X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=b4d2be0d73e10ed9f58d1189c84ed4bec3d0ad94;hb=db016cc07b5f0cb45a96022eb468e3080786078b;hp=e40579fa634dc51afc8eaf83ff5b3e68025180b5;hpb=98637c981445d35a061878923baf3ae4651ecb0b;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index e40579f..b4d2be0 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -91,34 +91,41 @@ def _basically_the_same(K1, K2): -def _rho(K, K2=None): +def _restrict_to_space(K, W): 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: _rho(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: _rho(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: _rho(K3).rays() + sage: K3_S = _restrict_to_space(K3, K3.span()).rays() + sage: K3_S N(1) in 1-d lattice N - sage: _rho(K2) == _rho(K3) + sage: K2_S == K3_S True TESTS: @@ -127,8 +134,7 @@ def _rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _rho(K) - sage: K_S.is_solid() + sage: _restrict_to_space(K, K.span()).is_solid() True And the resulting cone should live in a space having the same @@ -136,22 +142,22 @@ def _rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: K_S = _rho(K, K.dual() ) - sage: K_S.lattice_dim() == K.dual().dim() + 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_ambient_dim = 8) - sage: K.dim() == _rho(K).dim() + 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_ambient_dim = 8) - sage: K.lineality() == _rho(K).lineality() + sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() True No matter which space we restrict to, the lineality should not @@ -159,161 +165,71 @@ def _rho(K, K2=None): sage: set_random_seed() sage: K = random_cone(max_ambient_dim = 8) - sage: K.lineality() >= _rho(K).lineality() + sage: S = K.span(); P = K.dual().span() + sage: K.lineality() >= _restrict_to_space(K,S).lineality() True - sage: K.lineality() >= _rho(K, K.dual()).lineality() + 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_ambient_dim = 8, - ....: strictly_convex=False, - ....: solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, - ....: strictly_convex=True, - ....: solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, - ....: strictly_convex=False, - ....: solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim = 8, - ....: strictly_convex=True, - ....: solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() + 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: K_SP = _rho(K_S, K_S.dual()) + sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) sage: K_SP.is_proper() True - Test Proposition 7 in our paper concerning the duals and + 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 rho:: - - sage: set_random_seed() - sage: J = random_cone(max_ambient_dim = 8, - ....: solid=False, - ....: strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: - - sage: set_random_seed() - sage: J = random_cone(max_ambient_dim = 8, - ....: solid=True, - ....: strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: - - sage: set_random_seed() - sage: J = random_cone(max_ambient_dim = 8, - ....: solid=False, - ....: strictly_convex=True) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: + it. The operation of dual-taking should then commute with + _restrict_to_space:: sage: set_random_seed() - sage: J = random_cone(max_ambient_dim = 8, - ....: solid=True, - ....: strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) + 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 """ - if K2 is None: - K2 = K - - # First we project K onto the span of K2. This will explode if the - # rank of ``K2.lattice()`` doesn't match ours. - span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) - K = K.intersection(span_K2) - - # Cheat a little to get the subspace span(K2). The paper uses the - # rays of K2 as a basis, but everything is invariant under linear - # isomorphism (i.e. a change of basis), and this is a little - # faster. - W = span_K2.linear_subspace() + # 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) # We've already intersected K with the span of K2, so every # generator of K should belong to W now. - W_rays = [ W.coordinate_vector(r) for r in K.rays() ] + K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - L = ToricLattice(K2.dim()) - return Cone(W_rays, lattice=L) + L = ToricLattice(W.dimension()) + return Cone(K_W_rays, lattice=L) def discrete_complementarity_set(K): r""" - Compute the discrete complementarity set of this cone. + Compute a discrete complementarity set of this cone. - The complementarity set of a cone is the set of all orthogonal pairs - `(x,s)` such that `x` is in the cone, and `s` is in its dual. The - discrete complementarity set is a subset of the complementarity set - where `x` and `s` are required to be generators of their respective - cones. - - For polyhedral cones, the discrete complementarity set is always - finite. + A discrete complementarity set of `K` is the set of all orthogonal + pairs `(x,s)` such that `x \in G_{1}` and `s \in G_{2}` for some + generating sets `G_{1}` of `K` and `G_{2}` of its dual. Polyhedral + convex cones are input in terms of their generators, so "the" (this + particular) discrete complementarity set corresponds to ``G1 + == K.rays()`` and ``G2 == K.dual().rays()``. OUTPUT: A list of pairs `(x,s)` such that, * Both `x` and `s` are vectors (not rays). - * `x` is a generator of this cone. - * `s` is a generator of this cone's dual. + * `x` is one of ``K.rays()``. + * `s` is one of ``K.dual().rays()``. * `x` and `s` are orthogonal. REFERENCES: @@ -395,8 +311,7 @@ def discrete_complementarity_set(K): def LL(K): r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. + Compute a basis of Lyapunov-like transformations on this cone. OUTPUT: @@ -524,7 +439,7 @@ def LL(K): def lyapunov_rank(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Compute the Lyapunov rank (or bilinearity rank) of this cone. The Lyapunov rank of a cone can be thought of in (mainly) two ways: @@ -543,16 +458,7 @@ def lyapunov_rank(K): 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` + not possible (see [Orlitzky/Gowda]_). ALGORITHM: @@ -685,46 +591,6 @@ def lyapunov_rank(K): sage: lyapunov_rank(K1) == lyapunov_rank(K2) True - Just to be sure, test a few more:: - - sage: K1 = random_cone(max_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=True) - 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 - - :: - - sage: K1 = random_cone(max_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=False) - 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 - - :: - - sage: K1 = random_cone(max_ambient_dim=8, - ....: strictly_convex=False, - ....: solid=True) - 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 - - :: - - sage: K1 = random_cone(max_ambient_dim=8, - ....: strictly_convex=False, - ....: solid=False) - 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 et al.]_:: @@ -733,42 +599,6 @@ def lyapunov_rank(K): 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_ambient_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_ambient_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_ambient_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_ambient_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 @@ -802,52 +632,23 @@ def lyapunov_rank(K): sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() + 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 - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - 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_ambient_dim=8, - ....: strictly_convex=True, - ....: solid=False) - sage: lyapunov_rank(K) == len(LL(K)) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, - ....: strictly_convex=False, - ....: solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - True - - :: + The Lyapunov rank of any cone is just the dimension of ``LL(K)``:: sage: set_random_seed() - sage: K = random_cone(max_ambient_dim=8, - ....: strictly_convex=False, - ....: solid=False) + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == len(LL(K)) True - Test Theorem 3 in [Orlitzky/Gowda]_:: + We can make an imperfect cone perfect by adding a slack variable + (a Theorem in [Orlitzky/Gowda]_):: sage: set_random_seed() sage: K = random_cone(max_ambient_dim=8, @@ -867,19 +668,235 @@ def lyapunov_rank(K): if m < n: # K is not solid, restrict to its span. - K = _rho(K) + K = _restrict_to_space(K, K.span()) - # Lemma 2 - beta += m*(n - m) + (n - m)**2 + # Non-solid reduction lemma. + beta += (n - m)*n if l > 0: # K is not pointed, restrict to the span of its dual. Uses a # proposition from our paper, i.e. this is equivalent to K = # _rho(K.dual()).dual(). - K = _rho(K, K.dual()) + K = _restrict_to_space(K, K.dual().span()) - # Lemma 3 - beta += m * l + # Non-pointed reduction lemma. + beta += l * m beta += len(LL(K)) return beta + + + +def is_lyapunov_like(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. + + We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle + L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs + `\left\langle x,s \right\rangle` in the complementarity set of + ``K``. It is known [Orlitzky]_ that this property need only be + checked for generators of ``K`` and its dual. + + INPUT: + + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex cone. + + OUTPUT: + + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: + + If this function returns ``True``, then ``L`` is Lyapunov-like + on ``K``. However, if ``False`` is returned, that could mean one + of two things. The first is that ``L`` is definitely not + Lyapunov-like on ``K``. The second is more of an "I don't know" + answer, returned (for example) if we cannot prove that an inner + product is zero. + + REFERENCES: + + .. [Orlitzky] M. Orlitzky. The Lyapunov rank of an + improper cone (preprint). + + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True + + As is the "zero" transformation:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True + + Everything in ``LL(K)`` should be Lyapunov-like on ``K``:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: all([is_lyapunov_like(L,K) for L in LL(K)]) + True + + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in discrete_complementarity_set(K)]) + + +def random_element(K): + r""" + Return a random element of ``K`` from its ambient vector space. + + ALGORITHM: + + The cone ``K`` is specified in terms of its generators, so that + ``K`` is equal to the convex conic combination of those generators. + To choose a random element of ``K``, we assign random nonnegative + coefficients to each generator of ``K`` and construct a new vector + from the scaled rays. + + A vector, rather than a ray, is returned so that the element may + have non-integer coordinates. Thus the element may have an + arbitrarily small norm. + + EXAMPLES: + + A random element of the trivial cone is zero:: + + sage: set_random_seed() + sage: K = Cone([], ToricLattice(0)) + sage: random_element(K) + () + sage: K = Cone([(0,)]) + sage: random_element(K) + (0) + sage: K = Cone([(0,0)]) + sage: random_element(K) + (0, 0) + sage: K = Cone([(0,0,0)]) + sage: random_element(K) + (0, 0, 0) + + TESTS: + + Any cone should contain an element of itself:: + + sage: set_random_seed() + 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)) ] + + # Make sure we return a vector. Without the coercion, we might + # return ``0`` when ``K`` has no rays. + v = V(sum(scaled_gens)) + return v + + +def positive_operators(K): + r""" + Compute generators of the cone of positive operators on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``P`` in the list should have the property that ``P*x`` + is an element of ``K`` whenever ``x`` is an element of + ``K``. Moreover, any nonnegative linear combination of these + matrices shares the same property. + + EXAMPLES: + + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] + + Positive operators on the nonnegative orthant are nonnegative matrices:: + + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operators(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [ + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + + TESTS: + + A positive operator on a cone should send its generators into the cone:: + + 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() + + # 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) + + G1 = [ V(x) for x in K.rays() ] + G2 = [ V(s) for s in K.dual().rays() ] + + tensor_products = [ s.tensor_product(x) for x in G1 for s in G2 ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() + + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ M(v.list()) for v in pi_cone.rays() ]