X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=cd835a764f9baf7fd8dd2082aa948501610698f5;hb=dbef443b13d185940629eb870fc93f55cb5a70a3;hp=1a3ee3b2fad16e3a05938cb7b9c6a0beae9518d1;hpb=8dd2505bd26127a6704e0f16a50d93bda4da2fb3;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 1a3ee3b..cd835a7 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -76,11 +76,11 @@ def _basically_the_same(K1, K2): if K1.is_strictly_convex() != K2.is_strictly_convex(): return False - if len(LL(K1)) != len(LL(K2)): + if len(K1.lyapunov_like_basis()) != len(K2.lyapunov_like_basis()): return False - C_of_K1 = discrete_complementarity_set(K1) - C_of_K2 = discrete_complementarity_set(K2) + C_of_K1 = K1.discrete_complementarity_set() + C_of_K2 = K2.discrete_complementarity_set() if len(C_of_K1) != len(C_of_K2): return False @@ -211,233 +211,6 @@ def _restrict_to_space(K, W): return Cone(K_W_rays, lattice=L) - -def discrete_complementarity_set(K): - r""" - Compute a discrete complementarity set of this cone. - - A discrete complementarity set of `K` is the set of all orthogonal - pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some - generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral - convex cones are input in terms of their generators, so "the" (this - particular) discrete complementarity set corresponds to ``G1 - == K.rays()`` and ``G2 == K.dual().rays()``. - - OUTPUT: - - A list of pairs `(x,s)` such that, - - * Both `x` and `s` are vectors (not rays). - * `x` is one of ``K.rays()``. - * `s` is one of ``K.dual().rays()``. - * `x` and `s` are orthogonal. - - REFERENCES: - - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. - - 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) - [] - - Likewise when this cone is trivial (its dual is the entire space):: - - sage: L = ToricLattice(0) - sage: K = Cone([], ToricLattice(0)) - 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: set_random_seed() - sage: K1 = random_cone(max_ambient_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: sorted(actual) == sorted(expected) - True - - The pairs in the discrete complementarity set are in fact - complementary:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=6) - sage: dcs = discrete_complementarity_set(K) - sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) - 0 - - """ - V = K.lattice().vector_space() - - # Convert rays to vectors so that we can compute inner products. - xs = [V(x) for x in K.rays()] - - # We also convert the generators of the dual cone so that we - # return pairs of vectors and not (vector, ray) pairs. - 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 LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. - - OUTPUT: - - 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_ambient_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_ambient_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 ] - - # Sage doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) - - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] - - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() - - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) - - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] - - return matrix_basis - - - def lyapunov_rank(K): r""" Compute the Lyapunov rank (or bilinearity rank) of this cone. @@ -641,11 +414,12 @@ def lyapunov_rank(K): sage: actual == expected True - The Lyapunov rank of any cone is just the dimension of ``LL(K)``:: + The Lyapunov rank of any cone is just the dimension of + ``K.lyapunov_like_basis()``:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) - sage: lyapunov_rank(K) == len(LL(K)) + sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) True We can make an imperfect cone perfect by adding a slack variable @@ -683,7 +457,7 @@ def lyapunov_rank(K): # Non-pointed reduction lemma. beta += l * m - beta += len(LL(K)) + beta += len(K.lyapunov_like_basis()) return beta @@ -741,15 +515,16 @@ def is_lyapunov_like(L,K): sage: is_lyapunov_like(L,K) True - Everything in ``LL(K)`` should be Lyapunov-like on ``K``:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) - sage: all([is_lyapunov_like(L,K) for L in LL(K)]) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) True """ return all([(L*x).inner_product(s) == 0 - for (x,s) in discrete_complementarity_set(K)]) + for (x,s) in K.discrete_complementarity_set()]) def random_element(K): @@ -807,3 +582,194 @@ def random_element(K): # return ``0`` when ``K`` has no rays. v = V(sum(scaled_gens)) return v + + +def positive_operators(K): + r""" + Compute generators of the cone of positive operators on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``P`` in the list should have the property that ``P*x`` + is an element of ``K`` whenever ``x`` is an element of + ``K``. Moreover, any nonnegative linear combination of these + matrices shares the same property. + + EXAMPLES: + + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] + + Positive operators on the nonnegative orthant are nonnegative matrices:: + + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operators(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(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] + ] + + TESTS: + + A positive operator on a cone should send its generators into the cone:: + + sage: K = random_cone(max_ambient_dim = 6) + sage: pi_of_K = positive_operators(K) + sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + True + + """ + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() + + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ M(v.list()) for v in pi_cone.rays() ] + + +def Z_transformations(K): + r""" + Compute generators of the cone of Z-transformations on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``L`` in the list should have the property that + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the + discrete complementarity set of ``K``. Moreover, any nonnegative + linear combination of these matrices shares the same property. + + EXAMPLES: + + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: + + sage: K = Cone([(1,0),(0,1)]) + sage: Z_transformations(K) + [ + [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] + [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] + ] + sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) + sage: all([ z[i][j] <= 0 for z in Z_transformations(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) + True + + The trivial cone in a trivial space has no Z-transformations:: + + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformations(K) + [] + + Z-transformations on a subspace are Lyapunov-like and vice-versa:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ]) + sage: zs == lls + True + + TESTS: + + The Z-property is possessed by every Z-transformation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 6) + sage: Z_of_K = Z_transformations(K) + sage: dcs = K.discrete_complementarity_set() + sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K + ....: for (x,s) in dcs]) + True + + The lineality space of Z is LL:: + + sage: set_random_seed() + 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_transformations(K) ]) + sage: z_cone.linear_subspace() == lls + True + + """ + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + C_of_K = K.discrete_complementarity_set() + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. + L = ToricLattice(W.dimension()) + Sigma_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + Sigma_cone = Sigma_dual.dual() + + # And finally convert its rays back to matrix representations. + # But first, make them negative, so we get Z-transformations and + # not cross-positive ones. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ -M(v.list()) for v in Sigma_cone.rays() ]