X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=c71a24cbee0c9d0857fd3c8d8fe2ecbb0042238c;hb=a4109ab945f5d3ed94207c936e60b5b187ae450b;hp=6fb15ae21e242085b223eac729c857b8d04b8189;hpb=446018aef1279ff14866ae5e1d803a4a2a7c8024;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 6fb15ae..c71a24c 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,867 +1,771 @@ -# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we -# have to explicitly mangle our sitedir here so that "mjo.cone" -# resolves. -from os.path import abspath -from site import addsitedir -addsitedir(abspath('../../')) - from sage.all import * - -def drop_dependent(vs): +def is_lyapunov_like(L,K): r""" - Return the largest linearly-independent subset of ``vs``. - """ - if len(vs) == 0: - # ...for lazy enough definitions of linearly-independent - return vs - - result = [] - old_V = VectorSpace(vs[0].parent().base_field(), 0) + Determine whether or not ``L`` is Lyapunov-like on ``K``. - for v in vs: - new_V = span(result + [v]) - if new_V.dimension() > old_V.dimension(): - result.append(v) - old_V = new_V + 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. - return result - - -def iso_space(K): - r""" - Construct the space `W \times W^{\perp}` isomorphic to the ambient space - of ``K`` where `W` is equal to the span of ``K``. - """ - V = K.lattice().vector_space() + 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. - # Create the space W \times W^{\perp} isomorphic to V. - W_basis = drop_dependent(K.rays()) - W = V.subspace_with_basis(W_basis) - W_perp = W.complement() + INPUT: - return W.cartesian_product(W_perp) + - ``L`` -- A linear transformation or matrix. + - ``K`` -- A polyhedral closed convex cone. -def ips_iso(K): - r""" - Construct the IPS isomorphism and its inverse from our paper. + OUTPUT: - Given a cone ``K``, the returned isomorphism will split its ambient - vector space `V` into a cartesian product `W \times W^{\perp}` where - `W` equals the span of ``K``. - """ - V = K.lattice().vector_space() - V_iso = iso_space(K) - (W, W_perp) = V_iso.cartesian_factors() + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. - # A space equivalent to V, but using our basis. - V_user = V.subspace_with_basis( W.basis() + W_perp.basis() ) + .. WARNING:: - def phi(v): - # Write v in terms of our custom basis, where the first dim(W) - # coordinates are for the W-part of the basis. - cs = V_user.coordinates(v) + 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. - w1 = sum([ V_user.basis()[idx]*cs[idx] - for idx in range(0, W.dimension()) ]) - w2 = sum([ V_user.basis()[idx]*cs[idx] - for idx in range(W.dimension(), V.dimension()) ]) + REFERENCES: - return V_iso( (w1, w2) ) + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html + EXAMPLES: - def phi_inv( pair ): - # Crash if the arguments are in the wrong spaces. - V_iso(pair) + The identity is always Lyapunov-like in a nontrivial space:: - #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ]) - #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx] - # for idx in range(0,n-m) ]) + 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 - return sum( pair.cartesian_factors() ) + 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 - return (phi,phi_inv) + 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 unrestrict_span(K, K2=None): - if K2 is None: - K2 = K - _,phi_inv = ips_iso(K2) - V_iso = iso_space(K2) - (W, W_perp) = V_iso.cartesian_factors() +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. - rays = [] - for r in K.rays(): - w = sum([ r[idx]*W.basis()[idx] for idx in range(0,len(r)) ]) - pair = V_iso( (w, W_perp.zero()) ) - rays.append( phi_inv(pair) ) + INPUT: - L = ToricLattice(W.dimension() + W_perp.dimension()) + - ``K2`` -- (default: ``K1``) the codomain cone; the image of this + cone under the returned operators is a subset of ``K2``. - return Cone(rays, lattice=L) + 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. + REFERENCES: -def intersect_span(K1, K2): - r""" - Return a new cone obtained by intersecting ``K1`` with the span of ``K2``. - """ - L = K1.lattice() + .. [Orlitzky-Pi-Z] + M. Orlitzky. + Positive and Z-operators on closed convex cones. - if L.rank() != K2.lattice().rank(): - raise ValueError('K1 and K2 must belong to lattices of the same rank.') + .. [Tam] + B.-S. Tam. + Some results of polyhedral cones and simplicial cones. + Linear and Multilinear Algebra, 4:4 (1977) 281--284. - SL_gens = list(K2.rays()) - span_K2_gens = SL_gens + [ -g for g in SL_gens ] + EXAMPLES: - # The lattices have the same rank (see above) so this should work. - span_K2 = Cone(span_K2_gens, L) - return K1.intersection(span_K2) + 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] + ] -def restrict_span(K, K2=None): - r""" - Restrict ``K`` into its own span, or the span of another cone. + The trivial cone in a trivial space has no positive operators:: - INPUT: + sage: K = Cone([], ToricLattice(0)) + sage: positive_operator_gens(K) + [] - - ``K2`` -- another cone whose lattice has the same rank as this cone. + Every operator is positive on the trivial cone:: - OUTPUT: + sage: K = Cone([(0,)]) + sage: positive_operator_gens(K) + [[1], [-1]] - A new cone in a sublattice. + 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:: - sage: K = Cone([(1,)]) - sage: restrict_span(K) == K + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() True + sage: positive_operator_gens(K) + [[1], [-1]] - sage: K2 = Cone([(1,0)]) - sage: restrict_span(K2).rays() - N(1) - in 1-d lattice N - sage: K3 = Cone([(1,0,0)]) - sage: restrict_span(K3).rays() - N(1) - in 1-d lattice N - sage: restrict_span(K2) == restrict_span(K3) + 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] + ] - TESTS: + A non-obvious application is to find the positive operators on the + right half-plane:: - The projected cone should always be solid:: + 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] + ] - sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K_S = restrict_span(K) - sage: K_S.is_solid() - True + TESTS: - And the resulting cone should live in a space having the same - dimension as the space we restricted it to:: + Each positive operator generator should send the generators of one + cone into the other cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K_S = restrict_span( intersect_span(K, K.dual()), K.dual() ) - sage: K_S.lattice_dim() == K.dual().dim() + 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 - This function has ``unrestrict_span()`` as its inverse:: + Each positive operator generator should send a random element of one + cone into the other cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid=True) - sage: J = restrict_span(K) - sage: K == unrestrict_span(J,K) + 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 - This function should not affect the dimension of a cone:: + A random element of the positive operator cone should send the + generators of one cone into the other cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K.dim() == restrict_span(K).dim() + 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 - Nor should it affect the lineality of a cone:: + 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: K = random_cone(max_dim = 10) - sage: lineality(K) == lineality(restrict_span(K)) + 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 - No matter which space we restrict to, the lineality should not - increase:: + 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_dim = 10) - sage: J = intersect_span(K, K.dual()) - sage: lineality(K) >= lineality(restrict_span(J, K.dual())) + 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 - If we do this according to our paper, then the result is proper:: + 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_dim = 10) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + 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 - If ``K`` is strictly convex, then both ``K_W`` and - ``K_star_W.dual()`` should equal ``K`` (after we unrestrict):: + 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_dim = 10, strictly_convex=True) - sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: j1 = unrestrict_span(K_W, K.dual()) - sage: j2 = unrestrict_span(K_star_W_star, K.dual()) - sage: j1 == j2 - True - sage: j1 == K + 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 - sage: K; [ list(r) for r in K.rays() ] - Test the proposition in our paper concerning the duals, where the - subspace `W` is the span of `K^{*}`:: + The trivial cone, full space, and half-plane all give rise to the + expected dimensions:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid=False, strictly_convex=False) - sage: K_W = restrict_span(intersect_span(K,K.dual()), K.dual()) - sage: K_star_W_star = restrict_span(K.dual(), K.dual()).dual() - sage: K_W.nrays() == K_star_W_star.nrays() + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() True - sage: K_W.dim() == K_star_W_star.dim() + 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: lineality(K_W) == lineality(K_star_W_star) + sage: K = K.dual() + sage: K.is_full_space() True - sage: K_W.is_solid() == K_star_W_star.is_solid() + 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_W.is_strictly_convex() == K_star_W_star.is_strictly_convex() + 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 - """ - if K2 is None: - K2 = K - - phi,_ = ips_iso(K2) - (W, W_perp) = iso_space(K2).cartesian_factors() - - ray_pairs = [ phi(r) for r in K.rays() ] - - if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): - msg = 'Cone has nonzero components in W-perp!' - raise ValueError(msg) - - # Represent the cone in terms of a basis for W, i.e. with smaller - # vectors. - ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ] - - L = ToricLattice(W.dimension()) - - return Cone(ws, lattice=L) - - - -def lineality(K): - r""" - Compute the lineality of this cone. - - The lineality of a cone is the dimension of the largest linear - subspace contained in that cone. - - OUTPUT: - - A nonnegative integer; the dimension of the largest subspace - contained within this cone. - - REFERENCES: - - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. - - EXAMPLES: - - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 - - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: - - sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 - - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 - - TESTS: - - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: + The lineality of the cone of positive operators follows from the + description of its generators:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: l = lineality(K) - sage: l in ZZ - True - sage: (0 <= l) and (l <= K.lattice_dim()) + 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 - A strictly convex cone should have lineality zero:: + The trivial cone, full space, and half-plane all give rise to the + expected linealities:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, strictly_convex = True) - sage: lineality(K) - 0 - - """ - return K.linear_subspace().dimension() - - -def codim(K): - r""" - Compute the codimension of this cone. - - The codimension of a cone is the dimension of the space of all - elements perpendicular to every element of the cone. In other words, - the codimension is the difference between the dimension of the - ambient space and the dimension of the cone itself. - - OUTPUT: - - A nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. - - .. seealso:: - - :meth:`dim`, :meth:`lattice_dim` - - EXAMPLES: - - The codimension of the nonnegative orthant is zero, since the span of - its generators equals the entire ambient space:: - - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 - - However, if we remove a ray so that the entire cone is contained - within the `x-y`-plane, then the resulting cone will have - codimension one, because the `z`-axis is perpendicular to every - element of the cone:: - - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 - - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 - - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: - - sage: K = Cone([], lattice=ToricLattice(0)) - sage: codim(K) - 0 - - sage: K = Cone([(0,)]) - sage: codim(K) - 1 - - sage: K = Cone([(0,0)]) - sage: codim(K) - 2 - - TESTS: - - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: c = codim(K) - sage: c in ZZ + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() True - sage: (0 <= c) and (c <= K.lattice_dim()) + 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 solid cone should have codimension zero:: + A cone is proper if and only if its cone of positive operators + is proper:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) - 0 + 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 codimension of a cone is equal to the lineality of its dual:: + The positive operators of a permuted cone can be obtained by + conjugation:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) == lineality(K.dual()) + 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 - """ - return (K.lattice_dim() - K.dim()) - - -def discrete_complementarity_set(K): - r""" - Compute the discrete complementarity set of this cone. - - The complementarity set of this cone is the set of all orthogonal - pairs `(x,s)` such that `x` is in this cone, and `s` is in its - dual. The discrete complementarity set restricts `x` and `s` to be - generators of their respective cones. - - OUTPUT: - - A list of pairs `(x,s)` such that, - - * `x` is in this cone. - * `x` is a generator of this cone. - * `s` is in this cone's dual. - * `s` is a generator of this cone's dual. - * `x` and `s` are orthogonal. - - EXAMPLES: - - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: - - sage: K = Cone([(1,0),(0,1)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((0, 1), (1, 0))] - - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: - - sage: K = Cone([(1,0)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((1, 0), (0, -1))] - sage: K = Cone([(1,0,0)]) - sage: discrete_complementarity_set(K) - [((1, 0, 0), (0, 1, 0)), - ((1, 0, 0), (0, -1, 0)), - ((1, 0, 0), (0, 0, 1)), - ((1, 0, 0), (0, 0, -1))] - - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] - - TESTS: - - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + 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: K1 = random_cone(max_dim=6) - sage: K2 = K1.dual() - sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] - sage: actual = discrete_complementarity_set(K1) - sage: sorted(actual) == sorted(expected) + 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 - """ - V = K.lattice().vector_space() - - # Convert the rays to vectors so that we can compute inner - # products. - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] - - -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. - - OUTPUT: - - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. - - EXAMPLES: - - The trivial cone has no Lyapunov-like transformations:: - - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] - - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: - - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] - - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] - - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: LL(K) - [ - [1 0 0] [0 0 0] [0 0 0] - [0 0 0] [0 1 0] [0 0 0] - [0 0 0], [0 0 0], [0 0 1] - ] - - Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and - `L^{3}_{\infty}` cones [Rudolf et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: LL(L31) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] - - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] - - TESTS: - - The inner product `\left< L\left(x\right), s \right>` is zero for - every pair `\left( x,s \right)` in the discrete complementarity set - of the cone:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, max_rays=10) - sage: C_of_K = discrete_complementarity_set(K) - sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] - sage: sum(map(abs, l)) - 0 + 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-like transformations on a cone and its dual are related - by transposition, but we're not guaranteed to compute transposed - elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} - \right)` + 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 - sage: set_random_seed() - sage: K = random_cone(max_dim=8, max_rays=10) - sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] - sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) - sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] - sage: LL2_vecs = [ V(m.list()) for m in LL2 ] - sage: V.span(LL1_vecs) == V.span(LL2_vecs) + 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 """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + if K2 is None: + K2 = K1 - # Sage doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) + # 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() - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + tensor_products = [ s.tensor_product(x) for x in K1 for s in K2.dual() ] - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() + # Convert those tensor products to long vectors. + W = VectorSpace(F, n*m) + vectors = [ W(tp.list()) for tp in tensor_products ] - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) + 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 - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + # Create the dual cone of the positive operators, expressed as + # long vectors. + pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) - return matrix_basis + # 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 lyapunov_rank(K): +def Z_operator_gens(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. - - The Lyapunov rank of a cone can be thought of in (mainly) two ways: - - 1. The dimension of the Lie algebra of the automorphism group of the - cone. - - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. - - INPUT: - - A closed, convex polyhedral cone. + Compute generators of the cone of Z-operators on this cone. OUTPUT: - An integer representing the Lyapunov rank of the cone. If the - dimension of the ambient vector space is `n`, then the Lyapunov rank - will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). - - .. note:: - - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` - - ALGORITHM: - - The codimension formula from the second reference is used. We find - all pairs `(x,s)` in the complementarity set of `K` such that `x` - and `s` are rays of our cone. It is known that these vectors are - sufficient to apply the codimension formula. Once we have all such - pairs, we "brute force" the codimension formula by finding all - linearly-independent `xs^{T}`. + 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. REFERENCES: - .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper - cone and Lyapunov-like transformations, Mathematical Programming, 147 - (2014) 155-170. - - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. - - .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear - optimality constraints for the cone of positive polynomials, - Mathematical Programming, Series B, 129 (2011) 5-31. + M. Orlitzky. + Positive and Z-operators on closed convex cones. EXAMPLES: - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf et al.]_:: - - sage: positives = Cone([(1,)]) - sage: lyapunov_rank(positives) - 1 - sage: quadrant = Cone([(1,0), (0,1)]) - sage: lyapunov_rank(quadrant) - 2 - sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lyapunov_rank(octant) - 3 + Z-operators on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: - The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` - [Orlitzky/Gowda]_:: - - sage: R5 = VectorSpace(QQ, 5) - sage: gs = R5.basis() + [ -r for r in R5.basis() ] - sage: K = Cone(gs) - sage: lyapunov_rank(K) - 25 - - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one - [Rudolf et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 - - Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_:: - - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 - - A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n - + 1` [Orlitzky/Gowda]_:: + sage: K = Cone([(1,0),(0,1)]) + sage: Z_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([ z[i][j] <= 0 for z in Z_operator_gens(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) + True - sage: K = Cone([(1,0,0,0,0)]) - sage: lyapunov_rank(K) - 21 - sage: K.lattice_dim()**2 - K.lattice_dim() + 1 - 21 + The trivial cone in a trivial space has no Z-operators:: - A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_:: + sage: K = Cone([], ToricLattice(0)) + sage: Z_operator_gens(K) + [] - sage: e1 = (1,0,0,0,0) - sage: neg_e1 = (-1,0,0,0,0) - sage: e2 = (0,1,0,0,0) - sage: neg_e2 = (0,-1,0,0,0) - sage: z = (0,0,0,0,0) - sage: K = Cone([e1, neg_e1, e2, neg_e2, z, z, z]) - sage: lyapunov_rank(K) - 19 - sage: K.lattice_dim()**2 - K.dim()*codim(K) - 19 + Every operator is a Z-operator on the ambient vector space:: - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: Z_operator_gens(K) + [[-1], [1]] - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: K = L31.cartesian_product(octant) - sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() True + sage: Z_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] + ] - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. - The cone ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: + A non-obvious application is to find the Z-operators on the + right half-plane:: - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_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] + ] - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + Z-operators on a subspace are Lyapunov-like and vice-versa:: - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + 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 True TESTS: - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + The Z-property is possessed by every Z-operator:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: K = K1.cartesian_product(K2) - sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) + 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 - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + The lineality space of the cone of Z-operators is the space of + Lyapunov-like operators:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, max_rays=10) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + 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: 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 - Make sure we exercise the non-strictly-convex/non-solid case:: + The lineality of the Z-operators on a cone is the Lyapunov + rank of that cone:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_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() True - The Lyapunov rank of a proper polyhedral cone in `n` dimensions can - be any number between `1` and `n` inclusive, excluding `n-1` - [Gowda/Tao]_. By accident, the `n-1` restriction will hold for the - trivial cone in a trivial space as well. However, in zero dimensions, - the Lyapunov rank of the trivial cone will be zero:: + 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:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: b = lyapunov_rank(K) - sage: n = K.lattice_dim() - sage: (n == 0 or 1 <= b) and b <= n + 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: 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() + True + sage: pi_star = pi_cone.dual() + sage: z_star = Z_cone.dual() + sage: pi_star.linear_subspace() == z_star.linear_subspace() True - sage: b == n-1 - False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: + The trivial cone, full space, and half-plane all give rise to the + expected dimensions:: - sage: set_random_seed() - sage: K = random_cone(max_dim=10) - sage: b = lyapunov_rank(K) - sage: n = K.lattice_dim() - sage: b == n-1 - False + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + 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: 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: 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 + True - The calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky/Gowda]_:: + The Z-operators of a permuted cone can be obtained by conjugation:: sage: set_random_seed() - sage: K = random_cone(max_dim=10) - sage: actual = lyapunov_rank(K) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: l = lineality(K) - sage: c = codim(K) - sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 - sage: actual == expected + 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], + ....: 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], + ....: lattice=L, + ....: check=False) + sage: actual.is_equivalent(expected) True - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + 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:: sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) + 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: 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: M = MatrixSpace(F, n) + sage: L = M(Z_cone.random_element(ring=QQ).list()) + sage: Z_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) + True """ - K_orig = K - beta = 0 - - m = K.dim() + # 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() - l = lineality(K) - - if m < n: - # K is not solid, project onto its span. - K = restrict_span(K) - - # Lemma 2 - beta += m*(n - m) + (n - m)**2 - - if l > 0: - # K is not pointed, project its dual onto its span. - # Uses a proposition from our paper, i.e. this is - # equivalent to K = restrict_span(K.dual()).dual() - K = restrict_span(intersect_span(K,K.dual()), K.dual()) - #K = restrict_span(K.dual()).dual() - #Ks = [ list(r) for r in sorted(K.rays()) ] - #Js = [ list(r) for r in sorted(J.rays()) ] + # 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() ] - #if Ks != Js: - # print [ list(r) for r in K_orig.rays() ] - - # Lemma 3 - beta += m * l + # Turn our matrices into long vectors... + W = VectorSpace(F, n**2) + vectors = [ W(m.list()) for m in tensor_products ] - beta += len(LL(K)) - return beta + 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. + # 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 ] + + +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_operator_gens(K) + 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) + return Cone([ g.list() for g in gens ], lattice=L, check=False)