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     There are faster ways of checking this property. For example, we
  14     could compute a `lyapunov_like_basis` of the cone, and then test
  15     whether or not the given matrix is contained in the span of that
  16     basis. The value of this function is that it works on symbolic
  17     matrices.
  18
  19     INPUT:
  20
  21     - ``L`` -- A linear transformation or matrix.
  22
  23     - ``K`` -- A polyhedral closed convex cone.
  24
  25     OUTPUT:
  26
  27     ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``,
  28     and ``False`` otherwise.
  29
  30     .. WARNING::
  31
  32         If this function returns ``True``, then ``L`` is Lyapunov-like
  33         on ``K``. However, if ``False`` is returned, that could mean one
  34         of two things. The first is that ``L`` is definitely not
  35         Lyapunov-like on ``K``. The second is more of an "I don't know"
  36         answer, returned (for example) if we cannot prove that an inner
  37         product is zero.
  38
  39     REFERENCES:
  40
  41     M. Orlitzky. The Lyapunov rank of an improper cone.
  42     http://www.optimization-online.org/DB_HTML/2015/10/5135.html
  43
  44     EXAMPLES:
  45
  46     The identity is always Lyapunov-like in a nontrivial space::
  47
  48         sage: set_random_seed()
  49         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
  50         sage: L = identity_matrix(K.lattice_dim())
  51         sage: is_lyapunov_like(L,K)
  52         True
  53
  54     As is the "zero" transformation::
  55
  56         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
  57         sage: R = K.lattice().vector_space().base_ring()
  58         sage: L = zero_matrix(R, K.lattice_dim())
  59         sage: is_lyapunov_like(L,K)
  60         True
  61
  62         Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
  63         on ``K``::
  64
  65         sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
  66         sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
  67         True
  68
  69     """
  70     return all([(L*x).inner_product(s) == 0
  71                 for (x,s) in K.discrete_complementarity_set()])
  72
  73 def LL_cone(K):
  74     gens = K.lyapunov_like_basis()
  75     L = ToricLattice(K.lattice_dim()**2)
  76     return Cone([ g.list() for g in gens ], lattice=L, check=False)
  77
  78 def Sigma_cone(K):
  79     gens = K.cross_positive_operator_gens()
  80     L = ToricLattice(K.lattice_dim()**2)
  81     return Cone([ g.list() for g in gens ], lattice=L, check=False)
  82
  83 def Z_cone(K):
  84     gens = K.Z_operator_gens()
  85     L = ToricLattice(K.lattice_dim()**2)
  86     return Cone([ g.list() for g in gens ], lattice=L, check=False)
  87
  88 def pi_cone(K1, K2=None):
  89     if K2 is None:
  90         K2 = K1
  91     gens = K1.positive_operator_gens(K2)
  92     L = ToricLattice(K1.lattice_dim()*K2.lattice_dim())
  93     return Cone([ g.list() for g in gens ], lattice=L, check=False)