X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=ae3ec48cddc9700d4f63ae378fc01b178dee6e3b;hb=8353d776d562e16cdbccfd10881662fc542c8d6f;hp=23043381bca83acd6d7f17479ef4769980b9960f;hpb=012175f3a2591586099b4e955bb440958f2d63d7;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 2304338..ae3ec48 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -1,141 +1,480 @@ -# 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 lyapunov_rank(K): +def is_lyapunov_like(L,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. + Determine whether or not ``L`` is Lyapunov-like on ``K``. - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. + 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: - A closed, convex polyhedral cone. + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex 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). + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. - .. note:: + .. WARNING:: - In the references, the cones are always assumed to be proper. We - do not impose this restriction. + 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. - .. seealso:: + REFERENCES: - :meth:`is_proper` + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - ALGORITHM: + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: + + 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 + + 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 - 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}`. + 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 motzkin_decomposition(K): + r""" + Return the pair of components in the Motzkin decomposition of this 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``. + + OUTPUT: + + 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``. REFERENCES: - 1. M.S. Gowda and J. Tao. On the bilinearity rank of a proper cone - and Lyapunov-like transformations, Mathematical Programming, 147 - (2014) 155-170. + .. [Stoer-Witzgall] J. Stoer and C. Witzgall. Convexity and + Optimization in Finite Dimensions I. Springer-Verlag, New + York, 1970. + + EXAMPLES: + + The nonnegative orthant is strictly convex, so it is its own + strictly convex component and its subspace component is trivial:: + + 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 + + Likewise, full spaces are their own subspace components:: + + 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 + + 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) + sage: (P,S) = motzkin_decomposition(K) + sage: x = K.random_element() + sage: P.contains(x) or S.contains(x) + True + sage: x.is_zero() or (P.contains(x) != S.contains(x)) + True + + 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) + sage: (P,S) = motzkin_decomposition(K) + sage: P.is_strictly_convex() + True + sage: S.lineality() == S.dim() + True + + 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: (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 + """ + # 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()) + + # Since ``S`` is a subspace, the rays of its dual generate its + # orthogonal complement. + S_perp = Cone(S.dual(), K.lattice()) + P = K.intersection(S_perp) - 2. 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. + return (P,S) + + +def positive_operator_gens(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 nonnegative orthant in `\mathbb{R}^{n}` always has rank `n`:: + Positive operators on the nonnegative orthant are nonnegative matrices:: - 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 + sage: K = Cone([(1,)]) + sage: positive_operator_gens(K) + [[1]] - The `L^{3}_{1}` cone is known to have a Lyapunov rank of one:: + 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: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: lyapunov_rank(L31) - 1 + The trivial cone in a trivial space has no positive operators:: - Likewise for the `L^{3}_{\infty}` cone:: + sage: K = Cone([], ToricLattice(0)) + sage: positive_operator_gens(K) + [] - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: lyapunov_rank(L3infty) - 1 + Every operator is positive on the trivial cone:: - The Lyapunov rank should be additive on a product of cones:: + sage: K = Cone([(0,)]) + sage: positive_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([(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] + ] - Two isomorphic cones should have the same Lyapunov rank. The cone - ``K`` in the following example is isomorphic to the nonnegative - octant in `\mathbb{R}^{3}`:: + Every operator is positive on the ambient vector space:: - sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) - sage: lyapunov_rank(K) - 3 + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operator_gens(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operator_gens(K) + [ + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + + A non-obvious application is to find the positive operators on the + right half-plane:: + + sage: K = Cone([(1,0),(0,1),(0,-1)]) + sage: positive_operator_gens(K) + [ + [1 0] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + + TESTS: + + Each positive operator generator should send the generators of the + cone into the cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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 + + Each positive operator generator should send a random element of the + cone into the cone:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + sage: pi_of_K = positive_operator_gens(K) + sage: all([ K.contains(P*K.random_element()) for P in pi_of_K ]) + True - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself:: + A random element of the positive operator cone should send the + generators of the cone into the cone:: - sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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) + sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list()) + sage: all([ K.contains(P*x) for x in K ]) True + 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=5) + 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) + sage: P = matrix(K.lattice_dim(), pi_cone.random_element().list()) + sage: K.contains(P*K.random_element()) + True + + The dimension of the cone of positive operators is given by the + corollary in my paper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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: actual = Cone([p.list() for p in pi_of_K], lattice=L).dim() + sage: expected = n**2 - l*(m - l) - (n - m)*m + sage: actual == expected + True + + The lineality 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=5) + sage: n = K.lattice_dim() + sage: pi_of_K = positive_operator_gens(K) + sage: L = ToricLattice(n**2) + sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality() + sage: expected = n**2 - K.dim()*K.dual().dim() + sage: actual == expected + True + + The cone ``K`` is proper if and only if the cone of positive + operators on ``K`` is proper:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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) + sage: K.is_proper() == pi_cone.is_proper() + True """ - V = K.lattice().vector_space() + # 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() ] + + # Convert those tensor products to long vectors. + W = VectorSpace(F, n**2) + vectors = [ W(tp.list()) for tp in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + pi_dual = Cone(vectors, ToricLattice(W.dimension())) - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() - # WARNING: This isn't really C(K), it only contains the pairs - # (x,s) in C(K) where x,s are extreme in their respective cones. - C_of_K = [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + # And finally convert its rays back to matrix representations. + M = MatrixSpace(F, n) + return [ M(v.list()) for v in pi_cone.rays() ] - matrices = [x.column() * s.row() for (x,s) in C_of_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) +def Z_transformation_gens(K): + r""" + Compute generators of the cone of Z-transformations on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``L`` in the list should have the property that + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the + discrete complementarity set of ``K``. Moreover, any nonnegative + linear combination of these matrices shares the same property. - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) + EXAMPLES: + + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: + + sage: K = Cone([(1,0),(0,1)]) + 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_transformation_gens(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) + True + + The trivial cone in a trivial space has no Z-transformations:: + + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformation_gens(K) + [] + + 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_transformation_gens(K) ]) + sage: zs == lls + True - vectors = [phi(m) for m in matrices] + TESTS: - return (W.dimension() - W.span(vectors).rank()) + 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_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:: + + 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_transformation_gens(K) ]) + sage: z_cone.linear_subspace() == lls + True + + And thus, the lineality of Z is the Lyapunov rank:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=6) + 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) + sage: z_cone.lineality() == K.lyapunov_rank() + True + + The lineality spaces of pi-star and Z-star are equal: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=5) + 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_star = Cone([p.list() for p in pi_of_K], lattice=L).dual() + sage: z_star = Cone([ z.list() for z in Z_of_K], lattice=L).dual() + sage: pi_star.linear_subspace() == z_star.linear_subspace() + True + """ + # Matrices are not vectors in Sage, so we have to convert them + # to vectors explicitly before we can find a basis. We need these + # two values to construct the appropriate "long vector" space. + F = K.lattice().base_field() + n = K.lattice_dim() + + # These tensor products contain generators for the dual cone of + # the cross-positive 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.. + Sigma_dual = Cone(vectors, lattice=ToricLattice(W.dimension())) + + # 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-transformations and + # not cross-positive ones. + M = MatrixSpace(F, n) + return [ -M(v.list()) for v in Sigma_cone.rays() ] + + +def Z_cone(K): + gens = Z_transformation_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L) + +def pi_cone(K): + gens = positive_operator_gens(K) + L = None + if len(gens) == 0: + L = ToricLattice(0) + return Cone([ g.list() for g in gens ], lattice=L)