mjo/cone/cone.py

   1 from sage.all import *
   2
   3 def is_lyapunov_like(L,K):
   4     r"""
   5     Determine whether or not ``L`` is Lyapunov-like on ``K``.
   6
   7     We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle
   8     L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs
   9     `\left\langle x,s \right\rangle` in the complementarity set of
  10     ``K``. It is known [Orlitzky]_ that this property need only be
  11     checked for generators of ``K`` and its dual.
  12
  13     INPUT:
  14
  15     - ``L`` -- A linear transformation or matrix.
  16
  17     - ``K`` -- A polyhedral closed convex cone.
  18
  19     OUTPUT:
  20
  21     ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
  22     and ``False`` otherwise.
  23
  24     .. WARNING::
  25
  26         If this function returns ``True``, then ``L`` is Lyapunov-like
  27         on ``K``. However, if ``False`` is returned, that could mean one
  28         of two things. The first is that ``L`` is definitely not
  29         Lyapunov-like on ``K``. The second is more of an "I don't know"
  30         answer, returned (for example) if we cannot prove that an inner
  31         product is zero.
  32
  33     REFERENCES:
  34
  35     M. Orlitzky. The Lyapunov rank of an improper cone.
  36     http://www.optimization-online.org/DB_HTML/2015/10/5135.html
  37
  38     EXAMPLES:
  39
  40     The identity is always Lyapunov-like in a nontrivial space::
  41
  42         sage: set_random_seed()
  43         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
  44         sage: L = identity_matrix(K.lattice_dim())
  45         sage: is_lyapunov_like(L,K)
  46         True
  47
  48     As is the "zero" transformation::
  49
  50         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
  51         sage: R = K.lattice().vector_space().base_ring()
  52         sage: L = zero_matrix(R, K.lattice_dim())
  53         sage: is_lyapunov_like(L,K)
  54         True
  55
  56         Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
  57         on ``K``::
  58
  59         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
  60         sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
  61         True
  62
  63     """
  64     return all([(L*x).inner_product(s) == 0
  65                 for (x,s) in K.discrete_complementarity_set()])
  66
  67
  68 def motzkin_decomposition(K):
  69     r"""
  70     Return the pair of components in the Motzkin decomposition of this cone.
  71
  72     Every convex cone is the direct sum of a strictly convex cone and a
  73     linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
  74     such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
  75     is the direct sum of ``P`` and ``S``.
  76
  77     OUTPUT:
  78
  79     An ordered pair ``(P,S)`` of closed convex polyhedral cones where
  80     ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
  81     direct sum of ``P`` and ``S``.
  82
  83     REFERENCES:
  84
  85     .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
  86        Optimization in Finite Dimensions I. Springer-Verlag, New
  87        York, 1970.
  88
  89     EXAMPLES:
  90
  91     The nonnegative orthant is strictly convex, so it is its own
  92     strictly convex component and its subspace component is trivial::
  93
  94         sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
  95         sage: (P,S) = motzkin_decomposition(K)
  96         sage: K.is_equivalent(P)
  97         True
  98         sage: S.is_trivial()
  99         True
 100
 101     Likewise, full spaces are their own subspace components::
 102
 103         sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
 104         sage: K.is_full_space()
 105         True
 106         sage: (P,S) = motzkin_decomposition(K)
 107         sage: K.is_equivalent(S)
 108         True
 109         sage: P.is_trivial()
 110         True
 111
 112     TESTS:
 113
 114     A random point in the cone should belong to either the strictly
 115     convex component or the subspace component. If the point is nonzero,
 116     it cannot be in both::
 117
 118         sage: set_random_seed()
 119         sage: K = random_cone(max_ambient_dim=8)
 120         sage: (P,S) = motzkin_decomposition(K)
 121         sage: x = K.random_element()
 122         sage: P.contains(x) or S.contains(x)
 123         True
 124         sage: x.is_zero() or (P.contains(x) != S.contains(x))
 125         True
 126
 127     The strictly convex component should always be strictly convex, and
 128     the subspace component should always be a subspace::
 129
 130         sage: set_random_seed()
 131         sage: K = random_cone(max_ambient_dim=8)
 132         sage: (P,S) = motzkin_decomposition(K)
 133         sage: P.is_strictly_convex()
 134         True
 135         sage: S.lineality() == S.dim()
 136         True
 137
 138     The generators of the components are obtained from orthogonal
 139     projections of the original generators [Stoer-Witzgall]_::
 140
 141         sage: set_random_seed()
 142         sage: K = random_cone(max_ambient_dim=8)
 143         sage: (P,S) = motzkin_decomposition(K)
 144         sage: A = S.linear_subspace().complement().matrix()
 145         sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
 146         sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
 147         sage: P.is_equivalent(expected_P)
 148         True
 149         sage: A = S.linear_subspace().matrix()
 150         sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
 151         sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
 152         sage: S.is_equivalent(expected_S)
 153         True
 154     """
 155     # The lines() method only returns one generator per line. For a true
 156     # line, we also need a generator pointing in the opposite direction.
 157     S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
 158     S = Cone(S_gens, K.lattice())
 159
 160     # Since ``S`` is a subspace, the rays of its dual generate its
 161     # orthogonal complement.
 162     S_perp = Cone(S.dual(), K.lattice())
 163     P = K.intersection(S_perp)
 164
 165     return (P,S)
 166
 167
 168 def positive_operator_gens(K):
 169     r"""
 170     Compute generators of the cone of positive operators on this cone.
 171
 172     OUTPUT:
 173
 174     A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
 175     Each matrix ``P`` in the list should have the property that ``P*x``
 176     is an element of ``K`` whenever ``x`` is an element of
 177     ``K``. Moreover, any nonnegative linear combination of these
 178     matrices shares the same property.
 179
 180     EXAMPLES:
 181
 182     The trivial cone in a trivial space has no positive operators::
 183
 184         sage: K = Cone([], ToricLattice(0))
 185         sage: positive_operator_gens(K)
 186         []
 187
 188     Positive operators on the nonnegative orthant are nonnegative matrices::
 189
 190         sage: K = Cone([(1,)])
 191         sage: positive_operator_gens(K)
 192         [[1]]
 193
 194         sage: K = Cone([(1,0),(0,1)])
 195         sage: positive_operator_gens(K)
 196         [
 197         [1 0]  [0 1]  [0 0]  [0 0]
 198         [0 0], [0 0], [1 0], [0 1]
 199         ]
 200
 201     Every operator is positive on the ambient vector space::
 202
 203         sage: K = Cone([(1,),(-1,)])
 204         sage: K.is_full_space()
 205         True
 206         sage: positive_operator_gens(K)
 207         [[1], [-1]]
 208
 209         sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
 210         sage: K.is_full_space()
 211         True
 212         sage: positive_operator_gens(K)
 213         [
 214         [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
 215         [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
 216         ]
 217
 218     TESTS:
 219
 220     A positive operator on a cone should send its generators into the cone::
 221
 222         sage: set_random_seed()
 223         sage: K = random_cone(max_ambient_dim=5)
 224         sage: pi_of_K = positive_operator_gens(K)
 225         sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
 226         True
 227
 228     The dimension of the cone of positive operators is given by the
 229     corollary in my paper::
 230
 231         sage: set_random_seed()
 232         sage: K = random_cone(max_ambient_dim=5)
 233         sage: n = K.lattice_dim()
 234         sage: m = K.dim()
 235         sage: l = K.lineality()
 236         sage: pi_of_K = positive_operator_gens(K)
 237         sage: L = ToricLattice(n**2)
 238         sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
 239         sage: expected = n**2 - l*(m - l) - (n - m)*m
 240         sage: actual == expected
 241         True
 242
 243     The lineality of the cone of positive operators is given by the
 244     corollary in my paper::
 245
 246         sage: set_random_seed()
 247         sage: K = random_cone(max_ambient_dim=5)
 248         sage: n = K.lattice_dim()
 249         sage: pi_of_K = positive_operator_gens(K)
 250         sage: L = ToricLattice(n**2)
 251         sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
 252         sage: expected = n**2 - K.dim()*K.dual().dim()
 253         sage: actual == expected
 254         True
 255
 256     The cone ``K`` is proper if and only if the cone of positive
 257     operators on ``K`` is proper::
 258
 259         sage: set_random_seed()
 260         sage: K = random_cone(max_ambient_dim=5)
 261         sage: pi_of_K = positive_operator_gens(K)
 262         sage: L = ToricLattice(K.lattice_dim()**2)
 263         sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
 264         sage: K.is_proper() == pi_cone.is_proper()
 265         True
 266     """
 267     # Matrices are not vectors in Sage, so we have to convert them
 268     # to vectors explicitly before we can find a basis. We need these
 269     # two values to construct the appropriate "long vector" space.
 270     F = K.lattice().base_field()
 271     n = K.lattice_dim()
 272
 273     tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ]
 274
 275     # Convert those tensor products to long vectors.
 276     W = VectorSpace(F, n**2)
 277     vectors = [ W(tp.list()) for tp in tensor_products ]
 278
 279     # Create the *dual* cone of the positive operators, expressed as
 280     # long vectors..
 281     pi_dual = Cone(vectors, ToricLattice(W.dimension()))
 282
 283     # Now compute the desired cone from its dual...
 284     pi_cone = pi_dual.dual()
 285
 286     # And finally convert its rays back to matrix representations.
 287     M = MatrixSpace(F, n)
 288     return [ M(v.list()) for v in pi_cone.rays() ]
 289
 290
 291 def Z_transformation_gens(K):
 292     r"""
 293     Compute generators of the cone of Z-transformations on this cone.
 294
 295     OUTPUT:
 296
 297     A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
 298     Each matrix ``L`` in the list should have the property that
 299     ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
 300     discrete complementarity set of ``K``. Moreover, any nonnegative
 301     linear combination of these matrices shares the same property.
 302
 303     EXAMPLES:
 304
 305     Z-transformations on the nonnegative orthant are just Z-matrices.
 306     That is, matrices whose off-diagonal elements are nonnegative::
 307
 308         sage: K = Cone([(1,0),(0,1)])
 309         sage: Z_transformation_gens(K)
 310         [
 311         [ 0 -1]  [ 0  0]  [-1  0]  [1 0]  [ 0  0]  [0 0]
 312         [ 0  0], [-1  0], [ 0  0], [0 0], [ 0 -1], [0 1]
 313         ]
 314         sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)])
 315         sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K)
 316         ....:                    for i in range(z.nrows())
 317         ....:                    for j in range(z.ncols())
 318         ....:                    if i != j ])
 319         True
 320
 321     The trivial cone in a trivial space has no Z-transformations::
 322
 323         sage: K = Cone([], ToricLattice(0))
 324         sage: Z_transformation_gens(K)
 325         []
 326
 327     Z-transformations on a subspace are Lyapunov-like and vice-versa::
 328
 329         sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
 330         sage: K.is_full_space()
 331         True
 332         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
 333         sage: zs  = span([ vector(z.list()) for z in Z_transformation_gens(K) ])
 334         sage: zs == lls
 335         True
 336
 337     TESTS:
 338
 339     The Z-property is possessed by every Z-transformation::
 340
 341         sage: set_random_seed()
 342         sage: K = random_cone(max_ambient_dim=6)
 343         sage: Z_of_K = Z_transformation_gens(K)
 344         sage: dcs = K.discrete_complementarity_set()
 345         sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
 346         ....:                                  for (x,s) in dcs])
 347         True
 348
 349     The lineality space of Z is LL::
 350
 351         sage: set_random_seed()
 352         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
 353         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
 354         sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
 355         sage: z_cone.linear_subspace() == lls
 356         True
 357
 358     And thus, the lineality of Z is the Lyapunov rank::
 359
 360         sage: set_random_seed()
 361         sage: K = random_cone(max_ambient_dim=6)
 362         sage: Z_of_K = Z_transformation_gens(K)
 363         sage: L = ToricLattice(K.lattice_dim()**2)
 364         sage: z_cone  = Cone([ z.list() for z in Z_of_K ], lattice=L)
 365         sage: z_cone.lineality() == K.lyapunov_rank()
 366         True
 367
 368     The lineality spaces of pi-star and Z-star are equal:
 369
 370         sage: set_random_seed()
 371         sage: K = random_cone(max_ambient_dim=5)
 372         sage: pi_of_K = positive_operator_gens(K)
 373         sage: Z_of_K = Z_transformation_gens(K)
 374         sage: L = ToricLattice(K.lattice_dim()**2)
 375         sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual()
 376         sage: z_star  = Cone([ z.list() for z in Z_of_K], lattice=L).dual()
 377         sage: pi_star.linear_subspace() == z_star.linear_subspace()
 378         True
 379     """
 380     # Matrices are not vectors in Sage, so we have to convert them
 381     # to vectors explicitly before we can find a basis. We need these
 382     # two values to construct the appropriate "long vector" space.
 383     F = K.lattice().base_field()
 384     n = K.lattice_dim()
 385
 386     # These tensor products contain generators for the dual cone of
 387     # the cross-positive transformations.
 388     tensor_products = [ s.tensor_product(x)
 389                         for (x,s) in K.discrete_complementarity_set() ]
 390
 391     # Turn our matrices into long vectors...
 392     W = VectorSpace(F, n**2)
 393     vectors = [ W(m.list()) for m in tensor_products ]
 394
 395     # Create the *dual* cone of the cross-positive operators,
 396     # expressed as long vectors..
 397     Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
 398
 399     # Now compute the desired cone from its dual...
 400     Sigma_cone = Sigma_dual.dual()
 401
 402     # And finally convert its rays back to matrix representations.
 403     # But first, make them negative, so we get Z-transformations and
 404     # not cross-positive ones.
 405     M = MatrixSpace(F, n)
 406     return [ -M(v.list()) for v in Sigma_cone.rays() ]
 407
 408
 409 def Z_cone(K):
 410     gens = Z_transformation_gens(K)
 411     L = None
 412     if len(gens) == 0:
 413         L = ToricLattice(0)
 414     return Cone([ g.list() for g in gens ], lattice=L)
 415
 416 def pi_cone(K):
 417     gens = positive_operator_gens(K)
 418     L = None
 419     if len(gens) == 0:
 420         L = ToricLattice(0)
 421     return Cone([ g.list() for g in gens ], lattice=L)