X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b7456e21abc170d6479f9010a2b7ccdd5ab9438d;hb=1bbade9f41ffbfe366b15d0db657f666bc1f025d;hp=a327720132b3907562f269fc4872a2b829226a59;hpb=48a7ad09084a859da72e39e60312bfffb6b806e9;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index a327720..b7456e2 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,656 +1,23 @@ from sage.all import * -def is_lyapunov_like(L,K): - r""" - Determine whether or not ``L`` is Lyapunov-like on ``K``. - - We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle - L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs - `\left\langle x,s \right\rangle` in the complementarity set of - ``K``. It is known [Orlitzky]_ that this property need only be - checked for generators of ``K`` and its dual. - - INPUT: - - - ``L`` -- A linear transformation or matrix. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, - and ``False`` otherwise. - - .. WARNING:: - - If this function returns ``True``, then ``L`` is Lyapunov-like - on ``K``. However, if ``False`` is returned, that could mean one - of two things. The first is that ``L`` is definitely not - Lyapunov-like on ``K``. The second is more of an "I don't know" - answer, returned (for example) if we cannot prove that an inner - product is zero. - - REFERENCES: - - M. Orlitzky. The Lyapunov rank of an improper cone. - http://www.optimization-online.org/DB_HTML/2015/10/5135.html - - EXAMPLES: - - 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: 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: R = K.lattice().vector_space().base_ring() - sage: L = zero_matrix(R, K.lattice_dim()) - sage: is_lyapunov_like(L,K) - True - - 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: 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 K.discrete_complementarity_set()]) - - -def motzkin_decomposition(K): - r""" - Return the pair of components in the Motzkin decomposition of this cone. - - 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``. - - 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: - - .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and - Optimization in Finite Dimensions I. Springer-Verlag, New - York, 1970. - - EXAMPLES: - - The nonnegative orthant is strictly convex, so it is its own - strictly convex component and its subspace component is trivial:: - - 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: - - 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_ambient_dim=8) - sage: (P,S) = motzkin_decomposition(K) - sage: P.is_strictly_convex() - True - sage: S.lineality() == S.dim() - True - - A strictly convex cone should be equal to its strictly convex component:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) - sage: (P,_) = motzkin_decomposition(K) - sage: K.is_equivalent(P) - 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 - """ - # 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(), check=False) - - # Since ``S`` is a subspace, the rays of its dual generate its - # orthogonal complement. - S_perp = Cone(S.dual(), K.lattice(), check=False) - P = K.intersection(S_perp) - - return (P,S) - - -def positive_operator_gens(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: - - Positive operators on the nonnegative orthant are nonnegative matrices:: - - sage: K = Cone([(1,)]) - sage: positive_operator_gens(K) - [[1]] - - sage: K = Cone([(1,0),(0,1)]) - sage: positive_operator_gens(K) - [ - [1 0] [0 1] [0 0] [0 0] - [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,)]) - sage: K.is_full_space() - True - sage: positive_operator_gens(K) - [[1], [-1]] - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() - 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] - ] - - 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: - - 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=4) - 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=4) - sage: pi_of_K = positive_operator_gens(K) - 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=4) - 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, - ....: check=False) - 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=4) - 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, - ....: check=False) - 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=4) - 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, - ....: check=False) - 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=4) - 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, - ....: check=False) - 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=4) - 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, - ....: check=False) - sage: actual = pi_cone.dim() - sage: expected = n**2 - l*(m - l) - (n - m)*m - sage: actual == expected - True - - 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: pi_cone = Cone([p.list() for p in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = pi_cone.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: pi_cone = Cone([p.list() for p in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = pi_cone.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], check=False).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=4) - sage: n = K.lattice_dim() - 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, - ....: check=False) - sage: actual = pi_cone.lineality() - sage: expected = n**2 - K.dim()*K.dual().dim() - sage: actual == expected - True - - 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: pi_cone = Cone([p.list() for p in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = pi_cone.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: pi_cone = Cone([p.list() for p in pi_of_K], check=False) - sage: actual = pi_cone.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=4) - 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, - ....: check=False) - 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 - # two values to construct the appropriate "long vector" space. - F = K.lattice().base_field() - n = K.lattice_dim() - - tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] - - # Convert those tensor products to long vectors. - W = VectorSpace(F, n**2) - vectors = [ W(tp.list()) for tp in tensor_products ] - - check = True - if K.is_solid() or K.is_strictly_convex(): - # The lineality space of either ``K`` or ``K.dual()`` is - # trivial and it's easy to show that our generating set is - # minimal. I would love a proof that this works when ``K`` is - # neither pointed nor solid. - # - # Note that in that case we can get *duplicates*, since the - # tensor product of (x,s) is the same as that of (-x,-s). - check = False - - # Create the dual cone of the positive operators, expressed as - # long vectors. - pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) - - # Now compute the desired cone from its dual... - pi_cone = pi_dual.dual() - - # And finally convert its rays back to matrix representations. - M = MatrixSpace(F, n) - return [ M(v.list()) for v in pi_cone ] - - -def Z_transformation_gens(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_transformation_gens(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_transformation_gens(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_transformation_gens(K) - [] - - Every operator is a Z-transformation on the ambient vector space:: - - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() - True - sage: Z_transformation_gens(K) - [[-1], [1]] - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() - True - sage: Z_transformation_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] - ] - - A non-obvious application is to find the Z-transformations on the - right half-plane:: - - sage: K = Cone([(1,0),(0,1),(0,-1)]) - sage: Z_transformation_gens(K) - [ - [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0] - [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] - ] - - 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_transformation_gens(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=4) - 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 - ....: for (x,s) in dcs]) - True - - The lineality space of Z is LL:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: L = ToricLattice(K.lattice_dim()**2) - sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ], - ....: lattice=L, - ....: check=False) - sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ] - sage: lls = L.vector_space().span(ll_basis) - sage: z_cone.linear_subspace() == lls - True - - And thus, the lineality of Z is the Lyapunov rank:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - 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, - ....: check=False) - 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=4) - 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_cone = Cone([p.list() for p in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: pi_star = pi_cone.dual() - sage: z_cone = Cone([ z.list() for z in Z_of_K], - ....: lattice=L, - ....: check=False) - sage: z_star = z_cone.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 - # two values to construct the appropriate "long vector" space. - F = K.lattice().base_field() - n = K.lattice_dim() - - # These tensor products contain generators for the dual cone of - # the cross-positive transformations. - tensor_products = [ s.tensor_product(x) - for (x,s) in K.discrete_complementarity_set() ] - - # Turn our matrices into long vectors... - W = VectorSpace(F, n**2) - vectors = [ W(m.list()) for m in tensor_products ] - - check = True - if K.is_solid() or K.is_strictly_convex(): - # The lineality space of either ``K`` or ``K.dual()`` is - # trivial and it's easy to show that our generating set is - # minimal. I would love a proof that this works when ``K`` is - # neither pointed nor solid. - # - # Note that in that case we can get *duplicates*, since the - # tensor product of (x,s) is the same as that of (-x,-s). - check = False - - # Create the dual cone of the cross-positive operators, - # expressed as long vectors. - Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check) - - # 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(F, n) - return [ -M(v.list()) for v in Sigma_cone ] - +def LL_cone(K): + gens = K.lyapunov_like_basis() + L = ToricLattice(K.lattice_dim()**2) + return Cone(( g.list() for g in gens ), lattice=L, check=False) -def Z_cone(K): - gens = Z_transformation_gens(K) +def Sigma_cone(K): + gens = K.cross_positive_operators_gens() L = ToricLattice(K.lattice_dim()**2) - return Cone([ g.list() for g in gens ], lattice=L, check=False) + return Cone(( g.list() for g in gens ), lattice=L, check=False) -def pi_cone(K): - gens = positive_operator_gens(K) +def Z_cone(K): + gens = K.Z_operators_gens() L = ToricLattice(K.lattice_dim()**2) - return Cone([ g.list() for g in gens ], lattice=L, check=False) + return Cone(( g.list() for g in gens ), lattice=L, check=False) + +def pi_cone(K1, K2=None): + if K2 is None: + K2 = K1 + gens = K1.positive_operators_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) + return Cone(( g.list() for g in gens ), lattice=L, check=False)