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 407-439, 1996. 45 Ferris, Kanzow, Munson, "Feasible Descent Algorithms for Mixed 46 Complementarity Problems," Mathematical Programming, 86, 47 pages 475-497, 1999. 48 Fischer, "A Special Newton-type Optimization Method," Optimization, 49 24, pages 269-284, 1992 50 Munson, Facchinei, Ferris, Fischer, Kanzow, "The Semismooth Algorithm 51 for Large Scale Complementarity Problems," Technical Report 99-06, 52 University of Wisconsin - Madison, 1999. 53 */ 54 55 56 #undef __FUNCT__ 57 #define __FUNCT__ "TaoSetUp_ASILS" 58 PetscErrorCode TaoSetUp_ASILS(Tao tao) 59 { 60 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 61 PetscErrorCode ierr; 62 63 PetscFunctionBegin; 64 ierr = VecDuplicate(tao->solution,&tao->gradient);CHKERRQ(ierr); 65 ierr = VecDuplicate(tao->solution,&tao->stepdirection);CHKERRQ(ierr); 66 ierr = VecDuplicate(tao->solution,&asls->ff);CHKERRQ(ierr); 67 ierr = VecDuplicate(tao->solution,&asls->dpsi);CHKERRQ(ierr); 68 ierr = VecDuplicate(tao->solution,&asls->da);CHKERRQ(ierr); 69 ierr = VecDuplicate(tao->solution,&asls->db);CHKERRQ(ierr); 70 ierr = VecDuplicate(tao->solution,&asls->t1);CHKERRQ(ierr); 71 ierr = VecDuplicate(tao->solution,&asls->t2);CHKERRQ(ierr); 72 asls->fixed = NULL; 73 asls->free = NULL; 74 asls->J_sub = NULL; 75 asls->Jpre_sub = NULL; 76 asls->w = NULL; 77 asls->r1 = NULL; 78 asls->r2 = NULL; 79 asls->r3 = NULL; 80 asls->dxfree = NULL; 81 PetscFunctionReturn(0); 82 } 83 84 #undef __FUNCT__ 85 #define __FUNCT__ "Tao_ASLS_FunctionGradient" 86 static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr) 87 { 88 Tao tao = (Tao)ptr; 89 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 90 PetscErrorCode ierr; 91 92 PetscFunctionBegin; 93 ierr = TaoComputeConstraints(tao, X, tao->constraints);CHKERRQ(ierr); 94 ierr = VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff);CHKERRQ(ierr); 95 ierr = VecNorm(asls->ff,NORM_2,&asls->merit);CHKERRQ(ierr); 96 *fcn = 0.5*asls->merit*asls->merit; 97 98 ierr = TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre);CHKERRQ(ierr); 99 ierr = MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db);CHKERRQ(ierr); 100 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->db);CHKERRQ(ierr); 101 ierr = MatMultTranspose(tao->jacobian,asls->t1,G);CHKERRQ(ierr); 102 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->da);CHKERRQ(ierr); 103 ierr = VecAXPY(G,1.0,asls->t1);CHKERRQ(ierr); 104 PetscFunctionReturn(0); 105 } 106 107 #undef __FUNCT__ 108 #define __FUNCT__ "TaoDestroy_ASILS" 109 static PetscErrorCode TaoDestroy_ASILS(Tao tao) 110 { 111 TAO_SSLS *ssls = (TAO_SSLS *)tao->data; 112 PetscErrorCode ierr; 113 114 PetscFunctionBegin; 115 ierr = VecDestroy(&ssls->ff);CHKERRQ(ierr); 116 ierr = VecDestroy(&ssls->dpsi);CHKERRQ(ierr); 117 ierr = VecDestroy(&ssls->da);CHKERRQ(ierr); 118 ierr = VecDestroy(&ssls->db);CHKERRQ(ierr); 119 ierr = VecDestroy(&ssls->w);CHKERRQ(ierr); 120 ierr = VecDestroy(&ssls->t1);CHKERRQ(ierr); 121 ierr = VecDestroy(&ssls->t2);CHKERRQ(ierr); 122 ierr = VecDestroy(&ssls->r1);CHKERRQ(ierr); 123 ierr = VecDestroy(&ssls->r2);CHKERRQ(ierr); 124 ierr = VecDestroy(&ssls->r3);CHKERRQ(ierr); 125 ierr = VecDestroy(&ssls->dxfree);CHKERRQ(ierr); 126 ierr = MatDestroy(&ssls->J_sub);CHKERRQ(ierr); 127 ierr = MatDestroy(&ssls->Jpre_sub);CHKERRQ(ierr); 128 ierr = ISDestroy(&ssls->fixed);CHKERRQ(ierr); 129 ierr = ISDestroy(&ssls->free);CHKERRQ(ierr); 130 ierr = PetscFree(tao->data);CHKERRQ(ierr); 131 PetscFunctionReturn(0); 132 } 133 134 #undef __FUNCT__ 135 #define __FUNCT__ "TaoSolve_ASILS" 136 static PetscErrorCode TaoSolve_ASILS(Tao tao) 137 { 138 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 139 PetscReal psi,ndpsi, normd, innerd, t=0; 140 PetscInt iter=0, nf; 141 PetscErrorCode ierr; 142 TaoConvergedReason reason; 143 TaoLineSearchConvergedReason ls_reason; 144 145 PetscFunctionBegin; 146 /* Assume that Setup has been called! 147 Set the structure for the Jacobian and create a linear solver. */ 148 149 ierr = TaoComputeVariableBounds(tao);CHKERRQ(ierr); 150 ierr = TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao);CHKERRQ(ierr); 151 ierr = TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao);CHKERRQ(ierr); 152 153 /* Calculate the function value and fischer function value at the 154 current iterate */ 155 ierr = TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi);CHKERRQ(ierr); 156 ierr = VecNorm(asls->dpsi,NORM_2,&ndpsi);CHKERRQ(ierr); 157 158 while (1) { 159 /* Check the termination criteria */ 160 ierr = PetscInfo3(tao,"iter %D, merit: %g, ||dpsi||: %g\n",iter, (double)asls->merit, (double)ndpsi);CHKERRQ(ierr); 161 ierr = TaoMonitor(tao, iter++, asls->merit, ndpsi, 0.0, t, &reason);CHKERRQ(ierr); 162 if (TAO_CONTINUE_ITERATING != reason) break; 163 164 /* We are going to solve a linear system of equations. We need to 165 set the tolerances for the solve so that we maintain an asymptotic 166 rate of convergence that is superlinear. 167 Note: these tolerances are for the reduced system. We really need 168 to make sure that the full system satisfies the full-space conditions. 169 170 This rule gives superlinear asymptotic convergence 171 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 172 asls->rtol = 0.0; 173 174 This rule gives quadratic asymptotic convergence 175 asls->atol = min(0.5, asls->merit*asls->merit); 176 asls->rtol = 0.0; 177 178 Calculate a free and fixed set of variables. The fixed set of 179 variables are those for the d_b is approximately equal to zero. 180 The definition of approximately changes as we approach the solution 181 to the problem. 182 183 No one rule is guaranteed to work in all cases. The following 184 definition is based on the norm of the Jacobian matrix. If the 185 norm is large, the tolerance becomes smaller. */ 186 ierr = MatNorm(tao->jacobian,NORM_1,&asls->identifier);CHKERRQ(ierr); 187 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 188 189 ierr = VecSet(asls->t1,-asls->identifier);CHKERRQ(ierr); 190 ierr = VecSet(asls->t2, asls->identifier);CHKERRQ(ierr); 191 192 ierr = ISDestroy(&asls->fixed);CHKERRQ(ierr); 193 ierr = ISDestroy(&asls->free);CHKERRQ(ierr); 194 ierr = VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);CHKERRQ(ierr); 195 ierr = ISComplementVec(asls->fixed,asls->t1, &asls->free);CHKERRQ(ierr); 196 197 ierr = ISGetSize(asls->fixed,&nf);CHKERRQ(ierr); 198 ierr = PetscInfo1(tao,"Number of fixed variables: %D\n", nf);CHKERRQ(ierr); 199 200 /* We now have our partition. Now calculate the direction in the 201 fixed variable space. */ 202 ierr = TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1); 203 ierr = TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2); 204 ierr = VecPointwiseDivide(asls->r1,asls->r1,asls->r2);CHKERRQ(ierr); 205 ierr = VecSet(tao->stepdirection,0.0);CHKERRQ(ierr); 206 ierr = VecISAXPY(tao->stepdirection, asls->fixed,1.0,asls->r1);CHKERRQ(ierr); 207 208 /* Our direction in the Fixed Variable Set is fixed. Calculate the 209 information needed for the step in the Free Variable Set. To 210 do this, we need to know the diagonal perturbation and the 211 right hand side. */ 212 213 ierr = TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);CHKERRQ(ierr); 214 ierr = TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);CHKERRQ(ierr); 215 ierr = TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);CHKERRQ(ierr); 216 ierr = VecPointwiseDivide(asls->r1,asls->r1, asls->r3);CHKERRQ(ierr); 217 ierr = VecPointwiseDivide(asls->r2,asls->r2, asls->r3);CHKERRQ(ierr); 218 219 /* r1 is the diagonal perturbation 220 r2 is the right hand side 221 r3 is no longer needed 222 223 Now need to modify r2 for our direction choice in the fixed 224 variable set: calculate t1 = J*d, take the reduced vector 225 of t1 and modify r2. */ 226 227 ierr = MatMult(tao->jacobian, tao->stepdirection, asls->t1);CHKERRQ(ierr); 228 ierr = TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);CHKERRQ(ierr); 229 ierr = VecAXPY(asls->r2, -1.0, asls->r3);CHKERRQ(ierr); 230 231 /* Calculate the reduced problem matrix and the direction */ 232 if (!asls->w && (tao->subset_type == TAO_SUBSET_MASK || tao->subset_type == TAO_SUBSET_MATRIXFREE)) { 233 ierr = VecDuplicate(tao->solution, &asls->w);CHKERRQ(ierr); 234 } 235 ierr = TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);CHKERRQ(ierr); 236 if (tao->jacobian != tao->jacobian_pre) { 237 ierr = TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);CHKERRQ(ierr); 238 } else { 239 ierr = MatDestroy(&asls->Jpre_sub);CHKERRQ(ierr); 240 asls->Jpre_sub = asls->J_sub; 241 ierr = PetscObjectReference((PetscObject)(asls->Jpre_sub));CHKERRQ(ierr); 242 } 243 ierr = MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);CHKERRQ(ierr); 244 ierr = TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);CHKERRQ(ierr); 245 ierr = VecSet(asls->dxfree, 0.0);CHKERRQ(ierr); 246 247 /* Calculate the reduced direction. (Really negative of Newton 248 direction. Therefore, rest of the code uses -d.) */ 249 ierr = KSPReset(tao->ksp); 250 ierr = KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);CHKERRQ(ierr); 251 ierr = KSPSolve(tao->ksp, asls->r2, asls->dxfree);CHKERRQ(ierr); 252 253 /* Add the direction in the free variables back into the real direction. */ 254 ierr = VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);CHKERRQ(ierr); 255 256 /* Check the real direction for descent and if not, use the negative 257 gradient direction. */ 258 ierr = VecNorm(tao->stepdirection, NORM_2, &normd);CHKERRQ(ierr); 259 ierr = VecDot(tao->stepdirection, asls->dpsi, &innerd);CHKERRQ(ierr); 260 261 if (innerd <= asls->delta*pow(normd, asls->rho)) { 262 ierr = PetscInfo1(tao,"Gradient direction: %5.4e.\n", (double)innerd);CHKERRQ(ierr); 263 ierr = PetscInfo1(tao, "Iteration %D: newton direction not descent\n", iter);CHKERRQ(ierr); 264 ierr = VecCopy(asls->dpsi, tao->stepdirection);CHKERRQ(ierr); 265 ierr = VecDot(asls->dpsi, tao->stepdirection, &innerd);CHKERRQ(ierr); 266 } 267 268 ierr = VecScale(tao->stepdirection, -1.0);CHKERRQ(ierr); 269 innerd = -innerd; 270 271 /* We now have a correct descent direction. Apply a linesearch to 272 find the new iterate. */ 273 ierr = TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);CHKERRQ(ierr); 274 ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);CHKERRQ(ierr); 275 ierr = VecNorm(asls->dpsi, NORM_2, &ndpsi);CHKERRQ(ierr); 276 } 277 PetscFunctionReturn(0); 278 } 279 280 /* ---------------------------------------------------------- */ 281 /*MC 282 TAOASILS - Active-set infeasible linesearch algorithm for solving 283 complementarity constraints 284 285 Options Database Keys: 286 + -tao_ssls_delta - descent test fraction 287 - -tao_ssls_rho - descent test power 288 289 Level: beginner 290 M*/ 291 EXTERN_C_BEGIN 292 #undef __FUNCT__ 293 #define __FUNCT__ "TaoCreate_ASILS" 294 PetscErrorCode TaoCreate_ASILS(Tao tao) 295 { 296 TAO_SSLS *asls; 297 PetscErrorCode ierr; 298 const char *armijo_type = TAOLINESEARCHARMIJO; 299 300 PetscFunctionBegin; 301 ierr = PetscNewLog(tao,&asls);CHKERRQ(ierr); 302 tao->data = (void*)asls; 303 tao->ops->solve = TaoSolve_ASILS; 304 tao->ops->setup = TaoSetUp_ASILS; 305 tao->ops->view = TaoView_SSLS; 306 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 307 tao->ops->destroy = TaoDestroy_ASILS; 308 tao->subset_type = TAO_SUBSET_SUBVEC; 309 asls->delta = 1e-10; 310 asls->rho = 2.1; 311 asls->fixed = NULL; 312 asls->free = NULL; 313 asls->J_sub = NULL; 314 asls->Jpre_sub = NULL; 315 asls->w = NULL; 316 asls->r1 = NULL; 317 asls->r2 = NULL; 318 asls->r3 = NULL; 319 asls->t1 = NULL; 320 asls->t2 = NULL; 321 asls->dxfree = NULL; 322 323 asls->identifier = 1e-5; 324 325 ierr = TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);CHKERRQ(ierr); 326 ierr = TaoLineSearchSetType(tao->linesearch, armijo_type);CHKERRQ(ierr); 327 ierr = TaoLineSearchSetFromOptions(tao->linesearch);CHKERRQ(ierr); 328 329 ierr = KSPCreate(((PetscObject)tao)->comm, &tao->ksp);CHKERRQ(ierr); 330 ierr = KSPSetFromOptions(tao->ksp);CHKERRQ(ierr); 331 tao->max_it = 2000; 332 tao->max_funcs = 4000; 333 tao->fatol = 0; 334 tao->frtol = 0; 335 tao->gttol = 0; 336 tao->grtol = 0; 337 #if defined(PETSC_USE_REAL_SINGLE) 338 tao->gatol = 1.0e-6; 339 tao->fmin = 1.0e-4; 340 #else 341 tao->gatol = 1.0e-16; 342 tao->fmin = 1.0e-8; 343 #endif 344 PetscFunctionReturn(0); 345 } 346 EXTERN_C_END 347 348