X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=81698e467c603df3e57cb764c256a2e082df5ec6;hb=b966ec383942ef1ff7837786c29d1f3edc33b84e;hp=a5482b3aa95f7198938007c4be615c4e7a97e17d;hpb=b97553aaaf9734644bee13bf484014f817456b26;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index a5482b3..81698e4 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,6 +8,58 @@ addsitedir(abspath('../../')) from sage.all import * +def project_span(K, K2 = None): + r""" + Return a "copy" of ``K`` embeded in a lower-dimensional space. + + By default, we will project ``K`` into the subspace spanned by its + rays. However, if ``K2`` is not ``None``, we will project into the + space spanned by the rays of ``K2`` instead. + + EXAMPLES:: + + sage: K = Cone([(1,0,0), (0,1,0)]) + sage: project_span(K) + 2-d cone in 2-d lattice N + sage: project_span(K).rays() + N(1, 0), + N(0, 1) + in 2-d lattice N + + sage: K = Cone([(1,0,0), (0,1,0)]) + sage: K2 = Cone([(0,1)]) + sage: project_span(K, K2).rays() + N(1) + in 1-d lattice N + + """ + # Allow us to use a second cone to generate the subspace into + # which we're "projecting." + if K2 is None: + K2 = K + + # Use these to generate the new cone. + cs1 = K.rays().matrix().columns() + + # And use these to figure out which indices to drop. + cs2 = K2.rays().matrix().columns() + + perp_idxs = [] + + for idx in range(0, len(cs2)): + if cs2[idx].is_zero(): + perp_idxs.append(idx) + + solid_cols = [ cs1[idx] for idx in range(0,len(cs1)) + if not idx in perp_idxs + and not idx >= len(cs2) ] + + m = matrix(solid_cols) + L = ToricLattice(len(m.rows())) + J = Cone(m.transpose(), lattice=L) + return J + + def discrete_complementarity_set(K): r""" Compute the discrete complementarity set of this cone. @@ -87,15 +139,77 @@ def LL(K): OUTPUT: - A ``MatrixSpace`` object `M` such that every matrix `L \in M` is - Lyapunov-like on this cone. + 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) + [] + + The Lyapunov-like transformations on the nonnegative orthant are + simply diagonal matrices:: + + sage: K = Cone([(1,)]) + sage: LL(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: LL(K) + [ + [1 0] [0 0] + [0 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] + ] + + 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] + ] + + 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: 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 """ V = K.lattice().vector_space() C_of_K = discrete_complementarity_set(K) - matrices = [x.tensor_product(s) for (x,s) in C_of_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 @@ -108,7 +222,7 @@ def LL(K): W = VectorSpace(V.base_ring(), V.dimension()**2) # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in matrices ] + 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). @@ -118,9 +232,9 @@ def LL(K): # transformations. M = MatrixSpace(V.base_ring(), V.dimension()) - matrices = [ M(v.list()) for v in LL_vector.basis() ] + matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] - return matrices + return matrix_basis @@ -145,7 +259,7 @@ 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). + not possible for any cone. .. note:: @@ -171,6 +285,9 @@ def lyapunov_rank(K): 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. @@ -247,40 +364,44 @@ def lyapunov_rank(K): 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, this holds for the trivial cone in a - trivial space as well):: + [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: K = random_cone(max_dim=10, strictly_convex=True, solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() - sage: 1 <= b and b <= n + sage: (n == 0 or 1 <= b) and b <= n True sage: b == n-1 False - """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - matrices = [x.tensor_product(s) for (x,s) in C_of_K] + In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + Lyapunov rank `n-1` in `n` dimensions:: - # 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_dim=10, max_rays=16) + sage: b = lyapunov_rank(K) + sage: n = K.lattice_dim() + sage: b == n-1 + False - def phi(m): - r""" - Convert a matrix to a vector isomorphically. - """ - return W(m.list()) + The calculation of the Lyapunov rank of an improper cone can be + reduced to that of a proper cone [Orlitzky/Gowda]_:: - vectors = [phi(m) for m in matrices] + sage: K = random_cone(max_dim=15, max_rays=25) + sage: actual = lyapunov_rank(K) + sage: K_S = project_span(K) + sage: J_T1 = project_span(K_S.dual()).dual() + sage: J_T2 = project_span(K, K_S.dual()) + sage: J_T2 = Cone(J_T2.rays(), lattice=J_T1.lattice()) + sage: J_T1 == J_T2 + True + sage: J_T = J_T1 + sage: l = K.linear_subspace().dimension() + sage: codim = K.lattice_dim() - K.dim() + sage: expected = lyapunov_rank(J_T) + K.dim()*(l + codim) + codim**2 + sage: actual == expected + True - return (W.dimension() - W.span(vectors).rank()) + """ + return len(LL(K))