X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=28a84231d1eb461a1e12d0df258f980e48f21f2d;hb=b60f53acc5d274d07d836fc473ebeec77a3a953f;hp=a5f5f2f4fcf1ac603a9d8d48a75a0924003bd8cf;hpb=cd746d047748a9aa56c51bb78c3db9882ea16a16;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index a5f5f2f..28a8423 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -65,80 +65,104 @@ def is_lyapunov_like(L,K):
                 for (x,s) in K.discrete_complementarity_set()])
 
 
-def random_element(K):
+def motzkin_decomposition(K):
     r"""
-    Return a random element of ``K`` from its ambient vector space.
+    Return the pair of components in the Motzkin decomposition of this cone.
 
-    ALGORITHM:
+    Every convex cone is the direct sum of a strictly convex cone and a
+    linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones
+    such that ``P`` is strictly convex, ``S`` is a subspace, and ``K``
+    is the direct sum of ``P`` and ``S``.
 
-    The cone ``K`` is specified in terms of its generators, so that
-    ``K`` is equal to the convex conic combination of those generators.
-    To choose a random element of ``K``, we assign random nonnegative
-    coefficients to each generator of ``K`` and construct a new vector
-    from the scaled rays.
+    OUTPUT:
 
-    A vector, rather than a ray, is returned so that the element may
-    have non-integer coordinates. Thus the element may have an
-    arbitrarily small norm.
+    An ordered pair ``(P,S)`` of closed convex polyhedral cones where
+    ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the
+    direct sum of ``P`` and ``S``.
 
-    EXAMPLES:
+    REFERENCES:
 
-    A random element of the trivial cone is zero::
+    .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
+       Optimization in Finite Dimensions I. Springer-Verlag, New
+       York, 1970.
 
-        sage: set_random_seed()
-        sage: K = Cone([], ToricLattice(0))
-        sage: random_element(K)
-        ()
-        sage: K = Cone([(0,)])
-        sage: random_element(K)
-        (0)
-        sage: K = Cone([(0,0)])
-        sage: random_element(K)
-        (0, 0)
-        sage: K = Cone([(0,0,0)])
-        sage: random_element(K)
-        (0, 0, 0)
-
-    A random element of the nonnegative orthant should have all
-    components nonnegative::
+    EXAMPLES:
+
+    The nonnegative orthant is strictly convex, so it is its own
+    strictly convex component and its subspace component is trivial::
 
-        sage: set_random_seed()
         sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
-        sage: all([ x >= 0 for x in random_element(K) ])
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: K.is_equivalent(P)
+        True
+        sage: S.is_trivial()
+        True
+
+    Likewise, full spaces are their own subspace components::
+
+        sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+        sage: K.is_full_space()
+        True
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: K.is_equivalent(S)
+        True
+        sage: P.is_trivial()
         True
 
     TESTS:
 
-    Any cone should contain a random element of itself::
+    A random point in the cone should belong to either the strictly
+    convex component or the subspace component. If the point is nonzero,
+    it cannot be in both::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
-        sage: K.contains(random_element(K))
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: x = K.random_element()
+        sage: P.contains(x) or S.contains(x)
+        True
+        sage: x.is_zero() or (P.contains(x) != S.contains(x))
         True
 
-    A strictly convex cone contains no lines, and thus no negative
-    multiples of any of its elements besides zero::
+    The strictly convex component should always be strictly convex, and
+    the subspace component should always be a subspace::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim=8, strictly_convex=True)
-        sage: x = random_element(K)
-        sage: x.is_zero() or not K.contains(-x)
+        sage: K = random_cone(max_ambient_dim=8)
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: P.is_strictly_convex()
+        True
+        sage: S.lineality() == S.dim()
         True
 
-    The sum of random elements of a cone lies in the cone::
+    The generators of the components are obtained from orthogonal
+    projections of the original generators [Stoer-Witzgall]_::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8)
-        sage: K.contains(sum([random_element(K) for i in range(10)]))
+        sage: (P,S) = motzkin_decomposition(K)
+        sage: A = S.linear_subspace().complement().matrix()
+        sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A
+        sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice())
+        sage: P.is_equivalent(expected_P)
+        True
+        sage: A = S.linear_subspace().matrix()
+        sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A
+        sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice())
+        sage: S.is_equivalent(expected_S)
         True
-
     """
-    V = K.lattice().vector_space()
-    scaled_gens = [ V.base_field().random_element().abs()*V(r) for r in K ]
+    # The lines() method only returns one generator per line. For a true
+    # line, we also need a generator pointing in the opposite direction.
+    S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
+    S = Cone(S_gens, K.lattice())
+
+    # Since ``S`` is a subspace, the rays of its dual generate its
+    # orthogonal complement.
+    S_perp = Cone(S.dual(), K.lattice())
+    P = K.intersection(S_perp)
 
-    # Make sure we return a vector. Without the coercion, we might
-    # return ``0`` when ``K`` has no rays.
-    return V(sum(scaled_gens))
+    return (P,S)
 
 
 def positive_operator_gens(K):