mjo/cone/tests.py

   1 """
   2 Additional tests for the mjo.cone.cone module. These are extra
   3 properties that we'd like to check, but which are overkill for inclusion
   4 into Sage.
   5 """
   6
   7 # Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
   8 # have to explicitly mangle our sitedir here so that "mjo.cone"
   9 # resolves.
  10 from os.path import abspath
  11 from site import addsitedir
  12 addsitedir(abspath('../../'))
  13
  14 from sage.all import *
  15
  16 # The double-import is needed to get the underscore methods.
  17 from mjo.cone.cone import *
  18
  19 #
  20 # Tests for _restrict_to_space.
  21 #
  22 def _look_isomorphic(K1, K2):
  23     r"""
  24     Test whether or not ``K1`` and ``K2`` look linearly isomorphic.
  25
  26     This is a hack to get around the fact that it's difficult to tell
  27     when two cones are linearly isomorphic. Instead, we check a list of
  28     properties that should be preserved under linear isomorphism.
  29
  30     OUTPUT:
  31
  32     ``True`` if ``K1`` and ``K2`` look isomorphic, or ``False``
  33     if we can prove that they are not isomorphic.
  34
  35     EXAMPLES:
  36
  37     Any proper cone with three generators in `\mathbb{R}^{3}` is
  38     isomorphic to the nonnegative orthant::
  39
  40         sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
  41         sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
  42         sage: _look_isomorphic(K1, K2)
  43         True
  44
  45     Negating a cone gives you an isomorphic cone::
  46
  47         sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
  48         sage: _look_isomorphic(K, -K)
  49         True
  50
  51     TESTS:
  52
  53     Any cone is isomorphic to itself::
  54
  55         sage: K = random_cone(max_ambient_dim = 8)
  56         sage: _look_isomorphic(K, K)
  57         True
  58
  59     After applying an invertible matrix to the rows of a cone, the
  60     result should is isomorphic to the cone we started with::
  61
  62         sage: K1 = random_cone(max_ambient_dim = 8)
  63         sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
  64         sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
  65         sage: _look_isomorphic(K1, K2)
  66         True
  67
  68     """
  69     if K1.lattice_dim() != K2.lattice_dim():
  70         return False
  71
  72     if K1.nrays() != K2.nrays():
  73         return False
  74
  75     if K1.dim() != K2.dim():
  76         return False
  77
  78     if K1.lineality() != K2.lineality():
  79         return False
  80
  81     if K1.is_solid() != K2.is_solid():
  82         return False
  83
  84     if K1.is_strictly_convex() != K2.is_strictly_convex():
  85         return False
  86
  87     if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()):
  88         return False
  89
  90     C_of_K1 = K1.discrete_complementarity_set()
  91     C_of_K2 = K2.discrete_complementarity_set()
  92     if len(C_of_K1) != len(C_of_K2):
  93         return False
  94
  95     if len(K1.facets()) != len(K2.facets()):
  96         return False
  97
  98     return True
  99
 100
 101 """
 102 Apply _restrict_to_space according to our paper (to obtain our main
 103 result). Test all four parameter combinations::
 104
 105     sage: set_random_seed()
 106     sage: K = random_cone(max_ambient_dim = 8,
 107     ....:                 strictly_convex=False,
 108     ....:                 solid=False)
 109     sage: K_S = K._restrict_to_space(K.span())
 110     sage: K_SP = K_S.dual()._restrict_to_space(K_S.dual().span()).dual()
 111     sage: K_SP.is_proper()
 112     True
 113     sage: K_SP = K_S._restrict_to_space(K_S.dual().span())
 114     sage: K_SP.is_proper()
 115     True
 116
 117 ::
 118
 119     sage: set_random_seed()
 120     sage: K = random_cone(max_ambient_dim = 8,
 121     ....:                 strictly_convex=True,
 122     ....:                 solid=False)
 123     sage: K_S = K._restrict_to_space(K.span())
 124     sage: K_SP = K_S.dual()._restrict_to_space(K_S.dual().span()).dual()
 125     sage: K_SP.is_proper()
 126     True
 127     sage: K_SP = K_S._restrict_to_space(K_S.dual().span())
 128     sage: K_SP.is_proper()
 129     True
 130
 131 ::
 132
 133     sage: set_random_seed()
 134     sage: K = random_cone(max_ambient_dim = 8,
 135     ....:                 strictly_convex=False,
 136     ....:                 solid=True)
 137     sage: K_S = K._restrict_to_space(K.span())
 138     sage: K_SP = K_S.dual()._restrict_to_space(K_S.dual().span()).dual()
 139     sage: K_SP.is_proper()
 140     True
 141     sage: K_SP = K_S._restrict_to_space(K_S.dual().span())
 142     sage: K_SP.is_proper()
 143     True
 144
 145 ::
 146
 147     sage: set_random_seed()
 148     sage: K = random_cone(max_ambient_dim = 8,
 149     ....:                 strictly_convex=True,
 150     ....:                 solid=True)
 151     sage: K_S = K._restrict_to_space(K.span())
 152     sage: K_SP = K_S.dual()._restrict_to_space(K_S.dual().span()).dual()
 153     sage: K_SP.is_proper()
 154     True
 155     sage: K_SP = K_S._restrict_to_space(K_S.dual().span())
 156     sage: K_SP.is_proper()
 157     True
 158
 159 Test the proposition in our paper concerning the duals and
 160 restrictions. Generate a random cone, then create a subcone of
 161 it. The operation of dual-taking should then commute with rho. Test
 162 all parameter combinations::
 163
 164
 165     sage: set_random_seed()
 166     sage: J = random_cone(max_ambient_dim = 8,
 167     ....:                 solid=False,
 168     ....:                 strictly_convex=False)
 169     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 170     sage: K_W_star = K._restrict_to_space(J.span()).dual()
 171     sage: K_star_W = K.dual()._restrict_to_space(J.span())
 172     sage: _look_isomorphic(K_W_star, K_star_W)
 173     True
 174
 175 ::
 176
 177     sage: set_random_seed()
 178     sage: J = random_cone(max_ambient_dim = 8,
 179     ....:                 solid=True,
 180     ....:                 strictly_convex=False)
 181     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 182     sage: K_W_star = K._restrict_to_space(J.span()).dual()
 183     sage: K_star_W = K.dual()._restrict_to_space(J.span())
 184     sage: _look_isomorphic(K_W_star, K_star_W)
 185     True
 186
 187 ::
 188
 189     sage: set_random_seed()
 190     sage: J = random_cone(max_ambient_dim = 8,
 191     ....:                 solid=False,
 192     ....:                 strictly_convex=True)
 193     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 194     sage: K_W_star = K._restrict_to_space(J.span()).dual()
 195     sage: K_star_W = K.dual()._restrict_to_space(J.span())
 196     sage: _look_isomorphic(K_W_star, K_star_W)
 197     True
 198
 199 ::
 200
 201     sage: set_random_seed()
 202     sage: J = random_cone(max_ambient_dim = 8,
 203     ....:                 solid=True,
 204     ....:                 strictly_convex=True)
 205     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 206     sage: K_W_star = K._restrict_to_space(J.span()).dual()
 207     sage: K_star_W = K.dual()._restrict_to_space(J.span())
 208     sage: _look_isomorphic(K_W_star, K_star_W)
 209     True
 210
 211 """
 212
 213
 214 #
 215 # Lyapunov rank tests
 216 #
 217 """
 218
 219 The Lyapunov rank is invariant under a linear isomorphism. Check all
 220 combinations of parameters::
 221
 222     sage: K1 = random_cone(max_ambient_dim=8,
 223     ....:                  strictly_convex=True,
 224     ....:                  solid=True)
 225     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 226     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 227     sage: K1.lyapunov_rank() == K2.lyapunov_rank()
 228     True
 229
 230 ::
 231
 232     sage: K1 = random_cone(max_ambient_dim=8,
 233     ....:                  strictly_convex=True,
 234     ....:                  solid=False)
 235     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 236     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 237     sage: K1.lyapunov_rank() == K2.lyapunov_rank()
 238     True
 239
 240 ::
 241
 242     sage: K1 = random_cone(max_ambient_dim=8,
 243     ....:                  strictly_convex=False,
 244     ....:                  solid=True)
 245     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 246     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 247     sage: K1.lyapunov_rank() == K2.lyapunov_rank()
 248     True
 249
 250 ::
 251
 252     sage: K1 = random_cone(max_ambient_dim=8,
 253     ....:                  strictly_convex=False,
 254     ....:                  solid=False)
 255     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 256     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 257     sage: K1.lyapunov_rank() == K2.lyapunov_rank()
 258     True
 259
 260 The Lyapunov rank of a dual cone should be the same as the original
 261 cone. Check all combinations of parameters::
 262
 263     sage: set_random_seed()
 264     sage: K = random_cone(max_ambient_dim=8,
 265     ....:                 strictly_convex=False,
 266     ....:                 solid=False)
 267     sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
 268     True
 269
 270 ::
 271
 272     sage: set_random_seed()
 273     sage: K = random_cone(max_ambient_dim=8,
 274     ....:                 strictly_convex=False,
 275     ....:                 solid=True)
 276     sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
 277     True
 278
 279 ::
 280
 281     sage: set_random_seed()
 282     sage: K = random_cone(max_ambient_dim=8,
 283     ....:                 strictly_convex=True,
 284     ....:                 solid=False)
 285     sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
 286     True
 287
 288 ::
 289
 290     sage: set_random_seed()
 291     sage: K = random_cone(max_ambient_dim=8,
 292     ....:                 strictly_convex=True,
 293     ....:                 solid=True)
 294     sage: K.lyapunov_rank() == K.dual().lyapunov_rank()
 295     True
 296
 297 The Lyapunov rank of a cone ``K`` is the dimension of
 298 ``K.lyapunov_like_basis()``. Check all combinations of parameters::
 299
 300     sage: set_random_seed()
 301     sage: K = random_cone(max_ambient_dim=8,
 302     ....:                 strictly_convex=True,
 303     ....:                 solid=True)
 304     sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
 305     True
 306
 307 ::
 308
 309     sage: set_random_seed()
 310     sage: K = random_cone(max_ambient_dim=8,
 311     ....:                 strictly_convex=True,
 312     ....:                 solid=False)
 313     sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
 314     True
 315
 316 ::
 317
 318     sage: set_random_seed()
 319     sage: K = random_cone(max_ambient_dim=8,
 320     ....:                 strictly_convex=False,
 321     ....:                 solid=True)
 322     sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
 323     True
 324
 325 ::
 326
 327     sage: set_random_seed()
 328     sage: K = random_cone(max_ambient_dim=8,
 329     ....:                 strictly_convex=False,
 330     ....:                 solid=False)
 331     sage: K.lyapunov_rank() == len(K.lyapunov_like_basis())
 332     True
 333
 334 """