1"""Implement the GeometricProgram class""" 

2import sys 

3import warnings 

4from time import time 

5from collections import defaultdict 

6import numpy as np 

7from ..repr_conventions import lineagestr 

8from ..small_classes import CootMatrix, SolverLog, Numbers, FixedScalar 

9from ..keydict import KeyDict 

10from ..solution_array import SolutionArray 

11from .set import ConstraintSet 

12from ..exceptions import (InvalidPosynomial, Infeasible, UnknownInfeasible, 

13 PrimalInfeasible, DualInfeasible, UnboundedGP, 

14 InvalidLicense) 

15 

16 

17DEFAULT_SOLVER_KWARGS = {"cvxopt": {"kktsolver": "ldl"}} 

18SOLUTION_TOL = {"cvxopt": 1e-3, "mosek_cli": 1e-4, "mosek_conif": 1e-3} 

19 

20 

21class MonoEqualityIndexes: 

22 "Class to hold MonoEqualityIndexes" 

23 

24 def __init__(self): 

25 self.all = set() 

26 self.first_half = set() 

27 

28 

29def _get_solver(solver, kwargs): 

30 """Get the solverfn and solvername associated with solver""" 

31 if solver is None: 

32 from .. import settings 

33 try: 

34 solver = settings["default_solver"] 

35 except KeyError: 

36 raise ValueError("No default solver was set during build, so" 

37 " solvers must be manually specified.") 

38 if solver == "cvxopt": 

39 from ..solvers.cvxopt import optimize 

40 elif solver == "mosek_cli": 

41 from ..solvers.mosek_cli import optimize_generator 

42 optimize = optimize_generator(**kwargs) 

43 elif solver == "mosek_conif": 

44 from ..solvers.mosek_conif import optimize 

45 elif hasattr(solver, "__call__"): 

46 solver, optimize = solver.__name__, solver 

47 else: 

48 raise ValueError("Unknown solver '%s'." % solver) 

49 return solver, optimize 

50 

51 

52class GeometricProgram: 

53 # pylint: disable=too-many-instance-attributes 

54 """Standard mathematical representation of a GP. 

55 

56 Attributes with side effects 

57 ---------------------------- 

58 `solver_out` and `solve_log` are set during a solve 

59 `result` is set at the end of a solve if solution status is optimal 

60 

61 Examples 

62 -------- 

63 >>> gp = gpkit.constraints.gp.GeometricProgram( 

64 # minimize 

65 x, 

66 [ # subject to 

67 x >= 1, 

68 ], {}) 

69 >>> gp.solve() 

70 """ 

71 _result = solve_log = solver_out = model = v_ss = nu_by_posy = None 

72 choicevaridxs = integersolve = None 

73 

74 def __init__(self, cost, constraints, substitutions, *, checkbounds=True): 

75 self.cost, self.substitutions = cost, substitutions 

76 for key, sub in self.substitutions.items(): 

77 if isinstance(sub, FixedScalar): 

78 sub = sub.value 

79 if hasattr(sub, "units"): 

80 sub = sub.to(key.units or "dimensionless").magnitude 

81 self.substitutions[key] = sub 

82 if not isinstance(sub, (Numbers, np.ndarray)): 

83 raise TypeError("substitution {%s: %s} has invalid value type" 

84 " %s." % (key, sub, type(sub))) 

85 cost_hmap = cost.hmap.sub(self.substitutions, cost.vks) 

86 if any(c <= 0 for c in cost_hmap.values()): 

87 raise InvalidPosynomial("a GP's cost must be Posynomial") 

88 hmapgen = ConstraintSet.as_hmapslt1(constraints, self.substitutions) 

89 self.hmaps = [cost_hmap] + list(hmapgen) 

90 self.gen() # Generate various maps into the posy- and monomials 

91 if checkbounds: 

92 self.check_bounds(err_on_missing_bounds=True) 

93 

94 def check_bounds(self, *, err_on_missing_bounds=False): 

95 "Checks if any variables are unbounded, through equality constraints." 

96 missingbounds = {} 

97 for var, locs in self.varlocs.items(): 

98 upperbound, lowerbound = False, False 

99 for i in locs: 

100 if i not in self.meq_idxs.all: 

101 if self.exps[i][var] > 0: # pylint:disable=simplifiable-if-statement 

102 upperbound = True 

103 else: 

104 lowerbound = True 

105 if upperbound and lowerbound: 

106 break 

107 if not upperbound: 

108 missingbounds[(var, "upper")] = "." 

109 if not lowerbound: 

110 missingbounds[(var, "lower")] = "." 

111 if not missingbounds: 

112 return {} # all bounds found in inequalities 

113 meq_bounds = gen_meq_bounds(missingbounds, self.exps, self.meq_idxs) 

114 fulfill_meq_bounds(missingbounds, meq_bounds) 

115 if missingbounds and err_on_missing_bounds: 

116 raise UnboundedGP( 

117 "\n\n".join("%s has no %s bound%s" % (v, b, x) 

118 for (v, b), x in missingbounds.items())) 

119 return missingbounds 

120 

121 def gen(self): 

122 """Generates nomial and solve data (A, p_idxs) from posynomials. 

123 

124 k [posys]: number of monomials (rows of A) present in each constraint 

125 m_idxs [mons]: monomial indices of each posynomial 

126 p_idxs [mons]: posynomial index of each monomial 

127 cs, exps [mons]: coefficient and exponents of each monomial 

128 varlocs: {vk: monomial indices of each variables' location} 

129 meq_idxs: {all indices of equality mons} and {the first index of each} 

130 varidxs: {vk: which column corresponds to it in A} 

131 A [mons, vks]: sparse array of each monomials' variables' exponents 

132 

133 """ 

134 self.k = [len(hmap) for hmap in self.hmaps] 

135 self.m_idxs, self.p_idxs, self.cs, self.exps = [], [], [], [] 

136 self.varkeys = self.varlocs = defaultdict(list) 

137 self.meq_idxs = MonoEqualityIndexes() 

138 m_idx = 0 

139 row, col, data = [], [], [] 

140 for p_idx, (N_mons, hmap) in enumerate(zip(self.k, self.hmaps)): 

141 self.p_idxs.extend([p_idx]*N_mons) 

142 self.m_idxs.append(slice(m_idx, m_idx+N_mons)) 

143 if getattr(self.hmaps[p_idx], "from_meq", False): 

144 self.meq_idxs.all.add(m_idx) 

145 if len(self.meq_idxs.all) > 2*len(self.meq_idxs.first_half): 

146 self.meq_idxs.first_half.add(m_idx) 

147 self.exps.extend(hmap) 

148 self.cs.extend(hmap.values()) 

149 for exp in hmap: 

150 if not exp: # space out A matrix with constants for mosek 

151 row.append(m_idx) 

152 col.append(0) 

153 data.append(0) 

154 for var in exp: 

155 self.varlocs[var].append(m_idx) 

156 m_idx += 1 

157 self.p_idxs = np.array(self.p_idxs, "int32") # to use array equalities 

158 self.varidxs = {vk: i for i, vk in enumerate(self.varlocs)} 

159 self.choicevaridxs = {vk: i for i, vk in enumerate(self.varlocs) 

160 if vk.choices} 

161 for j, (var, locs) in enumerate(self.varlocs.items()): 

162 row.extend(locs) 

163 col.extend([j]*len(locs)) 

164 data.extend(self.exps[i][var] for i in locs) 

165 self.A = CootMatrix(row, col, data) 

166 

167 # pylint: disable=too-many-statements, too-many-locals 

168 def solve(self, solver=None, *, verbosity=1, gen_result=True, **kwargs): 

169 """Solves a GeometricProgram and returns the solution. 

170 

171 Arguments 

172 --------- 

173 solver : str or function (optional) 

174 By default uses a solver found during installation. 

175 If "mosek_conif", "mosek_cli", or "cvxopt", uses that solver. 

176 If a function, passes that function cs, A, p_idxs, and k. 

177 verbosity : int (default 1) 

178 If greater than 0, prints solver name and solve time. 

179 gen_result : bool (default True) 

180 If True, makes a human-readable SolutionArray from solver output. 

181 **kwargs : 

182 Passed to solver constructor and solver function. 

183 

184 

185 Returns 

186 ------- 

187 SolutionArray (or dict if gen_result is False) 

188 """ 

189 solvername, solverfn = _get_solver(solver, kwargs) 

190 if verbosity > 0: 

191 print("Using solver '%s'" % solvername) 

192 print(" for %i free variables" % len(self.varlocs)) 

193 print(" in %i posynomial inequalities." % len(self.k)) 

194 

195 solverargs = DEFAULT_SOLVER_KWARGS.get(solvername, {}) 

196 solverargs.update(kwargs) 

197 if self.choicevaridxs and solvername == "mosek_conif": 

198 solverargs["choicevaridxs"] = self.choicevaridxs 

199 self.integersolve = True 

200 starttime = time() 

201 solver_out, infeasibility, original_stdout = {}, None, sys.stdout 

202 try: 

203 sys.stdout = SolverLog(original_stdout, verbosity=verbosity-2) 

204 solver_out = solverfn(c=self.cs, A=self.A, meq_idxs=self.meq_idxs, 

205 k=self.k, p_idxs=self.p_idxs, **solverargs) 

206 except Infeasible as e: 

207 infeasibility = e 

208 except InvalidLicense as e: 

209 raise InvalidLicense("license for solver \"%s\" is invalid." 

210 % solvername) from e 

211 except Exception as e: 

212 raise UnknownInfeasible("Something unexpected went wrong.") from e 

213 finally: 

214 self.solve_log = "\n".join(sys.stdout) 

215 sys.stdout = original_stdout 

216 self.solver_out = solver_out 

217 

218 solver_out["solver"] = solvername 

219 solver_out["soltime"] = time() - starttime 

220 if verbosity > 0: 

221 print("Solving took %.3g seconds." % solver_out["soltime"]) 

222 

223 if infeasibility: 

224 if isinstance(infeasibility, PrimalInfeasible): 

225 msg = ("The model had no feasible points; " 

226 "you may wish to relax some constraints or constants.") 

227 elif isinstance(infeasibility, DualInfeasible): 

228 msg = ("The model ran to an infinitely low cost;" 

229 " bounding the right variables would prevent this.") 

230 elif isinstance(infeasibility, UnknownInfeasible): 

231 msg = ("Solver failed for an unknown reason. Relaxing" 

232 " constraints/constants, bounding variables, or" 

233 " using a different solver might fix it.") 

234 if (verbosity > 0 and solver_out["soltime"] < 1 

235 and hasattr(self, "model")): # fast, top-level model 

236 print(msg + "\nSince the model solved in less than a second," 

237 " let's run `.debug()` to analyze what happened.\n`") 

238 return self.model.debug(solver=solver) 

239 # else, raise a clarifying error 

240 msg += (" Running `.debug()` or increasing verbosity may pinpoint" 

241 " the trouble.") 

242 raise infeasibility.__class__(msg) from infeasibility 

243 

244 if not gen_result: 

245 return solver_out 

246 # else, generate a human-readable SolutionArray 

247 self._result = self.generate_result(solver_out, verbosity=verbosity-2) 

248 return self.result 

249 

250 @property 

251 def result(self): 

252 "Creates and caches a result from the raw solver_out" 

253 if not self._result: 

254 self._result = self.generate_result(self.solver_out) 

255 return self._result 

256 

257 def generate_result(self, solver_out, *, verbosity=0, dual_check=True): 

258 "Generates a full SolutionArray and checks it." 

259 if verbosity > 0: 

260 soltime = solver_out["soltime"] 

261 tic = time() 

262 # result packing # 

263 result = self._compile_result(solver_out) # NOTE: SIDE EFFECTS 

264 if verbosity > 0: 

265 print("Result packing took %.2g%% of solve time." % 

266 ((time() - tic) / soltime * 100)) 

267 tic = time() 

268 # solution checking # 

269 try: 

270 tol = SOLUTION_TOL.get(solver_out["solver"], 1e-5) 

271 self.check_solution(result["cost"], solver_out['primal'], 

272 solver_out["nu"], solver_out["la"], tol) 

273 except Infeasible as chkerror: 

274 chkwarn = str(chkerror) 

275 if not ("Dual" in chkwarn and not dual_check): 

276 print("Solution check warning: %s" % chkwarn) 

277 if verbosity > 0: 

278 print("Solution checking took %.2g%% of solve time." % 

279 ((time() - tic) / soltime * 100)) 

280 return result 

281 

282 def _generate_nula(self, solver_out): 

283 if "nu" in solver_out: 

284 # solver gave us monomial sensitivities, generate posynomial ones 

285 solver_out["nu"] = nu = np.ravel(solver_out["nu"]) 

286 nu_by_posy = [nu[mi] for mi in self.m_idxs] 

287 solver_out["la"] = la = np.array([sum(nup) for nup in nu_by_posy]) 

288 elif "la" in solver_out: 

289 la = np.ravel(solver_out["la"]) 

290 if len(la) == len(self.hmaps) - 1: 

291 # assume solver dropped the cost's sensitivity (always 1.0) 

292 la = np.hstack(([1.0], la)) 

293 # solver gave us posynomial sensitivities, generate monomial ones 

294 solver_out["la"] = la 

295 z = np.log(self.cs) + self.A.dot(solver_out["primal"]) 

296 m_iss = [self.p_idxs == i for i in range(len(la))] 

297 nu_by_posy = [la[p_i]*np.exp(z[m_is])/sum(np.exp(z[m_is])) 

298 for p_i, m_is in enumerate(m_iss)] 

299 solver_out["nu"] = np.hstack(nu_by_posy) 

300 else: 

301 raise RuntimeWarning("The dual solution was not returned.") 

302 return la, nu_by_posy 

303 

304 def _compile_result(self, solver_out): 

305 result = {"cost": float(solver_out["objective"])} 

306 primal = solver_out["primal"] 

307 if len(self.varlocs) != len(primal): 

308 raise RuntimeWarning("The primal solution was not returned.") 

309 result["freevariables"] = KeyDict(zip(self.varlocs, np.exp(primal))) 

310 result["constants"] = KeyDict(self.substitutions) 

311 result["variables"] = KeyDict(result["freevariables"]) 

312 result["variables"].update(result["constants"]) 

313 result["soltime"] = solver_out["soltime"] 

314 if self.integersolve: 

315 result["choicevariables"] = KeyDict( \ 

316 {k: v for k, v in result["freevariables"].items() 

317 if k in self.choicevaridxs}) 

318 result["warnings"] = {"No Dual Solution": [(\ 

319 "This model has the discretized choice variables" 

320 " %s and hence no dual solution. You can fix those variables" 

321 " to their optimal value and get sensitivities to the resulting" 

322 " continuous problem by updating your model's substitions with" 

323 " `sol[\"choicevariables\"]`." 

324 % sorted(self.choicevaridxs.keys()), self.choicevaridxs)]} 

325 return SolutionArray(result) 

326 if self.choicevaridxs: 

327 result["warnings"] = {"Freed Choice Variables": [(\ 

328 "This model has the discretized choice variables" 

329 " %s, but since the '%s' solver doesn't support discretization" 

330 " they were treated as continuous variables." 

331 % (sorted(self.choicevaridxs.keys()), solver_out["solver"]), 

332 self.choicevaridxs)]} 

333 

334 result["sensitivities"] = {"constraints": {}} 

335 la, self.nu_by_posy = self._generate_nula(solver_out) 

336 cost_senss = sum(nu_i*exp for (nu_i, exp) in zip(self.nu_by_posy[0], 

337 self.cost.hmap)) 

338 gpv_ss = cost_senss.copy() 

339 m_senss = defaultdict(float) 

340 for las, nus, c in zip(la[1:], self.nu_by_posy[1:], self.hmaps[1:]): 

341 while getattr(c, "parent", None) is not None: 

342 c = c.parent 

343 v_ss, c_senss = c.sens_from_dual(las, nus, result) 

344 for vk, x in v_ss.items(): 

345 gpv_ss[vk] = x + gpv_ss.get(vk, 0) 

346 while getattr(c, "generated_by", None): 

347 c = c.generated_by 

348 result["sensitivities"]["constraints"][c] = abs(c_senss) 

349 m_senss[lineagestr(c)] += abs(c_senss) 

350 # add fixed variables sensitivities to models 

351 for vk, senss in gpv_ss.items(): 

352 m_senss[lineagestr(vk)] += abs(senss) 

353 result["sensitivities"]["models"] = dict(m_senss) 

354 # carry linked sensitivities over to their constants 

355 for v in list(v for v in gpv_ss if v.gradients): 

356 dlogcost_dlogv = gpv_ss.pop(v) 

357 val = np.array(result["constants"][v]) 

358 for c, dv_dc in v.gradients.items(): 

359 with warnings.catch_warnings(): # skip pesky divide-by-zeros 

360 warnings.simplefilter("ignore") 

361 dlogv_dlogc = dv_dc * result["constants"][c]/val 

362 gpv_ss[c] = gpv_ss.get(c, 0) + dlogcost_dlogv*dlogv_dlogc 

363 if v in cost_senss: 

364 if c in self.cost.vks: 

365 dlogcost_dlogv = cost_senss.pop(v) 

366 before = cost_senss.get(c, 0) 

367 cost_senss[c] = before + dlogcost_dlogv*dlogv_dlogc 

368 result["sensitivities"]["cost"] = cost_senss 

369 result["sensitivities"]["variables"] = KeyDict(gpv_ss) 

370 result["sensitivities"]["constants"] = \ 

371 result["sensitivities"]["variables"] # NOTE: backwards compat. 

372 return SolutionArray(result) 

373 

374 def check_solution(self, cost, primal, nu, la, tol, abstol=1e-20): 

375 """Run checks to mathematically confirm solution solves this GP 

376 

377 Arguments 

378 --------- 

379 cost: float 

380 cost returned by solver 

381 primal: list 

382 primal solution returned by solver 

383 nu: numpy.ndarray 

384 monomial lagrange multiplier 

385 la: numpy.ndarray 

386 posynomial lagrange multiplier 

387 

388 Raises 

389 ------ 

390 Infeasible if any problems are found 

391 """ 

392 A = self.A.tocsr() 

393 def almost_equal(num1, num2): 

394 "local almost equal test" 

395 return (num1 == num2 or abs((num1 - num2) / (num1 + num2)) < tol 

396 or abs(num1 - num2) < abstol) 

397 # check primal sol # 

398 primal_exp_vals = self.cs * np.exp(A.dot(primal)) # c*e^Ax 

399 if not almost_equal(primal_exp_vals[self.m_idxs[0]].sum(), cost): 

400 raise Infeasible("Primal solution computed cost did not match" 

401 " solver-returned cost: %s vs %s." % 

402 (primal_exp_vals[self.m_idxs[0]].sum(), cost)) 

403 for mi in self.m_idxs[1:]: 

404 if primal_exp_vals[mi].sum() > 1 + tol: 

405 raise Infeasible("Primal solution violates constraint: %s is " 

406 "greater than 1" % primal_exp_vals[mi].sum()) 

407 # check dual sol # 

408 if self.integersolve: 

409 return 

410 # note: follows dual formulation in section 3.1 of 

411 # http://web.mit.edu/~whoburg/www/papers/hoburg_phd_thesis.pdf 

412 if not almost_equal(self.nu_by_posy[0].sum(), 1): 

413 raise Infeasible("Dual variables associated with objective sum" 

414 " to %s, not 1" % self.nu_by_posy[0].sum()) 

415 if any(nu < 0): 

416 minnu = min(nu) 

417 if minnu < -tol/1000: 

418 raise Infeasible("Dual solution has negative entries as" 

419 " large as %s." % minnu) 

420 if any(np.abs(A.T.dot(nu)) > tol): 

421 raise Infeasible("Dual: sum of nu^T * A did not vanish.") 

422 b = np.log(self.cs) 

423 dual_cost = sum( 

424 self.nu_by_posy[i].dot( 

425 b[mi] - np.log(self.nu_by_posy[i]/la[i])) 

426 for i, mi in enumerate(self.m_idxs) if la[i]) 

427 if not almost_equal(np.exp(dual_cost), cost): 

428 raise Infeasible("Dual cost %s does not match primal cost %s" 

429 % (np.exp(dual_cost), cost)) 

430 

431 

432def gen_meq_bounds(missingbounds, exps, meq_idxs): # pylint: disable=too-many-locals,too-many-branches 

433 "Generate conditional monomial equality bounds" 

434 meq_bounds = defaultdict(set) 

435 for i in meq_idxs.first_half: 

436 p_upper, p_lower, n_upper, n_lower = set(), set(), set(), set() 

437 for key, x in exps[i].items(): 

438 if (key, "upper") in missingbounds: 

439 if x > 0: 

440 p_upper.add((key, "upper")) 

441 else: 

442 n_upper.add((key, "upper")) 

443 if (key, "lower") in missingbounds: 

444 if x > 0: 

445 p_lower.add((key, "lower")) 

446 else: 

447 n_lower.add((key, "lower")) 

448 # (consider x*y/z == 1) 

449 # for a var (e.g. x) to be upper bounded by this monomial equality, 

450 # - vars of the same sign (y) must be lower bounded 

451 # - AND vars of the opposite sign (z) must be upper bounded 

452 p_ub = n_lb = frozenset(n_upper).union(p_lower) 

453 n_ub = p_lb = frozenset(p_upper).union(n_lower) 

454 for keys, ub in ((p_upper, p_ub), (n_upper, n_ub)): 

455 for key, _ in keys: 

456 needed = ub.difference([(key, "lower")]) 

457 if needed: 

458 meq_bounds[(key, "upper")].add(needed) 

459 else: 

460 del missingbounds[(key, "upper")] 

461 for keys, lb in ((p_lower, p_lb), (n_lower, n_lb)): 

462 for key, _ in keys: 

463 needed = lb.difference([(key, "upper")]) 

464 if needed: 

465 meq_bounds[(key, "lower")].add(needed) 

466 else: 

467 del missingbounds[(key, "lower")] 

468 return meq_bounds 

469 

470 

471def fulfill_meq_bounds(missingbounds, meq_bounds): 

472 "Bounds variables with monomial equalities" 

473 still_alive = True 

474 while still_alive: 

475 still_alive = False # if no changes are made, the loop exits 

476 for bound in set(meq_bounds): 

477 if bound not in missingbounds: 

478 del meq_bounds[bound] 

479 continue 

480 for condition in meq_bounds[bound]: 

481 if not any(bound in missingbounds for bound in condition): 

482 del meq_bounds[bound] 

483 del missingbounds[bound] 

484 still_alive = True 

485 break 

486 for (var, bound) in meq_bounds: 

487 boundstr = (", but would gain it from any of these sets: ") 

488 for condition in list(meq_bounds[(var, bound)]): 

489 meq_bounds[(var, bound)].remove(condition) 

490 newcond = condition.intersection(missingbounds) 

491 if newcond and not any(c.issubset(newcond) 

492 for c in meq_bounds[(var, bound)]): 

493 meq_bounds[(var, bound)].add(newcond) 

494 boundstr += " or ".join(str(list(condition)) 

495 for condition in meq_bounds[(var, bound)]) 

496 missingbounds[(var, bound)] = boundstr