X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=81698e467c603df3e57cb764c256a2e082df5ec6;hb=b966ec383942ef1ff7837786c29d1f3edc33b84e;hp=4b0193692edd7655f2880408d143bf141e6d567c;hpb=3a312d50b5aba08e72039f1ebcde7b12c62a1e9f;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 4b01936..81698e4 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, K2 = None):
+    r"""
+    Return a "copy" of ``K`` embeded in a lower-dimensional space.
+
+    By default, we will project ``K`` into the subspace spanned by its
+    rays. However, if ``K2`` is not ``None``, we will project into the
+    space spanned by the rays of ``K2`` instead.
+
+    EXAMPLES::
+
+        sage: K = Cone([(1,0,0), (0,1,0)])
+        sage: project_span(K)
+        2-d cone in 2-d lattice N
+        sage: project_span(K).rays()
+        N(1, 0),
+        N(0, 1)
+        in 2-d lattice N
+
+        sage: K = Cone([(1,0,0), (0,1,0)])
+        sage: K2 = Cone([(0,1)])
+        sage: project_span(K, K2).rays()
+        N(1)
+        in 1-d lattice N
+
+    """
+    # Allow us to use a second cone to generate the subspace into
+    # which we're "projecting."
+    if K2 is None:
+        K2 = K
+
+    # Use these to generate the new cone.
+    cs1 = K.rays().matrix().columns()
+
+    # And use these to figure out which indices to drop.
+    cs2 = K2.rays().matrix().columns()
+
+    perp_idxs = []
+
+    for idx in range(0, len(cs2)):
+        if cs2[idx].is_zero():
+            perp_idxs.append(idx)
+
+    solid_cols = [ cs1[idx] for idx in range(0,len(cs1))
+                            if not idx in perp_idxs
+                            and not idx >= len(cs2) ]
+
+    m = matrix(solid_cols)
+    L = ToricLattice(len(m.rows()))
+    J = Cone(m.transpose(), lattice=L)
+    return J
+
+
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -207,7 +259,7 @@ def lyapunov_rank(K):
     An integer representing the Lyapunov rank of the cone. If the
     dimension of the ambient vector space is `n`, then the Lyapunov rank
     will be between `1` and `n` inclusive; however a rank of `n-1` is
-    not possible (see the first reference).
+    not possible for any cone.
 
     .. note::
 
@@ -233,6 +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.
@@ -321,5 +376,32 @@ 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, max_rays=16)
+        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=15, max_rays=25)
+        sage: actual = lyapunov_rank(K)
+        sage: K_S = project_span(K)
+        sage: J_T1 = project_span(K_S.dual()).dual()
+        sage: J_T2 = project_span(K, K_S.dual())
+        sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice())
+        sage: J_T1 == J_T2
+        True
+        sage: J_T = J_T1
+        sage: l = K.linear_subspace().dimension()
+        sage: codim = K.lattice_dim() - K.dim()
+        sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2
+        sage: actual == expected
+        True
+
     """
     return len(LL(K))