X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b9e930e6819643710b82c06faa0b72b934298d96;hb=6bd30534d5aa984c73f511121efa8fda4386c51a;hp=2e3dc8afb78cd1f4e1d5f368eb5f75733d127e82;hpb=433b47ffeac9305beb9101afa302c7e924b60744;p=sage.d.git

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index 2e3dc8a..b9e930e 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -8,6 +8,311 @@ addsitedir(abspath('../../'))
 from sage.all import *
 
 
+def drop_dependent(vs):
+    r"""
+    Return the largest linearly-independent subset of ``vs``.
+    """
+    result = []
+    m = matrix(vs).echelon_form()
+    for idx in range(0, m.nrows()):
+        if not m[idx].is_zero():
+            result.append(m[idx])
+
+    return result
+
+
+def basically_the_same(K1,K2):
+    r"""
+    ``True`` if ``K1`` and ``K2`` are basically the same, and ``False``
+    otherwise.
+    """
+    if K1.lattice_dim() != K2.lattice_dim():
+        return False
+
+    if K1.nrays() != K2.nrays():
+        return False
+
+    if K1.dim() != K2.dim():
+        return False
+
+    if lineality(K1) != lineality(K2):
+        return False
+
+    if K1.is_solid() != K2.is_solid():
+        return False
+
+    if K1.is_strictly_convex() != K2.is_strictly_convex():
+        return False
+
+    if len(LL(K1)) != len(LL(K2)):
+        return False
+
+    C_of_K1 = discrete_complementarity_set(K1)
+    C_of_K2 = discrete_complementarity_set(K2)
+    if len(C_of_K1) != len(C_of_K2):
+        return False
+
+    if len(K1.facets()) != len(K2.facets()):
+        return False
+
+    return True
+
+
+
+def rho(K, K2=None):
+    r"""
+    Restrict ``K`` into its own span, or the span of another cone.
+
+    INPUT:
+
+    - ``K2`` -- another cone whose lattice has the same rank as this cone.
+
+    OUTPUT:
+
+    A new cone in a sublattice.
+
+    EXAMPLES::
+
+        sage: K = Cone([(1,)])
+        sage: rho(K) == K
+        True
+
+        sage: K2 = Cone([(1,0)])
+        sage: rho(K2).rays()
+        N(1)
+        in 1-d lattice N
+        sage: K3 = Cone([(1,0,0)])
+        sage: rho(K3).rays()
+        N(1)
+        in 1-d lattice N
+        sage: rho(K2) == rho(K3)
+        True
+
+    TESTS:
+
+    The projected cone should always be solid::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K)
+        sage: K_S.is_solid()
+        True
+
+    And the resulting cone should live in a space having the same
+    dimension as the space we restricted it to::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K_S = rho(K, K.dual() )
+        sage: K_S.lattice_dim() == K.dual().dim()
+        True
+
+    This function should not affect the dimension of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: K.dim() == rho(K).dim()
+        True
+
+    Nor should it affect the lineality of a cone::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: lineality(K) == lineality(rho(K))
+        True
+
+    No matter which space we restrict to, the lineality should not
+    increase::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: lineality(K) >= lineality(rho(K))
+        True
+        sage: lineality(K) >= lineality(rho(K, K.dual()))
+        True
+
+    If we do this according to our paper, then the result is proper::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: P.is_proper()
+        True
+        sage: P = rho(K_S, K_S.dual())
+        sage: P.is_proper()
+        True
+
+    Test the proposition in our paper concerning the duals, where the
+    subspace `W` is the span of `K^{*}`::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
+        sage: K_W = rho(K, K.dual())
+        sage: K_star_W_star = rho(K.dual()).dual()
+        sage: basically_the_same(K_W, K_star_W_star)
+        True
+
+    """
+    if K2 is None:
+        K2 = K
+
+    # First we project K onto the span of K2. This can be done with
+    # cones (i.e. without converting to vector spaces), but it's
+    # annoying to deal with lattice mismatches.
+    span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+    K = K.intersection(span_K2)
+
+    V = K.lattice().vector_space()
+
+    # Create the space W \times W^{\perp} isomorphic to V.
+    # First we get an orthogonal (but not normal) basis...
+    W_basis = drop_dependent(K2.rays())
+    W = V.subspace_with_basis(W_basis)
+
+    # We've already intersected K with the span of K2, so every
+    # generator of K should belong to W now.
+    W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
+
+    L = ToricLattice(K2.dim())
+    return Cone(W_rays, lattice=L)
+
+
+
+def lineality(K):
+    r"""
+    Compute the lineality of this cone.
+
+    The lineality of a cone is the dimension of the largest linear
+    subspace contained in that cone.
+
+    OUTPUT:
+
+    A nonnegative integer; the dimension of the largest subspace
+    contained within this cone.
+
+    REFERENCES:
+
+    .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton
+       University Press, Princeton, 1970.
+
+    EXAMPLES:
+
+    The lineality of the nonnegative orthant is zero, since it clearly
+    contains no lines::
+
+        sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        0
+
+    However, if we add another ray so that the entire `x`-axis belongs
+    to the cone, then the resulting cone will have lineality one::
+
+        sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)])
+        sage: lineality(K)
+        1
+
+    If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal
+    to the dimension of the ambient space (i.e. two)::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: lineality(K)
+        2
+
+    Per the definition, the lineality of the trivial cone in a trivial
+    space is zero::
+
+        sage: K = Cone([], lattice=ToricLattice(0))
+        sage: lineality(K)
+        0
+
+    TESTS:
+
+    The lineality of a cone should be an integer between zero and the
+    dimension of the ambient space, inclusive::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8)
+        sage: l = lineality(K)
+        sage: l in ZZ
+        True
+        sage: (0 <= l) and (l <= K.lattice_dim())
+        True
+
+    A strictly convex cone should have lineality zero::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim = 8, strictly_convex = True)
+        sage: lineality(K)
+        0
+
+    """
+    return K.linear_subspace().dimension()
+
+
 def discrete_complementarity_set(K):
     r"""
     Compute the discrete complementarity set of this cone.
@@ -62,11 +367,12 @@ def discrete_complementarity_set(K):
     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: set_random_seed()
+        sage: K1 = random_cone(max_dim=6)
         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
+        sage: sorted(actual) == sorted(expected)
         True
 
     """
@@ -87,15 +393,100 @@ 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]
+        ]
+
+    If our cone is the entire space, then every transformation on it is
+    Lyapunov-like::
+
+        sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)])
+        sage: M = MatrixSpace(QQ,2)
+        sage: M.basis() == LL(K)
+        True
+
+    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: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        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
+
+    The Lyapunov-like transformations on a cone and its dual are related
+    by transposition, but we're not guaranteed to compute transposed
+    elements of `LL\left( K \right)` as our basis for `LL\left( K^{*}
+    \right)`
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: LL2 = [ L.transpose() for L in LL(K.dual()) ]
+        sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2)
+        sage: LL1_vecs = [ V(m.list()) for m in LL(K) ]
+        sage: LL2_vecs = [ V(m.list()) for m in LL2   ]
+        sage: V.span(LL1_vecs) == V.span(LL2_vecs)
+        True
 
     """
     V = K.lattice().vector_space()
 
     C_of_K = discrete_complementarity_set(K)
 
-    tensor_products = [s.tensor_product(x) for (x,s) in C_of_K]
+    tensor_products = [ s.tensor_product(x) 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
@@ -171,6 +562,9 @@ 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 +584,15 @@ 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: gs = R5.basis() + [ -r for r in R5.basis() ]
+        sage: K = Cone(gs)
+        sage: lyapunov_rank(K)
+        25
+
     The `L^{3}_{1}` cone is known to have a Lyapunov rank of one
     [Rudolf et al.]_::
 
@@ -203,7 +606,30 @@ 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: z = (0,0,0,0,0)
+        sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z])
+        sage: lyapunov_rank(K)
+        19
+        sage: K.lattice_dim()**2 - K.dim()*codim(K)
+        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 +655,12 @@ 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: set_random_seed()
+        sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: K = K1.cartesian_product(K2)
         sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2)
         True
@@ -241,7 +668,36 @@ def lyapunov_rank(K):
     The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K``
     itself [Rudolf et al.]_::
 
-        sage: K = random_cone(max_dim=10, max_rays=10)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    Make sure we exercise the non-strictly-convex/non-solid case::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    Let's check the other permutations as well, just to be sure::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: lyapunov_rank(K) == lyapunov_rank(K.dual())
         True
 
@@ -251,7 +707,8 @@ def lyapunov_rank(K):
     trivial cone in a trivial space as well. However, in zero dimensions,
     the Lyapunov rank of the trivial cone will be zero::
 
-        sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True)
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
         sage: b = lyapunov_rank(K)
         sage: n = K.lattice_dim()
         sage: (n == 0 or 1 <= b) and b <= n
@@ -259,5 +716,82 @@ 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: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        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: set_random_seed()
+        sage: K = random_cone(max_dim=8)
+        sage: actual = lyapunov_rank(K)
+        sage: K_S = rho(K)
+        sage: P = rho(K_S.dual()).dual()
+        sage: l = lineality(K)
+        sage: c = codim(K)
+        sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
+        sage: actual == expected
+        True
+
+    The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    In fact the same can be said of any cone. These additional tests
+    just increase our confidence that the reduction scheme works::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
+    ::
+
+        sage: set_random_seed()
+        sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False)
+        sage: lyapunov_rank(K) == len(LL(K))
+        True
+
     """
-    return len(LL(K))
+    K_orig = K
+    beta = 0
+
+    m = K.dim()
+    n = K.lattice_dim()
+    l = lineality(K)
+
+    if m < n:
+        # K is not solid, project onto its span.
+        K = rho(K)
+
+        # Lemma 2
+        beta += m*(n - m) + (n - m)**2
+
+    if l > 0:
+        # K is not pointed, project its dual onto its span.
+        # Uses a proposition from our paper, i.e. this is
+        # equivalent to K = rho(K.dual()).dual()
+        K = rho(K, K.dual())
+
+        # Lemma 3
+        beta += m * l
+
+    beta += len(LL(K))
+    return beta