From 455255081db7cf2fbb9a221029a19d7d17310577 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 25 Sep 2016 15:41:41 -0400 Subject: [PATCH] Remove the Pi/Z stuff for inclusion into sage. --- mjo/cone/cone.py | 736 +---------------------------------------------- 1 file changed, 7 insertions(+), 729 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index fd63612..7e9c549 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -70,746 +70,24 @@ def is_lyapunov_like(L,K): return all([(L*x).inner_product(s) == 0 for (x,s) in K.discrete_complementarity_set()]) - -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. - - INPUT: - - - ``K2`` -- (default: ``K1``) the codomain cone; the image of this - cone under the returned operators is a subset of ``K2``. - - OUTPUT: - - 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. - - .. SEEALSO:: - - :meth:`cross_positive_operator_gens`, :meth:`Z_operator_gens`, - - REFERENCES: - - .. [Tam] - B.-S. Tam. - Some results of polyhedral cones and simplicial cones. - Linear and Multilinear Algebra, 4:4 (1977) 281--284. - - 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 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*x) for P in pi_K1_K2 for x in K1 ]) - 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 ]) - True - - A random element of the positive operator cone should send the - generators 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: 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 - - A random element of the positive operator cone 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: 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 - - 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: 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 - - 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 - - The positive operators of a permuted cone can be obtained by - conjugation:: - - 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 - - A transformation is positive on a cone if and only if its adjoint is - positive on the dual of that cone:: - - 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())) - 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 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 - 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 - - """ - if K2 is None: - K2 = K1 - - # 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() - - tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ] - - # Convert those tensor products to long vectors. - W = VectorSpace(F, n*m) - vectors = [ W(tp.list()) for tp in tensor_products ] - - check = True - if K1.is_proper() and K2.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 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, m, n) - return [ M(v.list()) for v in pi_cone ] - - -def cross_positive_operator_gens(K): - r""" - Compute generators of the cone of cross-positive operators 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 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:`Z_operator_gens`, - - EXAMPLES: - - 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 - - The trivial cone in a trivial space has no cross-positive operators:: - - sage: K = Cone([], ToricLattice(0)) - sage: cross_positive_operator_gens(K) - [] - - Every operator is a cross-positive operator on the ambient vector - space:: - - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() - True - sage: cross_positive_operator_gens(K) - [[1], [-1]] - - 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 - - TESTS: - - The cross-positive property is possessed by every cross-positive - operator:: - - 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]) - 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 - - The lineality of the cross-positive operators on a cone is the - Lyapunov rank of that cone:: - - 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() - 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:: - - 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() - 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: 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 - 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 - True - - The cross-positive operators of a permuted cone can be obtained by - conjugation:: - - 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 - - An operator is cross-positive on a cone if and only if its - adjoint is cross-positive on the dual of that cone:: - - 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: 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`, - - TESTS: - - The Z-property is possessed by every Z-operator:: - - 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) ] - - 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 Sigma_cone(K): - gens = cross_positive_operator_gens(K) + gens = K.cross_positive_operator_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_operator_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_operator_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) return Cone([ g.list() for g in gens ], lattice=L, check=False) -- 2.43.2