Coverage for picos/expressions/exp

1# ------------------------------------------------------------------------------

3# Based on the original picos.expressions module by Guillaume Sagnol.

5# This file is part of PICOS.

7# PICOS is free software: you can redistribute it and/or modify it under the

8# terms of the GNU General Public License as published by the Free Software

9# Foundation, either version 3 of the License, or (at your option) any later

10# version.

11#

12# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY

13# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR

14# A PARTICULAR PURPOSE. See the GNU General Public License for more details.

15#

16# You should have received a copy of the GNU General Public License along with

17# this program. If not, see <http://www.gnu.org/licenses/>.

18# ------------------------------------------------------------------------------

20"""Implements :class:`PowerTrace`."""

22import operator

23from collections import namedtuple

25import cvxopt

26import numpy

28from .. import glyphs

29from ..apidoc import api_end, api_start

30from ..constraints import Constraint, PowerTraceConstraint

31from .data import (convert_and_refine_arguments, convert_operands, cvx2np,

32 cvxopt_hpsd, make_fraction)

33from .exp_affine import AffineExpression, ComplexAffineExpression, Constant

34from .expression import Expression, refine_operands, validate_prediction

36_API_START = api_start(globals())

37# -------------------------------

40class PowerTrace(Expression):

41 r"""The trace of the :math:`p`-th power of a hermitian matrix.

43 :Definition:

45 Let :math:`p \in \mathbb{Q}`.

47 1. If the base expressions is a real scalar :math:`x` and no additional

48 constant :math:`m` is given, then this is the power :math:`x^p`.

50 2. If the base expressions is a real scalar :math:`x`,

51 :math:`p \in [0, 1]`, and a positive scalar constant :math:`m` is given,

52 then this is the scaled power :math:`m x^p`.

54 3. If the base expression is a hermitian matrix :math:`X` and no additional

55 constant :math:`M` is given, then this is the trace of power

56 :math:`\operatorname{tr}(X^p)`.

58 4. If the base expression is a hermitian matrix :math:`X`,

59 :math:`p \in [0, 1]`, and a hermitian positive semidefinite constant

60 matrix :math:`M` of same shape as :math:`X` is given, then this is the

61 trace of a scaled power :math:`\operatorname{tr}(M X^p)`.

63 No other case is supported. In particular, if :math:`p \not\in [0, 1]`, then

64 :math:`m`/:math:`M` must be undefined (:obj:`None`).

66 .. warning::

68 1. For a constraint of the form :math:`x^p \leq t` with :math:`p < 1`

69 and :math:`p \neq 0`, PICOS enforces :math:`x \geq 0` during solution

70 search.

72 2. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t` or

73 :math:`\operatorname{tr}(M X^p) \leq t` with :math:`p < 1` and

74 :math:`p \neq 0`, PICOS enforces :math:`X \succeq 0` during solution

75 search.

77 3. For a constraint of the form :math:`\operatorname{tr}(X^p) \leq t`

78 or :math:`\operatorname{tr}(M X^p) \leq t` with :math:`p > 1`, PICOS

79 enforces :math:`t \geq 0` during solution search.

80 """

82 # --------------------------------------------------------------------------

83 # Initialization and factory methods.

84 # --------------------------------------------------------------------------

86 @convert_and_refine_arguments("x")

87 def __init__(self, x, p, m=None, denominator_limit=1000):

88 """Construct a :class:`PowerTrace`.

90 :param x: The scalar or symmetric matrix to form a power of.

91 :type x: ~picos.expressions.AffineExpression

92 :param float p: The value for :math:`p`, which is cast to a limited

93 precision fraction.

94 :param m: An additional positive semidefinite constant to multiply the

95 power with.

96 :type m: :class:`~picos.expressions.AffineExpression` or anything

97 recognized by :func:`~picos.expressions.data.load_data`

98 :param int denominator_limit: The largest allowed denominator when

99 casting :math:`p` to a fraction. Higher values can yield a greater

100 precision at reduced performance.

101 """

102 # Validate x.

103 if not isinstance(x, ComplexAffineExpression):

104 raise TypeError("Can only form the power of an affine expression, "

105 "not of {}.".format(x.string))

106 elif not x.square:

107 raise TypeError(

108 "Can't form the power of non-square {}.".format(x.string))

109 elif not x.hermitian:

110 raise NotImplementedError("Taking {} to a power is not supported "

111 "as it is not necessarily hermitian.".format(x.string))

112

113 # Load p.

114 pNum, pDen, p, pStr = make_fraction(p, denominator_limit)

115

116 # Load m.

117 if m is not None:

118 mStr = "m" if len(x) == 1 else "M"

119

120 if p < 0 or p > 1:

121 raise ValueError(

122 "p-th power with an additional factor {} requires {}."

123 .format(mStr, glyphs.le(0, glyphs.le("p", 1))))

124

125 if not isinstance(m, ComplexAffineExpression):

126 try:

127 m = Constant(mStr, m, x.shape)

128 except Exception as error:

129 raise TypeError(

130 "Failed to load the additional factor {} as a matrix of"

131 " same shape as {}.".format(mStr, x.string)) from error

132 else:

133 m = m.refined

134

135 if not m.constant:

136 raise TypeError("The additional factor {} is not constant."

137 .format(m.string))

138 elif not cvxopt_hpsd(m.safe_value_as_matrix):

139 raise ValueError("The additional factor {} is not hermitian "

140 "positive semidefinite.".format(m.string))

141

142 self._x = x

143 self._num = pNum

144 self._den = pDen

145 self._m = m

146 self._limit = denominator_limit

147

148 if m is None:

149 if len(x) == 1:

150 typeStr = "Power"

151 if p == 2:

152 symbStr = glyphs.squared(x.string)

153 elif p == 3:

154 symbStr = glyphs.cubed(x.string)

155 else:

156 symbStr = glyphs.power(x.string, pStr)

157 else:

158 typeStr = "Trace of Power"

159 symbStr = glyphs.trace(glyphs.power(x.string, pStr))

160 else:

161 if len(x) == 1:

162 typeStr = "Scaled Power"

163 symbStr = glyphs.mul(m.string, glyphs.power(x.string, pStr))

164 else:

165 typeStr = "Trace of Scaled Power"

166 symbStr = glyphs.trace(glyphs.mul(

167 m.string, glyphs.power(x.string, pStr)))

168

169 Expression.__init__(self, typeStr, symbStr)

170

171 # --------------------------------------------------------------------------

172 # Abstract method implementations and method overridings, except _predict.

173 # --------------------------------------------------------------------------

174

175 def _get_refined(self):

176 if self._x.constant:

177 return Constant(self._symbStr, self.value)

178 elif self.p == 0:

179 if self._m is not None:

180 return self._m.tr

181 else:

182 return Constant(

183 glyphs.Fn("diaglen({})")(self._x.string), self._x.shape[0])

184 elif self.p == 1:

185 if self._m is not None:

186 # NOTE: No hermitian transpose as both m and x are hermitian.

187 return (self._m | self._x)

188 else:

189 return self._x.tr

190 elif self.p == 2 and self._x.scalar and self._m is None:

191 return (self._x | self._x)

192 else:

193 return self

194

195 Subtype = namedtuple("Subtype", ("diag", "num", "den", "hasM", "complex"))

196

197 def _get_subtype(self):

198 return self.Subtype(

199 self._x.shape[0], self._num, self._den, self._m is not None,

200 self._x.complex)

201

202 def _get_value(self):

203 x = cvx2np(self._x._get_value())

204 p = self.p

205

206 eigenvalues = numpy.linalg.eigvalsh(x)

207

208 if p != int(p) and any(value < 0 for value in eigenvalues):

209 raise ArithmeticError("Cannot evaluate {}: {} is not positive "

210 "semidefinite and the exponent is fractional."

211 .format(self.string, self._x.string))

212

213 if self._m is None:

214 trace = sum([value**p for value in eigenvalues])

215 else:

216 m = cvx2np(self._m._get_value())

217

218 U, S, V = numpy.linalg.svd(x)

219 power = U*numpy.diag(S**p)*V

220 trace = numpy.trace(m * power)

221

222 return cvxopt.matrix(trace)

223

224 def _get_mutables(self):

225 return self._x._get_mutables()

226

227 def _is_convex(self):

228 return self.p >= 1 or self.p <= 0

229

230 def _is_concave(self):

231 return self.p >= 0 and self.p <= 1

232

233 def _replace_mutables(self, mapping):

234 return self.__class__(

235 self._x._replace_mutables(mapping), self.p, self._m, self._limit)

236

237 def _freeze_mutables(self, freeze):

238 return self.__class__(

239 self._x._freeze_mutables(freeze), self.p, self._m, self._limit)

240

241 # --------------------------------------------------------------------------

242 # Python special method implementations, except constraint-creating ones.

243 # --------------------------------------------------------------------------

244

245 @classmethod

246 def _mul(cls, self, other, forward):

247 if isinstance(other, AffineExpression) and other.constant:

248 factor = other.safe_value

249

250 if not factor:

251 return AffineExpression.zero()

252 elif factor == 1:

253 return self

254 elif factor > 0 and self.p >= 0 and self.p <= 1:

255 if self._m is None:

256 m = other.dupdiag(self.n).renamed(other.string)

257 else:

258 m = other*self._m

259

260 return cls(self._x, self.p, m, self._limit)

261

262 if forward:

263 return Expression.__mul__(self, other)

264 else:

265 return Expression.__rmul__(self, other)

266

267 @convert_operands(scalarRHS=True)

268 @refine_operands()

269 def __mul__(self, other):

270 return PowerTrace._mul(self, other, True)

271

272 @convert_operands(scalarRHS=True)

273 @refine_operands()

274 def __rmul__(self, other):

275 return PowerTrace._mul(self, other, False)

276

277 # --------------------------------------------------------------------------

278 # Methods and properties that return expressions.

279 # --------------------------------------------------------------------------

280

281 @property

282 def x(self):

283 """The matrix concerned."""

284 return self._x

285

286 # --------------------------------------------------------------------------

287 # Methods and properties that describe the expression.

288 # --------------------------------------------------------------------------

289

290 @property

291 def n(self):

292 """Diagonal length of :attr:`x`."""

293 return self._x.shape[0]

294

295 @property

296 def p(self):

297 """The parameter :math:`p`.

298

299 This is a limited precision version of the parameter used when the

300 expression was constructed.

301 """

302 return float(self._num) / float(self._den)

303

304 @property

305 def num(self):

306 """The limited precision fraction numerator of :math:`p`."""

307 return self._num

308

309 @property

310 def den(self):

311 """The limited precision fraction denominator of :math:`p`."""

312 return self._den

313

314 @property

315 def m(self):

316 """An additional factor to multiply the power with."""

317 return self._m

318

319 # --------------------------------------------------------------------------

320 # Constraint-creating operators, and _predict.

321 # --------------------------------------------------------------------------

322

323 @classmethod

324 def _predict(cls, subtype, relation, other):

325 assert isinstance(subtype, cls.Subtype)

326

327 p = float(subtype.num) / float(subtype.den)

328

329 if relation == operator.__le__:

330 if p > 0 and p < 1: # Not convex.

331 return NotImplemented

332

333 if issubclass(other.clstype, AffineExpression) \

334 and other.subtype.dim == 1:

335 return PowerTraceConstraint.make_type(*subtype)

336 elif relation == operator.__ge__:

337 if p < 0 or p > 1: # Not concave.

338 return NotImplemented

339

340 if issubclass(other.clstype, AffineExpression) \

341 and other.subtype.dim == 1:

342 return PowerTraceConstraint.make_type(*subtype)

343

344 return NotImplemented

345

346 @convert_operands(scalarRHS=True)

347 @validate_prediction

348 @refine_operands()

349 def __le__(self, other):

350 if not self.convex:

351 raise TypeError("Cannot upper-bound a nonconvex (trace of) power.")

352

353 if isinstance(other, AffineExpression):

354 return PowerTraceConstraint(self, Constraint.LE, other)

355 else:

356 return NotImplemented

357

358 @convert_operands(scalarRHS=True)

359 @validate_prediction

360 @refine_operands()

361 def __ge__(self, other):

362 if not self.concave:

363 raise TypeError("Cannot lower-bound a nonconcave (trace of) power.")

364

365 if isinstance(other, AffineExpression):

366 return PowerTraceConstraint(self, Constraint.GE, other)

367 else:

368 return NotImplemented

369

370

371# --------------------------------------

372__all__ = api_end(_API_START, globals())

Coverage for picos/expressions/exp_powtrace.py: 77.46%

173 statements