X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=aeec0c90b5f582c8ba858d4b616fff191fb6d529;hb=c2d15c0884c8b92483f58826747887bd2bcdcdeb;hp=f2e8b2e9ee104e6cbd216e7473fd65dd76947d98;hpb=44802773ad9e5151890ed37e7bb2463ff9fc4135;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index f2e8b2e..aeec0c9 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,850 +1,455 @@ -# 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 * +from sage.geometry.cone import is_Cone - -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() - - # Create the space W \times W^{\perp} isomorphic to V. - # First we get an orthogonal (but not normal) basis... - M = matrix(V.base_field(), K.rays()) - W_basis,_ = M.gram_schmidt() - - W = V.subspace_with_basis(W_basis) - W_perp = W.complement() - - return W.cartesian_product(W_perp) - - -def ips_iso(K): - r""" - Construct the IPS isomorphism and its inverse from our paper. - - 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() - - # A space equivalent to V, but using our basis. - V_user = V.subspace_with_basis( W.basis() + W_perp.basis() ) - - 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) - - 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()) ]) - - return V_iso( (w1, w2) ) - - - def phi_inv( pair ): - # Crash if the arguments are in the wrong spaces. - V_iso(pair) - - #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) ]) - - return sum( pair.cartesian_factors() ) - - - return (phi,phi_inv) - - - -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() - - 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) ) - - L = ToricLattice(W.dimension() + W_perp.dimension()) - - return Cone(rays, lattice=L) - - - -def intersect_span(K1, K2): +def is_positive_on(L,K): r""" - Return a new cone obtained by intersecting ``K1`` with the span of ``K2``. - """ - L = K1.lattice() - - if L.rank() != K2.lattice().rank(): - raise ValueError('K1 and K2 must belong to lattices of the same rank.') - - SL_gens = list(K2.rays()) - span_K2_gens = SL_gens + [ -g for g in SL_gens ] + Determine whether or not ``L`` is positive on ``K``. - # The lattices have the same rank (see above) so this should work. - span_K2 = Cone(span_K2_gens, L) - return K1.intersection(span_K2) + We say that ``L`` is positive on a closed convex cone ``K`` if + `L\left\lparen x \right\rparen` belongs to ``K`` for all `x` in + ``K``. This property need only be checked for generators of ``K``. - - -def restrict_span(K, K2=None): - r""" - Restrict ``K`` into its own span, or the span of another cone. + To reliably check whether or not ``L`` is positive, its base ring + must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. INPUT: - - ``K2`` -- another cone whose lattice has the same rank as this cone. + - ``L`` -- A matrix over either an exact ring or ``SR``. + + - ``K`` -- A polyhedral closed convex cone. OUTPUT: - A new cone in a sublattice. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is positive on ``K``. - EXAMPLES:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - sage: K = Cone([(1,)]) - sage: restrict_span(K) == K - True + - ``True`` will be returned if it can be proven that ``L`` + is positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - 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) - True - - TESTS: + .. SEEALSO:: - The projected cone should always be solid:: + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K_S = restrict_span(K) - sage: K_S.is_solid() - True + EXAMPLES: - And the resulting cone should live in a space having the same - dimension as the space we restricted it to:: + Nonnegative matrices are positive operators on the nonnegative + orthant:: - 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: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = random_matrix(QQ,3).apply_map(abs) + sage: is_positive_on(L,K) True - This function has ``unrestrict_span()`` as its inverse:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid=True) - sage: J = restrict_span(K) - sage: K == unrestrict_span(J,K) - True + TESTS: - This function should not affect the dimension of a cone:: + The identity operator is always positive:: sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: K.dim() == restrict_span(K).dim() + sage: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_positive_on(L,K) True - Nor should it affect the lineality of a cone:: + The "zero" operator is always positive:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: lineality(K) == lineality(restrict_span(K)) + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_positive_on(L,K) True - No matter which space we restrict to, the lineality should not - increase:: + Everything in ``K.positive_operators_gens()`` should be + positive on ``K``:: - 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=5) + sage: all([ is_positive_on(L,K) # long time + ....: for L in K.positive_operators_gens() ]) # long time True - - If we do this according to our paper, then the result is proper:: - - 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: all([ is_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.positive_operators_gens() ]) # long time True - If ``K`` is strictly convex, then both ``K_W`` and - ``K_star_W.dual()`` should equal ``K`` (after we unrestrict):: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_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 - True - sage: K; [ list(r) for r in K.rays() ] + sage: K = [ vector([1,2,3]), vector([5,-1,7]) ] + sage: L = identity_matrix(3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - Test the proposition in our paper concerning the duals, where the - subspace `W` is the span of `K^{*}`:: + We can't give reliable answers over inexact rings:: - 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() - True - sage: K_W.dim() == K_star_W_star.dim() - True - sage: lineality(K_W) == lineality(K_star_W_star) - True - sage: K_W.is_solid() == K_star_W_star.is_solid() - True - sage: K_W.is_strictly_convex() == K_star_W_star.is_strictly_convex() - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. """ - 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()) + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - return Cone(ws, lattice=L) + if L.base_ring().is_exact(): + # This should be way faster than computing the dual and + # checking a bunch of inequalities, but it doesn't work if + # ``L*x`` is symbolic. For example, ``e in Cone([(1,)])`` + # is true, but returns ``False``. + return all([ L*x in K for x in K ]) + else: + # Fall back to inequality-checking when the entries of ``L`` + # might be symbolic. + return all([ s*(L*x) >= 0 for x in K for s in K.dual() ]) - -def lineality(K): +def is_cross_positive_on(L,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. + Determine whether or not ``L`` is cross-positive on ``K``. - REFERENCES: + We say that ``L`` is cross-positive on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \ge 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. + To reliably check whether or not ``L`` is cross-positive, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - EXAMPLES: + INPUT: - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: + - ``L`` -- A matrix over either an exact ring or ``SR``. - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 + - ``K`` -- A polyhedral closed convex cone. - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: + OUTPUT: - sage: K = Cone([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is cross-positive on ``K``. - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 + - ``True`` will be returned if it can be proven that ``L`` + is cross-positive on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not cross-positive on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: + .. SEEALSO:: - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 + :func:`is_positive_on`, + :func:`is_Z_operator_on`, + :func:`is_lyapunov_like_on` - TESTS: + EXAMPLES: - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: + The identity operator is always cross-positive:: 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=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_cross_positive_on(L,K) True - A strictly convex cone should have lineality zero:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, strictly_convex = True) - sage: lineality(K) - 0 - - """ - return K.linear_subspace().dimension() - + The "zero" operator is always cross-positive:: -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 + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_cross_positive_on(L,K) + True TESTS: - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: + Everything in ``K.cross_positive_operators_gens()`` should be + cross-positive on ``K``:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10) - sage: c = codim(K) - sage: c in ZZ + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_cross_positive_on(L,K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time True - sage: (0 <= c) and (c <= K.lattice_dim()) + sage: all([ is_cross_positive_on(L.change_ring(SR),K) # long time + ....: for L in K.cross_positive_operators_gens() ]) # long time True - A solid cone should have codimension zero:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) - 0 + sage: L = identity_matrix(3) + sage: K = [ vector([8,2,-8]), vector([5,-5,7]) ] + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - The codimension of a cone is equal to the lineality of its dual:: + We can't give reliable answers over inexact rings:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 10, solid = True) - sage: codim(K) == lineality(K.dual()) - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_cross_positive_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. """ - return (K.lattice_dim() - K.dim()) + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') + return all([ s*(L*x) >= 0 + for (x,s) in K.discrete_complementarity_set() ]) -def discrete_complementarity_set(K): +def is_Z_operator_on(L,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))] + Determine whether or not ``L`` is a Z-operator on ``K``. - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: + We say that ``L`` is a Z-operator on a closed convex cone``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle \le 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. It is known that this property need only be checked + for generators of ``K`` and its dual. - 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))] + A matrix is a Z-operator on ``K`` if and only if its negation is a + cross-positive operator on ``K``. - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: + To reliably check whether or not ``L`` is a Z operator, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. - 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:: - - 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) - 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()] + INPUT: - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + - ``L`` -- A matrix over either an exact ring or ``SR``. - -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. + - ``K`` -- A polyhedral closed convex 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) - [] + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is a Z-operator on ``K``. - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] + - ``True`` will be returned if it can be proven that ``L`` + is a Z-operator on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not a Z-operator on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] + .. SEEALSO:: - 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] - ] + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_lyapunov_like_on` - TESTS: + EXAMPLES: - 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:: + The identity operator is always a Z-operator:: 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: K = random_cone(max_ambient_dim=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_Z_operator_on(L,K) + 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 "zero" operator is always a Z-operator:: - 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) + sage: K = random_cone(max_ambient_dim=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_Z_operator_on(L,K) True - """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) + TESTS: - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + Everything in ``K.Z_operators_gens()`` should be a Z-operator + on ``K``:: - # 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) + sage: K = random_cone(max_ambient_dim=5) + sage: all([ is_Z_operator_on(L,K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True + sage: all([ is_Z_operator_on(L.change_ring(SR),K) # long time + ....: for L in K.Z_operators_gens() ]) # long time + True - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() + sage: L = identity_matrix(3) + sage: K = [ vector([-4,20,3]), vector([1,-5,2]) ] + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + We can't give reliable answers over inexact rings:: - return matrix_basis + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_Z_operator_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. + """ + return is_cross_positive_on(-L,K) -def lyapunov_rank(K): +def is_lyapunov_like_on(L,K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Determine whether or not ``L`` is Lyapunov-like on ``K``. - The Lyapunov rank of a cone can be thought of in (mainly) two ways: + We say that ``L`` is Lyapunov-like on a closed convex cone ``K`` if + `\left\langle L\left\lparenx\right\rparen,s\right\rangle = 0` for + all pairs `\left\langle x,s \right\rangle` in the complementarity + set of ``K``. This property need only be checked for generators of + ``K`` and its dual. - 1. The dimension of the Lie algebra of the automorphism group of the - cone. + An operator is Lyapunov-like on ``K`` if and only if both the + operator itself and its negation are cross-positive on ``K``. - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + To reliably check whether or not ``L`` is Lyapunov-like, its base + ring must be either exact (for example, the rationals) or ``SR``. An + exact ring is more reliable, but in some cases a matrix whose + entries contain symbolic constants like ``e`` and ``pi`` will work. INPUT: - A closed, convex polyhedral 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:: + - ``L`` -- A matrix over either an exact ring or ``SR``. - :meth:`is_proper` + - ``K`` -- A polyhedral closed convex cone. - ALGORITHM: + OUTPUT: - 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}`. + If the base ring of ``L`` is exact, then ``True`` will be returned if + and only if ``L`` is Lyapunov-like on ``K``. - REFERENCES: + If the base ring of ``L`` is ``SR``, then the situation is more + complicated: - .. [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. + - ``True`` will be returned if it can be proven that ``L`` + is Lyapunov-like on ``K``. + - ``False`` will be returned if it can be proven that ``L`` + is not Lyapunov-like on ``K``. + - ``False`` will also be returned if we can't decide; specifically + if we arrive at a symbolic inequality that cannot be resolved. - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + .. SEEALSO:: - .. [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. + :func:`is_positive_on`, + :func:`is_cross_positive_on`, + :func:`is_Z_operator_on` 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 - - 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([(1,0,0,0,0)]) - sage: lyapunov_rank(K) - 21 - sage: K.lattice_dim()**2 - K.lattice_dim() + 1 - 21 - - A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_:: - - 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 - - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: - - 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) - True - - 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}`:: + Diagonal matrices are Lyapunov-like operators on the nonnegative + orthant:: - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 - - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: - - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) + sage: L = diagonal_matrix(random_vector(QQ,3)) + sage: is_lyapunov_like_on(L,K) True TESTS: - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + The identity operator is always Lyapunov-like:: 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=8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + The "zero" operator is always Lyapunov-like:: - 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=8) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like_on(L,K) True - Make sure we exercise the non-strictly-convex/non-solid case:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - 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=5) + sage: all([ is_lyapunov_like_on(L,K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time 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:: - - 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: all([ is_lyapunov_like_on(L.change_ring(SR),K) # long time + ....: for L in K.lyapunov_like_basis() ]) # long time True - sage: b == n-1 - False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: + Technically we could test this, but for now only closed convex cones + are supported as our ``K`` argument:: - sage: set_random_seed() - sage: K = random_cone(max_dim=10) - sage: b = lyapunov_rank(K) - sage: n = K.lattice_dim() - sage: b == n-1 - False - - The calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky/Gowda]_:: - - 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 - True + sage: L = identity_matrix(3) + sage: K = [ vector([2,2,-1]), vector([5,4,-3]) ] + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + TypeError: K must be a Cone. - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: + We can't give reliable answers over inexact rings:: - sage: set_random_seed() - sage: K = random_cone(max_dim=10, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - True + sage: K = Cone([(1,2,3), (4,5,6)]) + sage: L = identity_matrix(RR,3) + sage: is_lyapunov_like_on(L,K) + Traceback (most recent call last): + ... + ValueError: The base ring of L is neither SR nor exact. """ - K_orig = K - beta = 0 - - m = K.dim() - n = K.lattice_dim() - l = lineality(K) + if not is_Cone(K): + raise TypeError('K must be a Cone.') + if not L.base_ring().is_exact() and not L.base_ring() is SR: + raise ValueError('The base ring of L is neither SR nor exact.') - if m < n: - # K is not solid, project onto its span. - K = restrict_span(K) + return all([ s*(L*x) == 0 + for (x,s) in K.discrete_complementarity_set() ]) - # 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() +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) - #Ks = [ list(r) for r in sorted(K.rays()) ] - #Js = [ list(r) for r in sorted(J.rays()) ] +def Sigma_cone(K): + gens = K.cross_positive_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) - #if Ks != Js: - # print [ list(r) for r in K_orig.rays() ] +def Z_cone(K): + gens = K.Z_operators_gens() + L = ToricLattice(K.lattice_dim()**2) + return Cone([ g.list() for g in gens ], lattice=L, check=False) - # Lemma 3 - beta += m * l - - beta += len(LL(K)) - return beta +def pi_cone(K1, K2=None): + if K2 is None: + K2 = K1 + gens = K1.positive_operators_gens(K2) + L = ToricLattice(K1.lattice_dim()*K2.lattice_dim()) + return Cone([ g.list() for g in gens ], lattice=L, check=False)