PetscCall(VecNorm(asls->dpsi, NORM_2, &ndpsi)); 271 } 272 PetscFunctionReturn(0); 273 } 274 275 /* ---------------------------------------------------------- */ 276 /*MC 277 TAOASFLS - Active-set feasible linesearch algorithm for solving 278 complementarity constraints 279 280 Options Database Keys: 281 + -tao_ssls_delta - descent test fraction 282 - -tao_ssls_rho - descent test power 283 284 Level: beginner 285 M*/ 286 PETSC_EXTERN PetscErrorCode TaoCreate_ASFLS(Tao tao) { 287 TAO_SSLS *asls; 288 const char *armijo_type = TAOLINESEARCHARMIJO; 289 290 PetscFunctionBegin; 291 PetscCall(PetscNewLog(tao, &asls)); 292 tao->data = (void *)asls; 293 tao->ops->solve = TaoSolve_ASFLS; 294 tao->ops->setup = TaoSetUp_ASFLS; 295 tao->ops->view = TaoView_SSLS; 296 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 297 tao->ops->destroy = TaoDestroy_ASFLS; 298 tao->subset_type = TAO_SUBSET_SUBVEC; 299 asls->delta = 1e-10; 300 asls->rho = 2.1; 301 asls->fixed = NULL; 302 asls->free = NULL; 303 asls->J_sub = NULL; 304 asls->Jpre_sub = NULL; 305 asls->w = NULL; 306 asls->r1 = NULL; 307 asls->r2 = NULL; 308 asls->r3 = NULL; 309 asls->t1 = NULL; 310 asls->t2 = NULL; 311 asls->dxfree = NULL; 312 asls->identifier = 1e-5; 313 314 PetscCall(TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch)); 315 PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1)); 316 PetscCall(TaoLineSearchSetType(tao->linesearch, armijo_type)); 317 PetscCall(TaoLineSearchSetOptionsPrefix(tao->linesearch, tao->hdr.prefix)); 318 PetscCall(TaoLineSearchSetFromOptions(tao->linesearch)); 319 320 PetscCall(KSPCreate(((PetscObject)tao)->comm, &tao->ksp)); 321 PetscCall(PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1)); 322 PetscCall(KSPSetOptionsPrefix(tao->ksp, tao->hdr.prefix)); 323 PetscCall(KSPSetFromOptions(tao->ksp)); 324 325 /* Override default settings (unless already changed) */ 326 if (!tao->max_it_changed) tao->max_it = 2000; 327 if (!tao->max_funcs_changed) tao->max_funcs = 4000; 328 if (!tao->gttol_changed) tao->gttol = 0; 329 if (!tao->grtol_changed) tao->grtol = 0; 330 #if defined(PETSC_USE_REAL_SINGLE) 331 if (!tao->gatol_changed) tao->gatol = 1.0e-6; 332 if (!tao->fmin_changed) tao->fmin = 1.0e-4; 333 #else 334 if (!tao->gatol_changed) tao->gatol = 1.0e-16; 335 if (!tao->fmin_changed) tao->fmin = 1.0e-8; 336 #endif 337 PetscFunctionReturn(0); 338 } 339