Actual source code: asils.c
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
14: Many other reformulations are possible and combinations of
15: feasible/infeasible and linesearch/trust region are possible.
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.
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.
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.
36: The theory for the feasible version follows from the feasible descent
37: algorithm framework.
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: */
55: static PetscErrorCode TaoSetUp_ASILS(Tao tao)
56: {
57: TAO_SSLS *asls = (TAO_SSLS *)tao->data;
59: VecDuplicate(tao->solution,&tao->gradient);
60: VecDuplicate(tao->solution,&tao->stepdirection);
61: VecDuplicate(tao->solution,&asls->ff);
62: VecDuplicate(tao->solution,&asls->dpsi);
63: VecDuplicate(tao->solution,&asls->da);
64: VecDuplicate(tao->solution,&asls->db);
65: VecDuplicate(tao->solution,&asls->t1);
66: VecDuplicate(tao->solution,&asls->t2);
67: asls->fixed = NULL;
68: asls->free = NULL;
69: asls->J_sub = NULL;
70: asls->Jpre_sub = NULL;
71: asls->w = NULL;
72: asls->r1 = NULL;
73: asls->r2 = NULL;
74: asls->r3 = NULL;
75: asls->dxfree = NULL;
76: return 0;
77: }
79: static PetscErrorCode Tao_ASLS_FunctionGradient(TaoLineSearch ls, Vec X, PetscReal *fcn, Vec G, void *ptr)
80: {
81: Tao tao = (Tao)ptr;
82: TAO_SSLS *asls = (TAO_SSLS *)tao->data;
84: TaoComputeConstraints(tao, X, tao->constraints);
85: VecFischer(X,tao->constraints,tao->XL,tao->XU,asls->ff);
86: VecNorm(asls->ff,NORM_2,&asls->merit);
87: *fcn = 0.5*asls->merit*asls->merit;
89: TaoComputeJacobian(tao,tao->solution,tao->jacobian,tao->jacobian_pre);
90: MatDFischer(tao->jacobian, tao->solution, tao->constraints,tao->XL, tao->XU, asls->t1, asls->t2,asls->da, asls->db);
91: VecPointwiseMult(asls->t1, asls->ff, asls->db);
92: MatMultTranspose(tao->jacobian,asls->t1,G);
93: VecPointwiseMult(asls->t1, asls->ff, asls->da);
94: VecAXPY(G,1.0,asls->t1);
95: return 0;
96: }
98: static PetscErrorCode TaoDestroy_ASILS(Tao tao)
99: {
100: TAO_SSLS *ssls = (TAO_SSLS *)tao->data;
102: VecDestroy(&ssls->ff);
103: VecDestroy(&ssls->dpsi);
104: VecDestroy(&ssls->da);
105: VecDestroy(&ssls->db);
106: VecDestroy(&ssls->w);
107: VecDestroy(&ssls->t1);
108: VecDestroy(&ssls->t2);
109: VecDestroy(&ssls->r1);
110: VecDestroy(&ssls->r2);
111: VecDestroy(&ssls->r3);
112: VecDestroy(&ssls->dxfree);
113: MatDestroy(&ssls->J_sub);
114: MatDestroy(&ssls->Jpre_sub);
115: ISDestroy(&ssls->fixed);
116: ISDestroy(&ssls->free);
117: PetscFree(tao->data);
118: return 0;
119: }
121: static PetscErrorCode TaoSolve_ASILS(Tao tao)
122: {
123: TAO_SSLS *asls = (TAO_SSLS *)tao->data;
124: PetscReal psi,ndpsi, normd, innerd, t=0;
125: PetscInt nf;
126: TaoLineSearchConvergedReason ls_reason;
128: /* Assume that Setup has been called!
129: Set the structure for the Jacobian and create a linear solver. */
131: TaoComputeVariableBounds(tao);
132: TaoLineSearchSetObjectiveAndGradientRoutine(tao->linesearch,Tao_ASLS_FunctionGradient,tao);
133: TaoLineSearchSetObjectiveRoutine(tao->linesearch,Tao_SSLS_Function,tao);
135: /* Calculate the function value and fischer function value at the
136: current iterate */
137: TaoLineSearchComputeObjectiveAndGradient(tao->linesearch,tao->solution,&psi,asls->dpsi);
138: VecNorm(asls->dpsi,NORM_2,&ndpsi);
140: tao->reason = TAO_CONTINUE_ITERATING;
141: while (1) {
142: /* Check the termination criteria */
143: PetscInfo(tao,"iter %D, merit: %g, ||dpsi||: %g\n",tao->niter, (double)asls->merit, (double)ndpsi);
144: TaoLogConvergenceHistory(tao,asls->merit,ndpsi,0.0,tao->ksp_its);
145: TaoMonitor(tao,tao->niter,asls->merit,ndpsi,0.0,t);
146: (*tao->ops->convergencetest)(tao,tao->cnvP);
147: if (TAO_CONTINUE_ITERATING != tao->reason) break;
149: /* Call general purpose update function */
150: if (tao->ops->update) {
151: (*tao->ops->update)(tao, tao->niter, tao->user_update);
152: }
153: tao->niter++;
155: /* We are going to solve a linear system of equations. We need to
156: set the tolerances for the solve so that we maintain an asymptotic
157: rate of convergence that is superlinear.
158: Note: these tolerances are for the reduced system. We really need
159: to make sure that the full system satisfies the full-space conditions.
161: This rule gives superlinear asymptotic convergence
162: asls->atol = min(0.5, asls->merit*sqrt(asls->merit));
163: asls->rtol = 0.0;
165: This rule gives quadratic asymptotic convergence
166: asls->atol = min(0.5, asls->merit*asls->merit);
167: asls->rtol = 0.0;
169: Calculate a free and fixed set of variables. The fixed set of
170: variables are those for the d_b is approximately equal to zero.
171: The definition of approximately changes as we approach the solution
172: to the problem.
174: No one rule is guaranteed to work in all cases. The following
175: definition is based on the norm of the Jacobian matrix. If the
176: norm is large, the tolerance becomes smaller. */
177: MatNorm(tao->jacobian,NORM_1,&asls->identifier);
178: asls->identifier = PetscMin(asls->merit, 1e-2) / (1 + asls->identifier);
180: VecSet(asls->t1,-asls->identifier);
181: VecSet(asls->t2, asls->identifier);
183: ISDestroy(&asls->fixed);
184: ISDestroy(&asls->free);
185: VecWhichBetweenOrEqual(asls->t1, asls->db, asls->t2, &asls->fixed);
186: ISComplementVec(asls->fixed,asls->t1, &asls->free);
188: ISGetSize(asls->fixed,&nf);
189: PetscInfo(tao,"Number of fixed variables: %D\n", nf);
191: /* We now have our partition. Now calculate the direction in the
192: fixed variable space. */
193: TaoVecGetSubVec(asls->ff, asls->fixed, tao->subset_type, 0.0, &asls->r1);
194: TaoVecGetSubVec(asls->da, asls->fixed, tao->subset_type, 1.0, &asls->r2);
195: VecPointwiseDivide(asls->r1,asls->r1,asls->r2);
196: VecSet(tao->stepdirection,0.0);
197: VecISAXPY(tao->stepdirection, asls->fixed,1.0,asls->r1);
199: /* Our direction in the Fixed Variable Set is fixed. Calculate the
200: information needed for the step in the Free Variable Set. To
201: do this, we need to know the diagonal perturbation and the
202: right hand side. */
204: TaoVecGetSubVec(asls->da, asls->free, tao->subset_type, 0.0, &asls->r1);
205: TaoVecGetSubVec(asls->ff, asls->free, tao->subset_type, 0.0, &asls->r2);
206: TaoVecGetSubVec(asls->db, asls->free, tao->subset_type, 1.0, &asls->r3);
207: VecPointwiseDivide(asls->r1,asls->r1, asls->r3);
208: VecPointwiseDivide(asls->r2,asls->r2, asls->r3);
210: /* r1 is the diagonal perturbation
211: r2 is the right hand side
212: r3 is no longer needed
214: Now need to modify r2 for our direction choice in the fixed
215: variable set: calculate t1 = J*d, take the reduced vector
216: of t1 and modify r2. */
218: MatMult(tao->jacobian, tao->stepdirection, asls->t1);
219: TaoVecGetSubVec(asls->t1,asls->free,tao->subset_type,0.0,&asls->r3);
220: VecAXPY(asls->r2, -1.0, asls->r3);
222: /* Calculate the reduced problem matrix and the direction */
223: if (!asls->w && (tao->subset_type == TAO_SUBSET_MASK || tao->subset_type == TAO_SUBSET_MATRIXFREE)) {
224: VecDuplicate(tao->solution, &asls->w);
225: }
226: TaoMatGetSubMat(tao->jacobian, asls->free, asls->w, tao->subset_type,&asls->J_sub);
227: if (tao->jacobian != tao->jacobian_pre) {
228: TaoMatGetSubMat(tao->jacobian_pre, asls->free, asls->w, tao->subset_type, &asls->Jpre_sub);
229: } else {
230: MatDestroy(&asls->Jpre_sub);
231: asls->Jpre_sub = asls->J_sub;
232: PetscObjectReference((PetscObject)(asls->Jpre_sub));
233: }
234: MatDiagonalSet(asls->J_sub, asls->r1,ADD_VALUES);
235: TaoVecGetSubVec(tao->stepdirection, asls->free, tao->subset_type, 0.0, &asls->dxfree);
236: VecSet(asls->dxfree, 0.0);
238: /* Calculate the reduced direction. (Really negative of Newton
239: direction. Therefore, rest of the code uses -d.) */
240: KSPReset(tao->ksp);
241: KSPSetOperators(tao->ksp, asls->J_sub, asls->Jpre_sub);
242: KSPSolve(tao->ksp, asls->r2, asls->dxfree);
243: KSPGetIterationNumber(tao->ksp,&tao->ksp_its);
244: tao->ksp_tot_its+=tao->ksp_its;
246: /* Add the direction in the free variables back into the real direction. */
247: VecISAXPY(tao->stepdirection, asls->free, 1.0,asls->dxfree);
249: /* Check the real direction for descent and if not, use the negative
250: gradient direction. */
251: VecNorm(tao->stepdirection, NORM_2, &normd);
252: VecDot(tao->stepdirection, asls->dpsi, &innerd);
254: if (innerd <= asls->delta*PetscPowReal(normd, asls->rho)) {
255: PetscInfo(tao,"Gradient direction: %5.4e.\n", (double)innerd);
256: PetscInfo(tao, "Iteration %D: newton direction not descent\n", tao->niter);
257: VecCopy(asls->dpsi, tao->stepdirection);
258: VecDot(asls->dpsi, tao->stepdirection, &innerd);
259: }
261: VecScale(tao->stepdirection, -1.0);
262: innerd = -innerd;
264: /* We now have a correct descent direction. Apply a linesearch to
265: find the new iterate. */
266: TaoLineSearchSetInitialStepLength(tao->linesearch, 1.0);
267: TaoLineSearchApply(tao->linesearch, tao->solution, &psi,asls->dpsi, tao->stepdirection, &t, &ls_reason);
268: VecNorm(asls->dpsi, NORM_2, &ndpsi);
269: }
270: return 0;
271: }
273: /* ---------------------------------------------------------- */
274: /*MC
275: TAOASILS - Active-set infeasible linesearch algorithm for solving
276: complementarity constraints
278: Options Database Keys:
279: + -tao_ssls_delta - descent test fraction
280: - -tao_ssls_rho - descent test power
282: Level: beginner
283: M*/
284: PETSC_EXTERN PetscErrorCode TaoCreate_ASILS(Tao tao)
285: {
286: TAO_SSLS *asls;
287: const char *armijo_type = TAOLINESEARCHARMIJO;
289: PetscNewLog(tao,&asls);
290: tao->data = (void*)asls;
291: tao->ops->solve = TaoSolve_ASILS;
292: tao->ops->setup = TaoSetUp_ASILS;
293: tao->ops->view = TaoView_SSLS;
294: tao->ops->setfromoptions = TaoSetFromOptions_SSLS;
295: tao->ops->destroy = TaoDestroy_ASILS;
296: tao->subset_type = TAO_SUBSET_SUBVEC;
297: asls->delta = 1e-10;
298: asls->rho = 2.1;
299: asls->fixed = NULL;
300: asls->free = NULL;
301: asls->J_sub = NULL;
302: asls->Jpre_sub = NULL;
303: asls->w = NULL;
304: asls->r1 = NULL;
305: asls->r2 = NULL;
306: asls->r3 = NULL;
307: asls->t1 = NULL;
308: asls->t2 = NULL;
309: asls->dxfree = NULL;
311: asls->identifier = 1e-5;
313: TaoLineSearchCreate(((PetscObject)tao)->comm, &tao->linesearch);
314: PetscObjectIncrementTabLevel((PetscObject)tao->linesearch, (PetscObject)tao, 1);
315: TaoLineSearchSetType(tao->linesearch, armijo_type);
316: TaoLineSearchSetOptionsPrefix(tao->linesearch,tao->hdr.prefix);
317: TaoLineSearchSetFromOptions(tao->linesearch);
319: KSPCreate(((PetscObject)tao)->comm, &tao->ksp);
320: PetscObjectIncrementTabLevel((PetscObject)tao->ksp, (PetscObject)tao, 1);
321: KSPSetOptionsPrefix(tao->ksp,tao->hdr.prefix);
322: KSPSetFromOptions(tao->ksp);
324: /* Override default settings (unless already changed) */
325: if (!tao->max_it_changed) tao->max_it = 2000;
326: if (!tao->max_funcs_changed) tao->max_funcs = 4000;
327: if (!tao->gttol_changed) tao->gttol = 0;
328: if (!tao->grtol_changed) tao->grtol = 0;
329: #if defined(PETSC_USE_REAL_SINGLE)
330: if (!tao->gatol_changed) tao->gatol = 1.0e-6;
331: if (!tao->fmin_changed) tao->fmin = 1.0e-4;
332: #else
333: if (!tao->gatol_changed) tao->gatol = 1.0e-16;
334: if (!tao->fmin_changed) tao->fmin = 1.0e-8;
335: #endif
336: return 0;
337: }