X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=21f9862c24a9e9e3d9cffc52a1e789018f59a153;hb=c9ab469a53dc691d2646e22bdbc0ea0b3f3961c1;hp=6d7d2d9b4d2b379bc6041fe089e5a6ea38b8e48a;hpb=342d9147356f7757bed2f9165a600a9e5ec0a5e2;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 6d7d2d9..21f9862 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,39 +1,34 @@ -# 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 _restrict_to_space(K, W): +def is_lyapunov_like(L,K): r""" - Restrict this cone (up to linear isomorphism) to a vector subspace. - - This operation not only restricts the cone to a subspace of its - ambient space, but also represents the rays of the cone in a new - (smaller) lattice corresponding to the subspace. The resulting cone - will be linearly isomorphic **but not equal** to the desired - restriction, since it has likely undergone a change of basis. - - To explain the difficulty, consider the cone ``K = Cone([(1,1,1)])`` - having a single ray. The span of ``K`` is a one-dimensional subspace - containing ``K``, yet we have no way to perform operations like - :meth:`dual` in the subspace. To represent ``K`` in the space - ``K.span()``, we must perform a change of basis and write its sole - ray as ``(1,0,0)``. Now the restricted ``Cone([(1,)])`` is linearly - isomorphic (but of course not equal) to ``K`` interpreted as living - in ``K.span()``. + Determine whether or not ``L`` is Lyapunov-like on ``K``. + + We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle + L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs + `\left\langle x,s \right\rangle` in the complementarity set of + ``K``. It is known [Orlitzky]_ that this property need only be + checked for generators of ``K`` and its dual. INPUT: - - ``W`` -- The subspace into which this cone will be restricted. + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex cone. OUTPUT: - A new cone in a sublattice corresponding to ``W``. + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: + + If this function returns ``True``, then ``L`` is Lyapunov-like + on ``K``. However, if ``False`` is returned, that could mean one + of two things. The first is that ``L`` is definitely not + Lyapunov-like on ``K``. The second is more of an "I don't know" + answer, returned (for example) if we cannot prove that an inner + product is zero. REFERENCES: @@ -42,608 +37,466 @@ def _restrict_to_space(K, W): EXAMPLES: - Restricting a solid cone to its own span returns a cone linearly - isomorphic to the original:: + The identity is always Lyapunov-like in a nontrivial space:: - sage: K = Cone([(1,2,3),(-1,1,0),(9,0,-2)]) - sage: K.is_solid() + 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 - sage: _restrict_to_space(K, K.span()).rays() - N(-1, 1, 0), - N( 1, 0, 0), - N( 9, -6, -1) - in 3-d lattice N - A single ray restricted to its own span has the same representation - regardless of the ambient space:: + As is the "zero" transformation:: - sage: K2 = Cone([(1,0)]) - sage: K2_S = _restrict_to_space(K2, K2.span()).rays() - sage: K2_S - N(1) - in 1-d lattice N - sage: K3 = Cone([(1,1,1)]) - sage: K3_S = _restrict_to_space(K3, K3.span()).rays() - sage: K3_S - N(1) - in 1-d lattice N - sage: K2_S == K3_S + 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 - Restricting to a trivial space gives the trivial cone:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - sage: K = Cone([(8,3,-1,0),(9,2,2,0),(-4,6,7,0)]) - sage: trivial_space = K.lattice().vector_space().span([]) - sage: _restrict_to_space(K, trivial_space) - 0-d cone in 0-d lattice N + 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 - TESTS: + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) - Restricting a cone to its own span results in a solid cone:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K_S.is_solid() - True +def motzkin_decomposition(K): + r""" + Return the pair of components in the Motzkin decomposition of this cone. - Restricting a cone to its own span should not affect the number of - rays in the cone:: + Every convex cone is the direct sum of a strictly convex cone and a + linear subspace [Stoer-Witzgall]_. Return a pair ``(P,S)`` of cones + such that ``P`` is strictly convex, ``S`` is a subspace, and ``K`` + is the direct sum of ``P`` and ``S``. - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K.nrays() == K_S.nrays() - True + OUTPUT: - Restricting a cone to its own span should not affect its dimension:: + An ordered pair ``(P,S)`` of closed convex polyhedral cones where + ``P`` is strictly convex, ``S`` is a subspace, and ``K`` is the + direct sum of ``P`` and ``S``. - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K.dim() == K_S.dim() - True + REFERENCES: - Restricting a cone to its own span should not affects its lineality:: + .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and + Optimization in Finite Dimensions I. Springer-Verlag, New + York, 1970. - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K.lineality() == K_S.lineality() - True + EXAMPLES: - Restricting a cone to its own span should not affect the number of - facets it has:: + The nonnegative orthant is strictly convex, so it is its own + strictly convex component and its subspace component is trivial:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: len(K.facets()) == len(K_S.facets()) + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(P) + True + sage: S.is_trivial() True - Restricting a solid cone to its own span is a linear isomorphism and - should not affect the dimension of its ambient space:: + Likewise, full spaces are their own subspace components:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, solid = True) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K.lattice_dim() == K_S.lattice_dim() + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: (P,S) = motzkin_decomposition(K) + sage: K.is_equivalent(S) + True + sage: P.is_trivial() True - Restricting a solid cone to its own span is a linear isomorphism - that establishes a one-to-one correspondence of discrete - complementarity sets:: + TESTS: + + A random point in the cone should belong to either the strictly + convex component or the subspace component. If the point is nonzero, + it cannot be in both:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, solid = True) - sage: K_S = _restrict_to_space(K, K.span()) - sage: dcs_K = K.discrete_complementarity_set() - sage: dcs_K_S = K_S.discrete_complementarity_set() - sage: len(dcs_K) == len(dcs_K_S) + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: x = K.random_element(ring=QQ) + sage: P.contains(x) or S.contains(x) + True + sage: x.is_zero() or (P.contains(x) != S.contains(x)) True - Restricting a solid cone to its own span is a linear isomorphism - under which the Lyapunov rank (the length of a Lyapunov-like basis) - is invariant:: + The strictly convex component should always be strictly convex, and + the subspace component should always be a subspace:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, solid = True) - sage: K_S = _restrict_to_space(K, K.span()) - sage: len(K.lyapunov_like_basis()) == len(K_S.lyapunov_like_basis()) + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() + True + sage: S.lineality() == S.dim() True - If we restrict a cone to a subspace of its span, the resulting cone - should have the same dimension as the space we restricted it to:: + A strictly convex cone should be equal to its strictly convex component:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: W_basis = random_sublist(K.rays(), 0.5) - sage: W = K.lattice().vector_space().span(W_basis) - sage: K_W = _restrict_to_space(K, W) - sage: K_W.lattice_dim() == W.dimension() + sage: K = random_cone(max_ambient_dim=8, strictly_convex=True) + sage: (P,_) = motzkin_decomposition(K) + sage: K.is_equivalent(P) True - Through a series of restrictions, any closed convex cone can be - reduced to a cartesian product with a proper factor [Orlitzky]_:: + The generators of the components are obtained from orthogonal + projections of the original generators [Stoer-Witzgall]_:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) - sage: K_SP.is_proper() + sage: K = random_cone(max_ambient_dim=8) + sage: (P,S) = motzkin_decomposition(K) + sage: A = S.linear_subspace().complement().matrix() + sage: proj_S_perp = A.transpose() * (A*A.transpose()).inverse() * A + sage: expected_P = Cone([ proj_S_perp*g for g in K ], K.lattice()) + sage: P.is_equivalent(expected_P) + True + sage: A = S.linear_subspace().matrix() + sage: proj_S = A.transpose() * (A*A.transpose()).inverse() * A + sage: expected_S = Cone([ proj_S*g for g in K ], K.lattice()) + sage: S.is_equivalent(expected_S) True """ - # We want to intersect ``K`` with ``W``. An easy way to do this is - # via cone intersection, so we turn the space ``W`` into a cone. - W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice()) - K = K.intersection(W_cone) + # The lines() method only returns one generator per line. For a true + # line, we also need a generator pointing in the opposite direction. + S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ] + S = Cone(S_gens, K.lattice(), check=False) - # We've already intersected K with W, so every generator of K - # should belong to W now. - K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] + # Since ``S`` is a subspace, the rays of its dual generate its + # orthogonal complement. + S_perp = Cone(S.dual(), K.lattice(), check=False) + P = K.intersection(S_perp) - L = ToricLattice(W.dimension()) - return Cone(K_W_rays, lattice=L) + return (P,S) -def lyapunov_rank(K): +def positive_operator_gens(K): r""" - Compute the Lyapunov rank of this cone. - - The Lyapunov rank of a cone is the dimension of the space of its - Lyapunov-like transformations -- that is, the length of a - :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the - dimension of the Lie algebra of the automorphism group of the cone. + Compute generators of the cone of positive operators on this cone. OUTPUT: - A nonnegative integer representing the Lyapunov rank of this cone. - - If the ambient space is trivial, the Lyapunov rank will be zero. - Otherwise, if the dimension of the ambient vector space is `n`, then - the resulting Lyapunov rank will be between `1` and `n` inclusive. A - Lyapunov rank of `n-1` is not possible [Orlitzky]_. - - 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}`. - - 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. - - M. Orlitzky. The Lyapunov rank of an improper cone. - http://www.optimization-online.org/DB_HTML/2015/10/5135.html - - 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. + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``P`` in the list should have the property that ``P*x`` + is an element of ``K`` whenever ``x`` is an element of + ``K``. Moreover, any nonnegative linear combination of these + matrices shares the same property. EXAMPLES: - The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf]_:: - - 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 - - The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` - [Orlitzky]_:: - - 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]_:: + Positive operators on the nonnegative orthant are nonnegative matrices:: - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 + sage: K = Cone([(1,)]) + sage: positive_operator_gens(K) + [[1]] - Likewise for the `L^{3}_{\infty}` cone [Rudolf]_:: + 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] + ] - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 + The trivial cone in a trivial space has no positive operators:: - A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n - + 1` [Orlitzky]_:: + sage: K = Cone([], ToricLattice(0)) + sage: positive_operator_gens(K) + [] - sage: K = Cone([(1,0,0,0,0)]) - sage: lyapunov_rank(K) - 21 - sage: K.lattice_dim()**2 - K.lattice_dim() + 1 - 21 + Every operator is positive on the trivial cone:: - A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_:: + sage: K = Cone([(0,)]) + sage: positive_operator_gens(K) + [[1], [-1]] - 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()*K.codim() - 19 + 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] + ] - The Lyapunov rank should be additive on a product of proper cones - [Rudolf]_:: + Every operator is positive on the ambient vector space:: - 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,),(-1,)]) + sage: K.is_full_space() True + sage: positive_operator_gens(K) + [[1], [-1]] - Two isomorphic cones should have the same Lyapunov rank [Rudolf]_. - The cone ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: - - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + 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] + ] - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf]_:: + A non-obvious application is to find the positive operators on the + right half-plane:: - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True + 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: - The Lyapunov rank should be additive on a product of proper cones - [Rudolf]_:: + Each positive operator generator should send the generators of the + cone into the cone:: sage: set_random_seed() - sage: K1 = random_cone(max_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=True) - sage: K2 = random_cone(max_ambient_dim=8, - ....: 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: pi_of_K = positive_operator_gens(K) + sage: all([ K.contains(P*x) for P in pi_of_K for x in K ]) True - The Lyapunov rank is invariant under a linear isomorphism - [Orlitzky]_:: + Each positive operator generator should send a random element of the + cone into the cone:: - sage: K1 = random_cone(max_ambient_dim = 8) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: lyapunov_rank(K1) == lyapunov_rank(K2) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=4) + sage: pi_of_K = positive_operator_gens(K) + sage: all([ K.contains(P*K.random_element(QQ)) for P in pi_of_K ]) True - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf]_:: + A random element of the positive operator cone should send the + generators of the cone into the cone:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8) - sage: lyapunov_rank(K) == lyapunov_rank(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: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list()) + sage: all([ K.contains(P*x) for x in K ]) 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:: + A random element of the positive operator cone should send a random + element of the cone into the cone:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, - ....: 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: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([ g.list() for g in pi_of_K ], + ....: lattice=L, + ....: check=False) + sage: P = matrix(K.lattice_dim(), pi_cone.random_element(QQ).list()) + sage: K.contains(P*K.random_element(ring=QQ)) True - sage: b == n-1 - False - In fact [Orlitzky]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: + 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=8) - sage: b = lyapunov_rank(K) - sage: n = K.lattice_dim() - sage: b == n-1 - False + 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 calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky]_:: + 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=8) - sage: actual = lyapunov_rank(K) - sage: K_S = _restrict_to_space(K, K.span()) - sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() + sage: K = random_cone(max_ambient_dim=4) + sage: n = K.lattice_dim() + sage: m = K.dim() sage: l = K.lineality() - sage: c = K.codim() - sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 + 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 Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`:: + 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=8) - sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) + 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 - We can make an imperfect cone perfect by adding a slack variable - (a Theorem in [Orlitzky]_):: + The trivial cone, full space, and half-plane all give rise to the + expected dimensions:: - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=True) - sage: L = ToricLattice(K.lattice_dim() + 1) - sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) - sage: lyapunov_rank(K) >= K.lattice_dim() + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() True - - """ - beta = 0 # running tally of the Lyapunov rank - - m = K.dim() - n = K.lattice_dim() - l = K.lineality() - - if m < n: - # K is not solid, restrict to its span. - K = _restrict_to_space(K, K.span()) - - # Non-solid reduction lemma. - beta += (n - m)*n - - if l > 0: - # K is not pointed, restrict to the span of its dual. Uses a - # proposition from our paper, i.e. this is equivalent to K = - # _rho(K.dual()).dual(). - K = _restrict_to_space(K, K.dual().span()) - - # Non-pointed reduction lemma. - beta += l * m - - beta += len(K.lyapunov_like_basis()) - return beta - - - -def is_lyapunov_like(L,K): - r""" - Determine whether or not ``L`` is Lyapunov-like on ``K``. - - We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle - L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs - `\left\langle x,s \right\rangle` in the complementarity set of - ``K``. It is known [Orlitzky]_ that this property need only be - checked for generators of ``K`` and its dual. - - INPUT: - - - ``L`` -- A linear transformation or matrix. - - - ``K`` -- A polyhedral closed convex cone. - - OUTPUT: - - ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, - and ``False`` otherwise. - - .. WARNING:: - - If this function returns ``True``, then ``L`` is Lyapunov-like - on ``K``. However, if ``False`` is returned, that could mean one - of two things. The first is that ``L`` is definitely not - Lyapunov-like on ``K``. The second is more of an "I don't know" - answer, returned (for example) if we cannot prove that an inner - product is zero. - - REFERENCES: - - M. Orlitzky. The Lyapunov rank of an improper cone. - http://www.optimization-online.org/DB_HTML/2015/10/5135.html - - EXAMPLES: - - The identity is always Lyapunov-like in a nontrivial space:: - - sage: set_random_seed() - sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) - sage: L = identity_matrix(K.lattice_dim()) - sage: is_lyapunov_like(L,K) + 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 - - As is the "zero" transformation:: - - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) - sage: R = K.lattice().vector_space().base_ring() - sage: L = zero_matrix(R, K.lattice_dim()) - sage: is_lyapunov_like(L,K) + sage: K = K.dual() + sage: K.is_full_space() True - - Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like - on ``K``:: - - sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) - sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) + 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 - """ - return all([(L*x).inner_product(s) == 0 - for (x,s) in K.discrete_complementarity_set()]) - - -def random_element(K): - r""" - Return a random element of ``K`` from its ambient vector space. - - ALGORITHM: - - The cone ``K`` is specified in terms of its generators, so that - ``K`` is equal to the convex conic combination of those generators. - To choose a random element of ``K``, we assign random nonnegative - coefficients to each generator of ``K`` and construct a new vector - from the scaled rays. - - A vector, rather than a ray, is returned so that the element may - have non-integer coordinates. Thus the element may have an - arbitrarily small norm. - - EXAMPLES: - - A random element of the trivial cone is zero:: - - sage: set_random_seed() - sage: K = Cone([], ToricLattice(0)) - sage: random_element(K) - () - sage: K = Cone([(0,)]) - sage: random_element(K) - (0) - sage: K = Cone([(0,0)]) - sage: random_element(K) - (0, 0) - sage: K = Cone([(0,0,0)]) - sage: random_element(K) - (0, 0, 0) - - TESTS: - - Any cone should contain an element of itself:: + The lineality of the cone of positive operators follows from the + description of its generators:: sage: set_random_seed() - sage: K = random_cone(max_rays = 8) - sage: K.contains(random_element(K)) + 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 - """ - V = K.lattice().vector_space() - F = V.base_ring() - coefficients = [ F.random_element().abs() for i in range(K.nrays()) ] - vector_gens = map(V, K.rays()) - scaled_gens = [ coefficients[i]*vector_gens[i] - for i in range(len(vector_gens)) ] - - # Make sure we return a vector. Without the coercion, we might - # return ``0`` when ``K`` has no rays. - v = V(sum(scaled_gens)) - return v - - -def positive_operators(K): - r""" - Compute generators of the cone of positive operators on this cone. - - OUTPUT: - - A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. - Each matrix ``P`` in the list should have the property that ``P*x`` - is an element of ``K`` whenever ``x`` is an element of - ``K``. Moreover, any nonnegative linear combination of these - matrices shares the same property. - - EXAMPLES: - - The trivial cone in a trivial space has no positive operators:: - - sage: K = Cone([], ToricLattice(0)) - sage: positive_operators(K) - [] - - Positive operators on the nonnegative orthant are nonnegative matrices:: - - sage: K = Cone([(1,)]) - sage: positive_operators(K) - [[1]] - - sage: K = Cone([(1,0),(0,1)]) - sage: positive_operators(K) - [ - [1 0] [0 1] [0 0] [0 0] - [0 0], [0 0], [1 0], [0 1] - ] - - Every operator is positive on the ambient vector space:: + The trivial cone, full space, and half-plane all give rise to the + expected linealities:: - sage: K = Cone([(1,),(-1,)]) - sage: K.is_full_space() + sage: n = ZZ.random_element().abs() + sage: K = Cone([[0] * n], ToricLattice(n)) + sage: K.is_trivial() True - sage: positive_operators(K) - [[1], [-1]] - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + 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: positive_operators(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: + 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 positive operator on a cone should send its generators into the cone:: + A cone is proper if and only if its cone of positive operators + is proper:: - sage: K = random_cone(max_ambient_dim = 6) - sage: pi_of_K = positive_operators(K) - sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + 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 """ - # 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. - V = K.lattice().vector_space() - 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 = K.lattice().base_field() + n = K.lattice_dim() tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] - - # Create the *dual* cone of the positive operators, expressed as - # long vectors.. - L = ToricLattice(W.dimension()) - pi_dual = Cone(vectors, lattice=L) + # Convert those tensor products to long vectors. + W = VectorSpace(F, n**2) + vectors = [ W(tp.list()) for tp in tensor_products ] + + check = True + if K.is_solid() or K.is_strictly_convex(): + # The lineality space of either ``K`` or ``K.dual()`` is + # trivial and it's easy to show that our generating set is + # minimal. I would love a proof that this works when ``K`` is + # neither pointed nor solid. + # + # Note that in that case we can get *duplicates*, since the + # tensor product of (x,s) is the same as that of (-x,-s). + check = False + + # Create the dual cone of the positive operators, expressed as + # long vectors. + pi_dual = Cone(vectors, ToricLattice(W.dimension()), check=check) # Now compute the desired cone from its dual... pi_cone = pi_dual.dual() # And finally convert its rays back to matrix representations. - M = MatrixSpace(V.base_ring(), V.dimension()) - - return [ M(v.list()) for v in pi_cone.rays() ] + M = MatrixSpace(F, n) + return [ M(v.list()) for v in pi_cone ] -def Z_transformations(K): +def Z_transformation_gens(K): r""" Compute generators of the cone of Z-transformations on this cone. @@ -661,13 +514,13 @@ def Z_transformations(K): That is, matrices whose off-diagonal elements are nonnegative:: sage: K = Cone([(1,0),(0,1)]) - sage: Z_transformations(K) + sage: Z_transformation_gens(K) [ [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] ] sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) - sage: all([ z[i][j] <= 0 for z in Z_transformations(K) + sage: all([ z[i][j] <= 0 for z in Z_transformation_gens(K) ....: for i in range(z.nrows()) ....: for j in range(z.ncols()) ....: if i != j ]) @@ -676,16 +529,43 @@ def Z_transformations(K): The trivial cone in a trivial space has no Z-transformations:: sage: K = Cone([], ToricLattice(0)) - sage: Z_transformations(K) + sage: Z_transformation_gens(K) [] + Every operator is a Z-transformation on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: Z_transformation_gens(K) + [[-1], [1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: Z_transformation_gens(K) + [ + [-1 0] [1 0] [ 0 -1] [0 1] [ 0 0] [0 0] [ 0 0] [0 0] + [ 0 0], [0 0], [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] + ] + + A non-obvious application is to find the Z-transformations on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: Z_transformation_gens(K) + [ + [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0] + [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1] + ] + Z-transformations on a subspace are Lyapunov-like and vice-versa:: sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) sage: K.is_full_space() True sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) - sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ]) + sage: zs = span([ vector(z.list()) for z in Z_transformation_gens(K) ]) sage: zs == lls True @@ -694,44 +574,142 @@ def Z_transformations(K): The Z-property is possessed by every Z-transformation:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 6) - sage: Z_of_K = Z_transformations(K) + sage: K = random_cone(max_ambient_dim=4) + sage: Z_of_K = Z_transformation_gens(K) sage: dcs = K.discrete_complementarity_set() sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K ....: for (x,s) in dcs]) True - The lineality space of Z is LL:: + The lineality space of the cone of Z-transformations is the space of + Lyapunov-like transformations:: sage: set_random_seed() - sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) - sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) - sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ]) - sage: z_cone.linear_subspace() == lls + sage: K = random_cone(max_ambient_dim=4) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ], + ....: lattice=L, + ....: check=False) + sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ] + sage: lls = L.vector_space().span(ll_basis) + sage: Z_cone.linear_subspace() == lls + True + + The lineality of the Z-transformations 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_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: Z_cone = Cone([ z.list() for z in Z_of_K ], + ....: lattice=L, + ....: check=False) + sage: Z_cone.lineality() == K.lyapunov_rank() True + The lineality spaces of the duals of the positive operator and + Z-transformation cones are equal. From this it follows that the + dimensions of the Z-transformation 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_transformation_gens(K) + sage: L = ToricLattice(K.lattice_dim()**2) + sage: pi_cone = Cone([p.list() for p in pi_of_K], + ....: lattice=L, + ....: check=False) + sage: 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 + + 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: Z_of_K = Z_transformation_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_transformation_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_transformation_gens(K) + sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False) + sage: Z_cone.dim() == 3 + True + + The Z-transformations 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_transformation_gens(pK) + sage: actual = Cone([t.list() for t in Z_of_pK], + ....: lattice=L, + ....: check=False) + sage: Z_of_K = Z_transformation_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 """ - # 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. - V = K.lattice().vector_space() - W = VectorSpace(V.base_ring(), V.dimension()**2) - - C_of_K = K.discrete_complementarity_set() - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_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 = K.lattice().base_field() + n = K.lattice_dim() + + # These tensor products contain generators for the dual cone of + # the cross-positive transformations. + tensor_products = [ s.tensor_product(x) + for (x,s) in K.discrete_complementarity_set() ] # Turn our matrices into long vectors... + W = VectorSpace(F, n**2) vectors = [ W(m.list()) for m in tensor_products ] - # Create the *dual* cone of the cross-positive operators, - # expressed as long vectors.. - L = ToricLattice(W.dimension()) - Sigma_dual = Cone(vectors, lattice=L) + check = True + if K.is_solid() or K.is_strictly_convex(): + # The lineality space of either ``K`` or ``K.dual()`` is + # trivial and it's easy to show that our generating set is + # minimal. I would love a proof that this works when ``K`` is + # neither pointed nor solid. + # + # Note that in that case we can get *duplicates*, since the + # tensor product of (x,s) is the same as that of (-x,-s). + check = False + + # Create the dual cone of the cross-positive operators, + # expressed as long vectors. + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension()), check=check) # Now compute the desired cone from its dual... Sigma_cone = Sigma_dual.dual() @@ -739,6 +717,16 @@ def Z_transformations(K): # And finally convert its rays back to matrix representations. # But first, make them negative, so we get Z-transformations and # not cross-positive ones. - M = MatrixSpace(V.base_ring(), V.dimension()) + M = MatrixSpace(F, n) + return [ -M(v.list()) for v in Sigma_cone ] + + +def Z_cone(K): + gens = Z_transformation_gens(K) + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) - return [ -M(v.list()) for v in Sigma_cone.rays() ] +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)