X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=8790c30673a25af96c29bf22ff9e84458516da09;hb=744bc420b47449500f0365e093467ffd268aeda2;hp=702472c30466f7cba79d4a79f1bf47a552079364;hpb=1ce1d354d9442cf64cb61d41b3354f55b8a4331d;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 702472c..8790c30 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -40,14 +40,14 @@ def is_lyapunov_like(L,K):
     The identity is always Lyapunov-like in a nontrivial space::
 
         sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
         sage: L = identity_matrix(K.lattice_dim())
         sage: is_lyapunov_like(L,K)
         True
 
     As is the "zero" transformation::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 8)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8)
         sage: R = K.lattice().vector_space().base_ring()
         sage: L = zero_matrix(R, K.lattice_dim())
         sage: is_lyapunov_like(L,K)
@@ -56,7 +56,7 @@ def is_lyapunov_like(L,K):
         Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like
         on ``K``::
 
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
         sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ])
         True
 
@@ -65,61 +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:
+
+    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``.
+
+    REFERENCES:
 
-    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.
+    .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and
+       Optimization in Finite Dimensions I. Springer-Verlag, New
+       York, 1970.
 
     EXAMPLES:
 
-    A random element of the trivial cone is zero::
+    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([], 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)
+        sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)])
+        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 an 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: (P,S) = motzkin_decomposition(K)
+        sage: x = K.random_element(ring=QQ)
+        sage: P.contains(x) or S.contains(x)
+        True
+        sage: x.is_zero() or (P.contains(x) != S.contains(x))
+        True
+
+    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_rays = 8)
-        sage: K.contains(random_element(K))
+        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 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: (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()
-    F = V.base_ring()
-    coefficients = [ F.random_element().abs() for i in range(K.nrays()) ]
-    vector_gens  = map(V, K.rays())
-    scaled_gens  = [ coefficients[i]*vector_gens[i]
-                         for i in range(len(vector_gens)) ]
+    # 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.
-    v = V(sum(scaled_gens))
-    return v
+    return (P,S)
 
 
 def positive_operator_gens(K):
@@ -136,12 +179,6 @@ def positive_operator_gens(K):
 
     EXAMPLES:
 
-    The trivial cone in a trivial space has no positive operators::
-
-        sage: K = Cone([], ToricLattice(0))
-        sage: positive_operator_gens(K)
-        []
-
     Positive operators on the nonnegative orthant are nonnegative matrices::
 
         sage: K = Cone([(1,)])
@@ -155,6 +192,27 @@ def positive_operator_gens(K):
         [0 0], [0 0], [1 0], [0 1]
         ]
 
+    The trivial cone in a trivial space has no positive operators::
+
+        sage: K = Cone([], ToricLattice(0))
+        sage: positive_operator_gens(K)
+        []
+
+    Every operator is positive on the trivial cone::
+
+        sage: K = Cone([(0,)])
+        sage: positive_operator_gens(K)
+        [[1], [-1]]
+
+        sage: K = Cone([(0,0)])
+        sage: K.is_trivial()
+        True
+        sage: positive_operator_gens(K)
+        [
+        [1 0]  [-1  0]  [0 1]  [ 0 -1]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
+        [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
+        ]
+
     Every operator is positive on the ambient vector space::
 
         sage: K = Cone([(1,),(-1,)])
@@ -172,21 +230,100 @@ def positive_operator_gens(K):
         [0 0], [ 0  0], [0 0], [ 0  0], [1 0], [-1  0], [0 1], [ 0 -1]
         ]
 
+    A non-obvious application is to find the positive operators on the
+    right half-plane::
+
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: positive_operator_gens(K)
+        [
+        [1 0]  [0 0]  [ 0  0]  [0 0]  [ 0  0]
+        [0 0], [1 0], [-1  0], [0 1], [ 0 -1]
+        ]
+
     TESTS:
 
-    A positive operator on a cone should send its generators into the cone::
+    Each positive operator generator should send the generators of the
+    cone into the cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: all([ K.contains(P*x) for P in pi_of_K for x in K ])
+        True
+
+    Each positive operator generator should send a random element of the
+    cone into the cone::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
         sage: pi_of_K = positive_operator_gens(K)
-        sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()])
+        sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ])
+        True
+
+    A random element of the positive operator cone should send the
+    generators of the cone into the cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+        sage: all([ K.contains(P*x) for x in K ])
+        True
+
+    A random element of the positive operator cone should send a random
+    element of the cone into the cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+        sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list())
+        sage: K.contains(P*K.random_element(ring=QQ))
+        True
+
+    The lineality space of the dual of the cone of positive operators
+    can be computed from the lineality spaces of the cone and its dual::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(K.lattice_dim()**2)
+        sage: pi_cone = Cone([ g.list() for g in pi_of_K ], lattice=L)
+        sage: actual = pi_cone.dual().linear_subspace()
+        sage: U1 = [ vector((s.tensor_product(x)).list())
+        ....:        for x in K.lines()
+        ....:        for s in K.dual() ]
+        sage: U2 = [ vector((s.tensor_product(x)).list())
+        ....:        for x in K
+        ....:        for s in K.dual().lines() ]
+        sage: expected = pi_cone.lattice().vector_space().span(U1 + U2)
+        sage: actual == expected
+        True
+
+    The lineality of the dual of the cone of positive operators
+    is known from its lineality space::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_ambient_dim=5)
+        sage: n = K.lattice_dim()
+        sage: m = K.dim()
+        sage: l = K.lineality()
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: L = ToricLattice(n**2)
+        sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+        sage: actual = pi_cone.dual().lineality()
+        sage: expected = l*(m - l) + m*(n - m)
+        sage: actual == expected
         True
 
     The dimension of the cone of positive operators is given by the
     corollary in my paper::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 5)
+        sage: K = random_cone(max_ambient_dim=5)
         sage: n = K.lattice_dim()
         sage: m = K.dim()
         sage: l = K.lineality()
@@ -197,8 +334,33 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
-    The lineality of the cone of positive operators is given by the
-    corollary in my paper::
+    The trivial cone, full space, and half-plane all give rise to the
+    expected dimensions::
+
+        sage: n = ZZ.random_element().abs()
+        sage: K = Cone([[0] * n], ToricLattice(n))
+        sage: K.is_trivial()
+        True
+        sage: L = ToricLattice(n^2)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+        sage: actual == n^2
+        True
+        sage: K = K.dual()
+        sage: K.is_full_space()
+        True
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim()
+        sage: actual == n^2
+        True
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K]).dim()
+        sage: actual == 3
+        True
+
+    The lineality of the cone of positive operators follows from the
+    description of its generators::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
@@ -210,8 +372,33 @@ def positive_operator_gens(K):
         sage: actual == expected
         True
 
-    The cone ``K`` is proper if and only if the cone of positive
-    operators on ``K`` is proper::
+    The trivial cone, full space, and half-plane all give rise to the
+    expected linealities::
+
+        sage: n = ZZ.random_element().abs()
+        sage: K = Cone([[0] * n], ToricLattice(n))
+        sage: K.is_trivial()
+        True
+        sage: L = ToricLattice(n^2)
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+        sage: actual == n^2
+        True
+        sage: K = K.dual()
+        sage: K.is_full_space()
+        True
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
+        sage: actual == n^2
+        True
+        sage: K = Cone([(1,0),(0,1),(0,-1)])
+        sage: pi_of_K = positive_operator_gens(K)
+        sage: actual = Cone([p.list() for p in pi_of_K]).lineality()
+        sage: actual == 2
+        True
+
+    A cone is proper if and only if its cone of positive operators
+    is proper::
 
         sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=5)
@@ -234,7 +421,9 @@ def positive_operator_gens(K):
     vectors = [ W(tp.list()) for tp in tensor_products ]
 
     # Create the *dual* cone of the positive operators, expressed as
-    # long vectors..
+    # long vectors. WARNING: check=True is necessary even though it
+    # makes Cone() take forever. For an example take
+    # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
     pi_dual = Cone(vectors, ToricLattice(W.dimension()))
 
     # Now compute the desired cone from its dual...
@@ -296,7 +485,7 @@ def Z_transformation_gens(K):
     The Z-property is possessed by every Z-transformation::
 
         sage: set_random_seed()
-        sage: K = random_cone(max_ambient_dim = 6)
+        sage: K = random_cone(max_ambient_dim=6)
         sage: Z_of_K = Z_transformation_gens(K)
         sage: dcs = K.discrete_complementarity_set()
         sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K
@@ -306,7 +495,7 @@ def Z_transformation_gens(K):
     The lineality space of Z is LL::
 
         sage: set_random_seed()
-        sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6)
+        sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=6)
         sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ])
         sage: z_cone  = Cone([ z.list() for z in Z_transformation_gens(K) ])
         sage: z_cone.linear_subspace() == lls
@@ -350,7 +539,9 @@ def Z_transformation_gens(K):
     vectors = [ W(m.list()) for m in tensor_products ]
 
     # Create the *dual* cone of the cross-positive operators,
-    # expressed as long vectors..
+    # expressed as long vectors. WARNING: check=True is necessary
+    # even though it makes Cone() take forever. For an example take
+    # K = Cone([(1,0,0),(0,0,1),(0,0,-1)]).
     Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()))
 
     # Now compute the desired cone from its dual...