From 9b3341192ec454ea7448eb547bf1d409ed67d2ad Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 23 Sep 2016 10:08:07 -0400 Subject: [PATCH] Implement Z-operators in terms of cross-positive ones. --- mjo/cone/cone.py | 232 +++++++++++++++++++++++++++++------------------ 1 file changed, 145 insertions(+), 87 deletions(-) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index c71a24c..124bf54 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -92,6 +92,10 @@ def positive_operator_gens(K1, K2 = None): 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: .. [Orlitzky-Pi-Z] @@ -501,19 +505,24 @@ def positive_operator_gens(K1, K2 = None): return [ M(v.list()) for v in pi_cone ] -def Z_operator_gens(K): +def cross_positive_operator_gens(K): r""" - Compute generators of the cone of Z-operators on this cone. + 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 + ``(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`, + REFERENCES: M. Orlitzky. @@ -521,124 +530,128 @@ def Z_operator_gens(K): EXAMPLES: - Z-operators on the nonnegative orthant are just Z-matrices. - That is, matrices whose off-diagonal elements are nonnegative:: + 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: Z_operator_gens(K) + 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] + [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_operator_gens(K) - ....: for i in range(z.nrows()) - ....: for j in range(z.ncols()) + 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 Z-operators:: + The trivial cone in a trivial space has no cross-positive operators:: sage: K = Cone([], ToricLattice(0)) - sage: Z_operator_gens(K) + sage: cross_positive_operator_gens(K) [] - Every operator is a Z-operator on the ambient vector space:: + Every operator is a cross-positive operator on the ambient vector + space:: sage: K = Cone([(1,),(-1,)]) sage: K.is_full_space() True - sage: Z_operator_gens(K) - [[-1], [1]] + 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: Z_operator_gens(K) + 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] + [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-operators on the - right half-plane:: + 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: Z_operator_gens(K) + 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] + [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-operators on a subspace are Lyapunov-like and vice-versa:: + 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: zs = span([ vector(z.list()) for z in Z_operator_gens(K) ]) - sage: zs == lls + sage: cs = span([ vector(c.list()) for c in cross_positive_operator_gens(K) ]) + sage: cs == lls True TESTS: - The Z-property is possessed by every Z-operator:: + The cross-positive property is possessed by every cross-positive + operator:: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=4) - sage: Z_of_K = Z_operator_gens(K) + sage: Sigma_of_K = cross_positive_operator_gens(K) sage: dcs = K.discrete_complementarity_set() - sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K + 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 Z-operators is the space of - Lyapunov-like operators:: + 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: Z_cone = Cone([ z.list() for z in Z_operator_gens(K) ], - ....: lattice=L, - ....: check=False) + 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: Z_cone.linear_subspace() == lls + sage: Sigma_cone.linear_subspace() == lls True - The lineality of the Z-operators on a cone is the Lyapunov - rank of that cone:: + 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: Z_of_K = Z_operator_gens(K) + sage: Sigma_of_K = cross_positive_operator_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() + 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 Z-operator - cones are equal. From this it follows that the dimensions of the - Z-operator cone and positive operator cone are equal:: + 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: Z_of_K = Z_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: Z_cone = Cone([ z.list() for z in Z_of_K], - ....: lattice=L, - ....: check=False) - sage: pi_cone.dim() == Z_cone.dim() + 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: z_star = Z_cone.dual() - sage: pi_star.linear_subspace() == z_star.linear_subspace() + 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 @@ -649,49 +662,50 @@ def Z_operator_gens(K): sage: K.is_trivial() True sage: L = ToricLattice(n^2) - sage: Z_of_K = Z_operator_gens(K) - sage: Z_cone = Cone([z.list() for z in Z_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = Z_cone.dim() + 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: Z_of_K = Z_operator_gens(K) - sage: Z_cone = Cone([z.list() for z in Z_of_K], - ....: lattice=L, - ....: check=False) - sage: actual = Z_cone.dim() + 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: Z_of_K = Z_operator_gens(K) - sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) - sage: Z_cone.dim() == 3 + 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 Z-operators of a permuted cone can be obtained by conjugation:: + 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: Z_of_pK = Z_operator_gens(pK) - sage: actual = Cone([t.list() for t in Z_of_pK], + 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: Z_of_K = Z_operator_gens(K) - sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K], + 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 a Z-operator on a cone if and only if its - adjoint is a Z-operator on the dual of that cone:: + 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) @@ -699,22 +713,22 @@ def Z_operator_gens(K): sage: n = K.lattice_dim() sage: L = ToricLattice(n**2) sage: W = VectorSpace(F, n**2) - sage: Z_of_K = Z_operator_gens(K) - sage: Z_of_K_star = Z_operator_gens(K.dual()) - sage: Z_cone = Cone([p.list() for p in Z_of_K], - ....: lattice=L, - ....: check=False) - sage: Z_star = Cone([p.list() for p in Z_of_K_star], - ....: lattice=L, - ....: check=False) + 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(Z_cone.random_element(ring=QQ).list()) - sage: Z_star.contains(W(L.transpose().list())) + 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: Z_cone.contains(L) == Z_star.contains(L_star) + sage: Sigma_cone.contains(L) == Sigma_star.contains(L_star) True """ # Matrices are not vectors in Sage, so we have to convert them @@ -749,10 +763,49 @@ def Z_operator_gens(K): Sigma_cone = Sigma_dual.dual() # And finally convert its rays back to matrix representations. - # But first, make them negative, so we get Z-operators and - # not cross-positive ones. M = MatrixSpace(F, n) - return [ -M(v.list()) for v in Sigma_cone ] + 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:: + + 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): @@ -760,6 +813,11 @@ def LL_cone(K): 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) + 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) L = ToricLattice(K.lattice_dim()**2) -- 2.44.2