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(TaoSolver tao) 59 { 60 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 61 PetscErrorCode ierr; 62 63 PetscFunctionBegin; 64 65 ierr = VecDuplicate(tao->solution,&tao->gradient); CHKERRQ(ierr); 66 ierr = VecDuplicate(tao->solution,&tao->stepdirection); CHKERRQ(ierr); 67 ierr = VecDuplicate(tao->solution,&asls->ff); CHKERRQ(ierr); 68 ierr = VecDuplicate(tao->solution,&asls->dpsi); CHKERRQ(ierr); 69 ierr = VecDuplicate(tao->solution,&asls->da); CHKERRQ(ierr); 70 ierr = VecDuplicate(tao->solution,&asls->db); CHKERRQ(ierr); 71 ierr = VecDuplicate(tao->solution,&asls->t1); CHKERRQ(ierr); 72 ierr = VecDuplicate(tao->solution,&asls->t2); CHKERRQ(ierr); 73 asls->fixed = PETSC_NULL; 74 asls->free = PETSC_NULL; 75 asls->J_sub = PETSC_NULL; 76 asls->Jpre_sub = PETSC_NULL; 77 asls->w = PETSC_NULL; 78 asls->r1 = PETSC_NULL; 79 asls->r2 = PETSC_NULL; 80 asls->r3 = PETSC_NULL; 81 asls->dxfree = PETSC_NULL; 82 83 PetscFunctionReturn(0); 84 } 85 86 #undef __FUNCT__ 87 #define __FUNCT__ "Tao_ASLS_FunctionGradient" 88 static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr) 89 { 90 TaoSolver tao = (TaoSolver)ptr; 91 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 92 PetscErrorCode ierr; 93 94 PetscFunctionBegin; 95 96 ierr = TaoComputeConstraints(tao, X, tao->constraints); CHKERRQ(ierr); 97 ierr = VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff); CHKERRQ(ierr); 98 ierr = VecNorm(asls->ff,NORM_2,&asls->merit); CHKERRQ(ierr); 99 *fcn = 0.5*asls->merit*asls->merit; 100 101 ierr = TaoComputeJacobian(tao, tao->solution, &tao->jacobian, &tao->jacobian_pre, &asls->matflag); CHKERRQ(ierr); 102 103 ierr = D_Fischer(tao->jacobian, tao->solution, tao->constraints, 104 tao->XL, tao->XU, asls->t1, asls->t2, 105 asls->da, asls->db); CHKERRQ(ierr); 106 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->db); CHKERRQ(ierr); 107 ierr = MatMultTranspose(tao->jacobian,asls->t1,G); CHKERRQ(ierr); 108 ierr = VecPointwiseMult(asls->t1, asls->ff, asls->da); CHKERRQ(ierr); 109 ierr = VecAXPY(G,1.0,asls->t1); CHKERRQ(ierr); 110 PetscFunctionReturn(0); 111 } 112 113 #undef __FUNCT__ 114 #define __FUNCT__ "TaoDestroy_ASILS" 115 static PetscErrorCode TaoDestroy_ASILS(TaoSolver tao) 116 { 117 TAO_SSLS *ssls = (TAO_SSLS *)tao->data; 118 PetscErrorCode ierr; 119 120 PetscFunctionBegin; 121 122 ierr = VecDestroy(&ssls->ff); CHKERRQ(ierr); 123 ierr = VecDestroy(&ssls->dpsi); CHKERRQ(ierr); 124 ierr = VecDestroy(&ssls->da); CHKERRQ(ierr); 125 ierr = VecDestroy(&ssls->db); CHKERRQ(ierr); 126 ierr = VecDestroy(&ssls->w); CHKERRQ(ierr); 127 ierr = VecDestroy(&ssls->t1); CHKERRQ(ierr); 128 ierr = VecDestroy(&ssls->t2); CHKERRQ(ierr); 129 ierr = VecDestroy(&ssls->r1); CHKERRQ(ierr); 130 ierr = VecDestroy(&ssls->r2); CHKERRQ(ierr); 131 ierr = VecDestroy(&ssls->r3); CHKERRQ(ierr); 132 ierr = VecDestroy(&ssls->dxfree); CHKERRQ(ierr); 133 ierr = MatDestroy(&ssls->J_sub); CHKERRQ(ierr); 134 ierr = MatDestroy(&ssls->Jpre_sub); CHKERRQ(ierr); 135 ierr = ISDestroy(&ssls->fixed); CHKERRQ(ierr); 136 ierr = ISDestroy(&ssls->free); CHKERRQ(ierr); 137 ierr = PetscFree(tao->data); CHKERRQ(ierr); 138 139 tao->data = PETSC_NULL; 140 PetscFunctionReturn(0); 141 142 } 143 #undef __FUNCT__ 144 #define __FUNCT__ "TaoSolve_ASILS" 145 static PetscErrorCode TaoSolve_ASILS(TaoSolver tao) 146 { 147 TAO_SSLS *asls = (TAO_SSLS *)tao->data; 148 PetscReal psi,ndpsi, normd, innerd, t=0; 149 PetscInt iter=0, nf; 150 PetscErrorCode ierr; 151 TaoSolverTerminationReason reason; 152 TaoLineSearchTerminationReason ls_reason; 153 154 PetscFunctionBegin; 155 156 /* Assume that Setup has been called! 157 Set the structure for the Jacobian and create a linear solver. */ 158 159 ierr = TaoComputeVariableBounds(tao); CHKERRQ(ierr); 160 ierr = TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao); CHKERRQ(ierr); 161 ierr = TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao); CHKERRQ(ierr); 162 163 164 /* Calculate the function value and fischer function value at the 165 current iterate */ 166 ierr = TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi); CHKERRQ(ierr); 167 ierr = VecNorm(asls->dpsi,NORM_2,&ndpsi); CHKERRQ(ierr); 168 169 while (1) { 170 171 /* Check the termination criteria */ 172 ierr = PetscInfo3(tao,"iter %D, merit: %G, ||dpsi||: %G\n",iter, asls->merit, ndpsi); CHKERRQ(ierr); 173 ierr = TaoMonitor(tao, iter++, asls->merit, ndpsi, 0.0, t, &reason); CHKERRQ(ierr); 174 if (TAO_CONTINUE_ITERATING != reason) break; 175 176 /* We are going to solve a linear system of equations. We need to 177 set the tolerances for the solve so that we maintain an asymptotic 178 rate of convergence that is superlinear. 179 Note: these tolerances are for the reduced system. We really need 180 to make sure that the full system satisfies the full-space conditions. 181 182 This rule gives superlinear asymptotic convergence 183 asls->atol = min(0.5, asls->merit*sqrt(asls->merit)); 184 asls->rtol = 0.0; 185 186 This rule gives quadratic asymptotic convergence 187 asls->atol = min(0.5, asls->merit*asls->merit); 188 asls->rtol = 0.0; 189 190 Calculate a free and fixed set of variables. The fixed set of 191 variables are those for the d_b is approximately equal to zero. 192 The definition of approximately changes as we approach the solution 193 to the problem. 194 195 No one rule is guaranteed to work in all cases. The following 196 definition is based on the norm of the Jacobian matrix. If the 197 norm is large, the tolerance becomes smaller. */ 198 ierr = MatNorm(tao->jacobian,NORM_1,&asls->identifier); CHKERRQ(ierr); 199 asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier); 200 201 ierr = VecSet(asls->t1,-asls->identifier); CHKERRQ(ierr); 202 ierr = VecSet(asls->t2, asls->identifier); CHKERRQ(ierr); 203 204 ierr = ISDestroy(&asls->fixed); CHKERRQ(ierr); 205 ierr = ISDestroy(&asls->free); CHKERRQ(ierr); 206 ierr = VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed); CHKERRQ(ierr); 207 ierr = ISCreateComplement(asls->fixed,asls->t1, &asls->free); CHKERRQ(ierr); 208 209 ierr = ISGetSize(asls->fixed,&nf); CHKERRQ(ierr); 210 ierr = PetscInfo1(tao,"Number of fixed variables: %d\n", nf); CHKERRQ(ierr); 211 212 /* We now have our partition. Now calculate the direction in the 213 fixed variable space. */ 214 ierr = VecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1); 215 ierr = VecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2); 216 ierr = VecPointwiseDivide(asls->r1,asls->r1,asls->r2); CHKERRQ(ierr); 217 ierr = VecSet(tao->stepdirection,0.0); CHKERRQ(ierr); 218 ierr = VecReducedXPY(tao->stepdirection,asls->r1, asls->fixed); CHKERRQ(ierr); 219 220 221 /* Our direction in the Fixed Variable Set is fixed. Calculate the 222 information needed for the step in the Free Variable Set. To 223 do this, we need to know the diagonal perturbation and the 224 right hand side. */ 225 226 ierr = VecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1); CHKERRQ(ierr); 227 ierr = VecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2); CHKERRQ(ierr); 228 ierr = VecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3); CHKERRQ(ierr); 229 ierr = VecPointwiseDivide(asls->r1,asls->r1, asls->r3); CHKERRQ(ierr); 230 ierr = VecPointwiseDivide(asls->r2,asls->r2, asls->r3); CHKERRQ(ierr); 231 232 /* r1 is the diagonal perturbation 233 r2 is the right hand side 234 r3 is no longer needed 235 236 Now need to modify r2 for our direction choice in the fixed 237 variable set: calculate t1 = J*d, take the reduced vector 238 of t1 and modify r2. */ 239 240 ierr = MatMult(tao->jacobian, tao->stepdirection, asls->t1); CHKERRQ(ierr); 241 ierr = VecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3); CHKERRQ(ierr); 242 ierr = VecAXPY(asls->r2, -1.0, asls->r3); CHKERRQ(ierr); 243 244 /* Calculate the reduced problem matrix and the direction */ 245 if (!asls->w && (tao->subset_type == TAO_SUBSET_MASK 246 || tao->subset_type == TAO_SUBSET_MATRIXFREE)) { 247 ierr = VecDuplicate(tao->solution, &asls->w); CHKERRQ(ierr); 248 } 249 ierr = MatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub); CHKERRQ(ierr); 250 if (tao->jacobian != tao->jacobian_pre) { 251 ierr = MatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub); CHKERRQ(ierr); 252 } else { 253 ierr = MatDestroy(&asls->Jpre_sub); CHKERRQ(ierr); 254 asls->Jpre_sub = asls->J_sub; 255 ierr = PetscObjectReference((PetscObject)(asls->Jpre_sub)); CHKERRQ(ierr); 256 } 257 ierr = MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES); CHKERRQ(ierr); 258 ierr = VecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree); CHKERRQ(ierr); 259 ierr = VecSet(asls->dxfree, 0.0); CHKERRQ(ierr); 260 261 /* Calculate the reduced direction. (Really negative of Newton 262 direction. Therefore, rest of the code uses -d.) */ 263 ierr = KSPReset(tao->ksp); 264 ierr = KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub, asls->matflag); CHKERRQ(ierr); 265 ierr = KSPSolve(tao->ksp, asls->r2, asls->dxfree); CHKERRQ(ierr); 266 267 /* Add the direction in the free variables back into the real direction. */ 268 ierr = VecReducedXPY(tao->stepdirection, asls->dxfree, asls->free); CHKERRQ(ierr); 269 270 271 /* Check the real direction for descent and if not, use the negative 272 gradient direction. */ 273 ierr = VecNorm(tao->stepdirection, NORM_2, &normd); CHKERRQ(ierr); 274 ierr = VecDot(tao->stepdirection, asls->dpsi, &innerd); CHKERRQ(ierr); 275 276 if (innerd <= asls->delta*pow(normd, asls->rho)) { 277 ierr = PetscInfo1(tao,"Gradient direction: %5.4e.\n", innerd); CHKERRQ(ierr); 278 ierr = PetscInfo1(tao, "Iteration %d: newton direction not descent\n", iter); CHKERRQ(ierr); 279 ierr = VecCopy(asls->dpsi, tao->stepdirection); CHKERRQ(ierr); 280 ierr = VecDot(asls->dpsi, tao->stepdirection, &innerd); CHKERRQ(ierr); 281 } 282 283 ierr = VecScale(tao->stepdirection, -1.0); CHKERRQ(ierr); 284 innerd = -innerd; 285 286 /* We now have a correct descent direction. Apply a linesearch to 287 find the new iterate. */ 288 ierr = TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0); CHKERRQ(ierr); 289 ierr = TaoLineSearchApply(tao->linesearch, tao->solution, &psi, 290 asls->dpsi, tao->stepdirection, &t, &ls_reason); CHKERRQ(ierr); 291 ierr = VecNorm(asls->dpsi, NORM_2, &ndpsi); CHKERRQ(ierr); 292 } 293 294 PetscFunctionReturn(0); 295 } 296 297 /* ---------------------------------------------------------- */ 298 EXTERN_C_BEGIN 299 #undef __FUNCT__ 300 #define __FUNCT__ "TaoCreate_ASILS" 301 PetscErrorCode TaoCreate_ASILS(TaoSolver tao) 302 { 303 TAO_SSLS *asls; 304 PetscErrorCode ierr; 305 const char *armijo_type = TAOLINESEARCH_ARMIJO; 306 307 PetscFunctionBegin; 308 ierr = PetscNewLog(tao,TAO_SSLS,&asls); CHKERRQ(ierr); 309 tao->data = (void*)asls; 310 tao->ops->solve = TaoSolve_ASILS; 311 tao->ops->setup = TaoSetUp_ASILS; 312 tao->ops->view = TaoView_SSLS; 313 tao->ops->setfromoptions = TaoSetFromOptions_SSLS; 314 tao->ops->destroy = TaoDestroy_ASILS; 315 tao->subset_type = TAO_SUBSET_SUBVEC; 316 asls->delta = 1e-10; 317 asls->rho = 2.1; 318 asls->fixed = PETSC_NULL; 319 asls->free = PETSC_NULL; 320 asls->J_sub = PETSC_NULL; 321 asls->Jpre_sub = PETSC_NULL; 322 asls->w = PETSC_NULL; 323 asls->r1 = PETSC_NULL; 324 asls->r2 = PETSC_NULL; 325 asls->r3 = PETSC_NULL; 326 asls->t1 = PETSC_NULL; 327 asls->t2 = PETSC_NULL; 328 asls->dxfree = PETSC_NULL; 329 330 asls->identifier = 1e-5; 331 332 ierr = TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch); CHKERRQ(ierr); 333 ierr = TaoLineSearchSetType(tao->linesearch, armijo_type); CHKERRQ(ierr); 334 ierr = TaoLineSearchSetFromOptions(tao->linesearch); CHKERRQ(ierr); 335 336 ierr = KSPCreate(((PetscObject)tao)->comm, &tao->ksp); CHKERRQ(ierr); 337 ierr = KSPSetFromOptions(tao->ksp); CHKERRQ(ierr); 338 tao->max_it = 2000; 339 tao->max_funcs = 4000; 340 tao->fatol = 0; 341 tao->frtol = 0; 342 tao->gttol = 0; 343 tao->grtol = 0; 344 tao->gatol = 1.0e-16; 345 tao->fmin = 1.0e-8; 346 347 PetscFunctionReturn(0); 348 } 349 EXTERN_C_END 350 351