X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=777d45e1a3e7a72a5ebbb167fd1ef6540c745282;hb=81a763e35b3e4322be6c60a815064be1f0dfcc3c;hp=23043381bca83acd6d7f17479ef4769980b9960f;hpb=012175f3a2591586099b4e955bb440958f2d63d7;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 2304338..777d45e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,6 +8,78 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
+def discrete_complementarity_set(K):
+    r"""
+    Compute the discrete complementarity set of this cone.
+
+    The complementarity set of this cone is the set of all orthogonal
+    pairs `(x,s)` such that `x` is in this cone, and `s` is in its
+    dual. The discrete complementarity set restricts `x` and `s` to be
+    generators of their respective cones.
+
+    OUTPUT:
+
+    A list of pairs `(x,s)` such that,
+
+      * `x` is in this cone.
+      * `x` is a generator of this cone.
+      * `s` is in this cone's dual.
+      * `s` is a generator of this cone's dual.
+      * `x` and `s` are orthogonal.
+
+    EXAMPLES:
+
+    The discrete complementarity set of the nonnegative orthant consists
+    of pairs of standard basis vectors::
+
+        sage: K = Cone([(1,0),(0,1)])
+        sage: discrete_complementarity_set(K)
+        [((1, 0), (0, 1)), ((0, 1), (1, 0))]
+
+    If the cone consists of a single ray, the second components of the
+    discrete complementarity set should generate the orthogonal
+    complement of that ray::
+
+        sage: K = Cone([(1,0)])
+        sage: discrete_complementarity_set(K)
+        [((1, 0), (0, 1)), ((1, 0), (0, -1))]
+        sage: K = Cone([(1,0,0)])
+        sage: discrete_complementarity_set(K)
+        [((1, 0, 0), (0, 1, 0)),
+          ((1, 0, 0), (0, -1, 0)),
+          ((1, 0, 0), (0, 0, 1)),
+          ((1, 0, 0), (0, 0, -1))]
+
+    When the cone is the entire space, its dual is the trivial cone, so
+    the discrete complementarity set is empty::
+
+        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+        sage: discrete_complementarity_set(K)
+        []
+
+    TESTS:
+
+    The complementarity set of the dual can be obtained by switching the
+    components of the complementarity set of the original cone::
+
+        sage: K1 = random_cone(max_dim=10, max_rays=10)
+        sage: K2 = K1.dual()
+        sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)]
+        sage: actual = discrete_complementarity_set(K1)
+        sage: actual == expected
+        True
+
+    """
+    V = K.lattice().vector_space()
+
+    # Convert the rays to vectors so that we can compute inner
+    # products.
+    xs = [V(x) for x in K.rays()]
+    ss = [V(s) for s in K.dual().rays()]
+
+    return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+
+
 def lyapunov_rank(K):
     r"""
     Compute the Lyapunov (or bilinearity) rank of this cone.
@@ -108,17 +180,29 @@ def lyapunov_rank(K):
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
+    TESTS:
+
+    The Lyapunov rank should be additive on a product of cones::
+
+        sage: K1 = random_cone(max_dim=10, max_rays=10)
+        sage: K2 = random_cone(max_dim=10, max_rays=10)
+        sage: K = K1.cartesian_product(K2)
+        sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
+        True
+
+    The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
+    itself::
+
+        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
     """
     V = K.lattice().vector_space()
 
-    xs = [V(x) for x in K.rays()]
-    ss = [V(s) for s in K.dual().rays()]
-
-    # WARNING: This isn't really C(K), it only contains the pairs
-    # (x,s) in C(K) where x,s are extreme in their respective cones.
-    C_of_K = [(x,s) for x in xs for s in ss if x.inner_product(s) == 0]
+    C_of_K = discrete_complementarity_set(K)
 
-    matrices = [x.column() * s.row() for (x,s) in C_of_K]
+    matrices = [x.tensor_product(s) for (x,s) in C_of_K]
 
     # Sage doesn't think matrices are vectors, so we have to convert
     # our matrices to vectors explicitly before we can figure out how