X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=421fb3c8f04abb23fd420271ed10e0a7e65815d4;hb=f8e779ee533992a940e1014d00960fc58c3c4b79;hp=2e3dc8afb78cd1f4e1d5f368eb5f75733d127e82;hpb=433b47ffeac9305beb9101afa302c7e924b60744;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 2e3dc8a..421fb3c 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,6 +8,58 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
+def project_span(K):
+    r"""
+    Project ``K`` into its own span.
+
+    EXAMPLES::
+
+        sage: K = Cone([(1,)])
+        sage: project_span(K) == K
+        True
+
+        sage: K2 = Cone([(1,0)])
+        sage: project_span(K2).rays()
+        N(1)
+        in 1-d lattice N
+        sage: K3 = Cone([(1,0,0)])
+        sage: project_span(K3).rays()
+        N(1)
+        in 1-d lattice N
+        sage: project_span(K2) == project_span(K3)
+        True
+
+    TESTS:
+
+    The projected cone should always be solid::
+
+        sage: K = random_cone(max_dim = 10)
+        sage: K_S = project_span(K)
+        sage: K_S.is_solid()
+        True
+
+    If we do this according to our paper, then the result is proper::
+
+        sage: K = random_cone(max_dim = 10)
+        sage: K_S = project_span(K)
+        sage: P = project_span(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+
+    """
+    L = K.lattice()
+    F = L.base_field()
+    Q = L.quotient(K.sublattice_complement())
+    vecs = [ vector(F, reversed(list(Q(r)))) for r in K.rays() ]
+
+    newL = None
+    if len(vecs) == 0:
+        newL = ToricLattice(0)
+
+    return Cone(vecs, lattice=newL)
+
+
+
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -87,8 +139,70 @@ def LL(K):
 
     OUTPUT:
 
-    A ``MatrixSpace`` object `M` such that every matrix `L \in M` is
-    Lyapunov-like on this cone.
+    A list of matrices forming a basis for the space of all
+    Lyapunov-like transformations on the given cone.
+
+    EXAMPLES:
+
+    The trivial cone has no Lyapunov-like transformations::
+
+        sage: L = ToricLattice(0)
+        sage: K = Cone([], lattice=L)
+        sage: LL(K)
+        []
+
+    The Lyapunov-like transformations on the nonnegative orthant are
+    simply diagonal matrices::
+
+        sage: K = Cone([(1,)])
+        sage: LL(K)
+        [[1]]
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: LL(K)
+        [
+        [1 0]  [0 0]
+        [0 0], [0 1]
+        ]
+
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        sage: LL(K)
+        [
+        [1 0 0]  [0 0 0]  [0 0 0]
+        [0 0 0]  [0 1 0]  [0 0 0]
+        [0 0 0], [0 0 0], [0 0 1]
+        ]
+
+    Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and
+    `L^{3}_{\infty}` cones [Rudolf et al.]_::
+
+        sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
+        sage: LL(L31)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+        sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)])
+        sage: LL(L3infty)
+        [
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
+        ]
+
+    TESTS:
+
+    The inner product `\left< L\left(x\right), s \right>` is zero for
+    every pair `\left( x,s \right)` in the discrete complementarity set
+    of the cone::
+
+        sage: K = random_cone(max_dim=8, max_rays=10)
+        sage: C_of_K = discrete_complementarity_set(K)
+        sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ]
+        sage: sum(map(abs, l))
+        0
 
     """
     V = K.lattice().vector_space()
@@ -171,6 +285,9 @@ def lyapunov_rank(K):
        cone and Lyapunov-like transformations, Mathematical Programming, 147
        (2014) 155-170.
 
+    .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an
+       Improper Cone. Work in-progress.
+
     .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear
        optimality constraints for the cone of positive polynomials,
        Mathematical Programming, Series B, 129 (2011) 5-31.
@@ -190,6 +307,15 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(octant)
         3
 
+    The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}`
+    [Orlitzky/Gowda]_::
+
+        sage: R5 = VectorSpace(QQ, 5)
+        sage: gens = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gens)
+        sage: lyapunov_rank(K)
+        25
+
     The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
     [Rudolf et al.]_::
 
@@ -203,7 +329,30 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(L3infty)
         1
 
-    The Lyapunov rank should be additive on a product of cones
+    A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n
+    + 1` [Orlitzky/Gowda]_::
+
+        sage: K = Cone([(1,0,0,0,0)])
+        sage: lyapunov_rank(K)
+        21
+        sage: K.lattice_dim()**2 - K.lattice_dim() + 1
+        21
+
+    A subspace (of dimension `m`) in `n` dimensions should have a
+    Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_::
+
+        sage: e1 = (1,0,0,0,0)
+        sage: neg_e1 = (-1,0,0,0,0)
+        sage: e2 = (0,1,0,0,0)
+        sage: neg_e2 = (0,-1,0,0,0)
+        sage: zero = (0,0,0,0,0)
+        sage: K = Cone([e1, neg_e1, e2, neg_e2, zero, zero, zero])
+        sage: lyapunov_rank(K)
+        19
+        sage: K.lattice_dim()**2 - K.dim()*(K.lattice_dim() - K.dim())
+        19
+
+    The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
         sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)])
@@ -229,11 +378,11 @@ def lyapunov_rank(K):
 
     TESTS:
 
-    The Lyapunov rank should be additive on a product of cones
+    The Lyapunov rank should be additive on a product of proper cones
     [Rudolf et al.]_::
 
-        sage: K1 = random_cone(max_dim=10, max_rays=10)
-        sage: K2 = random_cone(max_dim=10, max_rays=10)
+        sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True)
         sage: K = K1.cartesian_product(K2)
         sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
         True
@@ -259,5 +408,54 @@ def lyapunov_rank(K):
         sage: b == n-1
         False
 
+    In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have
+    Lyapunov rank `n-1` in `n` dimensions::
+
+        sage: K = random_cone(max_dim=10)
+        sage: b = lyapunov_rank(K)
+        sage: n = K.lattice_dim()
+        sage: b == n-1
+        False
+
+    The calculation of the Lyapunov rank of an improper cone can be
+    reduced to that of a proper cone [Orlitzky/Gowda]_::
+
+        sage: K = random_cone(max_dim=10)
+        sage: actual = lyapunov_rank(K)
+        sage: K_S = project_span(K)
+        sage: P = project_span(K_S.dual()).dual()
+        sage: l = K.linear_subspace().dimension()
+        sage: codim = K.lattice_dim() - K.dim()
+        sage: expected = lyapunov_rank(P) + K.dim()*(l + codim) + codim**2
+        sage: actual == expected
+        True
+
+    The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
+
+        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
     """
-    return len(LL(K))
+    beta = 0
+
+    m = K.dim()
+    n = K.lattice_dim()
+    l = K.linear_subspace().dimension()
+
+    if m < n:
+        # K is not solid, project onto its span.
+        K = project_span(K)
+
+        # Lemma 2
+        beta += m*(n - m) + (n - m)**2
+
+    if l > 0:
+        # K is not pointed, project its dual onto its span.
+        K = project_span(K.dual()).dual()
+
+        # Lemma 3
+        beta += m * l
+
+    beta += len(LL(K))
+    return beta