X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=5f058ca9a8c055f972ffffaa216975517e0dc723;hb=9bab3c89cc7d669e7b99295900c9f590e5525079;hp=87cdf704580e68b55fa72cd93b8dfa6c1d08a484;hpb=0bdf2bb8ca97eeb065e7dd3c36bdac6879a52116;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 87cdf70..5f058ca 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,152 +8,41 @@ addsitedir(abspath('../../')) from sage.all import * -def basically_the_same(K1,K2): +def _restrict_to_space(K, W): r""" - ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. - """ - if K1.lattice_dim() != K2.lattice_dim(): - return False - - if K1.nrays() != K2.nrays(): - return False - - if K1.dim() != K2.dim(): - return False - - if lineality(K1) != lineality(K2): - return False - - if K1.is_solid() != K2.is_solid(): - return False - - if K1.is_strictly_convex() != K2.is_strictly_convex(): - return False - - if len(LL(K1)) != len(LL(K2)): - return False - - C_of_K1 = discrete_complementarity_set(K1) - C_of_K2 = discrete_complementarity_set(K2) - if len(C_of_K1) != len(C_of_K2): - return False - - if len(K1.facets()) != len(K2.facets()): - return False - - return True - - - -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 restrict_span(K, K2=None): - r""" - Restrict ``K`` into its own span, or the span of another cone. + Restrict this cone a subspace of its ambient space. INPUT: - - ``K2`` -- another cone whose lattice has the same rank as this cone. + - ``W`` -- The subspace into which this cone will be restricted. OUTPUT: - A new cone in a sublattice. + A new cone in a sublattice corresponding to ``W``. - EXAMPLES:: + EXAMPLES: + + When this cone is solid, restricting it into its own span should do + nothing:: sage: K = Cone([(1,)]) - sage: restrict_span(K) == K + sage: _restrict_to_space(K, K.span()) == K True + A single ray restricted into its own span gives the same output + regardless of the ambient space:: + sage: K2 = Cone([(1,0)]) - sage: restrict_span(K2).rays() + sage: K2_S = _restrict_to_space(K2, K2.span()).rays() + sage: K2_S N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: restrict_span(K3).rays() + sage: K3_S = _restrict_to_space(K3, K3.span()).rays() + sage: K3_S N(1) in 1-d lattice N - sage: restrict_span(K2) == restrict_span(K3) + sage: K2_S == K3_S True TESTS: @@ -161,502 +50,511 @@ def restrict_span(K, K2=None): The projected cone should always be solid:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K) - sage: K_S.is_solid() + sage: K = random_cone(max_ambient_dim = 8) + sage: _restrict_to_space(K, K.span()).is_solid() True And the resulting cone should live in a space having the same dimension as the space we restricted it to:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = restrict_span(K, K.dual() ) - sage: K_S.lattice_dim() == K.dual().dim() - True - - This function has ``unrestrict_span()`` as its inverse:: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True) - sage: J = restrict_span(K) - sage: K == unrestrict_span(J,K) + sage: K = random_cone(max_ambient_dim = 8) + sage: K_P = _restrict_to_space(K, K.dual().span()) + sage: K_P.lattice_dim() == K.dual().dim() True This function should not affect the dimension of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.dim() == restrict_span(K).dim() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.dim() == _restrict_to_space(K,K.span()).dim() True Nor should it affect the lineality of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: lineality(K) == lineality(restrict_span(K)) + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() True No matter which space we restrict to, the lineality should not increase:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: lineality(K) >= lineality(restrict_span(K)) + sage: K = random_cone(max_ambient_dim = 8) + sage: S = K.span(); P = K.dual().span() + sage: K.lineality() >= _restrict_to_space(K,S).lineality() True - sage: lineality(K) >= lineality(restrict_span(K, K.dual())) + sage: K.lineality() >= _restrict_to_space(K,P).lineality() True If we do this according to our paper, then the result is proper:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() + sage: K = random_cone(max_ambient_dim = 8) + 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_SP.is_proper() True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) + sage: K_SP.is_proper() True - :: + Test the proposition in our paper concerning the duals and + restrictions. Generate a random cone, then create a subcone of + it. The operation of dual-taking should then commute with + _restrict_to_space:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() + sage: J = random_cone(max_ambient_dim = 8) + sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) + sage: K_W_star = _restrict_to_space(K, J.span()).dual() + sage: K_star_W = _restrict_to_space(K.dual(), J.span()) + sage: _basically_the_same(K_W_star, K_star_W) True - :: + """ + # First we want to intersect ``K`` with ``W``. The easiest way to + # do this is via cone intersection, so we turn the subspace ``W`` + # into a cone. + W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice()) + K = K.intersection(W_cone) - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() - True + # We've already intersected K with the span of K2, so every + # generator of K should belong to W now. + K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - :: + L = ToricLattice(W.dimension()) + return Cone(K_W_rays, lattice=L) - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: P.is_proper() - True - sage: P = restrict_span(K_S, K_S.dual()) - sage: P.is_proper() - True - Test the proposition in our paper concerning the duals, where the - subspace `W` is the span of `K^{*}`:: +def lyapunov_rank(K): + r""" + Compute the Lyapunov rank of this cone. - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True + 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. - :: + OUTPUT: - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True + 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]_. - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True + 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}`. - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True) - sage: K_W = restrict_span(K, K.dual()) - sage: K_star_W_star = restrict_span(K.dual()).dual() - sage: basically_the_same(K_W, K_star_W_star) - True + REFERENCES: - """ - if K2 is None: - K2 = K + .. [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. - phi,_ = ips_iso(K2) - (W, W_perp) = iso_space(K2).cartesian_factors() + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - ray_pairs = [ phi(r) for r in K.rays() ] + 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. - # Shouldn't matter? - # - #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]): - # msg = 'Cone has nonzero components in W-perp!' - # raise ValueError(msg) + EXAMPLES: - # 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 ] + The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` + [Rudolf]_:: - L = ToricLattice(W.dimension()) + 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 - return Cone(ws, lattice=L) + 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]_:: -def lineality(K): - r""" - Compute the lineality of this cone. + sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) + sage: lyapunov_rank(L31) + 1 - The lineality of a cone is the dimension of the largest linear - subspace contained in that cone. + Likewise for the `L^{3}_{\infty}` cone [Rudolf]_:: - OUTPUT: + sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) + sage: lyapunov_rank(L3infty) + 1 - A nonnegative integer; the dimension of the largest subspace - contained within this cone. + A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n + + 1` [Orlitzky]_:: - REFERENCES: + sage: K = Cone([(1,0,0,0,0)]) + sage: lyapunov_rank(K) + 21 + sage: K.lattice_dim()**2 - K.lattice_dim() + 1 + 21 - .. [Rockafellar] R.T. Rockafellar. Convex Analysis. Princeton - University Press, Princeton, 1970. + A subspace (of dimension `m`) in `n` dimensions should have a + Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_:: - EXAMPLES: + 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 - The lineality of the nonnegative orthant is zero, since it clearly - contains no lines:: + The Lyapunov rank should be additive on a product of proper cones + [Rudolf]_:: - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 0 + 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 - However, if we add another ray so that the entire `x`-axis belongs - to the cone, then the resulting cone will have lineality one:: + 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([(1,0,0), (-1,0,0), (0,1,0), (0,0,1)]) - sage: lineality(K) - 1 + sage: K = Cone([(1,2,3), (-1,1,0), (1,0,6)]) + sage: lyapunov_rank(K) + 3 - If our cone is all of `\mathbb{R}^{2}`, then its lineality is equal - to the dimension of the ambient space (i.e. two):: + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` + itself [Rudolf]_:: - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: lineality(K) - 2 + sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + True - Per the definition, the lineality of the trivial cone in a trivial - space is zero:: + TESTS: - sage: K = Cone([], lattice=ToricLattice(0)) - sage: lineality(K) - 0 + The Lyapunov rank should be additive on a product of proper cones + [Rudolf]_:: - TESTS: + 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) + True + + The Lyapunov rank is invariant under a linear isomorphism + [Orlitzky]_:: + + 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) + True + + The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` + itself [Rudolf]_:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) + 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_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 + True + sage: b == n-1 + False + + In fact [Orlitzky]_, no closed convex polyhedral cone can have + Lyapunov rank `n-1` in `n` dimensions:: + + 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 - The lineality of a cone should be an integer between zero and the - dimension of the ambient space, inclusive:: + The calculation of the Lyapunov rank of an improper cone can be + reduced to that of a proper cone [Orlitzky]_:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: l = lineality(K) - sage: l in ZZ + 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: l = K.lineality() + sage: c = K.codim() + sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 + sage: actual == expected True - sage: (0 <= l) and (l <= K.lattice_dim()) + + The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) True - A strictly convex cone should have lineality zero:: + We can make an imperfect cone perfect by adding a slack variable + (a Theorem in [Orlitzky]_):: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex = True) - sage: lineality(K) - 0 + 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() + True """ - return K.linear_subspace().dimension() + beta = 0 # running tally of the Lyapunov rank + m = K.dim() + n = K.lattice_dim() + l = K.lineality() -def codim(K): - r""" - Compute the codimension of this cone. + if m < n: + # K is not solid, restrict to its span. + K = _restrict_to_space(K, K.span()) - 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. + # Non-solid reduction lemma. + beta += (n - m)*n - OUTPUT: + 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()) - A nonnegative integer representing the dimension of the space of all - elements perpendicular to this cone. + # Non-pointed reduction lemma. + beta += l * m - .. seealso:: + beta += len(K.lyapunov_like_basis()) + return beta - :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:: +def is_lyapunov_like(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. - sage: K = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: codim(K) - 0 + 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. - 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:: + INPUT: - sage: K = Cone([(1,0,0), (0,1,0)]) - sage: codim(K) - 1 + - ``L`` -- A linear transformation or matrix. - If our cone is all of `\mathbb{R}^{2}`, then its codimension is zero:: + - ``K`` -- A polyhedral closed convex cone. - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: codim(K) - 0 + OUTPUT: - And if the cone is trivial in any space, then its codimension is - equal to the dimension of the ambient space:: + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. - sage: K = Cone([], lattice=ToricLattice(0)) - sage: K.lattice_dim() - 0 - sage: codim(K) - 0 + .. WARNING:: - sage: K = Cone([(0,)]) - sage: K.lattice_dim() - 1 - sage: codim(K) - 1 + 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. - sage: K = Cone([(0,0)]) - sage: K.lattice_dim() - 2 - sage: codim(K) - 2 + REFERENCES: - TESTS: + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - The codimension of a cone should be an integer between zero and - the dimension of the ambient space, inclusive:: + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: c = codim(K) - sage: c in ZZ - True - sage: (0 <= c) and (c <= K.lattice_dim()) + sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like(L,K) True - A solid cone should have codimension zero:: + As is the "zero" transformation:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) - 0 + 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) + True - The codimension of a cone is equal to the lineality of its dual:: + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, solid = True) - sage: codim(K) == lineality(K.dual()) + 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() ]) True """ - return (K.lattice_dim() - K.dim()) + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) -def discrete_complementarity_set(K): +def random_element(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. + Return a random element of ``K`` from its ambient vector space. - OUTPUT: + ALGORITHM: - A list of pairs `(x,s)` such that, + 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. - * `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. + 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: - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: - - sage: K = Cone([(1,0),(0,1)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((0, 1), (1, 0))] - - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: - - sage: K = Cone([(1,0)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((1, 0), (0, -1))] - sage: K = Cone([(1,0,0)]) - sage: discrete_complementarity_set(K) - [((1, 0, 0), (0, 1, 0)), - ((1, 0, 0), (0, -1, 0)), - ((1, 0, 0), (0, 0, 1)), - ((1, 0, 0), (0, 0, -1))] - - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: + A random element of the trivial cone is zero:: - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] + 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: - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: + Any cone should contain an element of itself:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=6) - sage: K2 = K1.dual() - sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] - sage: actual = discrete_complementarity_set(K1) - sage: sorted(actual) == sorted(expected) + sage: K = random_cone(max_rays = 8) + sage: K.contains(random_element(K)) 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)) ] - # Convert the rays to vectors so that we can compute inner - # products. - xs = [V(x) for x in K.rays()] - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] + # 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 LL(K): +def positive_operators(K): r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. + Compute generators of the cone of positive operators on this cone. OUTPUT: - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. + 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 has no Lyapunov-like transformations:: + The trivial cone in a trivial space has no positive operators:: - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) [] - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: + Positive operators on the nonnegative orthant are nonnegative matrices:: sage: K = Cone([(1,)]) - sage: LL(K) + sage: positive_operators(K) [[1]] sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) + sage: positive_operators(K) [ - [1 0] [0 0] - [0 0], [0 1] + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] ] - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: LL(K) - [ - [1 0 0] [0 0 0] [0 0 0] - [0 0 0] [0 1 0] [0 0 0] - [0 0 0], [0 0 0], [0 0 1] - ] + Every operator is positive on the ambient vector space:: - 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: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(K) [ - [1 0 0] - [0 1 0] - [0 0 1] + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] ] - If our cone is the entire space, then every transformation on it is - Lyapunov-like:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: M = MatrixSpace(QQ,2) - sage: M.basis() == LL(K) - True - TESTS: - The inner product `\left< L\left(x\right), s \right>` is zero for - every pair `\left( x,s \right)` in the discrete complementarity set - of the cone:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - 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 - - 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)` + A positive operator on a cone should send its generators into the cone:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - 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 = 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()]) True """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] - # 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. @@ -665,303 +563,124 @@ def LL(K): # 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) + 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 ] - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() - - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() - return matrix_basis + # 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() ] -def lyapunov_rank(K): +def Z_transformations(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. - - The Lyapunov rank of a cone can be thought of in (mainly) two ways: - - 1. The dimension of the Lie algebra of the automorphism group of the - cone. - - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. - - INPUT: - - A closed, convex polyhedral cone. + Compute generators of the cone of Z-transformations on this cone. OUTPUT: - An integer representing the Lyapunov rank of the cone. If the - dimension of the ambient vector space is `n`, then the Lyapunov rank - will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). - - .. note:: - - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` - - ALGORITHM: - - The codimension formula from the second reference is used. We find - all pairs `(x,s)` in the complementarity set of `K` such that `x` - and `s` are rays of our cone. It is known that these vectors are - sufficient to apply the codimension formula. Once we have all such - pairs, we "brute force" the codimension formula by finding all - linearly-independent `xs^{T}`. - - REFERENCES: - - .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper - cone and Lyapunov-like transformations, Mathematical Programming, 147 - (2014) 155-170. - - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. - - .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear - optimality constraints for the cone of positive polynomials, - Mathematical Programming, Series B, 129 (2011) 5-31. + 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. 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}`:: - - 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()) - True - - TESTS: - - The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: - - sage: set_random_seed() - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K = K1.cartesian_product(K2) - sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) - True - - The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - Make sure we exercise the non-strictly-convex/non-solid case:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - Let's check the other permutations as well, just to be sure:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - 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:: + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: - sage: set_random_seed() - sage: K = random_cone(max_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 = Cone([(1,0),(0,1)]) + sage: Z_transformations(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) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) True - sage: b == n-1 - False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have - Lyapunov rank `n-1` in `n` dimensions:: + The trivial cone in a trivial space has no Z-transformations:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: b = lyapunov_rank(K) - sage: n = K.lattice_dim() - sage: b == n-1 - False + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformations(K) + [] - The calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky/Gowda]_:: + Z-transformations on a subspace are Lyapunov-like and vice-versa:: - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: actual = lyapunov_rank(K) - sage: K_S = restrict_span(K) - sage: P = restrict_span(K_S.dual()).dual() - sage: l = lineality(K) - sage: c = codim(K) - sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2 - sage: actual == expected + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() True - - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) + 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 == lls True - In fact the same can be said of any cone. These additional tests - just increase our confidence that the reduction scheme works:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) - True + TESTS: - :: + The Z-property is possessed by every Z-transformation:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) + sage: K = random_cone(max_ambient_dim = 6) + sage: Z_of_K = Z_transformations(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(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) + 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 True """ - K_orig = K - beta = 0 + # 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) - m = K.dim() - n = K.lattice_dim() - l = lineality(K) + C_of_K = K.discrete_complementarity_set() + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] - if m < n: - # K is not solid, project onto its span. - K = restrict_span(K) + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] - # Lemma 2 - beta += m*(n - m) + (n - m)**2 + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. + L = ToricLattice(W.dimension()) + Sigma_dual = Cone(vectors, lattice=L) - 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, K.dual()) + # Now compute the desired cone from its dual... + Sigma_cone = Sigma_dual.dual() - # Lemma 3 - beta += m * l + # 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()) - beta += len(LL(K)) - return beta + return [ -M(v.list()) for v in Sigma_cone.rays() ]