X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ae3ec48cddc9700d4f63ae378fc01b178dee6e3b;hb=8353d776d562e16cdbccfd10881662fc542c8d6f;hp=1ab6b97c128cde3d1e176032cf0d90f601057166;hpb=c8c4a0927a7a83a5b9f0cd3c33ff450212b95b99;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 1ab6b97..ae3ec48 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_rays = 8) - sage: K.contains(random_element(K)) + sage: K = random_cone(max_ambient_dim=8) + 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 + + 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) + 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 +230,65 @@ 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*K.random_element()) 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().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: 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: 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().list()) + sage: K.contains(P*K.random_element()) 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() @@ -201,7 +303,7 @@ def positive_operator_gens(K): 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: pi_of_K = positive_operator_gens(K) sage: L = ToricLattice(n**2) @@ -209,6 +311,17 @@ def positive_operator_gens(K): sage: expected = n**2 - K.dim()*K.dual().dim() sage: actual == expected True + + The cone ``K`` is proper if and only if the cone of positive + operators on ``K`` is proper:: + + 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([p.list() for p in pi_of_K], lattice=L) + sage: K.is_proper() == pi_cone.is_proper() + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -285,7 +398,7 @@ 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 @@ -295,7 +408,7 @@ 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 @@ -304,10 +417,24 @@ def Z_transformation_gens(K): And thus, the lineality of Z is the Lyapunov rank:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) - sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ]) + sage: K = random_cone(max_ambient_dim=6) + sage: Z_of_K = Z_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: z_cone = Cone([ z.list() for z in Z_of_K ], lattice=L) sage: z_cone.lineality() == K.lyapunov_rank() True + + The lineality spaces of pi-star and Z-star are equal: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: Z_of_K = Z_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_star = Cone([p.list() for p in pi_of_K], lattice=L).dual() + sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual() + sage: pi_star.linear_subspace() == z_star.linear_subspace() + True """ # Matrices are not vectors in Sage, so we have to convert them # to vectors explicitly before we can find a basis. We need these @@ -336,3 +463,18 @@ def Z_transformation_gens(K): # not cross-positive ones. M = MatrixSpace(F, n) return [ -M(v.list()) for v in Sigma_cone.rays() ] + + +def Z_cone(K): + gens = Z_transformation_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L) + +def pi_cone(K): + gens = positive_operator_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L)