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 from mjo.cone.cone import _restrict_to_space
  19
  20 #
  21 # Tests for _restrict_to_space.
  22 #
  23 def _look_isomorphic(K1, K2):
  24     r"""
  25     Test whether or not ``K1`` and ``K2`` look linearly isomorphic.
  26
  27     This is a hack to get around the fact that it's difficult to tell
  28     when two cones are linearly isomorphic. Instead, we check a list of
  29     properties that should be preserved under linear isomorphism.
  30
  31     OUTPUT:
  32
  33     ``True`` if ``K1`` and ``K2`` look isomorphic, or ``False``
  34     if we can prove that they are not isomorphic.
  35
  36     EXAMPLES:
  37
  38     Any proper cone with three generators in `\mathbb{R}^{3}` is
  39     isomorphic to the nonnegative orthant::
  40
  41         sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)])
  42         sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)])
  43         sage: _look_isomorphic(K1, K2)
  44         True
  45
  46     Negating a cone gives you an isomorphic cone::
  47
  48         sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)])
  49         sage: _look_isomorphic(K, -K)
  50         True
  51
  52     TESTS:
  53
  54     Any cone is isomorphic to itself::
  55
  56         sage: K = random_cone(max_ambient_dim = 8)
  57         sage: _look_isomorphic(K, K)
  58         True
  59
  60     After applying an invertible matrix to the rows of a cone, the
  61     result should is isomorphic to the cone we started with::
  62
  63         sage: K1 = random_cone(max_ambient_dim = 8)
  64         sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
  65         sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
  66         sage: _look_isomorphic(K1, K2)
  67         True
  68
  69     """
  70     if K1.lattice_dim() != K2.lattice_dim():
  71         return False
  72
  73     if K1.nrays() != K2.nrays():
  74         return False
  75
  76     if K1.dim() != K2.dim():
  77         return False
  78
  79     if K1.lineality() != K2.lineality():
  80         return False
  81
  82     if K1.is_solid() != K2.is_solid():
  83         return False
  84
  85     if K1.is_strictly_convex() != K2.is_strictly_convex():
  86         return False
  87
  88     if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()):
  89         return False
  90
  91     C_of_K1 = K1.discrete_complementarity_set()
  92     C_of_K2 = K2.discrete_complementarity_set()
  93     if len(C_of_K1) != len(C_of_K2):
  94         return False
  95
  96     if len(K1.facets()) != len(K2.facets()):
  97         return False
  98
  99     return True
 100
 101
 102 """
 103 Apply _restrict_to_space according to our paper (to obtain our main
 104 result). Test all four parameter combinations::
 105
 106     sage: set_random_seed()
 107     sage: K = random_cone(max_ambient_dim = 8,
 108     ....:                 strictly_convex=False,
 109     ....:                 solid=False)
 110     sage: K_S = _restrict_to_space(K, K.span())
 111     sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
 112     sage: K_SP.is_proper()
 113     True
 114     sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
 115     sage: K_SP.is_proper()
 116     True
 117
 118 ::
 119
 120     sage: set_random_seed()
 121     sage: K = random_cone(max_ambient_dim = 8,
 122     ....:                 strictly_convex=True,
 123     ....:                 solid=False)
 124     sage: K_S = _restrict_to_space(K, K.span())
 125     sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
 126     sage: K_SP.is_proper()
 127     True
 128     sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
 129     sage: K_SP.is_proper()
 130     True
 131
 132 ::
 133
 134     sage: set_random_seed()
 135     sage: K = random_cone(max_ambient_dim = 8,
 136     ....:                 strictly_convex=False,
 137     ....:                 solid=True)
 138     sage: K_S = _restrict_to_space(K, K.span())
 139     sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
 140     sage: K_SP.is_proper()
 141     True
 142     sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
 143     sage: K_SP.is_proper()
 144     True
 145
 146 ::
 147
 148     sage: set_random_seed()
 149     sage: K = random_cone(max_ambient_dim = 8,
 150     ....:                 strictly_convex=True,
 151     ....:                 solid=True)
 152     sage: K_S = _restrict_to_space(K, K.span())
 153     sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual()
 154     sage: K_SP.is_proper()
 155     True
 156     sage: K_SP = _restrict_to_space(K_S, K_S.dual().span())
 157     sage: K_SP.is_proper()
 158     True
 159
 160 Test the proposition in our paper concerning the duals and
 161 restrictions. Generate a random cone, then create a subcone of
 162 it. The operation of dual-taking should then commute with rho. Test
 163 all parameter combinations::
 164
 165
 166     sage: set_random_seed()
 167     sage: J = random_cone(max_ambient_dim = 8,
 168     ....:                 solid=False,
 169     ....:                 strictly_convex=False)
 170     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 171     sage: K_W_star = _restrict_to_space(K, J.span()).dual()
 172     sage: K_star_W = _restrict_to_space(K.dual(), J.span())
 173     sage: _look_isomorphic(K_W_star, K_star_W)
 174     True
 175
 176 ::
 177
 178     sage: set_random_seed()
 179     sage: J = random_cone(max_ambient_dim = 8,
 180     ....:                 solid=True,
 181     ....:                 strictly_convex=False)
 182     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 183     sage: K_W_star = _restrict_to_space(K, J.span()).dual()
 184     sage: K_star_W = _restrict_to_space(K.dual(), J.span())
 185     sage: _look_isomorphic(K_W_star, K_star_W)
 186     True
 187
 188 ::
 189
 190     sage: set_random_seed()
 191     sage: J = random_cone(max_ambient_dim = 8,
 192     ....:                 solid=False,
 193     ....:                 strictly_convex=True)
 194     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 195     sage: K_W_star = _restrict_to_space(K, J.span()).dual()
 196     sage: K_star_W = _restrict_to_space(K.dual(), J.span())
 197     sage: _look_isomorphic(K_W_star, K_star_W)
 198     True
 199
 200 ::
 201
 202     sage: set_random_seed()
 203     sage: J = random_cone(max_ambient_dim = 8,
 204     ....:                 solid=True,
 205     ....:                 strictly_convex=True)
 206     sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice())
 207     sage: K_W_star = _restrict_to_space(K, J.span()).dual()
 208     sage: K_star_W = _restrict_to_space(K.dual(), J.span())
 209     sage: _look_isomorphic(K_W_star, K_star_W)
 210     True
 211
 212 """
 213
 214
 215 #
 216 # Lyapunov rank tests
 217 #
 218 """
 219
 220 The Lyapunov rank is invariant under a linear isomorphism. Check all
 221 combinations of parameters::
 222
 223     sage: K1 = random_cone(max_ambient_dim=8,
 224     ....:                  strictly_convex=True,
 225     ....:                  solid=True)
 226     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 227     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 228     sage: lyapunov_rank(K1) == lyapunov_rank(K2)
 229     True
 230
 231 ::
 232
 233     sage: K1 = random_cone(max_ambient_dim=8,
 234     ....:                  strictly_convex=True,
 235     ....:                  solid=False)
 236     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 237     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 238     sage: lyapunov_rank(K1) == lyapunov_rank(K2)
 239     True
 240
 241 ::
 242
 243     sage: K1 = random_cone(max_ambient_dim=8,
 244     ....:                  strictly_convex=False,
 245     ....:                  solid=True)
 246     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 247     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 248     sage: lyapunov_rank(K1) == lyapunov_rank(K2)
 249     True
 250
 251 ::
 252
 253     sage: K1 = random_cone(max_ambient_dim=8,
 254     ....:                  strictly_convex=False,
 255     ....:                  solid=False)
 256     sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular')
 257     sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice())
 258     sage: lyapunov_rank(K1) == lyapunov_rank(K2)
 259     True
 260
 261 The Lyapunov rank of a dual cone should be the same as the original
 262 cone. Check all combinations of parameters::
 263
 264     sage: set_random_seed()
 265     sage: K = random_cone(max_ambient_dim=8,
 266     ....:                 strictly_convex=False,
 267     ....:                 solid=False)
 268     sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
 269     True
 270
 271 ::
 272
 273     sage: set_random_seed()
 274     sage: K = random_cone(max_ambient_dim=8,
 275     ....:                 strictly_convex=False,
 276     ....:                 solid=True)
 277     sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
 278     True
 279
 280 ::
 281
 282     sage: set_random_seed()
 283     sage: K = random_cone(max_ambient_dim=8,
 284     ....:                 strictly_convex=True,
 285     ....:                 solid=False)
 286     sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
 287     True
 288
 289 ::
 290
 291     sage: set_random_seed()
 292     sage: K = random_cone(max_ambient_dim=8,
 293     ....:                 strictly_convex=True,
 294     ....:                 solid=True)
 295     sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
 296     True
 297
 298 The Lyapunov rank of a cone ``K`` is the dimension of
 299 ``K.lyapunov_like_basis()``. Check all combinations of parameters::
 300
 301     sage: set_random_seed()
 302     sage: K = random_cone(max_ambient_dim=8,
 303     ....:                 strictly_convex=True,
 304     ....:                 solid=True)
 305     sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
 306     True
 307
 308 ::
 309
 310     sage: set_random_seed()
 311     sage: K = random_cone(max_ambient_dim=8,
 312     ....:                 strictly_convex=True,
 313     ....:                 solid=False)
 314     sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
 315     True
 316
 317 ::
 318
 319     sage: set_random_seed()
 320     sage: K = random_cone(max_ambient_dim=8,
 321     ....:                 strictly_convex=False,
 322     ....:                 solid=True)
 323     sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
 324     True
 325
 326 ::
 327
 328     sage: set_random_seed()
 329     sage: K = random_cone(max_ambient_dim=8,
 330     ....:                 strictly_convex=False,
 331     ....:                 solid=False)
 332     sage: lyapunov_rank(K) == len(K.lyapunov_like_basis())
 333     True
 334
 335 """