X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=aeec0c90b5f582c8ba858d4b616fff191fb6d529;hb=c2d15c0884c8b92483f58826747887bd2bcdcdeb;hp=124bf5415f8841d36343e11c78ed5cd61b7ed276;hpb=9b3341192ec454ea7448eb547bf1d409ed67d2ad;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 124bf54..aeec0c9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,811 +1,435 @@ from sage.all import * +from sage.geometry.cone import is_Cone -def is_lyapunov_like(L,K): +def is_positive_on(L,K): r""" - Determine whether or not ``L`` is Lyapunov-like on ``K``. + Determine whether or not ``L`` is positive 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. + We say that ``L`` is positive on a closed convex cone ``K`` if + `L\left\lparen x \right\rparen` belongs to ``K`` for all `x` in + ``K``. This property need only be checked for generators of ``K``. - There are faster ways of checking this property. For example, we - could compute a `lyapunov_like_basis` of the cone, and then test - whether or not the given matrix is contained in the span of that - basis. The value of this function is that it works on symbolic - matrices. + To reliably check whether or not ``L`` is positive, its base ring + must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. INPUT: - - ``L`` -- A linear transformation or matrix. + - ``L`` -- A matrix over either an exact ring or ``SR``. - ``K`` -- A polyhedral closed convex cone. OUTPUT: - ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, - and ``False`` otherwise. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is positive on ``K``. - .. WARNING:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - 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. + - ``True`` will be returned if it can be proven that ``L`` + is positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - REFERENCES: + .. SEEALSO:: - M. Orlitzky. The Lyapunov rank of an improper cone. - http://www.optimization-online.org/DB_HTML/2015/10/5135.html + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` EXAMPLES: - The identity is always Lyapunov-like in a nontrivial space:: + Nonnegative matrices are positive operators on the nonnegative + orthant:: + + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = random_matrix(QQ,3).apply_map(abs) + sage: is_positive_on(L,K) + True + + TESTS: + + The identity operator is always positive:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: L = identity_matrix(K.lattice_dim()) - sage: is_lyapunov_like(L,K) + sage: is_positive_on(L,K) True - As is the "zero" transformation:: + The "zero" operator is always positive:: - sage: K = random_cone(min_ambient_dim=1, max_ambient_dim=8) + sage: K = random_cone(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) + sage: is_positive_on(L,K) True - Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like - on ``K``:: + Everything in ``K.positive_operators_gens()`` should be + positive 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() ]) + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_positive_on(L,K) # long time + ....: for L in K.positive_operators_gens() ]) # long time + True + sage: all([ is_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.positive_operators_gens() ]) # long time True - """ - return all([(L*x).inner_product(s) == 0 - for (x,s) in K.discrete_complementarity_set()]) + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: + sage: K = [ vector([1,2,3]), vector([5,-1,7]) ] + sage: L = identity_matrix(3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. -def positive_operator_gens(K1, K2 = None): - r""" - Compute generators of the cone of positive operators on this cone. A - linear operator on a cone is positive if the image of the cone under - the operator is a subset of the cone. This concept can be extended - to two cones, where the image of the first cone under a positive - operator is a subset of the second cone. + We can't give reliable answers over inexact rings:: - INPUT: + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - - ``K2`` -- (default: ``K1``) the codomain cone; the image of this - cone under the returned operators is a subset of ``K2``. + """ - OUTPUT: + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - A list of `m`-by-``n`` matrices where ``m == K2.lattice_dim()`` and - ``n == K1.lattice_dim()``. Each matrix ``P`` in the list should have - the property that ``P*x`` is an element of ``K2`` whenever ``x`` is - an element of ``K1``. Moreover, any nonnegative linear combination of - these matrices shares the same property. + if L.base_ring().is_exact(): + # This should be way faster than computing the dual and + # checking a bunch of inequalities, but it doesn't work if + # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])`` + # is true, but returns ``False``. + return all([ L*x in K for x in K ]) + else: + # Fall back to inequality-checking when the entries of ``L`` + # might be symbolic. + return all([ s*(L*x) >= 0 for x in K for s in K.dual() ]) - .. SEEALSO:: - :meth:`cross_positive_operator_gens`, :meth:`Z_operator_gens`, +def is_cross_positive_on(L,K): + r""" + Determine whether or not ``L`` is cross-positive on ``K``. - REFERENCES: + We say that ``L`` is cross-positive on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \ge 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - .. [Orlitzky-Pi-Z] - M. Orlitzky. - Positive and Z-operators on closed convex cones. + To reliably check whether or not ``L`` is cross-positive, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - .. [Tam] - B.-S. Tam. - Some results of polyhedral cones and simplicial cones. - Linear and Multilinear Algebra, 4:4 (1977) 281--284. + INPUT: - EXAMPLES: + - ``L`` -- A matrix over either an exact ring or ``SR``. - Positive operators on the nonnegative orthant are nonnegative matrices:: + - ``K`` -- A polyhedral closed convex cone. - sage: K = Cone([(1,)]) - sage: positive_operator_gens(K) - [[1]] + OUTPUT: - 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] - ] + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is cross-positive on ``K``. - The trivial cone in a trivial space has no positive operators:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - sage: K = Cone([], ToricLattice(0)) - sage: positive_operator_gens(K) - [] + - ``True`` will be returned if it can be proven that ``L`` + is cross-positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not cross-positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - Every operator is positive on the trivial cone:: + .. SEEALSO:: - sage: K = Cone([(0,)]) - sage: positive_operator_gens(K) - [[1], [-1]] + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - 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] - ] + EXAMPLES: - Every operator is positive on the ambient vector space:: + The identity operator is always cross-positive:: - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_cross_positive_on(L,K) True - sage: positive_operator_gens(K) - [[1], [-1]] - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() + The "zero" operator is always cross-positive:: + + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_cross_positive_on(L,K) 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 one - cone into the other cone:: + Everything in ``K.cross_positive_operators_gens()`` should be + cross-positive on ``K``:: - sage: set_random_seed() - sage: K1 = random_cone(max_ambient_dim=4) - sage: K2 = random_cone(max_ambient_dim=4) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: all([ K2.contains(P*x) for P in pi_K1_K2 for x in K1 ]) + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_cross_positive_on(L,K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time True - - Each positive operator generator should send a random element of one - cone into the other cone:: - - sage: set_random_seed() - sage: K1 = random_cone(max_ambient_dim=4) - sage: K2 = random_cone(max_ambient_dim=4) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: all([ K2.contains(P*K1.random_element(QQ)) for P in pi_K1_K2 ]) + sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time True - A random element of the positive operator cone should send the - generators of one cone into the other cone:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K1 = random_cone(max_ambient_dim=4) - sage: K2 = random_cone(max_ambient_dim=4) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) - sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], - ....: lattice=L, - ....: check=False) - sage: P = matrix(K2.lattice_dim(), - ....: K1.lattice_dim(), - ....: pi_cone.random_element(QQ).list()) - sage: all([ K2.contains(P*x) for x in K1 ]) - True + sage: L = identity_matrix(3) + sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ] + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - A random element of the positive operator cone should send a random - element of one cone into the other cone:: + We can't give reliable answers over inexact rings:: - sage: set_random_seed() - sage: K1 = random_cone(max_ambient_dim=4) - sage: K2 = random_cone(max_ambient_dim=4) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) - sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], - ....: lattice=L, - ....: check=False) - sage: P = matrix(K2.lattice_dim(), - ....: K1.lattice_dim(), - ....: pi_cone.random_element(QQ).list()) - sage: K2.contains(P*K1.random_element(ring=QQ)) - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - 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:: + """ + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - 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 + return all([ s*(L*x) >= 0 + for (x,s) in K.discrete_complementarity_set() ]) - The lineality of the dual of the cone of positive operators - is known from its lineality space:: +def is_Z_operator_on(L,K): + r""" + Determine whether or not ``L`` is a Z-operator on ``K``. - 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 + We say that ``L`` is a Z-operator on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \le 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. It is known that this property need only be checked + for generators of ``K`` and its dual. - The dimension of the cone of positive operators is given by the - corollary in my paper:: + A matrix is a Z-operator on ``K`` if and only if its negation is a + cross-positive operator on ``K``. - 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 + To reliably check whether or not ``L`` is a Z operator, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - The trivial cone, full space, and half-plane all give rise to the - expected dimensions:: + INPUT: - 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 + - ``L`` -- A matrix over either an exact ring or ``SR``. - The lineality of the cone of positive operators follows from the - description of its generators:: + - ``K`` -- A polyhedral closed convex cone. - 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 + OUTPUT: - The trivial cone, full space, and half-plane all give rise to the - expected linealities:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is a Z-operator on ``K``. - 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: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L) - sage: pi_cone.lineality() == 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 + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - A cone is proper if and only if its cone of positive operators - is proper:: + - ``True`` will be returned if it can be proven that ``L`` + is a Z-operator on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not a Z-operator on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - 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 + .. SEEALSO:: - The positive operators of a permuted cone can be obtained by - conjugation:: + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: L = ToricLattice(K.lattice_dim()**2) - sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() - sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) - sage: pi_of_pK = positive_operator_gens(pK) - sage: actual = Cone([t.list() for t in pi_of_pK], - ....: lattice=L, - ....: check=False) - sage: pi_of_K = positive_operator_gens(K) - sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: actual.is_equivalent(expected) - True + EXAMPLES: - A transformation is positive on a cone if and only if its adjoint is - positive on the dual of that cone:: + The identity operator is always a Z-operator:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: F = K.lattice().vector_space().base_field() - sage: n = K.lattice_dim() - sage: L = ToricLattice(n**2) - sage: W = VectorSpace(F, n**2) - sage: pi_of_K = positive_operator_gens(K) - sage: pi_of_K_star = positive_operator_gens(K.dual()) - sage: pi_cone = Cone([p.list() for p in pi_of_K], - ....: lattice=L, - ....: check=False) - sage: pi_star = Cone([p.list() for p in pi_of_K_star], - ....: lattice=L, - ....: check=False) - sage: M = MatrixSpace(F, n) - sage: L = M(pi_cone.random_element(ring=QQ).list()) - sage: pi_star.contains(W(L.transpose().list())) + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_Z_operator_on(L,K) True - sage: L = W.random_element() - sage: L_star = W(M(L.list()).transpose().list()) - sage: pi_cone.contains(L) == pi_star.contains(L_star) - True + The "zero" operator is always a Z-operator:: - The Lyapunov rank of the positive operator cone is the product of - the Lyapunov ranks of the associated cones if they're all proper:: - - sage: K1 = random_cone(max_ambient_dim=4, - ....: strictly_convex=True, - ....: solid=True) - sage: K2 = random_cone(max_ambient_dim=4, - ....: strictly_convex=True, - ....: solid=True) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: L = ToricLattice(K1.lattice_dim() * K2.lattice_dim()) - sage: pi_cone = Cone([ g.list() for g in pi_K1_K2 ], - ....: lattice=L, - ....: check=False) - sage: beta1 = K1.lyapunov_rank() - sage: beta2 = K2.lyapunov_rank() - sage: pi_cone.lyapunov_rank() == beta1*beta2 + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_Z_operator_on(L,K) True - The Lyapunov-like operators on a proper polyhedral positive operator - cone can be computed from the Lyapunov-like operators on the cones - with respect to which the operators are positive:: - - sage: K1 = random_cone(max_ambient_dim=4, - ....: strictly_convex=True, - ....: solid=True) - sage: K2 = random_cone(max_ambient_dim=4, - ....: strictly_convex=True, - ....: solid=True) - sage: pi_K1_K2 = positive_operator_gens(K1,K2) - sage: F = K1.lattice().base_field() - sage: m = K1.lattice_dim() - sage: n = K2.lattice_dim() - sage: L = ToricLattice(m*n) - sage: M1 = MatrixSpace(F, m, m) - sage: M2 = MatrixSpace(F, n, n) - sage: LL_K1 = [ M1(x.list()) for x in K1.dual().lyapunov_like_basis() ] - sage: LL_K2 = [ M2(x.list()) for x in K2.lyapunov_like_basis() ] - sage: tps = [ s.tensor_product(x) for x in LL_K1 for s in LL_K2 ] - sage: W = VectorSpace(F, (m**2)*(n**2)) - sage: expected = span(F, [ W(x.list()) for x in tps ]) - sage: pi_cone = Cone([p.list() for p in pi_K1_K2], - ....: lattice=L, - ....: check=False) - sage: LL_pi = pi_cone.lyapunov_like_basis() - sage: actual = span(F, [ W(x.list()) for x in LL_pi ]) - sage: actual == expected - True + TESTS: - """ - if K2 is None: - K2 = K1 + Everything in ``K.Z_operators_gens()`` should be a Z-operator + on ``K``:: - # 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 = K1.lattice().base_field() - n = K1.lattice_dim() - m = K2.lattice_dim() + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_Z_operator_on(L,K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True + sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True - tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ] + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - # Convert those tensor products to long vectors. - W = VectorSpace(F, n*m) - vectors = [ W(tp.list()) for tp in tensor_products ] + sage: L = identity_matrix(3) + sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ] + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - check = True - if K1.is_proper() and K2.is_proper(): - # All of the generators involved are extreme vectors and - # therefore minimal [Tam]_. If this cone is neither solid nor - # strictly convex, then the tensor product of ``s`` and ``x`` - # is the same as that of ``-s`` and ``-x``. However, as a - # /set/, ``tensor_products`` may still be minimal. - check = False - # Create the dual cone of the positive operators, expressed as - # long vectors. - pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) + We can't give reliable answers over inexact rings:: - # Now compute the desired cone from its dual... - pi_cone = pi_dual.dual() + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - # And finally convert its rays back to matrix representations. - M = MatrixSpace(F, m, n) - return [ M(v.list()) for v in pi_cone ] + """ + return is_cross_positive_on(-L,K) -def cross_positive_operator_gens(K): +def is_lyapunov_like_on(L,K): r""" - Compute generators of the cone of cross-positive operators on this - cone. - - OUTPUT: + Determine whether or not ``L`` is Lyapunov-like on ``K``. - 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 of - this cone's :meth:`discrete_complementarity_set`. Moreover, any - conic (nonnegative linear) combination of these matrices shares the - same property. + We say that ``L`` is Lyapunov-like on a closed convex cone ``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``. This property need only be checked for generators of + ``K`` and its dual. - .. SEEALSO:: + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. - :meth:`positive_operator_gens`, :meth:`Z_operator_gens`, + To reliably check whether or not ``L`` is Lyapunov-like, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - REFERENCES: + INPUT: - M. Orlitzky. - Positive and Z-operators on closed convex cones. + - ``L`` -- A matrix over either an exact ring or ``SR``. - EXAMPLES: + - ``K`` -- A polyhedral closed convex cone. - Cross-positive operators on the nonnegative orthant are negations - of Z-matrices; that is, matrices whose off-diagonal elements are - nonnegative:: - - sage: K = Cone([(1,0),(0,1)]) - sage: cross_positive_operator_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([ c[i][j] >= 0 for c in cross_positive_operator_gens(K) - ....: for i in range(c.nrows()) - ....: for j in range(c.ncols()) - ....: if i != j ]) - True + OUTPUT: - The trivial cone in a trivial space has no cross-positive operators:: + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is Lyapunov-like on ``K``. - sage: K = Cone([], ToricLattice(0)) - sage: cross_positive_operator_gens(K) - [] + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - Every operator is a cross-positive operator on the ambient vector - space:: + - ``True`` will be returned if it can be proven that ``L`` + is Lyapunov-like on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not Lyapunov-like on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() - True - sage: cross_positive_operator_gens(K) - [[1], [-1]] + .. SEEALSO:: - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: K.is_full_space() - True - sage: cross_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 cross-positive operators - on the right half-plane:: - - sage: K = Cone([(1,0),(0,1),(0,-1)]) - sage: cross_positive_operator_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] - ] - - Cross-positive operators 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: cs = span([ vector(c.list()) for c in cross_positive_operator_gens(K) ]) - sage: cs == lls - True + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` - TESTS: + EXAMPLES: - The cross-positive property is possessed by every cross-positive - operator:: + Diagonal matrices are Lyapunov-like operators on the nonnegative + orthant:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: dcs = K.discrete_complementarity_set() - sage: all([(c*x).inner_product(s) >= 0 for c in Sigma_of_K - ....: for (x,s) in dcs]) + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = diagonal_matrix(random_vector(QQ,3)) + sage: is_lyapunov_like_on(L,K) True - The lineality space of the cone of cross-positive operators is the - space of Lyapunov-like operators:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: L = ToricLattice(K.lattice_dim()**2) - sage: Sigma_cone = Cone([ c.list() for c in cross_positive_operator_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: Sigma_cone.linear_subspace() == lls - True + TESTS: - The lineality of the cross-positive operators on a cone is the - Lyapunov rank of that cone:: + The identity operator is always Lyapunov-like:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: L = ToricLattice(K.lattice_dim()**2) - sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], - ....: lattice=L, - ....: check=False) - sage: Sigma_cone.lineality() == K.lyapunov_rank() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - The lineality spaces of the duals of the positive and cross-positive - operator cones are equal. From this it follows that the dimensions of - the cross-positive operator cone and positive operator cone are equal:: + The "zero" operator is always Lyapunov-like:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: pi_of_K = positive_operator_gens(K) - sage: Sigma_of_K = cross_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: Sigma_cone = Cone([ c.list() for c in Sigma_of_K], - ....: lattice=L, - ....: check=False) - sage: pi_cone.dim() == Sigma_cone.dim() - True - sage: pi_star = pi_cone.dual() - sage: sigma_star = Sigma_cone.dual() - sage: pi_star.linear_subspace() == sigma_star.linear_subspace() + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - The trivial cone, full space, and half-plane all give rise to the - expected dimensions:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - sage: n = ZZ.random_element().abs() - sage: K = Cone([[0] * n], ToricLattice(n)) - sage: K.is_trivial() - True - sage: L = ToricLattice(n^2) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: Sigma_cone = Cone([c.list() for c in Sigma_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = Sigma_cone.dim() - sage: actual == n^2 - True - sage: K = K.dual() - sage: K.is_full_space() - True - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], - ....: lattice=L, - ....: check=False) - sage: actual = Sigma_cone.dim() - sage: actual == n^2 + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_lyapunov_like_on(L,K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - sage: K = Cone([(1,0),(0,1),(0,-1)]) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: Sigma_cone = Cone([ c.list() for c in Sigma_of_K ], check=False) - sage: Sigma_cone.dim() == 3 + sage: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - The cross-positive operators of a permuted cone can be obtained by - conjugation:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: L = ToricLattice(K.lattice_dim()**2) - sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix() - sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False) - sage: Sigma_of_pK = cross_positive_operator_gens(pK) - sage: actual = Cone([t.list() for t in Sigma_of_pK], - ....: lattice=L, - ....: check=False) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: expected = Cone([ (p*t*p.inverse()).list() for t in Sigma_of_K ], - ....: lattice=L, - ....: check=False) - sage: actual.is_equivalent(expected) - True + sage: L = identity_matrix(3) + sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ] + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - An operator is cross-positive on a cone if and only if its - adjoint is cross-positive on the dual of that cone:: + We can't give reliable answers over inexact rings:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: F = K.lattice().vector_space().base_field() - sage: n = K.lattice_dim() - sage: L = ToricLattice(n**2) - sage: W = VectorSpace(F, n**2) - sage: Sigma_of_K = cross_positive_operator_gens(K) - sage: Sigma_of_K_star = cross_positive_operator_gens(K.dual()) - sage: Sigma_cone = Cone([ p.list() for p in Sigma_of_K ], - ....: lattice=L, - ....: check=False) - sage: Sigma_star = Cone([ p.list() for p in Sigma_of_K_star ], - ....: lattice=L, - ....: check=False) - sage: M = MatrixSpace(F, n) - sage: L = M(Sigma_cone.random_element(ring=QQ).list()) - sage: Sigma_star.contains(W(L.transpose().list())) - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. - sage: L = W.random_element() - sage: L_star = W(M(L.list()).transpose().list()) - sage: Sigma_cone.contains(L) == Sigma_star.contains(L_star) - 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 operators. - 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_proper(): - # All of the generators involved are extreme vectors and - # therefore minimal. If this cone is neither solid nor - # strictly convex, then the tensor product of ``s`` and ``x`` - # is the same as that of ``-s`` and ``-x``. However, as a - # /set/, ``tensor_products`` may still be minimal. - 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. - M = MatrixSpace(F, n) - return [ M(v.list()) for v in Sigma_cone ] - - -def Z_operator_gens(K): - r""" - Compute generators of the cone of Z-operators on this cone. - - The Z-operators on a cone generalize the Z-matrices over the - nonnegative orthant. They are simply negations of the - :meth:`cross_positive_operators`. - - 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 of - this cone's :meth:`discrete_complementarity_set`. Moreover, any - conic (nonnegative linear) combination of these matrices shares the - same property. - - .. SEEALSO:: - - :meth:`positive_operator_gens`, :meth:`cross_positive_operator_gens`, - - REFERENCES: - - M. Orlitzky. - Positive and Z-operators on closed convex cones. - - TESTS: - - The Z-property is possessed by every Z-operator:: + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=4) - sage: Z_of_K = Z_operator_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 - """ - return [ -cp for cp in cross_positive_operator_gens(K) ] + return all([ s*(L*x) == 0 + for (x,s) in K.discrete_complementarity_set() ]) def LL_cone(K): @@ -814,16 +438,18 @@ def LL_cone(K): return Cone([ g.list() for g in gens ], lattice=L, check=False) def Sigma_cone(K): - gens = cross_positive_operator_gens(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) def Z_cone(K): - gens = Z_operator_gens(K) + gens = K.Z_operators_gens() L = ToricLattice(K.lattice_dim()**2) return Cone([ g.list() for g in gens ], lattice=L, check=False) -def pi_cone(K): - gens = positive_operator_gens(K) - L = ToricLattice(K.lattice_dim()**2) +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)